Creation of this text only version: 26 June 2016
(Minor corrections 6 Jul 2018; 2 Dec 2018)

Figures added (pp 287, 288, 307) : 5 Jan 2018



NEW DIGITISED VERSION

of


1962 Oxford University DPhil Thesis

KNOWING AND UNDERSTANDING

Relations between meaning and truth, meaning and
necessary truth, meaning and synthetic necessary truth

Aaron Sloman

Now at School of Computer Science University of Birmingham
http://www.cs.bham.ac.uk/~axs



Available at: http://goo.gl/9UNH81 in PDF, HTML and TXT formats,
with historical notes and comments and update information.

NOTE added 22 May 2016

The individual chapters are available on that site as .txt and .pdf
files, derived from a scanned (image only) version of the thesis
produced originally by the Bodleian Library in 2007. The original was a
carbon copy of the thesis with blurred, but easily readable text. This
proved too difficult for current OCR technology.

In 2014, the abstract, preface, table of contents and Chapter 1 were
(semi automatically) converted to machine readable .txt and pdf files,
but the process was very difficult and tedious. So, since May 2016,
thanks to manual transcription by Hitech Services, and a lot of help
from Luc Beaudoin, proof-reading and correcting the (mostly, but not
entirely, accurate) transcribed version (after conversion to files
readable in Libreoffice) .txt and .pdf versions of the remaining files
have been made available. They are assembled in a temporary draft book
form now. This contains everything apart from the index, which has not
been transcribed. As the text is now searchable, the index may not be
missed.

This concatenated version containing all the chapters is a temporary
version since there is still some checking to be done. Please,
therefore, do not save this copy. If it is accessed more than six months
after the date it was downloaded, please check whether a new version is
available, until this request is removed!

Figures were added to this HTML version on pages 287, 288 and 307
of the thesis on 5 Jan 2018.

Please report any errors or infelicities to: Aaron Sloman


The following pages are transcribed from the original scanned version of
the thesis produced in Oxford in 2007 (very bulky PDF).

===============================================================================


Front page of original thesis:
Form provided by Oxford University, stamped 28 May 1962:



* 1.(a) I give permission for my thesis entitled
      KNOWING AND UNDERSTANDING
      (Relations between meaning and truth, meaning and
      necessary truth, meaning and synthetic necessary truth.)

     to be made available to readers in the Library under the
     conditions determined by the Curators. 28 May 1962

   (b)I agree to my thesis, if asked for by another institution,
          being sent away on temporary loan under conditions
          determined by the Curators.

- Strike out the sentence or phrase which does not apply.


                    Signed     A.Sloman
                 Date       24th May 1962






THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
IN THE UNIVERSITY OF OXFORD



Abstract

of

KNOWING AND UNDERSTANDING
(Relations between meaning and truth, meaning and
necessary truth, meaning and synthetic necessary truth.)

A. Sloman


Stamp added:
BODLEIAN LIBRARY OXFORD
DEPOSITED THESIS
19.12.62



St. Antony's College
Oxford
Trinity Term
1962



Note added to online version 10 Feb 2014

       Some of the work for this had previously been done
       while I was at Balliol College 1957-1960. I originally
       came to Oxford as a mathematics graduate and gradually
       transferred to philosophy via Logic, supervised at
       first by Hao Wang. When I transferred to philosophy
       David Pears was assigned to me as supervisor.

       St Antony's College provided a two year Senior Scholarship
                      1960-1962 which allowed me to complete the thesis.
Page i

KNOWING AND UNDERSTANDING

Abstract


       The aim of the thesis is to show that there
are some synthetic necessary truths, or that synthetic
apriori knowledge is possible. This is really a pretext
for an investigation into the general connection between
meaning and truth. or between understanding and knowing,
which, as pointed out in the preface. is really the first
stage in a more general enquiry concerning meaning. (Not
all kinds of meaning are concerned with truth.) After
the preliminaries (chapter one). in which the problem is
stated and some methodological remarks made, the investi-
gation proceeds in two stages. First there is a detailed
inquiry into the manner in which the meanings or functions
of words occurring in a statement help to determine the
conditions in which that statement would be true (or false).
This prepares the way for the second stage, which is an
inquiry concerning the connection between meaning and
necessary truth (between understanding and knowing apriori).

The first stage occupies Part Two of the thesis, the second
stage Part Three. In all this. only a restricted class of
statements is discussed, namely those which contain nothing
but logical words and descriptive words, such as "Not all
round tables are scarlet" and "Every three-sided figure
is three-angled". (The reasons for not discussing
proper names and other singular definite referring ex-
pression as given in appendix I.)

Meaning and Truth.
Part two starts with some general remarks about
propositions and meanings. We can answer questions as
to what meanings and propositions are, by describing the


Page ii


criteria for deciding whether words are used with the same
meanings or whether sentences are understood to express the
same proposition. It turns out that there are various
levels at which criteria for identity are required, and
various kinds of criteria. (E.g. we need criteria for
identifying the functions of statements as opposed to
commands or questions, criteria for distinguishing the
functions of descriptive words and referring expressions,
criteria for identifying or distinguishing the meanings
of individual descriptive words.) In our language, and
others like it, the existence of a conceptual scheme
involving universals (observable properties and relations is
presupposed by the methods used for making the finest
distinctions between meanings of descriptive words.
(Section 2.C.)

i) Descriptive words.

After the general remarks in chapter two about criteria
for identity of meaning and the existence of universals,
chapter three goes on to show in some detail how descriptive
words (such as "scarlet", "round", "glossy", "table", and
"sticky") can be given their meanings by being correlated
with observable properties er combinations of properties.

These words can be classified according to how their
meanings are "synthesized" from properties. There are
logical syntheses and non-logical syntheses, and both
kinds may be further subdivided. (In 3.C a tentative
answer is given to the question: How does talking about
universals, i.e. properties and relations, explain our
use of descriptive words?) In this and the next chapter
many hidden complexities, including a number of different
kinds of indeterminateness (4.A and 4.B) are found even
in the meanings of innocent-looking words like "horse" and

Page iii

"red", but these complexities are taken account of within
the framework of a theory which does not assume that cor-
relations between words and universals must be of the simple
one-one type. The existence of "borderline cases" is due
to the existence of these complexities.

The importance of all this is that it shows how "sharp"
criteria may be used for identifying and distinguishing
meanings of descriptive words, and helps to explain why
the debate about the existence of synthetic necessary truths
has gone on for so long: namely, philosophers have un-
wittingly used loose and fluctuating criteria for identity
of meanings. Another cause has, of course, been unclarity
about the significance of the terms "analytic", "synthetic",
"necessary", etc. These are dealt with later on, their
application being illustrated by examples arising out of
the discussion of semantic correlations between descriptive
words and universals.

ii) Logical words.

Part Two concludes with chapter five, in which the role
of logical constants in sentences is explained by extending
and generalizing some ideas of Frege, Russell and
Wittgenstein (in "The Tractatus"). The explanation makes use
of the concept of what I call a rogator, which, like a
function, takes arguments and yields values; the differ-
ence is that to a function there corresponds a rule or
principle which fully determines its value for any given
argument-set, whereas to a rogator there corresponds a
principle or technique for finding out the value, the
outcome of which may depend on contingent facts, or how
things happen to be in the world. So the value of a
rogator for a given argument-set is not fully determined
Page iv


by the rogator and the argument-set, but depends on facts
which may have to be discovered by empirical observation,
and may change from time to time. The essential thing
is that there is a technique, which can be learnt, which,
together with the argument-set and the observable facts,
determines the value. A special type of rogator is a
"logical rogator", which corresponds to the logical form
of a proposition and may be represented by sentence-
matrices, such as "all P Q's are not R". A logical
rogator takes as arguments sets of descriptive words, such
as ('round', 'table', 'scarlet') and yields as values the
words "true" and "false". Which is the value depends on
the meanings of the descriptive words (the properties with
which they are correlated) and the facts. (In 5.B.18 a
variation on this is mentioned, in which sentences and
their negations are taken as values.) In learning to
speak, we learn general rules for the use of logical words
and constructions. and these are what determine which
logical technique (or which logical rogator) corresponds
to any sentence. This shows that the commonly held view
that the functions of logical words are explicable in
purely syntactical terms is either false or vague and
superficial. What lies behind it is the fact that the
distinguishing feature of logical constants is their
topic-neutrality (5.A): they are governed by rules which
are so general that from the occurrence of a logical word,
c.g. "or". in a sentence one can deduce nothing about the
subject-matter, or topic, of which it treats.

Thus, Part Two shows that the meanings or descriptive
words are given by correlations with universals, and the
meanings or functions of logical words by correlations
with logical rogators, or general logical techniques for
Page v

finding truth-values, and explains how these meanings or
functions determine the conditions in which sentences
composed of descriptive words and logical constants express
true, or false, propositions.

((Some by-products of this are mentioned in the thesis.
Logical relations. such as entailment and incompatibility,
are explained as arising out of relations between logical
rogators, or, more specifically, between techniques for
discovering truth-values. This explains the connection
between the geometrical forms of sentences and logical
properties of the propositions they express. and shows how
formal logic is possible. Secondly, we can clarify the
difference between the "implications" of a statement and
its "presuppositions", by pointing out that a rogator, like
a function. has a limited "domain of definition" and,
further, certain empirical conditions may have to he satis-
fied if its technique is to be applicable to finding out the
value corresponding to a given set of arguments. Thus,
the presuppositions of a statement are concerned with the
conditions which must be satisfied if it is to have a
truth-value at all, and its implications are concerned with
what must he the case if the techniques are applicable and
the truth-value comes out as "true". All this serves to
explain why apparently well-formed sentences may he sense-
less, and seems to provide the basis for a simpler and more
general theory of types and category rules than that which
uses the notion of the "range of significance" of a predicate.
This is suggested, but not developed, in 5.E.))


Meaning and Necessary Truth

Part Three explains. in chapter six, how it is possible
for a statement to he analytic and then goes on, in chapter
seven. to give a more general account of necessarily true
statements and show that some are synthetic.
Page vi

       Some uses of the concepts of "possibility" and
"necessity" are explained by drawing attention to certain
general and fundamental facts, but for which our thought
and language and experience could not be as they are, such
as the fact that universals (observable properties and
relations) are not essentially tied to those particular
objects which happen to instantiate then. (The table
on which I am writing is brown, but it might have had a
different colour, and the colour brown might have had
other instances than those which it does actually have,
without being a different colour: all this makes use of
some of the general remarks about conceptual schemes, in
chapter two.) This shows how it makes sense to talk
about "what might have been the case but is not". or "what
is possible though not actual". It is then noted that
although universals are not essentially tied to their
actual particular instances, nevertheless they may be
essentially tied to one another (or incompatible with
one another, etc.). The property of being bounded by four
plane surfaces cannot occur without the property of having
four vertices. These connections between properties can
justify our assertion of some kinds of subjunctive con-
ditional statements, such as "If this had had four sides,
than it would have had four angles", and therefore
enables us to assert that certain universal statements
*could* not have had any exceptions. This explains a
concept of "necessity", in terms of what would be the
case in any possible state of this world, where "this
world" is a world containing the same universals (observable
properties and relations) as our world.

The description of the connection between meaning and
Page vii

necessary truth follows on naturally from the general
description of the connection between meaning and truth.

Normally the value of a rogator for a given set of argu-
ments depends an how things are in the world, and has to
be discovered by applying the appropriate technique.

But in some "freak" cases the value is independent of
the facts and may be discovered by examining the tech-
nique and the arguments. or relations between the arguments.
In particular, the truth-value of a proposition, in "freak"
cases, may be discovered by examining the logical technique
corresponding to its logical form and noting relations
between the meanings of the non-logical words used to
express it. Since how things are in the world need not
be known, the truth-value would be the same in all possible
states of affairs. (But the truth value may also be
discovered in the normal way, by applying the technique instead of
examining it.

If one fails to notice that it is necessarily true that
every cube has twelve edges one may set out to discover
its truth by observing cubes. The fact that empirical
enquiries are relevant even where analytic propositions
are concerned brings out the defects in most accepted
definitions of "analytic".)

    So the truth-value of a necessarily true proposition
is determined by (a) its logical form, or the logical
techniques corresponding to its form and (b) relations
between the meanings of non-logical words, or, more
specifically, connections between the properties referred
to. The notion of a definition or partial definition
is examined and found to generate one kind of relation
between meanings or properties, called "identifying
relations". An "analytic" proposition may then be defined
as one whose truth-value can be determined only by its logical


Page viii

form and identifying relations between meanings. This
leaves open the question whether there are other sorts of
connections between properties, in virtue of which state-
ments may be necessarily true though not analytic. This
question is investigated in sections 7.C and 7.D, where
it is shown how simple geometrical proofs (using diagrams,
for example) may enable one to perceive connections between
geometrical properties in a manner which is quite different
from the way in which one draws logical conclusions from
identifying relations between the meanings of words. This
description of the workings of "informal proofs" shows,
therefore, how it is possible first of all to identify
universals by being acquainted with them and then, by
examining them, to have a further "insight" into their
interconnections. This helps to answer the question which
was left unanswered in chapter five, as to how one can
discover that logical rogators are connected in certain
ways (and hence that propositions have certain logical
properties) by examining their techniques.

All this shows that there are both analytic and syn-
thetic necessary truths. The former are true in virtue
of their logical form and identifying relations between
the meanings of non-logical words used to express them.
The latter are true in virtue of all this, and, in addition,
some non-identifying relations between meanings. In order
to know the truth-value of an analytic statement, it is
enough to know how the logical constants work and that some
of the descriptive words stand in certain identifying
relations with others, such as that some of them are used
as abbreviations for other expressions. But when the
statement is synthetic, one must, in addition to knowing
that the meanings of the words are identifyingly related
Page ix

in certain ways, also know what the meanings of some of
the descriptive words are, so as to be able to examine
the properties referred to and discover the connections
between them. ((It is assumed that all these statements
have truth-values. This cannot always be discovered
apriori. See remarks about applicability-conditions for
logical techniques.))

((The discussion of informal proofs is only a
beginning. and does not pretend to he conclusive. Com-
plications arising out of indeterminateness of meaning
and the fact that neither "absolutely specific" nor
"mathematically perfect" properties (e.g. the property
or being bounded by four perfectly plane sides) can be
described as "observable", are mentioned, but not discussed
in detail.))

Chapter eight is a concluding summary. It is
followed by appendices. The first explains why nothing
has been said about singular definite referring expressions.
The second describes some of the confusions which arise
out of too much concentration on symbolic logic. The third
discusses the notion of "implicit knowledge": knowledge
which one say be able to apply without being able to
formulate. The fourth makes some remarks about philoso-
phical analysis and suggests some further developments of
the thesis. The fifth appendix tentatively suggests that
examples of synthetic necessary truths may be found in
connection with other than geometrical properties. Finally,
the concept "apriori" is discussed, briefly.


Note on online version of Abstract:
This was originally transcribed from digitised PDF 9 Feb 2014.
Some errors may remain.
[Please send corrections to a.sloman AT cs.bham.ac.uk]

Note:
Replaced 'The avowed aim' with 'The aim'


====================================================
This is part of A.Sloman's 1962 Oxford DPhil Thesis
     "Knowing and Understanding"
====================================================


(Preface and contents)





THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
IN THE UNIVERSITY OF OXFORD





KNOWING AND UNDERSTANDING

(Relations between meaning and truth, meaning and
necessary truth, meaning and synthetic
necessary truth)

A. Sloman

St. Antony's College
Oxford



     .


                                                           Page i
PREFACE


       In this thesis I have tried to answer Kant's question:
"Are there any synthetic necessary truths?" by developing
a theory of meaning within which the question can be stated
clearly and given a decisive answer. However, I believe
the theory is of more general interest than this, since.
although it is formulated so as to deal only with the
connection between meaning and truth-conditions, it can be
extended quite naturally to include kinds of meaning which
have nothing to do with truth. This provides a framework
for the classification of types of relations between
meanings which treats relations between truth-conditions,
and, in particular, logical relations, as a special case.
My belief in the wider applicability of what I say in the
thesis is what explains the existence of many digressions,
not immediately relevant to the main question. Some of
these digressions are labelled as such by the word "note",
or by their occurrence as footnotes or appendices
( especially Appendix IV ).

       The main factor common to the theory developed within
the thesis and its proposed extension is the acceptance
of the existence of universals. The only kinds of
universals explicitly described as such (chapters two, thre
and seven) are observable properties of material objects,
but essentially the same concept of a universal is
implicitly involved in the notion of a "technique" for
discovering truth-values, illustrated in chapter five
(5.3). A full characterization of this wider concept
of a "universal" would require a detailed discussion of
the points made by Wittgenstein (in Investigations and
R.F.M) about the concept of "following a rule", in which


                                                          Page ii

he penetratingly criticizes his former beliefs. This
thesis could be regarded as a first step in the process
of patching up the Tractatus Logico Philosophicus so as
to meet some of those criticisms. At any rate, the point
I wish to make now is simply that the thesis is incomplete
not only insofar as its further developments are hardly
explored, but also, and more importantly, insofar as it rests
on a basis which still requires a great deal of investi-
gation. (This is hinted at in Appendix IV.8.a.)

       It will be clear from what I have said that my main
debts are to Kant and Wittgenstein: to the former for for-
mulating the main question and providing what seems to me
to be the right sort of answer, and to the latter for pro-
viding criticisms of the assumptions on which that answer is
based which throw their exact nature into much sharper
focus than ever before. (The reader may not find this
latter debt evident.)

       Now for some practical points. The order of develop-
ment in the thesis is not the most clear and logical one
possible, partly on account of the need for compression, and
partly on account of the fact that new ideas kept coming even
while the final draft was being written. (For example, a
great deal of chapters two and three - especially 2.C - is
intended to forestall objections to chapter seven, and ought,
ideally, to be preceded first by chapter seven and then the
objections. But that would have made the thesis much longer.)
For this reason the text is sprinkled with cross-references
either in parenthesis or in footnotes, as an aid to clarity.
It is hoped, however, that most of them can be ignored,
especially when they occur in footnotes, except when
the reader has forgotten an earlier definition or argu-
ment.


                                                         Page iii


         Finally, I should like to thank my supervisor, Mr.
D. F. Pears, for showing so much patience, and for
criticisms without which this thesis would have been far
more confused and obscure than it is.


                                                          Page iv


            C o n t e n t s

[Note: page numbers refer to original thesis]

               PART ONE: SOME PRELIMINARIES

Chapter one: Introduction                               1
           1.A. The problems                            1
           1.B. Methodological remarks                  5
           1.C. The programme                          13

       PART TWO: MEANING AND TRUTH

Chapter two: Propositions and meanings                 18
           2.A. Criteria of identity                   18
           2.B. General facts about language           24
           2.C. Universals and strict criteria         38
           2.D. The independence of universals         50

Chapter three: Semantic rules                          63
           Introduction                                63
           3.A. F-words                                64
           3.B. Logical syntheses                      70
           3.C. How properties explain                 83
           3.D. Non-logical syntheses                  93
           3.E. Concluding remarks and qualifications 102

Chapter four: Semantic rules and living languages     107
           4.A. Indefiniteness                        107
           4.B. Ordinary language works               117
           4.C. Purely verbal rules                   125

Chapter five: Logical form and logical truth          129
           Introduction                               129
           5.A. Logic and syntax                      130
           5.B. Logical techniques                    144
           5.C. Logical Truth                         166
           5.D. Some generalisations                  176
           5.E. Conclusions and qualifications        181
                                                           Page v

        PART THREE: MEANING AND NECESSARY TRUTH

Chapter six: Analytic propositions
     6.A. Introduction                                    194
     6.B. Some unsatisfactory accounts of the distinction 199
     6.C. Identifying relations between meanings          217
     6.D. Indefiniteness of meaning                       229
     6.E. Knowledge of analytic truth                     236
     6.F. Concluding remarks                              249

Chapter seven: Kinds of necessary truth                   260
     Introduction                                         260
     7.A. Possibility                                     261
     7.B. Necessity                                       272
     7.C. Synthetic necessary connections                 283
     7.D. Informal proofs                                 294
     7.E. Additional remarks                              319

Chapter eight: Concluding summary                         329

APPENDICES                                                335
(NOTE: the table of contents here expands the one in the thesis,
by including abstracts of appendices.)

 I. Singular referring expressions                        335

This appendix explains why sentences with singular referring
expressions are excluded from distinctions explained here.

II. Confusions of formal logicians                        340

This appendix presents arguments against the view that a natural
language must include a formal system, and that logic is just a
matter of syntax. One of the key points, also made by Frege, is
that semantics cannot emerge from syntax alone: we also need to
take account of the functions of the symbols used, not just
their form.

III. Implicit knowledge                                      357

This appendix gives examples of several kinds of implicit
knowledge, including allowing for the deployment of implicit
knowledge to be unreliable sometimes (Compare Chomsky's
Competence/Performance distinction, 1965). The ability to do
logic and mathematics, as well as many other kinds of things,
depends on the use of implicit knowledge, which can be very
difficult to make explicit.

NOTE:(At that point I knew nothing about the young science of AI
which was beginning to provide new techniques for articulating
implicit knowledge.)


                                                          Page vi

IV. Philosophical analysis                                   372

The ideas about implicit knowledge in Appendix III are used in
Appendix IV to explain some of the puzzling features of the
activity of conceptual analysis (disagreeing with R.M. Hare's
explanation). This leads to further discussion of the nature of
philosophical analysis and the claim that it cannot be concerned
merely with properties of concepts: it must also be concerned
with the world those concepts are used to describe, which may
support different sets of concepts.

NOTE added 9 Feb 2014:
This theme was taken up again many years later in my paper
distinguishing logical topography from logical geography in
http://www.cs.bham.ac.uk/research/projects/cosy/papers/#dp0703


 V. Further examples                                        381
VI. Apriori knowledge                                       386

Bibliography                                                389

END OF TABLE OF CONTENTS

[Transcribed from digitised PDF 10 Feb 2014. Some errors may remain.]
[Please send corrections to a.sloman @ cs.bham.ac.uk]



              ====================================================
               This is part of A.Sloman's 1962 Oxford DPhil Thesis
                           "Knowing and Understanding"
              ====================================================








                                 P a r t  O n e





                       S O M E   P R E L I M I N A R I E S




                                                           Page 1

Chapter One

INTRODUCTION

1.A.  The Problems

1.A.1. In order to know that some statement is true, one
must understand that statement, one must know what it
means. Sometimes understanding seems to be enough.

For example, I know that the statement "All bachelors are
unmarried" is true simply because I understand it, in
particular because I know the meaning at the word
"bachelor". In general, however, understanding is not
enough: one must do more than learn the meaning of a
statement in order to discover whether it is true or
false.

       In the simplest cases one must, in addition to
understanding, also carry out some sort of observation
at facts, or rely on the reports of others who have done
this. In these cases, the meaning of a statement does
not, on its own, suffice to determine whether it is true
or not, for facts, that is, the way things happen to be
in the world, may also be relevant.

       The main aim of this thesis is to inquire whether
the truth or falsity of a statement depends on how things
happen to be in the world in all cases where the meaning
does not suffice to determine this. Where understanding
a statement does not on its own enable one to know that
it is true, is some empirical observation of contingent
facts always necessary? In short, the thesis is con-
cerned with tun aid philosophical questions, first clearly
formulated by Kant, namely:


                                                           Page 2
Are there any synthetic necessary truths?
Is synthetic a priori knowledge possible?

1.A.2. These problems generate a whole family of prob-
lems, some of which will be tackled in this thesis. First,
the terms in which the questions are expressed must be
explained, and also many related terms, such as "analytic",
"contingent", "empirical", "factual", "meaning", "defini-
tion", "concept", "proposition", and so on.

       Although often used by philosophers, these words have
no precisely defined standard meanings. A large part of
this thesis will, therefore, be concerned with their
clarification.

1.A.3. In the course of this process of clarification,
a wide range of further problems will arise. Exactly
how is the meaning of a statement ever relevant to whether
it is true or not? What happens when one learns to
understand a statement, and what is the connection between
this and what happens when one comes to know that it is
true? How is it possible for a statement tn be true
simply in virtue of what it means? How is it possible
for a statement to be necessarily true, or to be known to
be true without empirical investigation? Are all nec-
essary truths analytic? Can all necessary truths be
known a priori? Are there different sorts of necessary
truths worth distinguishing from one another, even if the
distinctions are not the same as the analytic-synthetic
distinction? Are there different ways in which a state-
ment can be true in virtue of what it means? When a
proposition is true by definition in what sense can we
describe it as "true"?


                                                           Page 3


1.A.4. Even if we start with questions about language
or about linguistic entities, as some of these appear to
be, we soon find ourselves dealing with problems about
other entities, such as the persons who use words, or the
things and properties to which they intend their words
to refer. We shall find that it is impossible to answer
questions in logic without discussing a much wider
of topics. Logic seems to be inseparable from metaphysics,
from philosophical psychology, and from epistemology.

It is concerned with concepts and meanings and pro-
positions and truth; so it cannot be divorced from meta-
physics, the general study of the kinds of things which
can fall under concepts, which can be referred to in
propositions, and which can make statements true or false.

       We cannot discuss problems about meanings or concepts
or propositions without mentioning thinkers or speakers,
the persons who use with meanings, who understand
or intend propositions to be expressed by sentences.
So a philosopher of logic should be prepared to discuss
various mental statue or activities, such as meaning,
intending, thinking, or paying attention. This is why
I say that logic cannot be divorced from philosophical
psychology, the study of ordinary psychological concepts,
or at least those connected with our use of language.
(It may sometimes look as if logic can be done by talking
only about symbols, but there must always be some implicit
reference to persons who use these symbols or could use
them, for symbols have meanings only insofar as they are
taken to have meanings by some person or group of persons.
They do not have meanings in themselves.)

Finally, the discussion of concepts and propositions
leads us into the discussion of ways of coming to know that
objects fall under concepts, or that propositions are true



                                                           Page 4
or false, and this shows that there are connections
between logic and epistemology.

All this should help to explain the fact that although
this is primarily a logical investigation, a very wide
range of problems and topics must be dealt with. Unfor-
tunately, there will not be space to deal with all the
problems which are raised in the discussion.

1.A.5. Although the problems to he discussed were first
raised so long ago, they still seem to be of some interest.
Indeed, it is sometimes suggested that the question of
synthetic necessary truth is one of the most important of
the philosophical problems which remain unsolved, for
connected with it are problems about philosophical method
which ought to be of interest to philosophers engaged in
conceptual analysis, if they wish to be clear about what
sort of thing they are trying to do. Are they merely
producing reports on linguistic usage, or uttering analytic
statements. or demonstrating truths which are not analytic
but nevertheless necessary, or what?

       It has recently been pointed out that there are
difficulties in saying precisely what is meant by such
words as "analytic", "synonymous", "necessary" (mainly
by Quine), and in consequence it seems to have become
rather unfashionable to employ them in philosophical
discussions. Certainly a survey of recent publications
in philosophical Journals shows that people are not at all
clear as to what the analytic-synthetic and necessary-
contingent distinctions are. But they cannot really get
along without them, end and up talking about "absurdity",
"logical impossibility", "contradiction", "nonsense",
"inconsistent usage", "conflict with ordinary language",
"inference-licences", "rules of grammar", etc., often


                                                           Page 5

unaware that, in a groping sort of way, they are making
use of Kant's distinctions. It seems to me to be time
they faced up to this fact and tried to be clear about the
distinctions, taking seriously some of the problems
connected with them. This is what I shall try to do.

1.A.6. In the remainder of this chapter some remarks will
be made about the procedure to be followed in the rest of
the thesis.

1.B.    Methodological remarks

1.B.1. I have undertaken to explain the meanings of
certain terms and to answer some of the questions expressed
in these terms. In order to do this without circularity
I should have to avoid using words such as an "necessary",
"contingent" and "analytic" until after explaining their
meanings, but this would increase the length of the thesis
considerably. My excuse for using the words before
shewing that they correspond to real distinctions, apart
from the fact that it makes a great deal of compression
possible, is that philosophers and others do seem to have
some sort of intuitive understanding of them, and their
usage seems often to be in accord with the definition
which will be given later on, even though these philosophers
give very different definitions from mine when they try to
say precisely what such words mean. (This is an illustra-
tion of the familiar fact, to be discussed in the Appendix
on "Implicit Knowledge", that one may perfectly well know
how to use a word. without being able to say how it is
used.)

      When the meanings of these words are finally explained,
this will not be done by giving an explicit definition.



                                                           Page 6
Instead, the explanation is more nearly a process of
drawing attention to those aspects of our thought and
experience and our use of language which make it possible
for the words to be used. This process in some ways
resembles ostensive definition, except that here the
"pointing" is done with words.

1.B.2. The description of those aspects of our thought
and language and experience which make it possible to
distinguish analytic from synthetic propositions, and
necessary truths from contingent ones, will proceed from
a certain point of view, which I shall now try to des-
cribe, in order to reduce the possibility of misunderstandings.

Not everything that can be said about thinkers and
speakers has a content which can be exhausted by descriptions
from the point of view of experimental psychologists or
physiologists, or anthropologists who study human beings
as if they were only one kind of animal, to be observed
in a scientific way. The reason why statements about
persons cannot always be translated into the statements of
scientific observers is that they answer different kinds
of questions, they serve different kinds of purposes, they
are made and listened to by persons with different sorts
of interests and different kinds of curiosity.

When I make a statement about what a person thinks or
feels, or what he intends, or when I try to explain his
behaviour in terms of what he wants or how he reacts to
what he sees, or thinks he sees, then I may be trying to
say something which enables other persons to know what it
would have been like to be in his place: and this is not
the same thing as describing the physical state of his
brain or his dispositions to produce certain publicly
observable noises or movements in specified situations -


                                                           Page 7

though I may be talking about these things as well.

1.B.3. In order to understand descriptions or explana-
tions which refer to mental states or processes, it is
not enough to have observed their outward manifestations
or the concomitant physical or physiological processes
(the latter is not even necessary, let alone sufficient:
people are able to understand such statements as "I
jumped because I saw a face at the window", without
knowing anything about electromagnetic waves or what
goes on in the brain). In order to understand completely,
one must have had experiences sufficiently similar to
those described. One must be the same kind of being.
For example, unless I know what it is to act for reasons,
or perhaps to act because I want something, I cannot fully
understand a statement such as the following: "I climbed
on the chair because I wanted one of the biscuits on the
top shelf". Such "ignorance" would not, however, pre-
vent my coming to understand a scientific explanation
(e.g., at a physiological level) of this kind of behaviour.

       All this should be remembered in connection with
Chapters Three, Six and Seven, below (especially section
3.C). For example, when it is asserted that a property,
such as a shape or a colour, can explain how a person
uses a descriptive word, or how he groups things together,
this is not an assertion which might be made by an
experimental psychologist or a physiologist, but the sort
of statement which can be understood (fully) only by a
person, by one who knows what it is like to select objects
on the basis of their colour or shape or some other
visible property.


                                                           Page 8
1.B.4. It is sometimes suggested that the reason why
statements about conscious persons cannot be translated
into statements about publicly observable physical events
or states of affairs is that the two sorts of statements
describe things at different levels. For example,
statements about individual persons are at a different
"level" from statements about crowds or nations, and
statements about the positions of a moving point of light
are at a different "level" from statements about its
velocity, the radius of curvature of its path, and so
on. (Wittgenstein's comparison, on p.179 of "Philosophical
Investigations" might suggest that this was his view.
But see below: 1.B.6.(note).) This makes it look as if
it were just a matter of a difference of degree of
complexity, or a difference in which facts are counted as
relevant, or a difference in the ways in which the facts
or objects described are "organized" or "structured".

    The impossibility of translation, and the failure of
attempts to find logical connections between classes of
statements at the two levels might then be explained in
terms of the "open texture" of the concepts at a higher
level. However, if there were just a difference in
level, then there would surely be some logical implications
from one level to the other despite the "open texture".
For example: a complete description of the positions of
a dot at all times would logically entail statements
about its velocity, etc. In some cases a complete des-
cription of the behaviour of millions of individual per-
sons would entail a statement about a nation, such as
that the nation was at war with another. But no desc-
cription of physical and physiological states, however
exhaustive, can entail the statement that a person has a
toothache, or that he wants a drink. Something is always


                                                           Page 9
left out, namely a description of what is going on from
his point of view, how it feels to him.

1.B.5. The difference between statements to which
I am trying to draw attention, therefore, are not
merely differences in "level", or in the method of
organization of facts, but differences in the point or
view from which the descriptions are given. This is
what seems to me to be one of the most important reasons
for the failure of reductive programmes.

       This question cannot be dealt with in detail here.
I shall simply assume that there is a difference and that
it can be characterized thus: when describing things
from the "rational" or "personal" point view, one
assumes that it makes sense to wonder what it would be
like to be in the position of the person being described,
whereas, from the scientific (or "tough-minded") point
of view one tries only to describe what could be observed
by anyone at all, and seeks causal explanations for
human behaviour. For example, the experimental psycho-
logist investigating threshold levels is concerned only
with the subject's responses to stimuli, he is no more
concerned with what it would be like to be in the subject's
position, than a physicist wonders what it would be like
to be in the position of a magnet which attracts iron
filings, or an electron which is deflected by a magnetic
field. (In the early stages of development of a science,
of course, people may be concerned with what it feels
like to be set in motion by an external force, for
example. Here an intermediate point of view is adopted.)

1.B.6. The descriptions and explanations in the chapters


                                                          Page 10

which follow are given from this personal or rational
point of view, which means that I shall have to rely
heavily on the reader's ability to reflect on his own
experience while I try to draw attention ts certain
aspects of it. I am not writing from the point of view
of an anthropologist or psychologist or physiologist
of the sort who simply observes people "from the outside"
and records correlations between stimuli and responses.

(Compare the point of view adopted in Quine's book:
"Word and Object".) I am writing from the point of view
of a person who thinks and speaks and has experiences, and
I am trying to describe certain general features of his
thought and language and experience from that point
of view. This may be described, therefore, as a "pheno-
menological" essay.

       When, on occasion, I describe anything from the
point of view of a person who observes other persons, then
it is important to remember that the observer is the same
kind of being as the ones which he is observing, other-
wise he can only observe them, he cannot understand them.

That is, he can only record that they produce certain
marks on paper, or certain noises, or that they respond
in certain ways when vibrations in the air stimulate
their ear-drums, or when electromagnetic waves reach their retinas.
He will not be able to record that they are saying
anything, let alone what they are saying, or that they hear
sounds or see colours; or, if he can record these things,
then his records will be only short-hand descriptions of
patterns of observable behaviour. (Remember that we do
not see wavelengths nor hear vibrations when we see
colours and hear sounds.)


                                                          Page 11
(Note. Wittgenstein tried -- in "Philosophical
Investigations" -- to describe things from an inter-
mediate point of view: he talked from the point of
view of one who uses language, who is a person and can
communicate with other persons, but he concentrated
on publicly observable phenomena, on the kind of evi-
dence which makes people assume that they are communi-
cating successfully, on the publicly observable social
aspect: of our use at language. It seems to me that
he omitted a great deal that can be said about our
thought and language from the point of view of the one
who thinks and speaks and knows what he means, and I have
tried to fill that gap, or part of it.)

1.B.7. The point of view from which this thesis is
written is only one among several different possible
points; of view. It should not be thought to be more
correct or more important than any other (though perhaps
it might be argued that it is the only one which a
philosopher can adopt without laying himself open to
the charge that he is trying to he an arm-chair scientist!)
There are many different sorts of interest which one
may have in the world, and there is no reason why only
one of these interests should be fed, to the exclusion
of all others.

       This seems sometimes to be denied by so called
"tough-minded" philosophers. For example, Quine wrote,
in "Word and Object" (p. 264):

     "If there is a case for mental events and mental states,
     it must be just that the positing of them, like the
     positing of molecules, has some indirect systematic
     efficacy in the development of theory. But if a
     certain organisation is achieved by thus positing
     mental states and events behind physical behaviour,
     surely as much organization could be achieved by
     positing merely certain correlative physiological
     states and events instead."

But "organisation" and "systematic efficacy in the
development of theory" (presumably scientific theory)



                                                          Page 12

need not be the only aims of a philosophical enterprise.
There is another point of view, the one which I have
tried to characterise. It has not been arbitrarily
invented by me, nor is it anything mysterious or unfam-
iliar, for we are constantly asking people how they feel,
asking them what their intentions are, what they want,
Why they behave as they do; and we are not merely
requesting information which could be supplied by any
sufficiently clever and well-informed physiologist. If
we were to abandon this point of view in our dealings
with other people (e.g., when we ask "Is your toothache
very bad?"), our whole attitude to life and personal
relations would have to change. There would be no
more scope for such utterances as "I cannot understand
why he thinks his plan has the slightest chance of
working."

1.B.8. In what follows, it will therefore be taken for
granted that there is a point of view of the sort which
I have tried, somewhat too briefly, to characterize.
From this point of view, the point of view of a conscious
person who talks and thinks about the things he sees
about him, I believe that a coherent description can be
given of much of our language, thought and experience,
which shows that there is room for a distinction between
propositions which are analytic and propositions which
are synthetic. (Cf. Quine: "No systematic experimental
(sic) sense is to be made of a distinction between usage
due to meaning and usage due to generally shared col-
lateral information". Op.Cit., p.43.) Within this
general framework it will be shown that there is a clear
and interesting sense in which some statements which are
not analytic are nevertheless necessarily true.


                                                          Page 13

1.C. The Programme

1.C.1. So far the main problem has been stated, some
subsidiary problems mentioned, and remarks made about
the general framework within which these problems will
be treated. Now I shall briefly describe the programme
to be followed in succeeding chapters.

       After these preliminaries, the remainder of the thesis
will be divided into two parts. The first is concerned
with the general connection between meaning and truth,
the second with the connection between meaning and nec-
essary truth. (Some side issues and further developments
will be dealt with in appendices at the end.)

       Part Two, on meaning and truth, is divided into
four chapters. First I shall try to explain, in a
general way, how we can talk about meanings and pro-
positions, bringing out some presuppositions of such talk.
This lays the foundation for most of the remainder of the
thesis. Secondly, in Chapter Three, a detailed des-
cription will be given of some ways in which descriptive
words and expressions (e.g. adjectives) can have the
meanings which they do have, in virtue of their being
intended to refer to universals (i.e. observable pro-
perties and relations). This will involve a great deal
of over-simplification, and in Chapter Four some of the
complications in our ordinary use of words will be des-
cribed which are overlooked in Chapter Three. Finally,
Chapter Five describes our use of logical words and con-
structions (pointing out that attempts to reduce logic
to syntax are quite misguided), and shows how the way in
which logical constants and descriptive words are combined
to form a sentence determines the class of possible states


                                                          Page 14

of affairs in which that sentence would express a true
proposition. This shows how the truth of a statement
can depend both on what it means and on what the facts are,
and prepares the way for a description of cases in which
the truth depends only on what the statement means.

       Part Three, which is concerned with necessary truth,
starts with the explanation in Chapter Six of how it is
is possible for a statement to be analytic (true by defini-
tion), and how we can know that such statements are true
without knowing how things happen to be in the world.

Chapter Seven seeks to explain the meanings of "necessary"
and "contingent", and to show that there are good reasons
far saying that the necessary-contingent distinction is
different from the analytic-synthetic distinction. It
is hoped that some examples taken from elementary geo-
metry will illustrate the claim that there are some
necessary truths which are not analytic. (For example,
is it analytic that no solid object is bounded completely
by three plane surfaces? Further examples are given in
an appendix.)

       In all these chapters some general and sweeping
statements are made, followed by attempts to show how they
are oversimplified and ignore complications. In most
cases, however, there will be no room to go into these
qualifications in detail.

1.C.2. It will be noticed that no account is to be given
of our use of proper names and other singular definite
referring expressions which refer to particular material
objects. The reasons for this exclusion are stated in
Appendix I. The whole discussion will be restricted to
relatively simple statements about material objects and
their properties, such as "All red things are round", "No


                                                          Page 15

green things are glossy and pink" and "If a rectilinear
figure is three-sided, then it is three-angled".
These statements contain only logical constants and
descriptive words referring to observable features or
properties or combinations of properties (little will
be said about relations), and they are all universal
in form: that is, they contain no definite referring
expressions which presuppose the existence of particular
material objects.

       It is important to stick to relatively simple cases
at first, as it is much more difficult to avoid confusions
if one tries to discover things in a completely general way
right from the start, so as to take account of even the
most complex examples. It should not simply be taken
for granted, as it often is by philosophers, that it
obviously makes sense, or any clear sense, to apply the
analytic-synthetic distinction to almost every sort of
utterance. A failure to be clear about the conditions
in which the distinction can be applied leads to great
confusion. Some limitations on the applicability of
the distinction will be described later on, based on the
facts pointed out in Chapter Four.

1.C.3. The main conclusion of this essay, that there
exist synthetic necessary truths, or synthetic necessary
connections between concepts and propositions, was first
put forward by Kant. But my aim is not exegesis. I
shall not be concerned with whether Kant really was trying
to say the sort of thing which I shall be saying, or whether
he would have approved, and there will not be as much in
the way of reference to and comment on his texts as there
might have been in a more scholarly work. (Nevertheless,
It appears to me that most of my main arguments can be


                                                         Page  16

found in Kant's "Critique of Pure Reason", though often
expressed in an obscure and muddled fashion. He did
not, after all, have the benefit of advances in clarity
and insight achieved by philosophers during this century
and the last.)

The reader who desires an historical account of the
development of the problems discussed here, or criticism
or the views put forward by other philosophers, is referred
to "Semantics and Necessary Truth", by Arthur Pap. In
order to save space I shell refer to the writings of other
philosophers only when I think that this will help to
clarify what I am saying. (Many of my debts will have to
go unacknowledged.)

1.C.4. To summarize, I shall try, making use of the
assumptions and methods described in the previous section,
to describe the general connection between the meanings
of certain sorts of statements and the conditions in which
they are true, and then show how it is possible for a
proposition to be true solely in virtue of what it means,
that is, to be analytic. The question will then be
raised whether the class of analytic truths includes all
necessary truths, and the negative answer will be illus-
trated by the description of examples of necessary truths
which are synthetic. I hope that in the course of all
this it will become clear why other philosophers have
reached different conclusions, the most important reason
being, I think, that they have used much looser (and
fluctuating) criteria for identity of meanings and pro-
positions than I use. Failing to make fine discriminations,
they fail to notice interesting relationships. (See
section 2.C.)


                                                         Page  17


It is hoped that there will be something of interest
in the general picture that will be painted, even if the
details are neither new nor very interesting in themselves.
















NOTE
[This chapter was transcribed from digitised PDF 10 Feb 2014.
Revised 24th June 2016 Some errors may remain.]
[Please send corrections to a.sloman @ cs.bham.ac.uk]


















Part Two
MEANING AND TRUTH

_________________________________________________________

NOTE: This is part of A.Sloman's 1962 Oxford DPhil Thesis
     "Knowing and Understanding"



    18

Chapter Two

PROPOSITIONS AND MEANINGS

2.A.    Criteria of identity

2.A.1.  Before we can explain how the analytic-
synthetic distinction and the necessary-contingent
distinction are to be applied, and discuss the question
whether they divide things up differently or not, we
must be sure we know what sorts of things they are meant
to distinguish. This applies also to the true-false
distinction. Sometimes it is not clear whether
philosophers think these distinctions apply to sentences
or to statements or to ways of knowing, or something
else, (c.f. section 6.A) and this leads them into ambi-
guity and confusion. I shall apply the distinctions to
statements or propositions, which are expressed by
sentences. When I talk about statements, I am talking
about sentences together with the meanings they are
understood or intended to have. When I talk about
propositions, I shall be talking about the meanings of
sentences (as understood by some person or group of
persons). I shall often use the words "statement" and
"proposition" interchangeably, as the difference between
them is important only in contexts in which we are con-
cerned about the actual form of words used to express a
proposition.
But this leaves unanswered the question: what is
the meaning of a sentence, or the meaning which it is
taken by some person or persons to have? The only way
to answer this question is to describe the ways in which
words and sentences can be used with meanings or understood
    19

with meanings, and to say clearly how to tell whether
two words or sentences are used or understood with the
same meaning or not. That is to say, we must describe
criteria for identity of meanings and propositions.
I shall show presently how the failure to do this may
lead to confusion and the begging of questions.
It will not be possible to answer all questions
about identity of meanings in this chapter. A few
rather vague remarks, concerning very general facts
about languages, will be made in section 2.B. Section
2.C explains why it is necessary to use physical properties
to provide criteria for identity of meanings of descript-
tive words, and section 2.D attempts to show that this
is not a circular procedure, nor completely trivial.
But first of all a few general remarks about criteria
of identity will help to explain why all this discussion
is necessary.
(It should be noted that most of the general remarks
of this Chapter will be presupposed in all that follows.)

2.A.2.  Why should we talk about criteria for identity
of meanings? Talk about true or false statements, or
about meanings or propositions, is not merely talk about
sentences or words, for these are merely signs, and.
cannot, as such, be true or false, or uniquely identify
the sense which has been given to them. We cannot tell
simply by looking at the shape of a mark which someone
has drawn, or by listening to the sound he utters, what
he means by it, or how others will understand it. For
one hearer or reader may understand it in one way, while
another understands it differently, and both may have
failed to understand what the author meant by it. What
is more, one and the same person may understand different
    20

tokens of the same type of word or sentence differently
on different occasions, or in different contexts. (It
is sometimes suggested or implied that this is entirely
due to ambiguities in words or expressions which refer
to particular objects, expressions such as "John"" "the
tree on the corner", "you", and so on, but it is impor-
tant not to forget that descriptive words may also be
ambiguous, though perhaps less systematically.) Talking
about meanings presupposes that we know what it means to
talk about the absence of ambiguity, that we have some
way of telling when words or sentences are understood or
intended to have the same meanings. So we need some
way of identifying the meanings with which words are
used and the propositions which they are intended, or
taken, to express. How can this be done?

2.A.3.  There must be an answer to this question, for
we are quite used to talking about meanings: we can
ask what a word means in German, whether two words
mean the same in English, and whether two persons mean
the same by the word "tadpole". In learning to speak
we implicitly learn the answers to questions about
identification. We learn to apply tests for telling
whether two persons mean the same by a word, whether two
words mean the same in a language, and so on. We learn
how to pick out the occasions when we are using words
inconsistently (i.e. with changing meanings) and the
consequences of doing this. (We do not need to be
given some philosopher's criterion for synonymy. So
we are not troubled by the impossibility of breaking out
of Quine's "circle of intensional words". See "Two
Dogmas of Empiricism".)
Having learnt to apply tests, and having acquired
    21

much skill in applying them over the years, we can go
right ahead and say such things as: "The word 'red'
refers to the hue of that object over there", or "The
English word 'red' means the same as the German word
'rot'", without offering any further explanation. We
learn to say and understand things like "I said that he
had taken the money, but I did not mean that he was a
thief. You obviously misunderstood me." In using
each familiar language about meaning and referring and
translating, we presuppose the answers to many questions
about identity of meaning and make use of very general
facts about language and words and sentences. So in
order to state answers to those questions, we must make
explicit the general knowledge which is presupposed and
used in this way, and this involves making explicit some
of the things we learn when we learn to talk.

2.A.4.  By explaining how we ordinarily tell whether
two sentences are taken to express the sane proposition,
or whether two persons take the same proposition to be
expressed by some sentence, and so on, we remove much of
the obscurity which is involved in talking about meanings and
propositions. People sometimes object to talk about
propositions (and other intensional entities, such as
properties) because they do not wish to populate the
world with such mysterious things. But meanings and
propositions are not mysterious, if adequate criteria of
identity are available, and they do not "populate the
world" in the sense in which material objects do, any
more than directions or numbers do. We can talk about
the number of things in a class of material objects
without mystery, and straight lines can have directions,
because there are tests for identity of directions and


    22

numbers.
It may be objected that there are no universally
acceptable criteria for identity of propositions. But
are there universally applicable and acceptable criteria
for identity of physical objects, or shadows, or events,
or persons? (Think of the paradox of the twice-mended
axe, or paradoxes connected with immortality and re-incarnation.)
In general, the suitability of criteria
for identity depends on our purposes in identifying,
and, if purposes vary, then what counts as adequate
criteria of identity may vary. What counts as "the
same colour" for the purposes of the editor of a cheap
glossy magazine may not count as "the same colour" for the
purposes of an artist or a fashion expert. What counts
as "the same length" for a civil engineer may not do for
the physicist, or the mechanical engineer.
Almost any set of criteria may be shown to break
down in some conceivable situation or other. That is,
criteria may come into conflict with one another, or may
yield no answer, or an unsatisfactory answer to the
question "Are they the same?" That, however, need not
make us say that the things which these criteria serve to
identify do not exist, or that they are in any way mysterious
entities. Material objects, colours, shadows and
lengths all exist. But some sets of criteria are more
stable and widely accepted, because they are more useful,
than other sets of criteria. Criteria for identifying
material objects are simply of more general applicability
than criteria for identifying propositions, or meanings,
or shadows.

2.A.5.  It is a matter of fact that there are ordinarily
    23

accepted criteria for identifying propositions and
meanings, on which we rely when we talk about ambiguities
or the correctness of translations. But they are not
infallible: some kinds of ambiguities and misunder-
standing are very difficult to discover and to eli-
minate (see note at end of 2.B). In addition, it
should be noted that, as remarked above, which criteria
are employed may depend on the purposes for which judge-
ments of identity are made. This may be illustrated by
fluctuations in the criteria ordinarily adopted for
eliminating ambiguities.
Thus, if one is interested only in testing for and
eliminating flagrant ambiguities, the kinds which matter
for purposes of ordinary conversation about the weather,
about one's latest illness, or about Mrs Jones' son who
insists on bringing tadpoles into the house, then one
may employ fairly loose criteria. Conversations on such
topics may usually be reported in a wide variety of ways
without the charge of misrepresentation being incurred.
On the other hand, if one is discussing the weather,
in an airport control-tower, or if one is a doctor
recommending a patient for treatment, or a zoologist
writing about the breeding habits of frogs, one may have
to be more careful about what one means: one must look
not only for obvious ambiguities, but for more subtle ones
too. Someone reporting what is said in such cases has
to be more careful about the words which he uses. Here
stricter criteria of identity for meanings are employed,
not necessarily because the words used are different,
but because the purposes served by their utterance are
different. When engaged in logical enquiries, one may
use still stricter criteria for identity: a logician
may regard two propositions as different if he wishes to
    24

investigate the logical relation between them, such
as mutual entailment, although even a careful scientist
would regard them as one proposition.
We shall find (in Section 2.C) that the only way
to avoid begging questions by ignoring subtle ambiguities
is to use the strictest possible criteria for identity
of meanings. This means that we shall have to be more
careful than most logicians have been, and look for
ambiguities even where they would be quite unimportant
for most philosophical or non-philosophical purposes.
Our motto will have to be the following remark made by
Kant while discussing the role in philosophy of appeals
to common ideas (in the introduction to "Prolegomena"):
"Chisels and hammers may suffice to work a piece of wood,
but for steel-engraving we require an engraver's needle."
(Cf. 2.C.9.).

2.A.6.  All this shows that we must not expect any very
simple general answers to questions about meanings.
There are various ways of comparing and distinguishing
meanings, none of them intrinsically correct, each suitable
for some purpose or other. But there is another compli-
cation, which arises out of the fact that tests for
identifying meanings operate at several different levels.
This will come out in the next section, where I shall
discuss some of the general presuppositions of statements
about meanings.

2.B.    General facts about language

2.B.1.  So far I have merely said that it is important
to be clear as to what we mean by talking about "the same
meaning", or "the same proposition", if we are to be clear
    25

about applying the analytic-synthetic distinction and
related distinctions. A thorough treatment of the
subject would require a detailed investigation of what
goes on when children learn to speak, when a child or
adult learns a foreign language, or when we look for and
find ambiguities in our own familiar language. This is
the only way to answer all the questions raised in the
previous section. There is no room for such a detailed
investigation here, so, in this section I shall try only
to indicate some of the sorts of things which would pro-
bably come out of it, by making a few rather vague
generalizations.

2.B.2.  First of all, it will turn out that even in
employing simple tests for discovering the meanings with
which ordinary descriptive words are employed, we make
use of very general presuppositions about language and
linguistic activities. For example, knowing how to tell
whether two persons mean the same by some word or sentence
presupposes a knowledge of what kind of thing a language
or linguistic utterance is. For otherwise we should not
be able to tell whether the noises people were making, or
the things they were "writing" were part of a game, or a
religious ritual, for example. This is not a trivial
problem of identification, .to be solved by looking to
see what the marks they write down look like, or listening
to the sound of the noises they produce, since it is quite
possible for the same marks or noises to be produced as
moves in a complicated game, even where making such a move
is not using a language. Without presupposing an answer
to the question whether a person is using a language, we
cannot find out what he means by what he says. It follows
that there must be some means of identifying kinds of
    26

linguistic behaviour as such, and these methods must be
learnt, at least implicitly, by a child when it learns
to talk, and applied, explicitly or implicitly, by anthro-
pologists when they first decide that the grunts and
clicks and other noises produced by the members of some
newly-discovered tribe are linguistic utterances.

2.B.3.  This is not all. Not only must we snow what
it is for behaviour to be linguistic behaviour, in
addition we must know what sorts of things various kinds
of linguistic activities are, if we wish to talk about
the meanings of words and sentences. We must, for example,
know what it is to make a statement, and how this differs
from asking a question, giving a command, exclaiming, or
expressing jubilation, and so on. Knowing what state-
ments, questions, commands, etc., are cannot be simply
a matter of knowing the appropriate forms of words, for
trying to teach someone what a statement is, is not just
a matter of teaching him which form of words are called
"statement-making sentences". We have to teach him what
can be done with these forms of words. We must know how
to tell whether he is doing the right sorts of things with
them or not. Hence, being a statement, a command, a
question, etc., cannot be merely a matter of having cer-
tain syntactical properties. It is a matter of being
correlated somehow with certain purposes and activities.
There are rules for the correct use of various forms
of sentences, and they differ from language to language.
So the important thing in common between statements in
one language and statements in another, which they do not
share with questions in either, cannot be a type of form
of words, but rather a type of use to which such a form
may be put, for example. Knowing what this use is
    27

involves knowing what it is for a statement to give
information, to be true or false, to be believed or dis-
believed, to be contradicted or agreed with. Knowing
what a command is, involves knowing what it is to want
to get something done. It also involves knowing what
it is for a command to be obeyed or disobeyed. Perhaps
it involves knowing what it is to have authority. Such
things, and many more, must be learnt when we learn to
speak.
All this knowledge is presupposed even by simple
statements about the meanings of simple words, since we
cannot understand what a descriptive word like "smooth"
means without knowing what it would be for that word to
be used with that meaning in a statement, or a command -
such as, "Bring me the smooth block of wood". We have
therefore found two different levels at which tests are
required for identifying: meanings:
I. There must be criteria for identifying acti-
vities as linguistic.

II. There must be tests for identifying and dis-
tinguishing different kinds of linguistic
activity.

(We could put this by saying: I. we must know what sort
of thing a language is, and II, we must know what sorts
of things various kinds of linguistic utterances are,
if we wish to talk about meanings.)

2.B.4. It is time now to be a little more specific.
Knowing what it is for a statement to be true involves
knowing what sorts of things a statement can be about,
and how the words in the statement determine which
particular things are referred to in the statement and
what is said about them. What sorts of things a statement
can be about will depend on the particular language in

question, and different statements in the same language
may be about quite different sorts of things. Compare
a statement about the weather in Oxford, a statement
about a mathematical theorem, and a statement about the
morally best course of action in some situation.
But even if we restrict ourselves to the class of
statements to be discussed later, which may vaguely be
characterized as being "about the perceptible world", we
may find that in some languages a greater variety of
things may be said than in others. Or different lan-
guages may involve quite different ways of looking at
the world, or, what comes to the same thing almost, quite
different ways of talking about it. In particular, they
may employ quite different conceptual schemes. Thus,
one language may treat the world as consisting of
arrangements of enduring physical objects, as English
does, and permit the making of statements which say things
about the qualities or properties of such objects, or
their mutual relations, or the changes in such properties
and relations, whereas another language does not use
the concept of an enduring physical object, permitting
only statements to the effect that the speaker is aware
of certain features (hardness, roundness, furriness, and
so on) in his environment.
Unless we are sure that two persons do not use
languages which differ in this way, unless we are sure
that they employ the same sort of conceptual scheme, we
cannot be sure that they mean the same by the statements
they make, even if their statements would be true in the
same situations. For when statements employ different
conceptual schemes, there may be no way of translating one
into the other, or there may be several different systems
of translation, all equally satisfactory or equally
    29

unsatisfactory, there being no question of one translation
being better than others. In such a case, there can be
no clear sense to the question whether two persons who
use these different conceptual schemes, mean the same by
the words or sentences they utter. We can give it a
sense by selecting criteria for identity of meaning, but
in doing this we alter the sense of the words "mean the
same".
To sum up: knowing how any particular language
works involves knowing what sorts of conceptual schemes
it employs, and questions about identity of meaning pre-
suppose identity of conceptual schemes. So:
III.    There must be tests for identity of conceptual
schemes.

2.B.5.  As I am not trying to give a detailed account of
all the criteria for identity of meanings and propositions,
I shall not explain how we compare and distinguish con-
ceptual schemes, but will take it for granted that we
can, which is not unreasonable, since we can and do
successfully compare and distinguish the meanings with
which various persons use their words. For example, we
are reasonably sure about the translation of "red" into
German. This shows that Quine must be mistaken in his
assertion that we can never discover with certainty that
one conceptual scheme rather than another is employed by
some person or group of persons. (See "Word and Object",
sections 12, 15, 16, etc.) He must be wrong in any case,
for unless we had some method of identifying conceptual
schemes, we could not have learnt to understand the words
with which he describes the various conceptual schemes,
which he says we cannot distinguish! The method of making
such distinctions must be embodied in the way in which we
    30

learn to use words like "property", "unobserved", "the
same" (applied to persisting material objects at different
times), and so on. (We may have to rely on our shared
natural reactions to some extent.)
If we had no inkling of the kind of conceptual scheme
employed by certain people, then we could not ask or
answer questions about the meanings of individual words
or sentences in their language: indeed, we could not even
be sure that it was a language. We could, at most, learn,
by empirical observation, the conditions in which they
produced certain noises, the situations in which they
gave the appearance of "assenting" to statements, and
perhaps some idea of the causal connections between their
noise-producing habits and the smooth running of their
society. But to know all this is not to understand. We
could not say that we had understood them until we knew to
what aspects of their environment they were referring, or
whether they were referring to anything at all, when they
produced their noises as predicted. (Cf. 1.B.6 above).

2.B.6.  So, on the assumption that we can compare and
distinguish conceptual schemes, I shall restrict the
discussion to a language which is like our own in allowing
talk about particulars and universals. Particulars are
material objects, events, persons and other things which
are spatio-temporally located. Universals are the
properties which may be possessed and shared by these
particulars, and the relations in which they may stand
to one another. (I shall talk only about observable
properties and relations.) So knowing how a language
of this sort works, involves knowing what sorts of things
material objects and other particulars are, and what sorts
of things their properties and relations are. It involves
    31

knowing how words and combinations of words may be
correlated with such things and combined, perhaps with
other words, to form sentences which can be understood
as making statements, asking questions, and so on. The
assumption that there are observable properties and
relations, to which words can refer, does not seem to
be a very implausible assumption. The next section
but one (2.D) will be devoted to an explanation of what
the assumption means, and what justifies it. At present
I shall say only that knowing what sort of thing a
property is, involves knowing what it would be like for
that property to occur in other objects than the ones
in which it does in fact occur. Universals art not
essentially tied to those particular objects which
happen to instantiate them, and one who does not see
this has not fully mastered the conceptual scheme about
which I am talking. Much will be made of this in the
sequel (Cf. 2.B.9, 3.C.4, 3.E.5, and Chapter seven).
(Very little will be said about particulars, for reasons
explained in Appendix I.)
It should be noted that a language may be over-
determined as regards its conceptual schemes; for
example, our language say have some other conceptual
scheme built into it in addition to the one which I have
described. If so, then there are two or more quite
different ways in which we are able to look at the world
or talk about it. Perhaps, for example, in addition to
seeing it as made up of things and their properties, we
can see it as made up of facts, or instantaneous events,
etc. But we shall ignore such complications and problems,
and concentrate only on (a) the fact that identification
of meanings of words which can occur in statements about
    32

the world presupposes the identification of conceptual
schemes, and (b) the fact that we have at least the
conceptual scheme which allows the existence of material
objects and their observable properties and relations.
These are the only particulars and universals I shall
mention.

2.B.7.  We have seen that in order to find out what a
person means by his words and statements we must find out
what sort (or sorts) of conceptual scheme he employs, or
what sorts of entities he thinks of as making up the
world and how. In addition, we must understand how he
thinks of words as making up his sentences. For there
will not only be words which refer to the entities for
which there is room in his scheme (e.g. words referring
to particular material objects, or descriptive words
which refer to properties), but also other kinds of
words and types of logical and grammatical constructions
which help to determine what sort of statement is being
made about the things referred to. Examples are: the
word "is" in "My pencil is round", and the structure
which this statement shares with "Tom's hat is brown".
Neither the word nor the structure refers to any material
object or a property or relation, or anything else which
might be described as an observable entity, an object
of experience. But they help to determine the meanings
of sentences, and so we must know how they work if we
are to understand statements which employ them.
For example, we must understand the difference
between subjects and predicates if we are fully to under-
stand the statements quoted above. It is also necessary
for an understanding of the difference in meaning between
"round" and "roundness", which do have different meanings


    33

despite the fact that they refer to the same property,
owing to the fact that they have different roles in the
language. There might have been a language in which
the same word was capable of occurring in both sorts of
contexts, the difference in role being indicated by
something other than a difference in the word. In that
case it would not have quite the same meaning as either
"round" or "roundness". (Think of our word "red".)
In order fully to know the meaning of a word it is not
enough to know what things it refers to; one must know
also what kind of word it is meant to be, and how it
can be combined with other words to form sentences of
various kinds. So when we compare and distinguish the
meanings of individual words we take for granted a whole
system of logical and grammatical constructions, and if
we are to be able to identify the meanings of statements
we must know what sorts of logical systems are employed,
and what the functions are of individual logical words
and constructions. In Chapter five I shall discuss the
ways in which these "logical constants" help to determine
the meanings of sentences in which they occur, by deter-
mining the conditions in which statements are true,
commands are obeyed or disobeyed, and so on.

2.B.8.  From all this we can see that giving a full
account of tests for identity of meanings and propositions
would involve describing a great many different sorts of
criteria, operating at many different levels, and also
criteria for distinguishing things at the same level.
To sum up: there must be (I) tests for telling whether
certain behaviour should be described as linguistic,
(II) tests for identifying and comparing various kinds of
linguistic activity (statement making, questioning,
    34

commanding, etc.), (III) tests for identifying various
kinds of conceptual schemes, (IV) tests for identifying
various kinds of logical and grammatical functions of
words and constructions, and (V) tests for identifying
and distinguishing (if the language is like English) the
particular material objects or properties or relations
to which individual words may refer.
Our knowledge of how to apply all these tests is
presupposed not only when we talk about words and their
meanings, but also when we use words, in thinking or
talking. For we cannot use words without knowing what
are mean, and this involves knowing, for example, what
it would be like to mean the same or something different
at another time, or to be understood or misunderstood
by other persons. This requires some knowledge of how
to apply criteria of identity. Moreover, the existence
of criteria at all the levels described above is pre-
supposed, even when we use familiar "low-level" words, as
pointed out at the end of 2.B.3. (All this is very
much oversimplified. Some qualifications are made in a
note at the end of the section.)

2.B.9.  To give a detailed account of the criteria
required for identifying meanings at all these levels would
be a very complicated and lengthy task. I shall take
most of the answers for granted, concentrating explicitly
only on those aspects of meaning which are directly
relevant to my main problem, the problem of clarifying
and justifying the assertion of the existence of synthetic
necessary truths. Thus, several restrictions will be
imposed on the discussion.
I shall not, for example, try to say how we recognize
linguistic behaviour as such.
    35

Neither shall I try to describe the differences
between statements, commands, questions, exclamations
and other kinds of linguistic utterances, but will con-
centrate only on statements, with the further restriction
to statements containing only logical constants and
descriptive expressions referring to properties or
relations. (Cf. 1.C.2, above.) This eliminates the
need to discuss aspects of meaning which are not con-
cerned with the conditions in which statements are true or
false. For example, we need not discuss the conditions
in which it is appropriate to say "I advice you to leave
home" rather than "Please leave home", or "If you leave
home you will be happy". I use the notion of an
appropriateness-condition to cover a wide variety of
cases, including the conditions in which it is appropriate
to say "Ouch!" or "Alas!" or "Why?", or the conditions
in which it is appropriate to use statement-forms of
sentence rather than question-forms, and so on. The
identification of appropriateness-conditions presupposes
not only the identification of conceptual schemes and
logical systems (see III, and IV, above) but also the
identification of certain kinds of social institutions
and "forms of life". (Note, for example, how the use
of expressions such as "I advise ..", "You may ?",
"Please ?", "You ought ?", etc., presupposes the
existence of ways of life. Words whose use presupposes
the existence of social institutions and patterns of
social behaviour may, of course, be relevant to deter-
mining the truth-conditions of statements in which they
occur. The statement that someone made a promise, or
gave advice, or asked a question, may be true or false.)
The rules determining appropriateness-conditions for the
utterance of various forms of expression (e.g., questions
    36

or commands) may generate some so-called "pragmatic"
implications, such as the "implication" from "P is the
case", or "I assert that P is the case", to "I believe
that P is the case". I shall not go into this sort of
question, but it might be relevant when attempts are
made to generalize my account of the analytic-synthetic
distinction. (See Chapter six).
There is far more to meanings of words and statements
and other utterances than can be taken account of by
considering truth-conditions (see 5.A.11, for example),
but these other sorts of meanings can be ignored in a
discussion of analyticity or necessity, for this is a
matter of ways of being true or false.
In the next two chapters, three and four, I shall
discuss ways of identifying those aspects of the meanings
of descriptive words and expressions which help to deter-
mine truth-conditions of statements in which they occur.
Then I shall proceed to discuss the ways in which logical
words and constructions determine truth-conditions.
(Nothing will be said about proper names and other expressions
referring to particulars, for reasons given in
appendix I.) All this will prepare the way for a dis-
cussion of statements which are true in all conditions.
First, however, I shall try, in the remaining sections
of this chapter, to explain my we have to take account of
the existence of universals (observable properties and
relations) and then to explain what their existence amounts
to.

Note on section 2.B
In this section I have made many oversimplifying
assumptions, and now I should like to suggest a few quali-
fications to my remarks. I asserted that in order to
    37

understand talk about meanings of words, or even in
order to be able to use words with definite meanings, we
presuppose a large amount of general knowledge about
language, and, in particular, rely on the existence of
criteria for identity at various levels. It must not
be thought, however, that all this knowledge is explicit,
that we should be able to formulate it or describe the
criteria for identity which we presuppose. There is
much that we can do without being able to say how we do
it. (See Appendix III on "Implicit Knowledge".)
Secondly, it should not be assumed that all the
criteria for identity to which I have referred are
commonly applied, even when we are explicitly talking
about meanings. We take a great deal for granted in our
dealings with other persons (and ourselves). If I want
to teach someone how to ask a question in French, I may
simply assume that he knows what questions are, and say:
"This form of words and symbols is used for asking
questions". Similarly, when I am not sure whether some-
one is asking a question or making a statement, I do not
apply direct criteria, usually, but simply assume that he
knows how to apply them, and ask: "Are you asking me or
telling me?" In fact, we hardly ever apply criteria for
identity at the higher levels, since the things we take
for granted do not often lead us into trouble, owing to
great regularities in human behaviour: we cross our
bridges only when we come to them, and we don't often
come to them. Nevertheless, it seems that it makes
sense to talk about meanings only because it is possible
to test our assumptions by applying criteria, even at the
highest levels. (But they are not infallible. See
2.A.5.)
Finally, many of my remarks must be modified in
    38

order to apply to a person whose linguistic training is
incomplete, owing either to an unfortunate environment
and bad teaching, or to his own constitutional inability
to pick up concepts, or his young age. Children who
cannot yet form sentences and make statements on their
own initiative may be able to respond with correct
answers to questions like "Is this red?" or "Is this
round?", saying "Yes" or "No". But nothing very defin-
ite is likely to come out if we apply tests to find out
exactly how they understand the question (e.g. in the
sense of "Is this object red?" or "Is redness here?").
Their conceptual schemes may be still too underdeveloped.
The process of development and elimination of indefiniteness
continues even in later life. (See 4.B.4, below.)

2.C.    Universals and strict criteria
2.C.1.  I have stated that all mystery can be removed
from talk about meanings and propositions by making
criteria for identity of meanings explicit. (In 2.A.)
I went on to describe some general presupposition of
talk about meanings, showing how criteria for identity had
to apply at several different levels. In particular,
the identification of meanings of individual words pre-
supposes the identification of some conceptual scheme.
In this section, taking for granted the existence of a
conceptual scheme in which words may refer to universals
or particulars (cf. 2.B.6, above), I shall try to show
how observable properties and relations (i.e., universals)
can provide sharp criteria for identity of meanings of
descriptive words, at least from the point of view of a
person who uses such words. Unless we use sharp criteria
for identity of meanings we are likely to find ourselves
confusing issues and begging questions when we try to
    39

apply the analytic-synthetic distinction. This will
be illustrated with the aid of some controversial examples,
about which more will be said in Chapter seven. These
and other examples help to demonstrate that criteria for
identity of meaning which are normally employed are too
loose for our purposes. (It may be recalled that there
are no "correct" sets of criteria: their adequacy depends
on the purposes for which they are chosen. (2.A.5)).
Many complications in our ordinary language will be
ignored, at present, attempts being made in chapters three
and four to remedy this deficiency.

2.C.2.  What are we to make of the statement that two
descriptive words mean the same, or are taken to have the
same meaning, by some person or group of persons? How
are we to answer questions about synonymy? Philosophers
are sometimes inclined to deal with such questions not by
looking to see how we in fact decide whether to say that
two words or two sentences mean the same or not, but by
proposing neat tests, like "substitutability salva
veritate". In search of a slogan they ignore our every-
day practice. When they have found that their slogans
will not work, they give up, demanding that the notion
of synonymy be rejected, or they turn to nominalism, or
some such thing. Of course we do not and need not
decide whether we mean the same by two words by substi-
tuting one for another in all possible sentences and
seeing whether the truth-value of the statement expressed
is changed by the substitution. How could we possibly
apply such a test? How could we know that the truth-
value would or would not be changed unless we knew whether
the words had the same meanings?
Surely in order to settle questions about the meanings
    40

of words we cannot merely talk about relations between
words and sentences, without ever mentioning the things
to which those words refer? I shall try to show that
talking about the properties to which words are under-
stood to refer may help to explain talk about synonymy
of descriptive words.

2.C.3.  Let us look at some ordinary ways of eliminating
ambiguities. If it is possible for a sequence of sounds
or marks produced by some person to be taken as either a
sentence in English or a sentence in French, then we can
find out which of two possible meanings he intends his
utterance to have by asking which of the two languages he
was intending to speak. But this may still leave some
questions unanswered, for ambiguities may persist within
a language.
In some cases, remaining ambiguities may be eliminated
by simple re-interpretation, by saying in other less
ambiguous words precisely what was meant. Thus we may
ask: "When you said 'I saw three tadpoles yesterday' did
you mean you saw three of the army's new amphibious craft,
or were you talking about animals which are the larvae of
frogs?" Alternatively, it may be possible to eliminate
the ambiguity by pointing to objects which the words are
supposed to describe. For example, by pointing to frog-
larvae and saying "I was talking about those things", one
may enable others to identify one's meaning. But this
eliminates only flagrant ambiguities. There may be
remaining ambiguities which are more subtle, as is brought
out by the question: "Did you intend the word 'tadpole'
to mean 'animals with this shape colour and habitat', or
did you intend it to mean 'animals which are the larvae
of frogs', or did you mean the conjunction of the two?"
    41

In most cases, there will be no definite answer to this
question, and for normal purposes it does not matter
(for reasons which will be explained below, and in
Chapter four), but it may matter for our purposes, for
example if we want to know whether the statement "All
tadpoles are frog-larvae" is analytic or synthetic.

2.C.4.  Normally the difference between "I saw three
frog-larvae" and "I saw three of the army's new amphibious
craft" is important because it is very likely that when one
is true the other in false. Similarly, the difference
between the corresponding two senses of the word "tadpole"
will be important because it makes a difference to whether
a particular object is correctly described by that word
or not. But the difference between "I saw three frog-
larvae" and "I saw three animals with the shape colour
and habitat of tadpoles" does not matter for ordinary
purposes since it is true (or let us so assume for the
sake of illustration), and generally believed, that whenever
one of these statements would be true the other would be
true too. The assertion of either would enable the hearer
to know what had been seen by the speaker. So, to report
"I saw three tadpoles" as "He said he saw three frog-
larvae" is to report accurately enough for normal pur-
poses, and it would be equally accurate to say "He said
he saw three animals of such and such an appearance and
habitat." In other words, we are often quite content to
use extensional criteria for identity of meanings of des-
criptive words: where it can be taken for granted and is
true that the same objects are correctly describable by
two words, then, for many normal purposes, those words are
synonymous. There is often no point in distinguishing
their meanings, or in objecting to reports such as the
    42

first one above, by saying: "You are misrepresenting me,
as I merely intended to say I saw three animals with a
certain appearance, without implying anything about
their parentage". Such an objection would often provoke
complete bafflement.
This helps to explain way Malcolm wrote (in Mind,
1940, p.339, et.seq.) that if two persons would take the
same states of affairs as verifying what they take to be
expressed by certain sentences, then we should say that
they understand the same thing by those sentences, that
they take them to express the same proposition. It also
gives some support for Frege's decision to take identity
in extension as a criterion for identity of concepts:
"? coincidence in extension is a necessary and sufficient
criterion for the occurrence between concepts of the
relation corresponding to identity between objects."
(See "Translations", p.80.)

2.C.5.  Despite all this, identity in extension is not a
universally acceptable criterion for identity of meanings.
For it is possible that there might be an object which had
the appearance and habitat of a tadpole which was not in
fact the larva of a frog, and such an object, if it
existed, would be describable by the word "tadpole" in one
of the two senses explained above, but not in the other.
So although there would not normally be a point in making
the objection mentioned above to the report of the stat-
ement "I saw three tadpoles", there would be an objection
if the speaker thought that this possibility should be
taken seriously, and did not want to assume that animals
were frog-larvae just because they had certain recognizable
features. So the mere possibility of a state of affairs
    43

in which with one sense a sentence would express a false-
hood while with the other sense it expressed a truth
can count against the identification of the two senses,
for some purposes, despite the fact that the possibility
is not actually realized. We acknowledge this when we
ask the question "What would you have said if such and
such had been the case?" in order to discover exactly
what a person means by some word or sentence. So exten-
sional criteria of identity may be too loose to pick out
relatively subtle ambiguities of a kind which do not
matter for ordinary purposes, but might matter. We
apply a sharper criterion when we talk about possibilities.

2.C.6.  But how do we apply the sharper criterion? How
do we tell that there might have been an animal with the
characteristic appearance of a tadpole which was not the
larva of a frog? How do we perform the activity of
considering possibilities? The only possible states
of affairs which we can observe and examine in order to
find out which statements would be true and which false
if they were actual states of affairs are those which
are actual states of affairs. We cannot perceive the
set of all possible worlds, we can perceive only the
actual one. So when we decide that two meanings of a
word or sentence should be distinguished on account of
the possibility of its making a difference to whether the
word correctly describes something or whether the state-
ment is true or false, possibilities cannot be the funda-
mental explanation of our decision. I shall argue in
Chapter seven that it is by paying attention to observable
properties that we are able to think and talk about
unactualized possibilities. Although in fact the
extension of the word "W" is the same whether it has the
    44

meaning M1, or the meaning M2 we can tell nevertheless
that it is possible for "W" correctly to describe an
object when it has one meaning while it does not des-
cribe it when it has the other meaning, because the
word refers to different properties when it has these
different meanings, and the properties may "come apart".
We can examine the characteristic appearance of a tad-
pole to see if there is anything involved in the possession
of this property (i.e., the appearance) which necessitates
being the off-spring of frogs, and find that there is
not: the properties may come apart, an animal could
have one property without the other. So we are able to
distinguish two (or more) possible meanings of "tadpole",
even though there would usually be no point in dis-
tinguishing them, for normal non-philosophical purposes.

2.C.7.  It might be thought that we could avoid talking
about the properties to which descriptive words are
intended to refer, if we concentrated our attention
instead on the process in which people learn to use those
words. Then we might discover whether two persons meant
the same by "tadpole" or not, by finding out whether
they had learnt its use in the same way. But this is
open to two objections.
First of ail, it would be a difficult test to apply,
since the process in which we learn to speak is very
gradual and extended, and subject to an enormous amount
of possible variation. This means that it would not be
easy to say what counted as "learning in the same way", or
to discover whether two persons had learnt some word in the
same way. Secondly, even if two persons are given exactly
the same instructions, they may understand them differently.
Two children may both be shown tadpoles and told that they
    45

are produced by frogs and will, if they survive, them-
selves grow into frogs. One may take this as merely
an additional fact about those animals which are des-
cribable as "tadpoles", while the other regards it as
a necessary condition for being correctly so describable.
The latter regards being a frog-larva as part of what
is meant by being a tadpole, the former does not. The
facts being what they are, this difference in the way
they understand may never happen to come out (though it
could do so).
So even a careful examination of the teaching process
may fail to reveal subtle ambiguities, unless we take
into account what goes on in the pupil's mind, and this
means asking which properties he takes the word to refer
to, or which properties were drawn to his attention by the
teaching process.

2.C.8.  But there is another more important reason why
we must look to properties rather than mere possibilities
for a criterion for identity of meanings of descriptive
words. Suppose the word "tetrahedral" to mean "solid
figure bounded by planes and having four vertices", while
the word "tetralateral" means "bounded by four plane
surfaces". Since it is impossible for a plane-sided
object to have four vertices without being a plane-sided
object with four faces and vice versa, it is impossible
for either of the following statements to be true unless
the other is (in the same circumstances): namely (1) "The
paperweight on the table is tetrahedral" and (2) "The
paperweight on the table is tetralateral".
Despite this equivalence, it seems intelligible to
say that they are different statements, that they have
different meanings, since the words "tetrahedral" and
    46

"tetralateral" refer to different properties. (In the
terminology of Chapter three, their meanings are
"synthesized" differently.) It is possible to notice,
attend to,  have  in mind,  think about, or
recognize one of these properties without being aware
of the existence of the other - so why can one not
intend a word to refer to one of them and not the other?
After all, if to understand what a person means by des-
cribing something as "tetrahedral" is to know what it
would be like to do so for the same reason as he does
(Cf. 1.B.6 and 3.C.5), then we shall not have understood
if he intends the word to refer to the property of
having four vertices, etc., and we think it refers to
the other property, despite the impossibility of there
being any object which is correctly described by the word
in one sense and not in the other.
This brings out another reason why the teaching process
cannot serve infallibly as a criterion for identity of
meaning, for once again the same method of teaching may give
two different pupils two different concepts, if one of
them happens to notice one aspect of the illustrative
examples, while the other notices another aspect. And
here it is even less likely that the ambiguity will be
detected than in the case of "tadpole" - though it is also
less likely that it will matter, for most normal purposes.
This example shows also that yet another criterion for
identity of statements will not always do, though it is
sometimes appealed to, namely that two sentences express
the same statement if the statement made by each of them
entails the statement made by the other, or if each entails
and is entailed by the same statements as the other.
This is too loose because it would fail to distinguish the
statements (1) and (2) above, since, owing to the
    47

impossibility that either of them is false while the
other is true, each entails the other.

2.C.9.  Now it may be objected that there la a good
reason for saying that the words "tetrahedral" and
"tetralateral" refer to one and the same property, since
they are analytically equivalent, in a sense to be
explained below. This may be so, but it should not be
taken as obvious that there is no point in distinguishing
the meanings of the words, or the properties to which
they refer, especially when we are discussing the question
whether there are any necessary connections which are
not analytic. If we wish to take this question seriously,
we must be prepared to distinguish meanings where for
most other purposes meanings do not need to be distinguished,
and this means looking for the sharpest possible
criteria for identity of meanings. (We must use Kant's
engraver's needle. Cf. 2.A.5.) For otherwise we shall
be in danger of begging questions.
If we do not distinguish meanings as finely as
possible, we may fail to separate two propositions, or
two concepts, thinking there is only one. Hence we may
fail to notice the relation between these propositions,
or concepts, and say simply overlook a possible candidate
for the title of "synthetic necessary connection", or we
may fail to distinguish some necessarily true proposition
which is analytic from one which is synthetic, if there are
such things. In this way we settle the question at
issue merely by selecting such criteria for identity of
meanings as ensure that analytic connections between
meanings cannot be distinguished from necessary ones and
that no proposition can be necessarily true without
    48

being identical with an analytic one. In order to avoid
this question-begging procedure we must look for the
sharpest possible criteria. Only then can we hope to
understand what people mean, or think they mean, when
they assert that there are synthetic necessary truths.
It seems very likely that the apparently irresolvable
disagreement amongst philosophers on this topic can be
traced to a failure to make explicit the sets of criteria
for identity which are implicitly used, so that one lot
uses sharp criteria while the others use looser ones,
and arguments proceed at cross purposes, without any hope
of agreement. I do not wish to imply that any one set
of criteria is correct. (See section 2.A.) It all
depends on the purposes for which they are chosen. But
for the purposes of this discussion we must, for the
time being, use what seem to be the sharpest criteria for
identity of meanings, and say that "tetrahedral" and
"tetralateral" have different meanings, for the reasons
given in 2.C.8.

2.C.10. We shall therefore reject as superficial and
question-begging, arguments such as the following: "It
is analytic that every figure bounded by three straight
lines has exactly three vertices, for the concept of a
triangle can be defined either in terms of being bounded
by three sides, or in terms of having three angles.
So the statement 'Every figure bounded by three straight
sides has exactly three vertices' is the same as the
statement 'Every triangle has exactly three vertices',
which is analytic, by definition of 'triangle'. Q.E.D."
(Cf. 3.C.10.)
I have put the argument in a very crude form, but
    49

it can appear in more subtle guise. (see 7.C.6.) The essential
thing to note in it is the use of the phrase "can be
defined ?" with the implication that one and the same
word can be given its meaning in more than one way.
This is not uncommon usage in mathematics (see, for
example, how many different "definitions" mathematicians
may use of the property of being a conic section), but
it presupposes the use of relatively loose criteria
for identity of meanings, too loose for our purposes.
For we are not concerned with whether words can be so
defined as to make certain statements analytic, but
whether they have to be. We wish to ask whether they
can be distinguished, as having different meanings, and
yet be necessarily connected.

2.C.11. All this should be recalled if it appears that
some of the techniques employed in later chapters for
describing and distinguishing meanings are too nice and
artificial. We cannot avoid them, if we are to use the
sharpest possible criteria for identity of meanings of
descriptive words.
We have seen that various more or lees familiar
tests for identity of meaning are not sufficiently
stringent, for our purposes, such as comparing extensions,
comparing methods of instruction, and so on. I have
suggested that stricter tests are possible if we ask
always to which properties descriptive words are taken to
refer. We shall see, in Chapter four, that there are
sometimes no answers to such questions: our ordinary
statements are not made with sufficiently definite meanings.
This will have the consequence that it is not possible to
apply the analytic-synthetic distinction to all ordinary
    50

statements, but that need not trouble us, so long as
it can be applied in some cases, at least in principle.
It may be objected that talk about properties and
other universals does not help, because they cannot be
used to explain anything, since they do not exist as
"complete and independent entities" (see Price,
"Thinking and Experience"), and in any case their
existence depends on the existence of words with meanings,
so they cannot be used to identify and distinguish
meanings, without circularity. I shall try to answer
this in the next section, by showing how, at least from
one point of view (cf. 1.B.2), the existence of universals
is independent of language, and so may be used to explain
our use of descriptive words. The sense in which it
explains will become clearer in Chapter three.

2.D.    The independence of universals

2.D.1.  Meanings, concepts and propositions are not
things which exist in their own right as objects of
experience, for we cannot see them, hear them, feel them,
or in any other way perceive their existence. We might
say that talk about meanings, concepts or propositions
is a sort of circumlocution for talk about entities of
other kinds, such as particulars referred to by proper
names, or properties, qualities or relations referred to
by descriptive words, the types of situations and verbal
contexts in which the use of a word is appropriate,
the purposes with which sentences are uttered, and so
on. In short: talk about meanings, etc., is a circum-
locution for talk about the things which we examine in
applying criteria for identity of meanings.
    51

On the other hand, universals, that is observable
properties and relations, do exist in their own right
as objects of experience, for they can be seen, attended
to, thought about, imagined or referred to, without
the mediation of any other kinds of things out of which
they have to be "logically constructed". That is why
they can be said to explain our use of descriptive words.
It is only because observable properties and relations
exist that we can use descriptive words as we do, just
as it is only because their referents exist that we can
use proper names and other definite referring expressions
as we do.
Not only the use of referring expressions, but also
the use of descriptive words and expressions requires that
conditions of existence and identifiability be satisfied.
The mere fact that a word occurs in a predicate-position
in an utterance does not guarantee that it has a meaning,
that it predicates successfully. What more is required?
In some cases at least, what is required is that there
should be some property or combination of properties to
which it refers. So, if universals are the things
referred to by such words and expressions, then their
existence cannot be reduced to the existence of words in
a language, since their existence is presupposed by the
use of those words. This will now be amplified.

2.D.2.  Descriptive words may occur in sentenced express-
ing statements, questions, commands, etc., and knowing
their meanings involves knowing how they contribute to
the meanings of these sentences. (See section 2.B.)
A descriptive word contributes to the meaning of a state-
ment, by helping to determine the conditions in which
that statement would be true. The conditions in which
    52

the statement would be true depend on the conditions in
which particular objects are correctly describable by the
word in question. So knowing in general how to tell
whether statements including the word "P" are true or
false requires the ability to tell whether simple state-
ments of the form "That is a P" or "That is P" are true,
for example, "That is a cube" or "That is red". So
understanding a descriptive word involves knowing how to
tell which objects fall within its extension and which
do not. How does one tell?
In general the answer is very complicated, as will
be seen in the next two chapters, but in the simplest cases
one tells whether an object is correctly describable by
a word "P" by looking to see whether that object has a
property which one has learnt to correlate with "P", or
not. (Here we have a sense in which the meaning of a
word can determine an application "in advance".) In many
cases the examination is visual, but it needn't be: consider
how we tell whether an object is describable as
"sticky" or "cool".
I shall try to show that what determines the des-
cribability of particular objects by descriptive words
and expressions may, at least in some cases, be correlations
between words and recognizable properties. I call these
semantic correlations, because they correlate words with
non-linguistic entities. They may also be described as
"semantic rules" since, in virtue of them, some descriptions
are correct and some are incorrect. They are to be dis-
tinguished from what I shall describe as "purely verbal
rules", which merely correlate words with other words.
By providing us with "assertion licences", that is, by
fixing the conditions in which assertions may be made
truthfully, semantic correlations help to ensure that
    53

concepts have boundaries, or rather that the extensions
of concepts have boundaries, for they enable us to decide
whether objects fall under those concepts or not. (I
have been talking about observable properties, but similar
remarks could be made about relations.)

2.D.3.  There must be these correlations, or something
similar, if words are to have meanings, if concepts are
to have boundaries, if there is to be a difference between
describing something correctly and describing it incor-
rectly. It may look as if we do not need anything more
than correlations between words and other words, such as
definitions of words in terms of other words, since we
can sometimes teach the meaning of a word by giving a
definition, or discover whether two persons mean the
same by their words by asking them for definitions. But
this assumes that the meanings of the words used in those
definitions have been taught and understood. The process
of defining words in terms of other words must start
somewhere, with the setting up of correlations between
words and other things. Words alone will not do. For
example, to say that two words are to be incompatible
descriptions, or synonyms, does not even begin to tell us
which things are and which things are not describable by
either of them unless the meaning of the other is already
known. It seems that Hampshire overlooked this when he
wrote: "In all cases, clarifying the use of a descriptive
word or phrase is a process of drawing attention to its
established links with other descriptions." (In
Philosophy 1950, p.243.)
Correlations between words and other words may, of
course, enable us to decide that some statements are
necessarily true and that others are necessarily false, and
    54

they may license us to make inferences, or to substitute
one expression for another in a true statement.
(Cf. 4.C.3.) But they cannot, of their own accord,
enable us to decide in which circumstances contingent
statements are true or false. They may enable us to
assert "Nothing is both round and square", but they
cannot tell us when we may say things like "There is a
square piece of paper on my typewriter". For such
"assertion licences" we require not just rules relating
words and words, but semantic rules correlating words
and non-linguistic entities, such as properties. We
need something more than mere verbal rules, and when
we have the "something more", it may turn out that we
do not need correlations between words and words as
well: Hampshire's "links between descriptive expressions"
may turn out to be superfluous.1 (Cf. 2.D.4 & 3.B.4.c.)
____________
1.  I said that correlations between words and words may
tell us which statements are necessarily true or
necessarily false, though they cannot yield
assertion-licences for contingent statements. But
this needs qualification. For if there are only
verbal rules, and no rules correlating words with
non-linguistic entities, how can a statement be
about anything? And if it is not about anything,
how can it be a statement? How can it be true,
or false? The mere fact that someone is uttering
sounds according to rules which permit some sounds
and not others does not guarantee that he is using
a language, or that he is saying anything which is
true or false, or which conveys any information.
This shows that when philosophers talk about truth in
connection with formal systems, or when they describe
them as "languages", they are just muddled and talking
well-disguised nonsense. They are muddled, because,
although formulae in such a system may represent cer-
tain aspects of statements for purposes of classific-
ation,they are not themselves statements, and cannot
be true or false, for the symbols of a formal system
cannot be used to make contingent statements about
anything. (Some further remarks are made on this topic
in Appendix II, and in section 5.A.)
    55

2.D.4.  In "Analytic/Synthetic I" (Analysis, Dec. 1949),
Waismann pointed out that certain kinds of linguistic
rules, namely explicit definitions, or what I call
"verbal rules", serve as "substitution licences", which
enable us to make inferences from one proposition to
another (see pp. 33 and 37). Later on (in "Analytic/
Synthetic II", p. 31) he remarked that ostensive defini-
tions do not serve as substitution licences. We may
now point out that ostensive definitions are procedures
for teaching the semantic rules which govern the use of
descriptive words, and therefore, although they do not
directly set up substitution licences, they do serve as
assertion licences: that is, they help to determine
whether contingent statements are true or false. But
we may add also that, in some cases, when words have been
correlated with properties it may turn out that two words
or expressions are thereby rendered synonymous. For
example, in virtue of the semantic correlations it may
turn out that "gleen" is synonymous with "glossy and
green". In that case, even semantic rules, as taught
ostensively, may, indirectly, provide us with substitution
licences, or inference licences. (This is one of the
ways in which Hampshire's "links between descriptive
expressions", mentioned above, may turn out to be super-
fluous, once semantic correlations have been taken into
account.)

2.D.5.  All this may seem obvious and trivial. But it
is not yet quite clear what my assertion that universals
exist in their own right and can explain our use of
descriptive words, comes to. The best way for me to
clarify this further is to say where I disagree with other
philosophers.
    56

First there are those who say that the existence of
properties is merely a matter of the existence of their
instances. (It would be foolish to deny outright that
there are properties, for we are all aware that there are
shapes, colours, sizes, textures, kinds of feel, and so on.)
This seems to me to be quite wrong, for, as remarked above
(2.B.6.) properties are not essentially tied to their
actual instances. For example, I can assert that a cer-
tain complicated shape is not the shape of my table, and
in doing so I presuppose that there is such a shape, but
I certainly do not presuppose that anything ever did have
or will have that shape, since I know quite well which
shape I mean without thinking about any particular object
or objects with that shape. The existence of a property
does not imply the existence of actual instances, only that
it is possible for instances to exist. There may be many
complicated shapes which never have been or will be
instantiated, but they nevertheless exist. Perhaps there
are colours which would be instantiated for the first time,
if only someone would put the correct combination of
chemicals together. Perhaps someone with a sufficiently
good imagination can "think up" a colour which has hither-
to not been instantiated, and even decide to associate
a word with it.
Anyone who disagrees with me and thinks that the
existence of properties involves the existence of par-
ticular instances, will, of course, deny that correlation
with properties can explain the use of descriptive words,
since such a word can be used, as remarked above, without
presupposing that it refers to a property which is
instantiated. Thus confusion about the sense in which
universals exist can lead to confusion as to whether they
explain.
    57

2.D.5. (Note).  Strawson seems to think that when we
assert that a universal exists all we mean is to imply
that it has instances,
"? as when one says, for example, that saintli-
ness exists, or that there is such a thing as
saintliness, and means by this the same as we mean
by saying that there exist, or that there are,
saintly people ?" ("Individuals", p. 241.)

I do not think we often do mean this sort of thing, any
more than when one denies that there is such a thing as
saintliness he means simply to deny that there are have
been or will be saintly people. (To say "There is no
such thing as X-ness" would normally be to imply that
there couldn't be anything which was an X.)
Strawson does, of course, allow that there is
another sense in which existence may be ascribed to
universals (p.239-241), but this is a purely formal sense,
and seems to imply only that there are formal analogies
between the word "saintliness" and other substantive words.
I want to say that even the use of "red" in contexts
like "My notebook is red", where it is not a substantive,
presupposes the existence of a property, and does so in
a "metaphysically charged" way (op. cit. p.239), since
the existence of such a property is a fact about the world,
which must usually be learnt through experience. It is
a fact about the world because there might have been a
world in which the property did not exist, a world in
which nothing could be red. (It is clear from his
remarks on pp. 183, 184, 185, 186, 193, 238, 239-241,
that Strawson did not consider this way of looking at
the existence of universals. Had he done so I cannot
see how he could have tried to relate the grammatical
subject-predicate distinction to the categorial particular-
universal distinction, via a distinction between expressions
    58

which do and expressions which do not presuppose facts
about the world.)

2.D.6.  This explains how I disagree with those who
would reduce the existence of universals to the existence
of their instances. Secondly, I disagree with philo-
sophers who say that universals depend for their exi-
stence on language, or imply that we must know what a
language is in order to know what universals are, or
deny that universals can explain any aspects of our use
of language. Usually, such philosophers really mean
only to reject bad theories of universals, which over-
simplify things. For example, they assume that universals
can explain our use of words only if there is
exactly one universal for every descriptive word, such
as a single property common to all the objects descri-
bable by that word. But this "one-one" model leads
people into grave difficulties, when they try to find
some one thing common to all the objects describable by
a word with a complicated meaning. [Their failure to
find a common property leads them to say, for example,
that universals are intangible to the senses, being
apprehended only in thought (Cf. Lazerowitz in Mind 1946,
p.1.), or that universals are "partial realizations of
the specific forms, existing only as the thought of
them" (Cf. Blanshard, "The Nature of Thought", Vol.I,
p.609). Or they may be led to say that universals
cannot be sensed because things sensed are many and
different where there is only one universal. (Cf.
Austin, in P.A.S. Supp, 1939, p.85).]
However, when the intolerable implications of the
"one-one" model lead philosophers to give up universals
    59

altogether, they give up too much, for then they are
left only with words and no way of explaining how words
describe or have meanings. By taking into account
complexities in our use of descriptive words, as in the
next two chapters, we can preserve a theory of universals
while rejecting the one-one model. (If we insist on
looking for one property corresponding to every descrip-
tive word, then some must be "improper" properties, con-
structed or synthesized out of other "proper" properties
which are tangible to the senses. (Cf. 3.B.5.))

2.D.7.  A stubborn insistence on the one-one model is
not the only thing which accounts for the refusal to
acknowledge the existence of universals as independent
entities. Another is the fact that which properties
a person sees in the world, and the way in which he
classifies things as having something in common, may
depend to some extent on the society in which he has
been reared, and the classifications made in its language.
(See, for example, Waismann, in "Verifiability", p.137-9,
etc.) This may provide good reason for saying that the
existence of a property is a fact about people not a
fact about the world, but only if we take up a different
point of view from the one adopted in this essay. (See
Chapter one, section 1.B.). For from the point of view
of a person who can see properties, they certainly exist
in their own right, as things which he can perceive,
attend to, recognize, bear in mind, etc., without having
to think about people or language. What is more, from
his point of view, or the point of view of a person who
understands him, the existence of these properties which
he can see explains how he classifies things and what he
says about objects in his environment (as will be shown
    60

in more detail in section 3.C.) This, however, leaves
open the possibility of explaining his behaviour from
some other point of view, such as a psychological or
anthropological point of view, a possibility with
which we are not concerned, since we are looking for
rational explanations, not causal ones.

2.D.8.  From the point of view of one who talks and can
see properties, therefore, we must disagree with the
attitudes underlying remarks made by philosophers to
the effect that universals are somehow generated by
language, or that their existence is to be explained by
talking about language. Here are some examples:

(i) "To say that a property exists is to say that
a general word has been or could be introduced, to
characterize the things which possess it."
(Quinton, in "Properties and Classes", P.A.S.,
1957-8.)
(ii)    "Saying what meaning a symbol has involves
describing the relevant experiences, and this
brings us back into the realm of symbols."
(Ayer, in "Thinking and Meaning", p.27.)
(iii)   "And in the end the kind of similarity which
is meant can be specified only by a backward
reference to the name". (Pears, in LL.II,
p.56-7.)
(iv)    "It is fatally easy to talk carelessly about
things in a way which suggests that they stand
out there already labelled in a way which indicates
their properties ?. One refers airily to
THE TWO classes as if one could say WHICH TWO
classes without using the words." (Pears,
LL.II, p.116.)
(v) "The sense of a predicate expression (e.g. 'is a
rose') generates its referent ('rosehood') if
there is one. It could not fail to refer."
(J. R. Searle, D. Phil, thesis, p.208.)
    61

(vi)    "The notion that an entity stands to a
predicate as an object stands to a singular
referring expression must be finally abandoned."
(Searle, p.188.) "Universals ? do not lie
in the world" (p.192) and "? propositions
asserting their existence are tautologies"
(p.191). (See also Strawson's "individuals";
p.184: "But now we no longer have an empirical
proposition, a fact about the world. We
have a tautology ?. It is a fact about
language.")

Concerning (ii), (iii) and (iv), consider the
following question: When I say what person or thing a
proper name refers to, by using words, does this bring
us back to the "realm of symbols"? Is this a "backward
reference" to a name?
Concerning (v) and (vi), recall the muddles, e.g.,
about "substance" into which philosophers have been led
by thinking that the existence of proper names and other
referring expressions somehow suffices to guarantee
the existence of the particulars referred to. (Cf.
Wittgenstein, "Tractatus", 2.02 to 2.03, especially
2.0211 and 2.024.)

2.D.9.  I have so far been labouring the point that if
our words and sentences are to have any meaning, to be
capable of being used to make statements which can be
true or false, then there must be semantic rules correl-
ating words with non-linguistic entities, and I have tried
to show that universals, that is, observable properties
and relations, are suitable non-linguistic entities.
Not only are they non-linguistic, but their existence
cannot be reduced to the existence of sets of objects
which resemble one another, since a universal (such as
a shape) can exist even though it has no instances.
    62

This means that anyone who tries to explain our
use of descriptive words such as "red", "round" or
"smooth", in terms of actual particular objects or
sets of particular objects whose existence is pre-
supposed by the use of these words, has gone wrong
somewhere. (Cf. A. Pap, in S.N.T., chapters 9 and
13. See also Körner, in "Conceptual Thinking", where
he talks about sets of "exemplars".) Properties are
not essentially tied to actual instances, and can be
thought about or referred to independently of their
instances. (See circa 3.C.4 & 3.E.5.)
It may be objected that it is still not clear in
what sense the existence of universals can explain our
use of descriptive words. It explains because, by
memorizing the properties (etc.) correlated with des-
criptive words we can learn to distinguish states of
affairs in which statements using those words are true
from those in which they are false. This will be made
clearer in the next chapter, which describes some of the
ways in which we may correlate descriptive words with
observable properties. In addition, it should show in
detail how we can make sharp distinctions between the
meanings of words by taking into account the properties
to which they refer, as required by the programme adopted
in the previous section. (See also the motto at the
end of 2.A.5.) In order to avoid digressions into
philosophical psychology, I shall not deal with questions
as to how we can tell which properties or things a par-
ticular person takes or intends his words to refer to,
but will assume, for the time being, that we can. At
any rate, each of us knows what he means his words to
refer to, in most cases. This omission means that, from
a certain point of view, my account of how to apply sharp
criteria for identity of meanings is essentially incomplete.

                                                        Page    63

====================================================
This is part of A.Sloman's 1962 Oxford DPhil Thesis
     "Knowing and Understanding"

Note (24/06/2016): When this chapter was written I knew nothing
about programming and Artificial Intelligence. In retrospect,
much of the discussion of procedures for applying concepts is
directly relevant to the problems of designing human-like
intelligent machines. References to "morons" can be interpreted
as references to computer-models.
====================================================



Chapter Three

SEMANTIC RULES

Introduction
Chapter two contained an argument to show that
in order to avoid begging questions we must look for
the sharpest possible criteria for identity of meanings,
and it was suggested that only by taking note of the
universals (i.e. observable properties and relations)
to which words are intended to refer could we find
sufficiently sharp criteria. (See 2.C.) The way had
been prepared for this in section 2.B., where it was
shown how conceptual schemes were important in connection
with identification of meanings, and how our own con-
ceptual scheme had provision for a distinction between
material objects and the universals which they instan-
tiate. Section 2.D contained arguments to show that
talk about universals can explain since their existence
is a fact about the world, independent of the existence
of instances or of our use of language. In this chapter
an attempt will be made to show in more detail how
properties may be used to give descriptive words their
meanings, and how we may compare and distinguish meanings
by examining the ways in which words refer to properties.
This will provide many interesting examples to which the
analytic-synthetic distinction may be applied later on.
   The programme for the chapter will be roughly as
follows. First of all the simplest type of correlation
between words and properties will be discussed, and then
it will be shown how more complicated correlations are
possible, firstly by means of logical syntheses of con-
cepts and secondly by means of non-logical syntheses.

    64

This will help to justify my claim that universals
explain our use of descriptive words.
    There will be many oversimplifications in this
chapter, since it ignores the fact that words are
ordinarily used with relatively indefinite meanings,
but it is hoped that this will be compensated for by
the discussion in chapter four. In addition, this
chapter will be concerned only to show how we decide
whether or not a particular object is describable by
some word. In order to explain how descriptive words
can contribute to the meanings of whole sentences, we
must wait for the discussion of logical words and con-
structions in chapter five.
Finally, notice that although the discussion is
restricted to words which refer to properties, never-
theless similar remarks could be made about words
referring to observable relations.

3.A.    F-words
3.A.1.  The simplest sort of semantic rule, though by
no means the only sort, is one which correlates a des-
criptive word with only one property, which must be
possessed by objects correctly describable by that word.
I describe this sort of word as an "f-word" (or feature-
word), and shall say that it is governed by an f-rule.
Such words describe objects in virtue of something which
they have in common, some respect in which they are all
alike. If, for example, the word "scarlet" refers to a
specific shade of colour, then we may say that it is an
f-word, and all the things which it describes, since
they have exactly the same shade of colour, are alike in
some respect.

    65

    The word "red", as used by normal persons, also
refers to one property, not a shade, but a hue, which
may be common to objects of different shades. When
we look at the white light spectrum (or a rainbow),
we see a continuous range of continuously varying
shades of colour. Yet despite this continuity, the
spectrum is divided into fairly definite bands, each
containing a range of specific shades which are differ-
ent from one another, yet have something in common.
All the shades in the red band, for example, have some-
thing in common which they do not share with shades in
the orange band, or the yellow band, despite the possi-
bility that shades of red and shades of orange may
resemble one another closely, if they are near the
red-orange boundary.
    Hampshire wrote, in "Thought and Action", on p.35:
"there are a definite number of discriminable shades,
to each one of which a definite name can be allotted".
He must surely have meant hues rather than specific
shades, for there seem to be indefinitely many different
specific shades. Nevertheless his remarks illustrate
what I mean by an f-word. I shall ignore for the time
being, the fact that the boundaries between bands may
be more or less indeterminate, and the fact that differ-
ent persons may see their bands in different places.
(Contrast what I have said with Wittgenstein's remarks,
in the "Blue and Brown Books", p.133-5.)

3.A.2.  Just as normal persons can learn to see the hue
common to objects with different shades of red, and
associate it with the word "red", so can most normal
persons learn to perceive the property common to objects

    66

which are all triangular, even though they have different
specific triangular shapes. Such persons may adopt an
f-rule, correlating the word "triangular" with that common
property. In addition, each of the many different
specific triangular shapes may be memorized and correlated
with a descriptive word by an f-rule. (E.g., the shape
of an equilateral triangle, or a triangle whose sides
meet at angles of 90°, 60° and 30°.)
    It should be noticed that I am not talking about
so-called "perfect" triangles. I am talking about
shapes which we can all recognize and which a child can
learn to distinguish long before it learns to prove
geometrical theorems or talk about "perfectly" straight
lines. We all know how to distinguish triangular
pieces of cardboard, or diagrams, from round or square
ones, for example. In chapter seven something will be
said about "perfect" geometrical concepts and other
idealized concepts, such as the concept of a perfectly
specific shade of colour. But this chapter is not con-
cerned with such things.

3.A.3.  The examples "triangle" and "red", illustrate
an ambiguity in talking about a word which is correlated
with just one property. This does not mean that there
may not be a whole range of different properties which
correspond to the word. For example, there are very
many different shades of red which may be possessed by
red objects, and different triangular shapes which may
be possessed by triangles. Nevertheless, in each case,
if the word is an f-word, than there is only one property
in virtue of which all those objects are correctly
describable by it. (Cf. 3.C.5.)

    67

    Neither do I wish to rule out the possibility that
there may be other less specific properties common to
all the objects described by an f-word. For example,
even if the word "triangular" refers to only one property,
there are nevertheless several other properties common
to all objects which it describes. For example, all are
bounded by straight lines, may be inscribed in circles,
and have no reflex angles. These properties may be
possessed by other objects too, such as square or hexa-
gonal objects. But there are other properties common
only to triangles, such as the property of being recti-
linear and having angles which add up to a straight line.

3.A.4.  It may be objected that there is not just one
feature or property associated with the word "triangular"
since a definition can be given in terms of simpler
notions. But anyone who talks about the possibility of
analysing such a concept in terms of simpler ones, or
about criteria for telling whether an object has the
property or not, must at least admit that at some stage
we simply have to recognize something, be it a criterion
or one of the "simpler" properties. Then a word could
be correlated with that "something" by means of an f-rule
and would illustrate what I am talking about. However,
since triangularity is a feature which most of us can
perceive and take in at a glance, why not allow that the
word "triangular" can be used as an f-word, if there are
f-words at all? I do not wish to settle this here.
(One person may regard some property simple or unanalys-
able, while another regards it as built out of simpler
properties. Are there two properties, or only one?
Cf. "tetrahedral" example in 2.C.8.)

    68

3.A.5.  F-words need not describe only continuously
existing material objects. A sound which starts, lasts
a few minutes, then stops is a particular, and may be
described as a sort of physical object with physical
properties. It can be located in time, and sometimes
in space too. It may be a sound of a definite pitch, and
this property may be shared with other sounds. Or it
may have a definite timbre, such as the tone of a flute,
or clarinet, or electronic organ, and share this pro-
perty with other sounds quite different in pitch. It
may be the sound of a major chord, and share this pro-
perty with other sounds in different keys, or with
different dynamic distributions (e.g. the tonic may be
louder than the dominant in one, but not the other).
Each of these properties common to different sounds can
be memorized, associated with a descriptive f-word, and
recognized again later on.
    A sound may also change. If it changes in pitch,
then the pattern of changes may be recognizable, and we
can speak of a "tune", and other sounds may have the same
tune. Some persons may be able to memorize the sound of
a whole symphony, and associate that property with an
f-word. Less fortunate beings can merely recognize
parts of symphonies, or the styles in which they are
written, such as Beethoven's style, or Hindemith's.
These are properties of enduring objects or events, and
have to be perceived during an interval of time. But
they may all be correlated with descriptive f-words, by
means of f-rules.

3.A.6.  The important thing about all the examples is
that they involve properties which can be perceived by

    69

means of the senses, memorized, and recognized in new
instances. A property which is not observable by means
of the senses, such as the property of being magnetized,
or of having a certain electrical resistivity, cannot
be correlated thus with a descriptive word and provide
a rational explanation of our use of the word. Words
may, of course, refer to such "inferred" properties
(e.g. "dispositional" properties), but not in the same
way. (There may be some intermediate cases.)

3.A.7.  These observable properties are the basic enti-
ties out of which the meanings of many kinds of descriptive
words are constructed. I have so far described only
the very simplest kind of descriptive word, governed by
the very simplest kind of semantic rule, namely a rule
which correlates one property with one word.
    It is commonly denied that descriptive words cor-
respond to single entities which are their meanings,
or account for their having meanings (see, for example,
remarks in 2.D.6 and 7, etc., to the effect that the
"one-one" model will not do). Unfortunately, this
denial is usually much too vague to be of use to anyone.
By showing that there are other kinds of descriptive
words than f-words, and why they fail to fit the "one-
one" model, I shall be describing one clear sense in
which the denial is justified, though relatively trivial.
But it is important to distinguish the thesis that the
one-one model is inadequate to account for most of our
descriptive words from the thesis that descriptive words
do not refer to properties or other universals which
can explain their use. It is very easy to confuse these
theses. (I think Wittgenstein's discussion of the notion
of "following a rule" in "Philosophical Investigations"

    70

was intended to support something like the latter
thesis. I shall not explicitly argue against him,
but my account can be construed as an attempt to show
that an alternative picture can be coherently constructed.)
    The time has now come to turn to more complicated
types of semantic correlations.

3.B. Logical syntheses

3.B.1.  Some one-one correlations between descriptive
words and properties have been described, and now we must
see how more complicated correlations are possible if
new semantic correlations are constructed out of the
simplest ones. Three methods of construction will be
described in this section, namely disjunction, conjunction
end negation. These correspond to the use of the
logical connectives "or", "and" and "not" in explicit
definitions. They may be thought of not only as pro-
positional connectives, but also as meaning-functions,
which take words as arguments and yield expressions
whose meanings are simple functions of the meanings of
the arguments. I shall simply assume that we under-
stand these logical words, and will not try to explain
how they work. (See chapter five.)
   The construction of new semantic correlations of the
sorts about to be described may be called a process of
"logical synthesis". Later, we shall contrast it with
processes of "non-logical synthesis".

3.B.2. D-words
   The first sort of rule which does not fit the simple
one-one model is a semantic rule which correlates a word
with more than one property, disjunctively. I shall call

    71

such a rule a d-rule, and the word it governs a d-word.
For example, the word "ored" may be correlated with the
two hues, red and orange, so that the word describes
an object if and only if it has one or other of these two
properties. If the words "red" and "orange" are f-words
which refer to these two properties, then the word "ored"
means the same as "red or orange".
    A more interesting kind of disjunctive rule is one
which correlates a word with a whole range of properties,
such as a range of specific shades of colour. The word
"red" may be used as a d-word of this sort, instead of
as an f-word. For there may be persons who can see and
discriminate and memorize specific shades of colour,
though quite unable to see hues in the way in which most
normal persons can, as described in 3.A.1, above. Such
a person will see the spectrum as a single band of
continuously varying shades of colour, much as we see one
of the bands of the spectrum. This hue-blind (but not
colour-blind) person will not see the spectrum divided
up into different bands, so he cannot learn to use the
word "red" in the normal way. If presented with pieces
of coloured paper all of different shades, and instructed
to arrange them in groups with a common feature, he will
be unable to do so, even if there are several red pieces,
several yellow pieces, and so on. To him they all simply
look different. (They look different to normal persons
too, but they also have respects of similarity, which is
why we can group them.) Though unable to learn to use
the word "red" in the normal way, such a hue-blind person
may learn to use it as a d-word, by memorizing all the
different shades in the spectrum which lie in the red
band, and then describing an object as "red" if and only
if it has one of the specific shades of colour which he

    72

has learnt to associate with the word. Similarly,
a person who is not hue-blind, but sees the spectrum
divided up differently from the way we do (his "hues"
are different because he sees bands in different places)
may learn to use our word "red" as a d-word, by memorizing
specific shades of colour. All we require of such
persons is that they agree with normal persons as to
whether objects are exactly the same shade of colour or
not.

3.B.2.a.    In the same way, there may be a person who is
unable to see anything common to all those shapes which
are triangular, although he can see and discriminate
specific shapes and tell, for example, whether two
objects are both equilaterally triangular, or not.
Perhaps he is unable to count up to three - but the
explanation of his inability to perceive triangularity
need not concern us. Such a person cannot use the words
"triangle", "quadrilateral", etc., as f-words, for he
cannot see any common property with which they may be
correlated. But if he can see and memorize specific
triangular shapes, such as the shape of a right-angled
isosceles triangle, and distinguish them from other
specific shapes, such as the shape of a square or a
regular pentagon, then he can memorize a whole range of
specific triangular shapes and adopt a d-rule correlating
them with the word "triangular". He then uses the word
to describe objects if and only if they have one of the
many shapes which he has memorized, as in the case of
"ored" or the d-word "red". (As before, I am not talking
about "perfect" mathematical, shapes, but shapes which we
can all learn to recognize and discriminate with greater
and lesser degrees of accuracy.)

    73

    Of course, these examples are highly artificial,
since there are indefinitely many different specific
shades of red, and indefinitely many specific triangular
shapes and nobody could memorize them all. But the
essential point could as well be illustrated by a person
who merely memorized very many different shades of red,
or triangular shapes, enough to get by with in most
ordinary circumstances. (Later, a procedure for picking
out a whole range of properties without memorizing them
all will be described.) Notice that a person who
memorizes a set of properties and correlates them with a
word need not have a name for each of them. His d-word
need not, therefore, be definable in his vocabulary.

3.B.3.  C-words
    The next type of semantic rule is one which correlates
a word with a combination of properties. This is a
c-word, and refers to a set of properties conjunctively.
For example, the word "gleen" might be defined so as to
refer to the combination of the hue, green, and the
surface-property, glossiness. It would then describe
objects which possessed both of these properties, and
would be synonymous with the expression "green and
glossy". (As before, someone might learn to use a
c-word to refer to a combination of properties without
being taught names for those different properties.
Then, in his vocabulary, the word would be indefinable,
despite the possibility of defining it in a richer
vocabulary.)
    We may think of such c-rules, like d-rules, as being
logically constructed out of f-rules, just as we can
think of the meanings of c-words and d-words as logically
constructed out of the meanings of f-words (although, of

    74

course, the language in which they are used need not
include the required f-words, for the reason just stated).
    We need not restrict the notion to combinations of
only two properties. A word might describe a sequence
of sounds if and only if it possessed the three properties
of being in the key of E-major, of being the sound of a
piano, and of being in the style of Beethoven. This
would then be a c-word referring conjunctively to three
properties.

3.B.4.  N-rules
    Semantic correlations involving negation can be very
confusing as there are several different ways in which
negation may come in, and it is important to be clear
about them.
    I shall describe a strong n-rule as a rule which
correlates a word "W" with a property P negatively, as
follows: the word "W" does not describe an object if
that object has the property P. In such a case, the
possession of the property is a sufficient condition for
not being describable by the word, and the absence of the
property is a necessary condition for being describable.
Whether it is also a sufficient condition, will depend
on the other rules, if any with which the n-rule is con-
joined. Thus, the expression "scalene-triangle" is
correlated negatively with the property of symmetry, and
requires the absence of that property in objects which
it describes. But the absence of the property is not
sufficient, for in addition the object must be triangular.
Usually there are other rules and absence of the negatively
correlated property is not sufficient to ensure
describability.

    75

3.B.4.a.    These "strong" n-rules, specify inapplicability-
conditions for words. They are to be distinguished
from "weak" n-rules, which merely limit the applicability-
conditions of words, thereby helping to make the meanings
of indefinite words more definite. The difference may
be illustrated by means of an example.
    I have hitherto ignored doubts which may arise over
the possession or non-possession of a property by an
object, but it is sometimes difficult to decide whether
an object possesses some property or not, where this
is not an empirical difficulty arising out of the diffi-
culty of seeing the object clearly or the difficulty
of remembering what the property looked like. I may
have plenty of red objects around to remind me of the
hue associated with the word "red" (an f-word) and be
able to see an object quite clearly in a good light, and
yet be undecided as to whether it has the same hue as the
other red objects or not. In this case I am undecided
about the redness of the object, though I may be able
to see its specific shade quite clearly and recognize
it again in other objects. We may say that the word
"red" refers to an indefinite property, and that its
extension has an indefinite boundary. (Many more kinds
of indeterminateness will be described in chapter four.)
    In such a case, the indefiniteness may he eliminated,
or at least reduced, by the adoption of an additional
rule. Suppose we call the difficult shade of colour,
of the doubtful object, "redange" (if it is on the red-
orange boundary). Then we may decide to adopt an
additional rule correlating the word "red" with the shade
redange positively, or an n-rule correlating it negatively.
In either case the decision would make the word more
definite. In the former case, we should have a new

    76

word "RED", say, governed by a disjunctive rule: it
describes objects if they definitely have the hue red-
ness, or if they have the specific shade, redange.
In the latter case we should have a new word "RED"
which does not refer to the shade redange, and means,
roughly, "red and not redange".
    Now, however, there is an important ambiguity to
be noticed. Does this new n-rule specify that not
being redange is a necessary condition for being RED, or
does it merely specify that being redange is not a
sufficient condition for being RED? In the former case,
the n-rule is a strong one, in the latter case we have
a weak n-rule.

3.B.4.b.    The weak n-rule, unlike the strong one, leaves
open the question whether objects which are redange in
colour may not have some other feature in virtue of
which they are RED. That is, the strong rule takes
"not-redange" to be part of the meaning of "RED", while
the weak n-rule merely specifies that "redange" is not
part of the meaning of "RED". Something else may make
it impossible for any object which is redange also to
be RED, such as the impossibility of its having some
other specific shade of colour which is definitely a
shade of red. But the impossibility does not have its
origin solely in the weak n-rule. Indeed, the weak
rule leaves open the possibility that the word "RED" is
conjunctively correlated with the property of being
glossy, in which case a glossy and redange object would
definitely be RED, despite the weak negative correlation
between "RED" and the shade, redange.
    The weak rule specifies a sort of irrelevance
condition: being redange is irrelevant to being RED, and

    77

other factors must settle the matter. If there are
definitely no other factors, then the object which is
redange is definitely not to be described as "RED":
this is how even a weak n-rule may help to eliminate
borderline cases and so reduce indefiniteness.

3.B.4.c.    It might be thought that weak n-rules were
always necessary to specify that words are neither
incompatible nor stand in a relation of entailment, but
this is not so. We can learn to correlate the word
"red" with a recognizable hue, and the word "glossy"
with a recognizable property of surfaces, without the
need for any explicit rule to the effect that the pro-
perty referred to by one of them is irrelevant to
describability by the other. This is because we can
tell whether an object is red, or glossy, without ever
having to notice whether it has the other property or not.
We can therefore learn to understand either word without having
to be told anything about its connection with the property
referred to by the other, since each refers to a property
which is sufficiently definite without any rule cor-
relating it with the other. The mere fact that a thing
is glossy does not, on its own, raise the slightest
doubt as to whether it is red or not, so there is no
doubt to resolve by adopting- an n-rule, even a weak one.
Only where there is some kind of indefiniteness, as in
the case of redange objects, can there be a point in
adopting a weak n-rule (and even then there is a point
only insofar as there is a point in removing the indefinite-
ness: see chapter four). This is another illustration
of the remark made in 2.D.3. and 2.D.4 to the effect that
"links between descriptive expressions" may be rendered

    78

superfluous by semantic correlations between descriptive
expressions and properties.

3.B.4.d.    The importance of all this is that it shows
that sometimes correlations between words and properties
are enough to determine the uses of the words without
the aid of additional correlations between words and
words. This shows that when people argue that the
incompatibility of determinates in the same range of
determinables is due to linguistic rules which make
descriptive expressions incompatible, then this must be
defended by an argument to show that such rules are
necessary. Perhaps correlations between words and pro-
perties can suffice to give the words the meanings they
have, and the incompatibilities are due to something
other than the rules which fix their meanings. What is
more, even if weak n-rules are required, in order to
remove certain kinds of indefiniteness, the argument
shows that these n-rules do not on their own make des-
criptions incompatible: strong n-rules are needed for
that. But philosophers who so blithely say that it is
analytic that nothing can be red and yellow all over at
the same time owing to linguistic rules which make the
words "red" and "yellow" incompatible descriptions, are
not usually even aware of the difference between weak
and strong n-rules, and so do not notice that an argument
in support of the need for weak n-rules does not establish
that we need strong n-rules too. More will be said about
this below. (All this helps to illustrate the application
of sharp criteria for identifying and distinguishing
meanings.)

3.B.4.e.    It might be argued that there is no difference

    79

between f-rules and strong n-rules since every f-rule
correlating a word positively with a property is equi-
valent to a strong n-rule correlating that word negatively
with the absence of that property. Thus, the f-word
"red" would be correlated negatively with the property
of not having the hue, red. This is irrelevant to our
purposes, since the important thing is that given a
word and a property with which it is correlated we must
know whether it is positively correlated with the
property if we are to know its meaning, and it doesn't
matter if we find out the answer by discovering whether
the word is negatively correlated with the absence of
the property. In any case, it is unreasonable to
argue that in general there is a symmetry between the
possession of a property and the non-possession of a
property, since the perception and identification of,
for example, redness, is quite different from the per-
ception and identification of the "property" of not being
red. For example, when I look at the surface of an
object, I see one colour, but if the absence of a colour
is also a perceptible property, then I perceive indefin-
itely many different properties of this sort in any one
object. (There are, however, intermediate cases. For
example, some rectilinear shapes are regular and some
are irregular: which is the perceptible property and
which the absence of a perceptible property, regularity
or irregularity? It doesn't matter.)

3.B.5. Reiterated constructions
    It should not be thought that the logical operations
of disjunction, conjunction and negation can be applied
only to f-rules. For the process of constructing new

    80

semantic rules out of old ones is a process which can
be reiterated, like the process of constructing new
propositions out of old ones, using truth-functional
connectives. So not only f-rules can be disjoined,
conjoined or negated, but also d-rules, c-rules and
n-rules.
    For example, if P, Q and R are three different
properties, and S represents the range of properties
(S1, S2,?.), then a word may be governed by the follow-
ing semantic rule: The word "W" describes an object
correctly if and only if the object either has the
property P and not the combination of properties Q and
R, or it has the property R and not one of the properties
in the range S, or it has the property Q and not the
property P. The word therefore refers to the following
complex property, which is logically synthesized out of
simpler properties:
  P & not-(Q&R) .v. R & not-(S1v S2v?) .v. Q & not-P
There is clearly not just one property correlated with
the word "W". Nevertheless the correlation between the
word and the observable properties mentioned serves to
explain how the word means what it does: it determines
the boundaries of the extension of the word. So we see
that universals can explain even if the one-one model
is rejected.
    There is no need to say that there is one property
to which such a word refers, or that there is any one
thing common to all the objects which it describes, to
be discovered by abstracting from their specific differ-
ences. What on earth could abstraction yield in the case
of objects describable by a word like "W"? We may, if
we wish, make it true by definition of "property" that

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there is a property correlated with such a word, but
then we should have to distinguish some properties (or
universals) as "improper" properties,1 as they are not
perceptible objects of experience, but mere logical
constructions out of other perceptible properties.
(This should be clear even from a consideration of
simple d-rules. One word may be disjunctively cor-
related with two or more properties which have absolutely
nothing to do with one another. What point could there
be in saying that this created a new property common to
all the objects it describes? The point would be merely
verbal.)
    Another thing shown by this example, is that
demonstrating that the possession of a property by an
object is neither a necessary nor a sufficient condition
for describability of that object by some word, does
not establish that there is no definite semantic
correlation between the word and the property. For
neither possession of the property P, nor possession of
S2 is either a necessary or a sufficient condition for
describability of an object by "W". This is sometimes
overlooked by philosophers who try to show that there is
no logical connection between concepts by showing that
there are neither necessary; nor sufficient connections.
(Cf. "Goodness and Choice", by Mrs Foot, in P.A.S.Supp.
1961. See also all the talk about necessary and suf-
ficient conditions in Hart's essay: "The Ascription of
Responsibility and Rights" in L.L.I.)

3.B.6.  We have seen how words may be correlated by means
______________
1.  Cf. 2.D.6.

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of f-rules with single properties, and how repeated
application of logical methods of construction may
yield more and more complicated kinds of semantic
correlations. When words whose meanings are syn-
thesized in these ways occur in a proposition, then
it is possible to analyse that proposition into a
truth-functional complex constructed out of simpler
propositions, in the manner of Wittgenstein's "Tractatus".
   There are, however, more complicated kinds of
logical synthesis than those mentioned so far, since
quantifiers may be used too. For example, I might
define a word to mean the same as the expression "as
big as the biggest of the mammals", and this would
involve a sort of logical synthesis of a new descriptive
word in terms of old ones, namely "mammal", and "big".
Or, to take a slightly more complicated example, someone
might use the word "lawnmower" as a synonym for "machine
which has most of the properties common to things which
can cut grass". Here we have a complicated logical
synthesis involving quantification over properties. A
full discussion of such complicated cases would require
us to go into the "Ramified theory of types" of "Principia
Mathematica", which would really be unnecessary for the
main purposes of this essay.
    In addition to these more complicated types of
logical synthesis, there are also non-logical methods
of synthesizing meanings of descriptive words, some of
which will be discussed presently. But first we must
see what light all of this sheds on my claim that talk
about universals can explain our use of descriptive words.

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3.C. How properties explain

3.C.1.  The discussion of logically synthesized semantic
rules in the previous section puts us in a position to
see how talking about observable properties can account
for our use of descriptive words. We have already
noticed (in 3.B.5.) that it explains because the posses-
sion and non-possession of properties may determine
whether objects are or are not describable by a word,
and this is an explanation because, as pointed out in
section 2.D, universals are independent entities, not
essentially tied to their actual particular instances.
   This is why pointing to an observable property common
to a set of objects can explain why they are classified
together or why the extension of a concept has the bound-
aries which it does have. As shown by the examples of
the previous section, talk about universals can explain,
even when correlations are not of the simple one-one
type. In such cases, it is not the complex, logically
synthesized "improper" property which explains, but the
observable ones out of which it is synthesized, as will
be shown also by the discussion of this section. (See
3.C.6, for example.)

3.C.2.  In this section, I wish to try to show how talking
about universals can provide explanations of the sort
which were described in chapter one as "rational", or
"personal", explanations of linguistic behaviour such
as describing and classifying. By describing a person's
behaviour from his own point of view, we can explain
what it would be like to be in his position (to act for
his reasons) and this can remove certain kinds of puzzle-
ment. (The description may also serve, partly, as a

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causal explanation, from a slightly different point of
view: explanations from different points of view may
overlap to some extent.)
    It should be noted that talking about observable
properties can explain not only linguistic matters, but
other things too, such as how one recognizes a person,
for example by the sound of his voice, the shape of his
head, the colour of his hair or some other feature or
combination of features. It may explain one's reaction
to a work of art: "It is not so much because of the
pattern of shapes that I like the painting, as because
of the distribution of colours." Similarly, mention
of an observable property explains how a person recog-
nizes an object as being of a certain kind, and the
fact that he intends a word to refer to that property
explains why he describes the object as he does.
    Of course, we do not feel puzzled or curious con-
cerning many familiar types of behaviour, since we know
what it is like to produce that kind of behaviour.
Hence we do not feel the need for explanations of the
kind which I am talking about. But this is only because
of the familiarity of the situations, which ensures
that we already possess the necessary explanations.
Talk about universals can make explicit the reasons for
which we do not regard certain things as puzzling, as well
as providing explanations which remove puzzlement.

3.C.3.  We must now be more precise. Talk about pro-
perties can explain because we can have them in mind
without having any of their instances in mind. We can
perceive properties (they are tangible to the senses);
we can pay attention to them or draw attention to them
("Look at the colour of her dress!"); we can bear them

    85

in mind ("Think of the colour of our wall-paper when
you choose the curtains"); we can think about them and
imagine them in the absence of any instances ("Try to
imagine a shape for a suitable frame for this picture");
we can memorize them and recognize them again in new
instances, or in new contexts ("Look, that roof is the
same shape as ours!" or "I'll never forget the sound
of his walk, I'll always be able to recognize his
approach by the sound of his footsteps").
    All these are ways of having a property in mind,
and it should be noted that this need not involve having
any sort of "mental image" of the same sort as after-images.
I can think of the way a tune goes without
actually hearing it in my head, or remember what some-
one's face looks like without actually having a visual
image. Of course, in a sense I hear the tune or see the
face "in my mind" (e.g. "in my mind's eye"), but this
may be quite different from, for example, seeing ghost
pictures. In addition, one may be quite sure, and
correctly so, that one can recognize a face or a tune
or the sound of a word when one next meets it, even
though one cannot at present remember how it goes.
("I've got his name on the tip of my tongue ?")

3.C.4.  Since I can have a universal in mind, I can
decide to associate a word with it, so that the word
describes objects if and only if they are instances of it.
Or I may associate two universals with the word, so that
it describes anything which is an instance of at least
one of them. Or I may decide that the word is to
describe only objects which instantiate both of them.
And so on. Alternatively, instead of deciding, I may

    86

simply acquire the habit of making the association, as
a result of my environment or education.
   If we say that the correlation of a descriptive
word with a property or set of properties, in one of these
ways, gives the word its meaning, then, since it is
possible to have a universal in mind without having any
particular instances in mind, there is a clear sense
in which the meaning which is given to the word deter-
mines the way in which it is to be applied in particular
cases. The decision to associate a word with a uni-
versal or set of universals does not require the word
to be correlated with any actual particular instances,
so this decision is independent of and distinct from
any later decision to say that some particular object is
correctly described by the word. (E.g. it may be an
object which one has never previously seen.) Neverthe-
less, the later decision is justified, or explained, or
determined, by the earlier one, together with the fact
that the object has such and such observable properties.
    This shows that there is room for a distinction
between the intended use of a word, or its meaning, and
its actual use. The meaning, or intended use, explains
the actual use and is therefore not constituted by it.
(It seemed that Hampshire wished to deny this, in
"Scepticism and Meaning". See also Bennett: "On
Being Forced to a Conclusion", in P.A.S.Supp., 1961.)
Here is the main reason why arguments from paradigm
cases are likely to be fallacious. Of course, since the
meaning of a word may be indefinite in some respects, it
need not fully determine the whole use, as unexpected
borderline cases may turn up. (This fact seems to have
obsessed Wittgenstein, in his discussion of "following
a rule", in "Investigations", so that he overlooked the

    87

fact that part of the use may be determined "in advance"
by the meaning of a word.)

3.C.5.  We can see more clearly in what sense the
properties correlated with a word can determine its
application or explain our use of the word, by going
back to some of the examples of the previous section.
    It was shown that the word "red" could be used
either as an f-word, referring to a single property, the
observable hue, redness, or as a d-word, disjunctively
correlated with many specific shades of red. Let us
distinguish these two cases by talking about "f-red"
and "d-red". A person who can see and memorize hues,
may learn to use the word "f-red", while a hue-blind
person, who can only see specific shades, has to use the
word "d-red". Now suppose each of them comes across
an object with a specific shade which he has never
previously seen, though it is a shade of red. The
first person, who can see the hue exhibited by the
object, is able to recognize it as being describable as
"f-red". The hue-blind person, however, since he cannot
see the hue, and does not recognize the specific shade,
will say that the object is not d-red. Here it is clear
that the different ways in which they correlate the word
"red" with properties can explain the difference in their
behaviour in the case described. In the same way, the
way in which each of them correlates words and properties
explains his behaviour in cases where their classifications
do not diverge. The behaviour is the same in such cases,
but the explanations are not, and the difference is a real
one even if, as a matter of fact, no shade of colour ever
happens to turn up which would show up the difference.

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In other words, the difference may be of a kind which can
be described only by saying what it would be like to
be in the position of each of them, to be deciding in
the same way as they do how to describe the things they
see. They may describe the same objects with the same
words, but in virtue of different facts. (Cf. "tetra-
hedral" example, in 2.C.8.)

3.C.6.  It should be noted that it is possible for these
two persons to agree in their behaviour even when they
come across new shades of colour, for in such a case, the
hue-blind person may somehow simply guess that people
who can see hues would describe the object as "red".
Or he may simply "decide" to enlarge the scope of his
d-word "red" by including the new shade as one of the
properties henceforth to be disjunctively correlated
with the word. Or he may simply happen to call the
object "red" without even noticing that its shade is
not familiar to him, though if asked why he used the
word he is baffled. In all these cases the explanation
of what he does, if there is one, is quite different
from that of the person who sees the hue. The hue-
blind person is not, from his own point of view, using
the word rationally, or intelligently, or according to
a rule in the cases described. His decision to describe
the object as "red" is quite arbitrary, as it is not
explained or justified in any way by the meaning with
which he understands the word, despite the fact that
other persons may be able to see a justification for
classifying the object with those which he calls "red".
Here is a case where meaning does not determine use.
    There may, of course, be a causal explanation for
this non-rational behaviour, such as a psychological

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or physiological explanation. But this does not make
his decision any the less arbitrary, from his point of
view. Talk about what goes on in his brain does not
give a rational explanation. It does not describe
from his point of view what it would be like to act as
he does, for he is as unaware of anything going on in
his brain as we are.
    (In some cases, it may be difficult to tell whether
a decision is arbitrary or not (determined by a meaning
or not).)

3.C.7.  This difference between a person who uses a
word like "red" as an f-word referring to a single
property and a person who uses it as a d-word referring
to a whole range of properties disjunctively, gives some
point to the assertion that they do not really under-
stand each other, despite the fact that they describe
things in the same way in general. For similar reasons
we may say that persons who are totally blind or colour-
blind cannot fully understand what others mean by colour-
words, even though they can, in a way, use the words;
for example, when a blind man asks whether the sky is
red because he wants to know whether it is likely to be
raining the next day or not. A blind person cannot use
the word "red" in the same way as one who can see, for
he cannot discover in the same way whether things are
red or not. This inability is explained by the fact
that he cannot see the property or properties to which the
word refers. By contrast, the sighted person's use of
the word is explained by the fact that he can see the
property.
    If everything that explained our use of descriptive
words were in the "realm of symbols" (see 2.D.8), then

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the blind or colour-blind person would be able to use
colour words in the same way as normal persons, since
they have as much access to symbols.

3.C.8.  Of course, talk about universals does not
explain everything, it does not answer all questions.
For example, we cannot explain the fact that the hue-
blind person chooses just this range of specific shades
of colour to correlate with his d-word "red" in terms of
properties which he can see. Similarly, the fact that
words such as "horse" and "rod" refer to those ranges
of specific shapes to which they do refer is not expli-
cable in terms of some common visible property. (See
next chapters 4.A.4.a & 4.A.6.) Quite a different
level of explanation is required. It may be an his-
torical explanation, referring to some arbitrary decision
made in the past, or there may be a sociological or
anthropological explanation, in terms of our environment,
or in terms of certain natural reactions which we all
share, or in terms of our purposes in classifying things.
One way of looking for these explanations is to consider
factors which could cause changes in usage.
    But even if universals do not explain how words
have the meanings which they do have, nevertheless, they
explain how their having these meanings determines what
we say about the world.

3.C.9.  It should not be thought that all this talk about
the way in which properties can explain our use of des-
criptive words is irrelevant to the main purpose of this
essay. For the difference in the explanations of the
two kinds of use of the word "red" gives us good reason

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to say that the word has different meanings when it is
used as an f-word and when it is used as a d-word, even
if the difference does not affect the class of objects
which the word happens to describe correctly. Thus,
the d-word is synonymous with a long disjunction (or
would be if there were enough names for specific shades
of colour in the vocabulary), whereas the f-word is not.
But here we are obviously applying very sharp criteria
for identity of meanings, for it is obvious that for most
normal purposes the difference would not matter in the
least and the words would be regarded as synonymous
(see chapter Two, especially 2.C and 2.A.). Similarly,
when "triangular" is used as a d-word correlated with a
range of specific shapes it does not have the same meaning
as when it is used as an f-word correlated with a single
property common to all triangular objects, if sharp
criteria of identity are employed. In 3.B.4.b, etc., we
saw that the word "RED" might be negatively correlated
with the specific shade redange, by either a strong n-rule
or a weak n-rule. Here again, for similar reasons, we
must say that these rules give the word two different
meanings, if we are to use the strictest possible criteria
for identity of meanings. (Cf. example in 2.C.8.)
    So all these examples help to illustrate the claim
that by considering correlations between words and pro-
perties, we can apply strict criteria for identity.

3.C.10. The importance of all this for the purpose of
this essay is that it provides us with a whole host of
potential candidates for the title of "synthetic necessary
truth", which we should not be able to discuss if we used
more familiar loose criteria. Thus, since the meanings

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of "d-red" and "f-red" are so different (they are
correlated with different properties), it seems unlikely
that it is analytic that all d-red objects are f-red.
But is it necessarily true? Is it necessarily true
that anything which has one of the specific shades
correlated with "d-red" has the hue correlated with
"f-red"? And what about the converse? We are armed
against the slick, question-begging argument which
demonstrates that the necessity is analytic by sliding
from one meaning of "red" to another: our sharp cri-
teria will not let this go undetected. (Cf. 2.C.10.)
Similarly, we can open an interesting question of the
form: Is it analytic that nothing is RED and redange?
(Cf. 3.B.4.d.)

3.C.11. It may be objected that my account is incomplete,
since I have failed to describe how we can tell which
sort of rule is being followed in some of these examples.
This is, of course, only one aspect of a general problem:
How do we tell which properties are the objects of a
person's mental acts? How can we be sure which property
a person is thinking about, or looking at, or trying to
draw attention to, or surprised by? I think I have given
the beginnings of an answer to this by describing how in
some cases correlating words with properties in different
ways may lead persons to behave in different ways. But
I do not wish to solve these problems of mind and body
here. This is a phenomenological enquiry, and I am
trying only to describe the use of words and sentences
from the point of view of the person who uses them, and
from his point of view there is certainly a difference
between being; able to see the redness common to all red
things and having to memorize a whole range of different

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specific shades of red, no matter how difficult it
may be for other persons to detect the difference in him.
    The fact that we have words in English which enable
us to describe the difference is strong evidence for the
existence of ways of detecting it. But I shall not look
for them.

3.C.12. I have so far illustrated the application of
sharp criteria for identity of meanings only by comparing
and distinguishing different methods of logical synthesis.
But, as already remarked, there are other ways in which
the meanings of descriptive words may be synthesized, and
they also yield interesting examples of connections between
concepts which are apparently not analytically related.
Some of these will now be described.

3.D. Non-logical syntheses

3.D.1.  So far, only logical methods of constructing new
concepts out of old ones have bean described. In this
section it will be shown that there are other types of
construction, which involve more or less complicated
procedures for picking out properties or for deciding
whether an object is describable by a word. Where such
a procedure is involved in the application of a word,
I shall refer to it as a "p-word", and say that it is
governed by a "p-rule".

3.D.2.  We have seen that a person who is hue-blind,
but can see and memorize specific shades of colour, may
learn to use the word "red" as a d-word, disjunctively
correlated with a range of specific shades. If he finds
it difficult to memorize so many specific properties, such

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a person may adopt a procedure for picking out the right
shades.
    For this we require that he should be able to arrange
specific shades of colour in the order in which they
occur in the white-light spectrum, or to tell whether
three given shades are in the right order or not. There
are many different ways of doing this. For example,
he may simply be able to see which of three given shades
lies between the other two. Or he may simply memorize
the order in which the shades occur in the spectrum
(though this would again raise the memory difficulty).
Or he may simply memorize the appearance of the whole
spectrum and then tell whether the shade of colour of
one object lies between the shades of two others by
looking to see whether it does so in the spectrum as he
remembers it. Perhaps he can arrange bits of coloured
paper in the right order by experimenting with them
until their colours vary in the least discontinuous way
along the row. The differences between these various ways
of judging the order of shades of colour may, as before,
be detectable in cases where new shades turn up. Which
of these methods is employed will make a difference to the
procedure I am about to describe, but that need not
concern us.
    If a person is capable of making judgements of the
sort "This object has a shade of colour between the shades
of those two", then he can learn to use the word "red"
as a p-word by memorizing two shades which lie as near as
possible to the boundaries of the red band of the spectrum
and then applying the word to objects if they have shades
lying between the two which he has memorized.
In this way he avoids having to memorize ail the
individual shades, though he must, of course, memorize the

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two boundary shades and the procedure to be employed.
So, unlike the person who simply follows a d-rule, he
can deal with specific shades of red which he has never
seen before. (See 3.C.5.)

3.D.2. (Note).  There are, of course, far more com-
plicated and indirect procedures which may be used for
applying colour words. Thus, a person who is completely
colour-blind, and cannot even distinguish specific shades
of colour, may have to employ a spectroscope, and make
use of a correlation between spectroscopic readings and
colour-words, in order to decide how to describe objects.
Or instead he may take a "colour slave" around with him,
that is someone who can perceive colours and has been
trained to give the right answers to questions about
colours of objects. (See Smart, in Philosophy, April
and July, 1961, circa p.140.) Or he may simply ask
other people, without bothering to acquire a slave.
    A person who can distinguish shades, but cannot
memorize them easily may have to carry a colour-chart
around with him, for comparison. Even so, he must
memorize the correct procedure for using the chart, such
as what it is about the samples that he has to compare
with the objects he wishes to describe. This may be
compared with our use of metre rules and standard weights,
which we require on account of our inability to memorize
lengths and weights accurately. I call these standard
particulars. Of course, they aid not only our memory,
but also our rather limited powers of discrimination and
comparison (rulers and pieces of string help us to compare
lengths, weights and balances help us to compare weights
accurately). That is to say, one and the same thing may

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serve both as a standardized particular and also as
an instrument (e.g. graduated rulers, spring balances).
Notice that even with these aids, there is still always
some point at which something or other has to be per-
ceived and recognized by the observer, even if they are
only numbers or letters flashed on a dial.

3.D.3.  Procedures may be used in connection with shape
concepts too. For example, the person who is unable to
perceive and recognize the feature common to all tri-
angles, and has trouble memorizing all the specific
triangular shapes, may learn to use a procedure for
picking out triangular objects.
    Suppose, for example, that such a person is able
to tell, by examining two objects, whether it is possible
to deform the shape of one of them into the other by
using only stretches and shears. Such a deformation
will turn triangles into triangles, quadrilaterals into
quadrilaterals, and so on, since it preserves straight-
ness of lines and does not turn corners into straight
angles. Since any triangular shape can be turned into any
other triangular shape by two stretches and a shear (one
stretch to get the base right, another to get the height
right, and then a shear to get the vertex in the right
position relative to the base) it is possible for the
partially shape-blind person to memorize just one specific
triangular shape, and then decide whether objects are
triangular or not by seeing whether their shapes are
deformable into the one which he has memorized by a
succession of stretches and shears. In this way he could
use the word "triangular" according to a procedure, as a
p-word. Similar sorts of procedures could be used for

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words like "rectangle", "quadrilateral", "parallelogram",
if suitable kinds of deformations are allowed.

3.D.4.  There are, of course, other sorts of procedures
which might be used for applying the word "triangular",
by a person who could neither see triangularity, nor
discriminate and memorize specific triangular shapes.
He might apply the word to objects by looking to see if
they had an outline bounded by straight lines, and then
uttering the sounds "bing" "bang" "bong" in sequence as
he pointed to each side in turn. If he could do this
without leaving out any side, and without pointing to any
side more than once or uttering any of the sounds more
than once, then he would describe the object as "tri-
angular". He need not know how to count, or read any-
thing more into the ceremony than I have described in it.
(Compare also, Nicod: "Foundations of Geometry and
Induction", Part III.)

3.D.5.  Still more geometrical examples are available.
The word "star" may be used to describe rectilinear
plane figures in which alternate angles are reflex and
acute, and the word "starlike" to describe objects with
this shape. A person who could not perceive and memorize
this sort of shape might pick out objects to be described
by the word, by seeing whether all the sides were straight
and the angles came in the order: bending in, bending
out, bending in, bending out ? etc., as he ran his
attention round the boundary.

3.D.6.  In each of these cases, a new geometrical concept
is synthesized out of other geometrical concepts by means
of a geometrical construction. It is not a logical

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construction, since, for example, the procedure for
picking out "starlike" objects does not involve looking
to see whether objects have certain combinations of
properties, or whether certain properties are absent,
etc. The notion of a shape built up by adding straight
lines one after the other, bending first one way then
another and finally closing up, is different from the
notion of a shape which is a certain combination of
shapes or other properties. We do not, in employing
this sort of procedure, look to see which properties
an object has and then apply truth-tables. We use
notions which do not correspond to properties of the
object as a whole, in order to build up a property of
the object as a whole. (So we have a kind of complexity
which cannot be analysed truth-functionally, in the
manner of the "Tractatus". Compare 3.B.6, above, and
3.D.9, below.)

3.D.7.  There are also many musical examples. A person
who can listen to a triad (sound made up of three notes)
and tell whether it is a major chord or not just by its
sound, can use the expression "major chord" according to
an f-rule correlating it with a single property. A
person who cannot do this can nevertheless use the word
according to a p-rule provided that he can hear the
three notes separately (some can do this, some cannot),
and can sing, aloud or "to himself" a major scale starting
on any given note. (One may be able to recognize the
sound of a major scale without being able to recognize
the sound of a major chord.) The following procedure
could then be used for picking out major chords: sing
the major scale starting on each of the three notes
in the triad, and if one of the scales is such that the

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other two notes occur as the third and the fifth notes
of the scale, then the triad is a major chord.
    It is conceivable that a person may be able to
recognize the sound of a major chord as a whole without
being able to hear the three notes separately, in which
case he could not apply this procedure. Thus we should
have two different concepts of "major chord", and
familiar questions would arise about the relation between
them. (See 3.C.9, etc.) (Compare: most of us can
recognize the characteristic timbre of a flute without
hearing the harmonics separately. Perhaps some persons
can recognize the sound only by listening for harmonics
and seeing how they are distributed.)
    Here again, the synthesis is non-logical, because
the object (a sound) has the synthesized property not in
virtue of having or not having several different pro-
perties, but in virtue of the fact that its various
"parts" stand in some non-logical relation to one another,
or some of its properties stand in non-logical relations
to other properties.

3.D.8.  Just as logical operations can be applied
recursively, so as to construct complex semantic cor-
relations, so also is it possible to apply logical
operations to p-rules, yielding "mixed" rules. For
example, a p-rule may be conjoined with a d-rule, and
the whole may be negated and conjoined with a c-rule.
Or a procedure may start with a property which has
already been synthesized to some extent. (See 3.B.5.)
    All this helps to illustrate the way in which the
one-one model for semantic correlations is inadequate.
We could, as pointed out above (3.B.5.) say that every
descriptive word referred to one property (or, more

    100

generally, one universal), but then not all properties
could be used to explain the use of descriptive words,
since some of them would be "improper" properties con-
structed logically or otherwise out of simpler pro-
perties, and only the simpler observable properties can
explain (e.g. by explaining in detail how the procedure
for applying a word works).

3.D.9. Remarks
     There are several points to notice about these
examples. First of all, although a procedure may help
someone to pick out something which he cannot perceive or
memorize, it is always necessary for him to be able to
perceive some features or properties of the objects which
he wishes to describe, and he must be able to memorize
something, including the type of procedure to be employed
(which is, of course, a complicated universal).
    Secondly, as before, we have found that two different
p-rules, or a p-rule and an f-rule may both give a word
very similar uses (e.g. the extension may be the same in
both cases). Once more we can apply sharp criteria for
identity and say that they then have different meanings
(though they are the same for normal purposes), thereby
leaving open interesting questions about synthetic
necessary connections.
    Thirdly, it should be noted that a person may learn
to follow a procedure without being able to describe it
in words, for one reason or another. (See appendix on
"implicit knowledge".) Hence he may have difficulty in
saying what he means by a word, though he can use it as
a matter of course. This helps to account for the fact
that people may fail to notice how all these different
kinds of rules may lie behind one and the same familiar

    101

word, such as "red".
    Words which do not refer to observable properties,
such as "magnetic", cannot be used according to f-rules,
but must be governed by p-rules or rules constructed in
a complicated way out of f-rules. These words may be
described as referring to "inferred" properties.

3.D.10. Finally, it may be remarked that the point
in describing the syntheses of the present chapter as
non-logical is that in each case the type of synthesis
is restricted in its applicability to special kinds of
features or properties, whereas logical methods of
synthesis are quite generally applicable. They are
topic-neutral. (In chapter five, topic-neutrality
will be selected as the main distinguishing character-
istic of logical constants, as opposed to non-logical
words.) For example, when someone picks out specific
shades corresponding to the word "red" by seeing whether
they lie between two shades which he has memorized, the
relation between the shade picked out and the ones
memorized is not a logical relation; it is a relation
which holds specifically between shades of colour, and
in order to know what the relation is one must be acquainted
with colours and know what it is like for one shade to be
between two others. Acquaintance with this kind of
property is required. (In Kant's terminology: an appeal
to intuition is required. See chapter seven.) Contrast
this with knowing what it is like for one "property" to
be the combination or disjunction of two or more others:
here we have a very general kind of knowledge, for the
relation in question can hold between any sorts of pro-
perties, so acquaintance with no particular kind of

    102

property is presupposed.
     (Problem: could these examples of procedures be
reduced to a kind of logical synthesis by talking about
"properties" which are logically synthesized in a com-
plicated way out of both properties and relations?
E.g. the property p-redness is synthesized out of the
two boundary shades of colour and the relation of
"betweenness" holding between shades of colours, the
synthesis being logical. This does not matter much
for my purposes, as the main aim was to show how to
distinguish different concepts where they are not nor-
mally distinguished owing to the use of loose criteria
for identity of meanings.)
     Where necessary, we may describe these non-logical
types of synthesis as "geometrical synthesis", "musical
synthesis", and so on.

3.E.    Concluding remarks and qualifications

3.E.1.  It may be thought odd that most of my examples
to illustrate the various kinds of semantic rules des-
cribed in this chapter should be so contrived and arti-
ficial, and that in several oases I had to invent new
words to illustrate a point, instead of using words we
all know. This is because I have oversimplified many
features of our use of descriptive words in order to
illustrate the principles which are to be employed for
making sharp distinctions between meanings. It is only
to be expected that there should be some oversimplification
in the early stages of the description of any system of
classification. But because most of our ordinary concepts
are very complicated, in ways which will be described

    103

presently, they cannot be used without modification
to illustrate oversimplified schemes of classification.
    It is necessary to oversimplify at first, in the
interests of clarity. Normally people start right off
talking about complicated cases, and then they fail to
sort out all the various complexities, having nothing
with which to contrast them, and this, I think, helps
to account for the fact that controversies concerning
the analytic-synthetic distinction and related distinc-
tions have gone on for so long, without any progress being
made.
    Thus, the importance of these oversimplified
examples, as will appear presently, is that they show
that it is possible for concepts to stand in definite
relations, even if, owing to the complexities which we
have so far ignored, and will discuss in chapter four,
most of our ordinary concepts do not, a fact which
sometimes leads philosophers to think that there is no
clear distinction between analytic propositions and
synthetic propositions.

3.E.2.  One way in which my descriptions oversimplify
what goes on when we ordinarily use descriptive words
has been by disregarding some of the complexities in the
ways in which various rules may be synthesized. For
example, the ordinary word "red" is probably used partly
as an f-word, by those who can see hues, partly as a
d-word, by those who can memorize shades of colour,
partly as a p-word, by those who can memorize boundary
shades and tell whether a given colour lies between them
or not, partly as a word correlated with a scientific
procedure for measuring wave-lengths of light, and so on;

    104

and all these different kinds of rules or concepts
may be "superimposed" in one concept "red" without
being combined definitely as a conjunction, or a dis-
junction, or a disjunction of conjunctions, or anything
as simple as these. (This sort of thing helps to
account for so called "open texture".) The meanings
with which we use words are far less definite than has
been suggested by the descriptions of this chapter.
(This is connected with the fact that, for normal pur-
poses, there is no need to apply strict criteria for
identity of meanings. See section 2.C.) Some of these
oversimplifications will be eliminated in the next
chapter.

3.E.3.  In addition to ignoring complications in the way
in which we correlate words with universals, I have
oversimplified other matters. For example, I have
assumed that colours vary only in one dimension, so that
all shades can be arranged along a continuous spectrum
in a definite order. I have failed to take note of
the difficulties in saying that the same colour (whether
a hue, or a specific shade) may be present in ordinary
opaque objects, in transparent objects (solid or liquid),
in objects with various sorts of surface textures, or
even in phosphorescent objects and neon signs. Is it
the same property in all these cases? I have ignored
the fact that there may be limits to our powers of dis-
criminating specific shades of colour, or specific shapes.
(Something will be said about this in Chapter Seven).
    It is not possible for all problems to be solved at
once. Many of my remarks are idealizations which require
qualifications of one form or another, but the quali-
fications do not usually affect the main argument.

    105

3.E.4.  One of the main points of the discussion has
been to show how people who oversimplify things even
more than I have done may be led to adopt intolerably
obscure and confused theories of meaning and universals,
or to make sweeping generalizations in rejecting such
bad theories, so that they overlook the element of
truth behind them. The main oversimplification is to
ignore the possibility that the use of a descriptive
word may be explained in terms of complex correlations
between words and universals. Insistence on the one-
one model, or a determination to say that one property
or universal corresponds to each word, leads people to
say such things as that universals are "intangible to
the senses, apprehended in thought alone, the potential-
ity of their differentiations, the identity to be found
in variety", etc. (See 2.D.6.) Or it prevents their
seeing clearly how talk about universals can explain.

3.E.5.  In showing now talk about properties and other
universals can explain, we stressed the fact that the
ability to use words presupposes the ability to perceive
and attend to features or properties, to memorize them
and recognize them again later. This explains how we
can learn the use of a word from examples and then go on
and use it in quite different contexts. It explains how,
having learnt to use the word, we can understand its use
even in false statements which ascribe properties to
objects when those objects do not have those properties.
All this is possible because in memorizing a property
one need not bear in mind any particular object or objects
having that property (cf. section 2.D) (The particulars
used in teaching the meaning of a word do not thereafter
have any special role in connection with it: after they

    106

have provided the required illustration they drop out
as irrelevant and may change their properties or
relations without this having any effect on the meaning
with which the word is understood.)
    This loose connection between universals and their
actual instances, or between descriptive words and the
particulars which they describe, which has been so
important in our explanations so far, will turn out to
be very important once again in chapter seven, where it
will provide the basis for an explanation of the meanings
of "possible" and "necessary".

3.E.6.  Despite all the oversimplifications, the dis-
cussion of this chapter has shown in a general way how
correlations between descriptive words and properties can
help to determine which objects are correctly describable
by which words, or at least the conditions in which objects
are describable by words. In chapter five, the dis-
cussion of logical constants will show how descriptive
words may be combined with other words to form sentences
expressing statements. So this chapter and chapter five
will together have shown how correlations between des-
criptive words and properties can help to fix the con-
ditions in which statements are true. The importance of
this for our main problem is that it helps to explain how
the analytic synthetic distinction works by showing how it
is possible for a statement to be analytic, or true in
virtue of its meaning. We shall see that analytic state-
ments form merely a special case of the class of all state-
ments which are true in virtue of both what they mean and
what the facts are.


====================================================
This is part of A.Sloman's 1962 Oxford DPhil Thesis
     "Knowing and Understanding"
====================================================


Chapter four
SEMANTIC RULES AND LIVING LANGUAGES

4.A.    Indefiniteness

4.A.1.  In chapter three an attempt was made to describe
various ways in which descriptive words may be correlated
with universals by semantic rules. It was pointed out
in section 3.E that our ordinary use of words is much
more complex than the uses described in that chapter,
and the purpose of this chapter is to describe some of
those complexities.
There are many respects in which the description of
semantic correlations and logical and non-logical syntheses
of meanings provided an oversimplified model.
For example, it took no account of descriptive words
which refer to tendencies or dispositions or unobservable
properties or theoretical notions of the sciences, or
those words, such as "angry", "hopes", "intends" which
may be used to talk about conscious beings. However, even
if we leave out these complicated concepts, and concern
ourselves only with words which are correlated with
observable properties in something like the manner described
in the previous chapter, we shall find complications
which have not been accounted for, though very briefly
mentioned near the end.
Philosophers sometimes draw attention to these complications
by saying that ordinary empirical concepts
have "open texture", are vague, have indefinite boundaries,
or admit of difficult borderline cases. Sometimes the
point is made by saying that concepts do not stand in
exact logical relations to one another, or that it is
    108

impossible to make a clear distinction between usage
due to meaning and usage due to generally shared collateral
beliefs. (Cf. Quine: "Word and Object",
p.43). Sometimes they are carried away by all this
and say that words in a language are not used according
to rules, or that logical laws do not apply to ordinary
languages, or that there is no clear distinction between
analytic and synthetic statements, or between necessary
and contingent truths.
Unfortunately, no one seems to have given a very
clear and systematic account of all these complications
and difficulties, nor explained how a language is able
to work despite many kinds of indeterminateness in it.
I suspect that this is because people have no clear
notion of what it would be like for these indeterminacies
to be absent: so they do not have any model for the
missing simplicity with which to contrast the actual
complexity and provide a basis for systematic discussion.
I believe that the account of semantic rules in the
previous chapter provides at least part of such a model
and hope to illustrate this by contrasting some of the
complication in ordinary usage with its relative
simplicity.

4.A.2.  The kinds of indeterminateness which will be
described in this chapter fall into two main classes:
(i) those due to indefiniteness of properties themselves
and (ii) those due to indefinite semantic correlations
between words and properties. It will not be possible
to describe all possible cases: there is room for only
an incomplete and somewhat condensed sketch.


109
4.A.3.  Indefiniteness of properties.
It is sometimes remarked that properties can be
indefinite, as if it were perfectly clear what this means
or how it is possible. It is certainly not clear to
me. The point seems to be that when we try to decide
which objects have and which have not got some property,
we may come across a borderline case where it is difficult
to decide. (Compare 3.B.4.a.) There seem to
be several different cases in which one stay have this
sort of doubt about an object in one's field of perception.
1)  The doubt may be empirical, and due to abnormal
circumstances. For example, the light may be bad, or
one may be too far away to see clearly, or one may be
temporarily unable to concentrate, owing to tiredness,
a headache or emotional problem. Or one may simply
have forgotten what the property looked like. Such
doubts can be eliminated by eliminating the abnormal
circumstances, and are of little interest.
2)  The doubt may be due to permanent psychological or
physiological limitations, such as an inability to make
fine discriminations. This can cause doubt whether two
visible objects are exactly alike in some respect (e.g.
shape, or colour), or whether an object which is present
has exactly the same shade of colour as the shade which
one has in mind (e.g. a shade seen on the previous day).
Other examples are: a permanent inability to memorize
fine shades of colour, or an inability to "take in" or
"survey" complicated properties, like the property of
being a figure bounded by 629 sides. (In some of these
cases, the use of procedures, such as counting, may help
to resolve the doubt. This raises interesting problems
as to which doubt it resolves.)
3)  It is possible that there is a third sort of doubt
    110

to be described as being due to indefiniteness of a
property. An object and all its properties can be seen
plainly, and yet one may be in doubt as to whether its
hue is red or not despite the presence of many red objects
to ensure that memory is not at fault. This does not
seem to be an empirical doubt, to be settled by closer
examination in a better light, for example, or under a
spectroscope. (If I let the spectroscopic readings
settle the question, then I have taken a decision to use
the word "red" in a new way.) For what the spectroscope
tells me cannot remove any doubt about how the object
looks to me. Notice that although the doubt concerns
the way the object looks, nevertheless there is a sense
in which I am in no doubt as to how it looks, for I can
memorize its appearance, and recognize other objects as
having exactly the same shade of colour. This is what
suggests that it is a doubt about a property: is that
hue (red) present in this shade (e.g. redange - see
3.B.4.a.)?
It is not at all clear to me that there is a difference
between cases 2) and 3). Perhaps it depends on whether
there is only one person who is unable to decide or
whether everyone is unable to decide. At any rate, I
shall be content to leave this and carry on with discussion
of indeterminate semantic correlations.

4.A.4.  Indefinite rules
Not only may the properties to which words refer be
indefinite, but in addition the way in which they are
referred to may be indefinite, or indeterminate. (Though
it is not clear that these two cases can always be distinguished.)
Here again, there are various ways in which
doubt as to the describability of an object by a word may arise.
    111

First consider an f-rule, correlating a word with
one observable property. It may be unclear which is
the property with which the word is meant to be correlated,
and this, of course, may cause doubt in the
application of the word. No more need be said, as
this is just a special case of the next sort of doubt.
Secondly, if a word is correlated with a set or
range of properties disjunctively, by a d-rule (see
3.B.2,ff.), then the boundaries of the set of properties
may be indefinitely specified. This is one of the
things which may be meant by the word "vagueness".
Notice that although it is a range of properties which
has indeterminate boundaries, this may have the consequence
that the class of objects with those properties
has indeterminate boundaries, in which case the word has
an extension with indeterminate boundaries. (Part of
the indeterminateness may be due to indefiniteness of
the individual properties, of course.) The previous
case is clearly an example of this, for it involves a
unit-set of properties, with indeterminate boundaries,
so that it is not clear which is the property in that set.

4.A.4.a.    It is difficult to illustrate this by means of
an example, because our ordinary words are too complicated
and illustrate too many things at once. But we can come
close to seeing what sort of thing is meant by noting that
the word "rod" is correlated with a whole range of shapes,
each more or leas straight, with a fairly uniform cross-section,
and a length somewhat greater than its diameter.
But how much greater? How much longer must a rod be
than it is wide? It should be clear that the range of
shapes correlated with the word "rod" has somewhat
indeterminate boundaries, for although the ratio of length
    112

to diameter must not be too small or too great (else the
word "disc" or "filament" or "wire" may be more appropriate),
nevertheless there are no definite limits. So
there is a range of shapes which may definitely be
possessed by rods, and a range which may definitely not
be possessed by rods, but there are no determinate boundaries
between them. (Compare also the words "heap", "few",
"small", "many", "giant", etc. As with "rod", caution is
required, since these words illustrate more than one kind
of indeterminateness.)

4.A.4.B.    Next we have indefinite conjunctive correlations,
Two or (usually) more properties may be conjointly referred
to by a descriptive word in an indefinite way. (Cf. 3.B.3.)
For example, if the properties are as a matter of fact
always found together and never separately, then the word
"W" may be used to describe objects which have all the
properties without its ever being decided whether possession
of all of them is necessary for describability by the word,
or whether only some subset need be possessed, and if the
possession of only a subset is sufficient, in that case.
We can describe this sort of indefiniteness in terms of
the last, as follows. Form the set of all possible combinations
of one two or more of the properties in question.
Then we regard the word "W" as correlated disjunctively with
some subset of this set of complex properties, the boundaries
of the subset being indeterminate. When, in such a case,
the original set of properties (and so also the set of
possible combinations of those properties) is infinite,
or at least indefinitely extensible, we seem to have an
illustration of what Waismann called "open texture".
(In "Verifiability".) I shall postpone illustrations
till later, for the reason already mentioned: ordinary
    113

words are too complex and illustrate too many different
things.,

4.A.4.c.    The next kind of indefiniteness involves
n-rules, as described in 3.B.4,ff. It was shown that
words might be negatively correlated with properties by
either strong or weak n-rules. The correlation may
be indefinite in some cases, where for example, it is
not certain whether the n-rule actually does govern
the word or not. Thus, before the discovery of black
swans, there might have been an indeterminate negative
correlation between "swan" and blackness. Perhaps the
more interesting case is that in which the indefiniteness
is due to its not being clearly specified whether a weak
or a strong n-rule correlates some word with a property
(see 3.B.4.b.). This has the consequence that it is
not definitely analytic nor definitely synthetic that
nothing with the property is correctly describable by
that word.

4.A.4.d.    Not only logical syntheses, but non-logical
ones also may be indefinite. Thus a word governed by a
p-rule (see section 3.D) may have a meaning which is
indefinite in a way analogous to that discussed above,
in connection with d-rules. For example, in 3.D.2. we
described a procedure for using the word "red". Two
boundary-shades are selected and memorized, and then the
word is applied to an object if and only if it has a
specific shade of colour lying between the two boundary-shades.
If, however, there is something indeterminate
in the selection of the boundaries, or in the decision
whether specific shades lie between the boundary shades
or not, then the procedure considered as a whole will
    114

be partly indeterminate. For example, the word "red"
my be correlated with a set of properties with indeterminate
boundaries by such a procedure.

4.A.5.  We have seen how each of the types of synthesis
described in chapter three may be indeterminate, giving
rise to concepts whose boundaries are not clearly defined.
If we recall that all these operations for constructing
new semantic rules can be reiterated, to yield very
complicated correlations between words and properties,
involving both logical and non-logical syntheses, we see
that the final product may be indeterminate in many
different ways all at once, and even more so if we allow
that properties themselves may be indefinite (see 4.A.3.).
In 3.B.5, we saw that a word may be correlated with
the following combination of properties:
     P & not-(Q&R) .v. R & not-(S1 v S2 v ?) .v. Q & not-P.
In such a case, each of the properties P, Q and R may be
indefinite in the manner of 4.A.3, the range S may have
indeterminate boundaries in the scanner of 4.A.4 or 4.A.4.d,
and it may not be certain that any one of the main disjuncts
is a sufficient condition for the applicability of
the word, though any two together do definitely provide a
sufficient condition. This illustrates the way in which
several different kinds of indefiniteness may simultaneously
contribute to the indeterminateness of the boundary
of the extension of a word.
It should be stressed that we must distinguish
borderline cases due to difficulty in deciding whether
certain particular objects do or do not exhibit certain
properties, and those which arise out of indecision as to
whether those properties which are quite evidently possessed
    115

or not possessed by objects are correlated with a word
in one way or in another. This shears that concepts may
have indeterminate boundaries in two quite different senses:
it may mean that the extension, the set of particular
objects falling under the concept, has indeterminate
boundaries, or it may mean that the set of properties,
or combinations of properties, sufficient to guarantee
inclusion in the extension may have indefinite boundaries.
In either case borderline cases are possible, that is,
particular objects which are neither definitely describable,
nor definitely not describable, by some word. We
may say that in these cases the application of the word
is not determined by or explained by the meaning of the
word, or by the universals correlated with it. (See
3.C.4, 2.D.2.)

4.A.6.  I have remarked that it is difficult to find
words in a living language which illustrate only one kind
of indeterminateness. It is much easier to find words
which simultaneously illustrate several kinds. The
word "horse" is a familiar example. There is a range
of shapes which may be possessed by horses, but the range
has no definite boundaries, for the shape of a horse may
change continuously into that of an elephant or giraffe
without definitely ceasing to be a possible shape for a
horse at some one point.
Similarly, there is a range of possible colours for
horses, and here it is not even clear whether there is any
boundary at all, since whether a colour is possible may
depend on other factors, such as whether the horse has
been painted that colour. If some "horse" were born
bright blue and produced off-spring with red white and
blue stripes, we might not be sure whether to say that it
    116

was a horse after all. Similar remarks may be made
about the textures of the skins of horses. They must
not he metallic, but there is no definite boundary.
In addition, it is likely that even within the
ranges of permissible shapes colours and textures, there
are some which must not occur together. Some odd
colours may be allowed, but not if the animal also has
too odd a shape and texture too. However, there is
surely no definite limit to the kinds of combinations
which we should allow in objects correctly describable
as "horses". Further, we may allow the possibility
that biological investigations will provide an "explanation"
of the existence of freaks and so persuade us once
more to call them "horses". It is very likely that
no explanation at all would redeem some cases, yet there
is surely no clear boundary between those cases which
may be explained away and those which may not. Investigation
would doubtless reveal further complexities here.

4.A.7.  These remarks help to illustrate the claim that
the account given in chapter three was hopelessly inadequate
to explain the use of all kinds of descriptive
words. But there are still many kinds of complexity and
indeterminateness which have not been mentioned. For
example, we discover empirical regularities in the world,
and construct scientific (or non-scientific) theories
based on these regularities. Then in some cases the
theories may be built into the definitions of some of
the words used to state them. This may occur in an
indeterminate way so that, for example, it is not clear
whether it is a matter of definition that gases at constant
pressure have a linear coefficient of increase in volume
with rise in temperature, or a contingent fact. The
    117

correspondence between mercury column readings, and
gas-thermometer readings is, of course, a matter of
experience, not a matter of definition. The indeter-
minateness consists in the fact that it would not be
clear how to describe the situation in which the cor-
respondence broke down. (In some cases, further inves-
tigation might make it clear, by yielding explanations
in terms of accepted theories.) So we can say that
increase in length of a mercury column is neither
definitely merely evidence nor a defining criterion
for the applicability of the expression "a rise in
temperature".
But there is no room for a detailed discussion of
all kinds of indefiniteness. Many cases are already
familiar (see, for example, the chapter on "Reduction
and Open Concepts", in "Semantics and Necessary Truth",
by Pap). I shall leave the description and classi-
fication of examples now, and make some general remarks
about indefiniteness.

4.B.    Ordinary language works

4.B.1.  The previous section showed how it is possible
to take account of various sorts of indefiniteness
within the framework of a theory which attempts to explain
our use of descriptive words in terms of correlations
with observable properties and other universals. It
brings out more clearly than ever some of the inade-
quacies of the one-one model, which assumes that there
is one universal correlated with each descriptive word,
and simultaneously shows why there is no need to give
up talking about universals altogether just because the
one-one model will not work.
    118

For example, one sort of objection to talking about
meanings as explained by properties, is that the pro-
perties to which words seem to correspond may change
over the years while there is no need to say that their
meanings change, or that the concepts corresponding to
them change: concepts may have a history. But we can
easily take account of this, for, owing to the com-
plexities in the correlations between words and properties,
it is possible for small changes to take place while most
of the correlations remain unaltered. A concept may
become more or less definite in some respect, a bound-
ary may shift along a range of properties without
becoming more definite, and so on. But, for normal
purposes, or when people talk about the "history of
ideas", the fact that so much remains unaltered while
these changes take place is regarded as a sufficient
reason for talking about "the same concept", or "the
same meaning".
Here loose criteria for identity of meanings are
used, and work as well as loose criteria of other sorts,
as when we talk about "the same car", despite the minor
repairs and replacements which it has suffered. In
either case, where changes are too drastic, we may be
unsure whether to say they are "the same" or not: any
system of criteria for identity may work well in some cir-
cumstances and break down in others (see section 2.A).
In short, the complexity of semantic correlations,
their indeterminateness, and the fact that in normal
parlance we use looser criteria for identity of meanings
or concepts than for identity of properties explains
why oversimplified theories of universals will not work.
But the argument of 2.D shows that there must be semantic
correlations if there is to be anything definite about
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the meanings of words, if concepts are to have any
boundaries at all, however indeterminate, and that is
we need some explanation of the sort which I have
tried to give, of various ways in which descriptive words
may be correlated with observable properties.

4.B.2.  The question now arises: how can we get away
with so indefinite and imprecise a use of words? How
can our ordinary language be "in order as it is"?
(Cf. "Tractatus", 5.5563, and "Philosophical Investigations",
I.98). To answer this, we must examine the purposes for
which we speak and write, and the circumstances in which
we do so, and this will show why it does not matter for
our normal purposes that what we mean by our words is
not perfectly precise and definite, no matter how dis-
tasteful logicians may find it. Of course, when our
purposes change, or the circumstances change, (e.g.
black swans are discovered, or unexpected counter-
examples turn up to accepted mathematical theorems),
then we may have to eliminate some of the indeterminate-
ness. New scientific discoveries, or new purposes, may
reveal to us previously unnoticed indeterminateness and
force us to eliminate it. (How to eliminate it may be
a matter for arbitrary decision, or it may be determined
by questions of convenience.) The indeterminateness
does not matter because until the new possibilities turn
up (such as horses which produce giraffe-shaped off-spring)
we need not consider how to describe them: indeed it
would not only be futile, but quite impossible, to consider
all such possibilities and adopt definite linguistic
rules for dealing with them. (Read Wittgenstein's
"Philosophical Investigations", I.65-88, 97-100, inclusive.)
    120

4.B.3.  It may help to explain why the meanings of our
words contain no much indefiniteness, if we notice that
our desires and intentions may be indefinite in the same
sort of way. When a farmer instructs his employee:
"Go to the market and buy me a horse", can we say that
there is a definite kind of thing that he wants? Can
he say honestly that there are definite limits to what
would satisfy him? Could he be expected to take
seriously the request for a precise specification of the
range of possible shapes, colours, textures and kinds
of behaviour the animal may have? Must he have an
answer to the question: "Will you mind if the only
horse I can find has a neck as long as a giraffe's?"
If we can see why, in normal circumstances, this would
be a stupid question for the employee to ask before
setting out, then we shall see also why it would be
stupid to expect people to use words with more precise
meanings for normal purposes than they do in fact.
(It is important that so little of our linguistic
behaviour consists in merely making statements.)
If something works, then this is a justification
for using it. Of course, something else may serve the
same purposes better: but that is a question to be
settled by experience, not by appeals to logical ideals
of exactness and rigour.

4.B.4.  All this may explain why it doesn't matter for
normal purposes that words and sentences are used with
indefinite meanings, but it does not really explain how
this comes about. That can be seen by examining the
conditions in which we first learn to speak.
Our initial lessons in the use of language must
give us all the apparatus described in section 2.B. We
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must learn what a language is, what it is to ask a
question, make a statement, give a command, etc. We
must learn what it is for statements to be true or
false, for a word to describe an object correctly or
incorrectly. We must learn to employ a conceptual
scheme in which the world is seen as made up of enduring
physical objects with shared properties and relations.
General knowledge at all these different levels has to
be picked up in addition to more specific knowledge of
the rules correlating individual words with things and
properties. a child cannot learn to use words with
definite meanings without learning all these things.
But he cannot acquire the general knowledge without
first having learnt to speak: so, many different things
must be learnt simultaneously, in a gradual and com-
plicated process.
Progress must be made not only at different levels
simultaneously, but also along a wide front. We do
not first learn the precise meaning of one word, then the
precise meaning of another, as if we could already speak
and were learning the vocabulary of a new language by
memorizing a dictionary. The clarity and definiteness
of our understanding of many different words increases
in one slow process. Only when we have already acquired
a fair vocabulary and a considerable mastery of linguistic
techniques, are we able to use meta-linguistic concepts
(such as "meaning", "refers to", etc.) and thereby give
our words relatively precise, simple and definite meanings.
Not until an advanced stage has been reached can people
intend their words to be governed by definite semantic
rules of the kinds described in chapter three. Even so,
the process of making our meanings more definite and
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precise does not proceed beyond the point at which it
serves our purposes to do so. We have already remarked
(in 2.A.5.) that persons with special purposes, such
as scientists, lawyers, or philosophers, may give
ordinary words meanings more precise than in ordinary
speech. The process can go on almost indefinitely.
(See note at end of 2.B.)

4.B.5.  Two reasons have been found why words are first
of all used with indefinite meanings and only later on
made more precise, namely because first of all a higher-
level conceptual apparatus is required for making mean-
ings precise, and secondly because many words are learnt
simultaneously in a gradual process. But there is
another reason, which is important if we wish to under-
stand some of the things said by philosophers about the
analytic-synthetic distinction.
The third reason is that much of the child's voca-
bulary is picked up from things said by people around
him such as "Here comes Daddy", "Look, there's a kitten",
"Would you like some more jam?" "Don't splash your milk",
and these are complete statements using words, not defin-
itions explaining them. The child does not hear things
like: "The word 'red' is a descriptive word, referring
to the hue common to those three objects". The child
learns which statements are true and which are false:
but this cannot teach him the precise meanings of the
words occurring in those statements. Knowing the mean-
ings involves more than knowing that statements are true
in certain conditions: it requires a knowledge of why
the statements are true in those conditions, what it is
about those circumstances which makes the statements true,
or whether, for example, the statement would be true in
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any circumstances at all.
For example, when a child hears someone utter a
statement that is analytic or necessarily true, all that
the child can learn is that it is true. More has to
be said before he can learn that it is analytic, and
usually not enough is said for the child to be able to
answer all questions about precise meanings of words
(not that he formulates these questions, of course).
How much evidence is available to the child for him to
learn whether "tadpole" is used to refer to the property
of being a frog-larva, so that it is analytic that all
tadpoles are frog-larvae, or whether it is merely a
contingent, generally accepted fact?
In view of all this, it is perhaps remarkable that
children do learn to use words with meanings at all.
It is certain that they must take many a leap into the
dark, extrapolating beyond what they can learn from the
linguistic utterances of their elders. (Of course, they
do not do this consciously.)

4.B.6.  In view of all this, and also the fact that our
powers of discrimination, etc., are limited in the way
described in 4.A.3., it is not at all surprising that
people learn to associate only relatively indeterminate
meanings with words. In order to understand all the
words in a statement in a definite way, one would have to
know how to decide in all possible conditions whether that
statement would be true or false (cf. 2.C.5.). But the
child can only observe the use of words in actually exist-
ing conditions, and so it understands things in an
indefinite way, until new experiences force it to remove
some of the indefiniteness. As already pointed out, some
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kinds of indefiniteness are never removed, and this does
not matter for normal purposes (which is why loose
criteria for identity of meanings are employed). But
we shall see later on that it does matter when we try
to apply the analytic-synthetic distinction.

4.B.7.  The indefiniteness of which I have been speaking
may manifest itself in many ways. There may be fluctua-
tions in usage which can be observed in a whole society
at any one time, or over a period of time. It may mani-
fest itself in fluctuations in the usage of only one
person over a period of time. Even at one time, the
meaning with which a person understands a statement may be
indefinite in any of the ways described, as is easily
shown by asking someone how much sand is required in
order to amount to a heap, or now long a cylinder has
to be relative to its diameter in order to be describable
as a "rod".

4.B.8.  We can now summarize the main points made so far.
(a) Indeterminateness, or indefiniteness in meaning, or
the existence of borderline cases, may be a consequence of
indefiniteness of the properties with which words are
correlated, or it may be due to indeterminateness in the
semantic rules correlating words with observable proper-
ties, or features.
(b) It may not always be possible clearly to distinguish
these two causes of indefiniteness from each other, or
from indeterminateness in usage which is due to limitations
in our abilities to make fine discriminations, to survey
complicated patterns or structures, or to memorize very
specific features or complex properties. But in some
cases the distinctions can be made.
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(c) As we shall see later on, the indefiniteness of the
meanings with which ordinary words are used, may make
it impossible to apply sharp criteria for identity of
meanings to ordinary statements, and so impossible to
apply the analytic-synthetic distinction to some of
those statements.
(d) Finally, we have seen how, within the framework of
a theory of universals, to take account of complexities
which could not be coped with by the one-one model.
Universals can explain our use of descriptive words, and
the existence of boundaries to empirical concepts, but
not in a simple way.

4.C.    Purely verbal rules

4.C.1.  The description of various sorts of indefinite-
ness in the meanings with which words are normally
understood, has helped to bring out one of the ways in
which the description in chapter three gave an over-
simplified picture of correlations between words and
universals. But there is another oversimplification,
which goes back to section 2.D, where it was argued
(see 2.D.3.) that there was something wrong with
Hampshire's claim that "In all cases, clarifying the
use of a descriptive word or phrase is a process of
drawing attention to its established links with other
descriptions." This claim ignored the fact that if
words are to have meanings, if they are to be able to
occur in statements which are about anything, then they
must be correlated not just with other words, but with
non-linguistic entities. However, there is some truth
in the claim, for it is possible for some aspects of the
use of words to be determined by rules which merely
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correlate words with other words, and this has been over-
looked so far, engrossed as we have been with semantic
correlations. This oversimplification must now be
eliminated.

4.C.2.  In many cases, setting up a correlation between
words and other words has the effect of setting up a
semantic correlation, between one word and the properties
referred to by the other words, for example. Thus, if I
rule that the word "gleen" is to describe objects if and
only if they are correctly describable by the English
words "glossy" and "green", then this correlates the
word "gleen" with the combination of the properties
referred to by the other two words, namely glossiness
and greenness. Similarly, as shown in 3.B.4.a-b, the
adoption of an incompatibility rule, correlating the
word "RED", which primarily refers to a hue, with the
word "redange", which refers to a specific shade of
colour on the red-orange boundary, may help to make the
meaning of "RED" less indefinite than it would be other-
wise, by correlating it negatively with a specific shade.
Sometimes, however, we may adopt a purely verbal
rule, which merely correlates a word with another word,
in cases where indeterminateness of meaning gives rise
to indeterminateness of relations between descriptive
words. Thus, suppose the relation between the words
"red" and "orange" is indeterminate, owing to the fact
that there are difficult borderline cases which are
neither definitely red nor definitely not red, and at the
same time neither definitely orange nor definitely not
orange. (It does not matter for the illustration,
whether this is due to indefiniteness of the hues referred
to by the two words, or to indeterminateness of the
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boundaries of ranges of specific shades, if they are
used as d-words.) In such a case a rule may be adopted
to the effect that "red" and "orange" are incompatible
descriptions. We may call this a purely verbal
(strong) n-rule.

4.C.3.  Such a rule, unlike the one correlating "RED"
and "redange", leaves the two concepts it correlates
as indefinite as they were without it, for it does not
say which objects are to be described as "red" and
which as "orange": it leaves borderline cases as
undecided as ever. But it does mean that if one
makes a decision about the use of the word "red" in
some of these cases, then one may be committed to a
decision concerning "orange": the two lots of problems
about borderline cases may not be settled quite inde-
pendently. So it rules out the possibility of the
truth of some statements, such as "This box is red
and orange all over" and the falsity of some others
such as "Nothing is red and orange all over at the same
time". This will turn out to be important when we come
to look for a definition of "analytic", for no definition
will do which does not make this last statement analytic
in cases where the words are governed by the incompat-
ibility rule under discussion.

4.C.4.  Since such a rule does not provide more deter-
minate boundaries to the extension of either word, it
might be thought to be a completely useless sort of rule,
and so it almost is, for no situation can be described
any more precisely after it has been adopted than before.
But it does have the advantage of making a "second-order"
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concept more definite: the concept of a hue. If we
have no way of telling what sorts of things hues are,
except by saying that hues are referred to by hue-words,
then, if adopt the linguistic convention that hue-
words must be incompatible with one another, this helps
to make the concept of a hue more definite, since it
has the effect that no two hues may occur together,
despite the fact that it does not make any of the
individual hue-concepts any more definite. (Compare
this with Dummett's example, in Phil.Rev., July, 1959,
p.328.) All this may be of some use in enabling us to
formulate some scientific theory or other kind of theory
about hues in a simple or elegant way, without fear of
embarrassing counter-examples in borderline cases. At
present I am not concerned to show what the point of
such an incompatibility convention might be, so much as
to show that there might be such a convention, and thus
to illustrate the fact that there is some truth in the
claim that "links between descriptive expressions" may
have to be described if the full meanings of some words
are to be described.

4.C.5.  This completes my account of ways in which chapter
three contained an oversimplified description of the
linguistic conventions giving our descriptive words their
meanings. Not all oversimplifications have been pointed
out and the description contains many omissions, but it
will have to do.
The next chapter attempts to snow how it is that once
the semantic rules have been adopted, which determine which
objects are describable by which words, those words may be
combined with other words to form statements which may be
true or false.

__________________________________________________________
NOTE: This is part of A.Sloman's 1962 Oxford DPhil Thesis
"Knowing and Understanding"
__________________________________________________________


    129
Chapter Five
LOGICAL FORM AMD LOGICAL TRUTH

Introduction
We are now ready to set out upon the last lap of
Part Two, in which our main aim has been to explain how
certain kinds of words and sentences can have the meanings
they do have, and how their having these meanings helps
to determine the conditions in which propositions which
they are used to express are true. This explanation
serves two important purposes. First of all, it pro-
vides an answer to the question: what sorts of things
are propositions, the entities to which the analytic-
synthetic and necessary-contingent distinctions are to
be applied? (Cf. 2.A.1.) Secondly it helps to display
the general connection between truth and meaning, between
knowing and understanding, at least in a certain class
of cases. This prepares the way for the discussion of
some more restricted kinds of connection, in Part Three.
(Part of that discussion will be anticipated in the present chapter.)
So far, except for a few rather vague and general
remarks in chapter two, we have been concerned only with
descriptive words, and have seen how semantic correlations
between them and universals (observable properties and
relations) can determine which particular objects they
describe correctly, and which they do not describe cor-
rectly, depending on whether those objects are or are not
instances of the universals referred to. This, however,
is not the full story of what happens when such words are
put together with other words to form sentences expressing
propositions. In addition, we have to describe the

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functions of the other words. Even if there were no
other sorts of words, even if it were possible to make
statements just by combining descriptive words, we should
have to discuss the way in which the method of construction
of a sentence contributed towards its having a certain
meaning in cases where the meaning depends on how the
words are arranged. In short, we must explain what the
logical form of a proposition is and how logical words
and constructions work. This will now be done.
First of all, an attempt will be made to say what
logical words and constructions are, that is, to char-
acterize the notion of a "logical constant". Then the
way in which the logical constants occurring in a stat-
ement help to determine its truth-conditions will be des-
cribed. Finally a discussion of what makes a proposition
a formal truth (i.e. true in virtue of its logical form)
will serve as an introduction to some of the problems of
Part Three.

5.A.1.  Logic and syntax

5 A.1.  In the sentences "Fido is black", "All cubes have
plane faces", we seem to be able to distinguish words which
refer to entities, whether particular objects or universals,
and words which do not. Among the former are "Fido",
"black", "cubes". "All" and "is" are in the latter class.
The latter are commonly described as "logical" words, or
"logical constants", and in this section I wish to discuss
the rationale behind our selection of some words to be
described as "logical", while others are "non-logical".
What is so special about the words "all", "is", "not",
"some", "and", etc.?
One answer which is sometimes given to this question
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is that these logical words are governed by linguistic
rules which are purely syntactical. That is to say,
unlike the semantic rules which correlate "Fido" and
"black" with non-linguistic entities (material objects or
properties), the rules for the use of logical words
merely correlate words with other expressions, never
with non-linguistic entities. After all, such words
can occur in statements which can be seen to be true merely
by examining their structure, that is, merely by examining
the way in which logical words occur in them, for they can
occur in formal truths, such as "It is raining or it is
not raining", i.e. in statements which are true in virtue
of their logical form. It is claimed that all that is
relevant to their being true is their syntax, or their
verbal structure, whence it follows that the linguistic
rules which give the logical words their meanings, since
they permit structure to generate truth, must surely be
rules which do not refer to anything other than verbal
structure. That is, they must be purely syntactical
rules.
In addition, it is sometimes argued that formal
systems of axioms and rules of inference, such as any
standard formulation of the propositional calculus,
serve to define the primitive symbols occurring in the
axioms and rules, and that these primitive symbols are
our ordinary logical words. Since the axioms and rules
of a formal system are concerned only with symbols and
relations between symbols, no mention being made of any-
thing non-linguistic, it appears that the rules which
"define" logical words are purely syntactical.
Despite all this, I think it can be shown that the
assertion that logical words are governed by purely
syntactical rules is either false or so vague as to be
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quite misleading.

5.A.2.  Let us now see what is wrong with saying that the
rules governing logical words are purely syntactical.
When we learn to use the truth-functional connectives,
Such as "or", and other logical words, we do not learn
to use them only in logically true propositions, such
as "It is raining or it is not raining", for they may
occur also in sentences like "My book is on the table or
you have moved it" and "Dawn is breaking or the moon is
still shining". Now, how can the meaning of "or" con-
tribute to the meanings of these sentences? How do we
learn the principles according to which logical words
work? This is something which has often worried people.
For example, Pap, who was sure that logical words could
not be ostensively defined, wrote:
"The analogy between interpretation of descriptive
constants and interpretation of logical constants
seems to break down: in the case of descriptive
constants we can, after having reached the primitives,
go on to ostensive definition, since there is some-
thing 'in the world' which they designate. But
what would it be like to show the semantic meaning
of the primitive LOGICAL constants of a natural
language, such as the English word 'or'?"
("Semantics and Necessary Truth", p.364. Cf.
p.366.)
Faced with this problem, some philosophers have been
driven to talk about subjective feelings of "hesitation"
or "indecision" which are correlated with such logical
words. Others, rightly rejecting this, have gone to the
other extreme and abandoned the search for anything which
can be correlated with such words, taking refuge in the
thesis that logic can be reduced to syntax.
Surely the correct answer is that learning the
meaning or function of a logical word involves learning
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how to recognize the states of affairs in which state-
ments using the word are true or false, just as learning
the meaning of a descriptive word involves learning to
recognize the states of affairs in which sentences
employing it are true or false?

5.A.3.  Consider the word "is", in the statement Fido
is black". In order to understand how it works one
must, in effect, learn the following: "A sentence may
be made up of a referring expression (i.e. an expression
referring to a particular object), the word 'is' and a
descriptive expression, in that sequence. In order to
discover whether the statement expressed by the sentence
is true or false, examine the particular object referred
to and see whether it has the property (or combination
of properties) correlated with the descriptive word."
This rule does not correlate the word "is" with any one
entity, but it certainly is concerned with non-linguistic
entities, though in a very general way. In order to
discover whether statements using the word are true, it
is not enough to examine the structure of the sentences
expressing those statements. The same goes for contingent
statements using the word "or". In order to understand
its role in a sentence such as "Dawn is breaking or the
moon is still shining", one must (at least implicitly)
learn the following rule: "If S and S' are sentences
expressing statements, then a new statement may be
expressed by the sentence consisting of S followed by
the word 'or' followed by S'. In order to discover
whether the new statement is true or false, examine the
facts (i.e. look to see how things are in the world) and
see whether a state of affairs obtains in which at least
one of the statements expressed by S and S' is true or
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whether both are false". (An understanding of the
two sentences S and S' is, of course, presupposed.)
In this latter case, as in the former, whether
the statement is true or false does not depend merely
on the structure of the sentence expressing it, and the
rule for the use of "or" does not refer only to syn-
tactical properties of sentences, but also to states of
affairs, which are non-linguistic entities. It is
concerned with how things are in the world, with the
facts in virtue of which the disjuncts express true or
false propositions. What, therefore, is left of the
assertion that the rules for the use of logical words
are purely syntactical?

5.A.4.  We can see what has happened here. Rules for
the use of logical constants are extremely general. The
rule for "is" does not correlate it with any particular
object or set of objects, nor with any particular pro-
perty or set of properties (or set of describability
conditions). The rule for "or" does not correlate it
with any specific state of affairs, but with all kinds
of states of affairs. The rules are highly non-specific:
they concern objects, but no specific kinds of objects;
properties, but no specific kinds of properties; and states
of affairs, but no specific kinds of states of affairs.
The rules are "topic-neutral". They allow logical
constants to occur in statements which are about anything
at all: they are not restricted to statements concerning
certain topics. The word "or" has the same function
in "That table is wet or highly polished" and in "She
is unhappy or unwell". Since the rules governing the
use of logical words are topic-neutral, one cannot
discover anything about the specific subject-matter of a
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statement from the fact that such a word occurs in it,
as one could if the word "table" occurred in it. This
makes it seem that such words are not correlated with
anything non-linguistic, that they are governed by
purely syntactical rules.

5.A.5.  It was argued that the rules for the use of
logical constants must be purely syntactical since they
had the consequence that statements like "It is raining
or it is not raining" can be seen to be true without
examining anything non-linguistic. But in order to know
that such a sentence expresses a truth, it is not enough
to see the marks of which it is made up: one must know
also how they contribute to the meaning of the sentence
as a whole, and this involves knowing in general when
a sentence containing these words expresses a truth,
including cases where the truth is contingent. It is
not enough to know the visible structure of the sentence:
one must know the functions of the various things which
make up this structure. The function of a symbol is
not a syntactical property, even when it is as general a
function as that of the word "or". I shall show later on
how a knowledge of these general functions may enable us
to discover truth-values without empirical enquiry, in
"freak" cases.

5.A.6.  We can now see also that the argument in terms
of definability of logical words by means of formal
systems (see 5.A.1.) falls away, for, since a formal
system cannot define the use of words in contingent
statements, it cannot fully define the function of
symbols like "or". (It cannot define the use of words
in logically true statements either. See 2.D.3, and
    136

2.D.3. note. Compare Appendix II.) Truth tables
can, of course, be used to explain the use of some
logical words, but only for someone who knows what
truth is, and, as already remarked, since propositions
are, in general, true or false in virtue of how things
are in the world, truth-table definitions do not
merely correlate symbols with other symbols.

5.A.7. It may be objected that I have missed the point
of the assertion that logic can be reduced to syntax.
It is true that I have in fact used the very arguments
employed by some people to defend the assertion. But
then the word "syntax" is used in a sense which must be
clearly defined if it is not to lead to confusion.
For example, when Wittgenstein argued (in "Tractatus")
that the truth of a logical proposition can be perceived
in the symbol alone (6.126,6.113), he did not regard a
symbol as just a sign (e.g. a mark on paper or a sound),
but a sign with a use, i.e. a sign standing in a "pro-
jective relation to the world". (See 3.12, 3.32, 3.321,
3.327, 3.262.) So when he talked about "logical syntax",
he was not concerned only with what is now often meant
by "syntax", namely something concerned only with com-
binatorial geometrical properties of signs. This,
unfortunately, is not realized by some who think that
the "Tractatus" supports their belief that logic can be
reduced to syntax.

5.A.8.  If I explain what it would be like for a logical
constant to be defined by purely syntactical rules, this
may make it clear exactly what I am denying. A "purely
syntactical" rule for the use of a word, as I understand
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the term, would specify that the truth or falsity of
statements making use of that word could in all cases
be determined entirely from a consideration of its verbal
form, without considering the meanings of any of the words
in it which were correlated with non-linguistic entities,
and without considering how things are in the world.
The following rule introduces into the English
language two new words, "plit" and "plat". Each word
may occur at most once in any sentence, right at the
beginning or right at the end. If "plit" occurs at the
beginning, or "plat" at the end, the sentence expresses
a truth. If "plit" occurs at the end, or "plat" at the
beginning, the sentence expresses a false proposition.
So "Plit the sun is shining", "It is dark plat" and
"Plit it is raining plat" all expresses true propositions,
and "It is raining plit" and "Plat all men are mortal"
both express false propositions. In all cases this can
be seen merely by examining the structures of the
sentences.
We can tell whether the propositions expressed by
such sentences are true or false: but what propositions
are they? Do such sentences say anything? Is there any
difference in meaning between "Plit the sun is shining"
and "It is dark plat"? Do these sentences say the same
thing or something different? The mere fact that the
words "true" and "false" are used in formulating rules
for the use of "plit" and "plat" is not enough to guar-
antee that they have anything to do with true or false
propositions. Similar remarks may be made about theorems
in a formal system, considered simply as theorems in a
formal system: what has being a theorem, that is a
formula derivable according to fixed rules from a specified
    138

combination of symbols, to do with truth or falsehood?
What can mere syntactical properties have to do with
truth?

5.A.9.  This example shows that although there may be
words whose "use" is governed by purely syntactical
rules, our ordinary logical words, such as "not", "and",
"is", "or", etc., are not like that, for, unlike "plit"
and "plat", they can occur in sentences expressing
contingent propositions, whose truth has to be established
by observation. They can occur in formal truths or
formal falsehoods, whose truth-value can be discovered
without considering how things are in the world, but only
because they have a more general employment can we speak
of truths or propositions in such cases. We shall see
later on, that such formal truths are merely degenerate
cases and that empirical enquiries can be used to show
that they are true, despite the fact that they are not
necessary. (See 5.0.7.) Thus, if normal observations
show that dawn is breaking, then this establishes the
truth of "Dawn is breaking or dawn is not breaking",
though it could be established otherwise. How could
empirical observation even be relevant if such a state-
ment were true in virtue of syntactical properties?
(Compare: "Concepts which occur in 'necessary' proposi-
tions must also occur and have a meaning in non-necessary
ones". Wittgenstein, "Remarks on the Foundations of
Mathematics", part IV, 41. Compare also "Tractatus"
6.124.)

5.A.10. All this arose out of the questions: "What is
special about the logical words, that distinguishes them
from descriptive words or proper names?" We have found
    139

that it is not true in any easily intelligible sense
that they are distinguished by being governed by purely
syntactical rules.
One peculiarity which has turned up (see 5.A.4.)
is that their rules are topic-neutral, that is, so
general as not to mention any specific kind of subject-
matter. For example, rules for the use of truth-
functional connectives are concerned only with whether
something or other is the case, without being concerned
with what kind of thing is the case. Other logical
words also exhibit this topic-neutrality. From the
mere fact that the word "is" occurs in the sentence
"My car is green", one cannot discover what the sentence
is about, or what sort of thing it is about. One can
tell only that if it states a truth then the particular
object referred to in it satisfies the describability-
conditions for the descriptive word occurring in it.
This tells us nothing about which object it is, nor
what kinds of properties are referred to by the des-
criptive word.
Topic-neutrality is not only displayed by logical
words. For example, in the sentence "All red horses
are red", the fact that the same word occurs in the
second and fifth places is a topic-neutral aspect of the
sentence which helps to specify its meaning. The fact
that the second word is a descriptive expression rather
than a referring expression is likewise a topic-neutral
aspect. We may say that all these topic-neutral aspects
of propositions are logical constants.

5.A.11. I shall take topic-neutrality of words or
constructions as a necessary condition for being a logical
    140

constant. But it is not sufficient. For example,
there is a sense in which the word "good" is topic-
neutral, but I do not wish to say that it is a logical
constant in "He has a good appetite". In addition,
there are words, such as "alas", which may occur in
sentences without restricting their subject-matter in
any way, but which will not be described as logical
constants, since they are not relevant to the question
of truth or falsity of the propositions expressed. As
already remarked (in 2.B.9), the study of truth-conditions
is only one aspect of the study of meanings.1

5.A.12. The following, therefore, will serve as our
definition of "logical constant":
The expression "logical constant" describes any word or
feature or aspect of a sentence which helps to determine
whether the sentence expresses a true or a false pro-
position, and which is topic-neutral, so that from the
fact that it occurs in the sentence one cannot discover
what things or kinds of things that statement is about.

1.  There are many expressions which are vaguely like
logical constants, at least in so far as they are
topic-neutral, but are not immediately relevant to
questions of truth and falsity. Examples are
"incidentally", "however", "moreover", "perhaps",
"probably", "nevertheless", "it seems that ...",
"of course," "obviously", etc. These all have some
kind of "pragmatic" meaning. That is, they concern
some relation between the speaker or hearer and what
is being said (e.g. surprise, or absence of surprise,
hesitancy, etc.). They are of the same general
nature as the following: "As you would expect ...",
"Much to my surprise ....", "It is clear to me
that ....", "In my opinion, for what it's worth ....",
"I am inclined to think that ....", "The evidence
available to me seems to show that ....", "I con-
fidently predict that ...", and so on. (See remark
about appropriateness-conditions in 2.B.9.)
    141

(It should be recalled that I am talking only about
relatively simple statements using words which can refer
to or describe material objects in virtue of their
observable properties and relations. Cf. 1.C.2.)
In so far as two or more statements have logical
constants in common, we can describe them as having a
common "logical form". The logical form of a proposi-
tion, therefore, is determined by the topic-neutral
words or constructions used to express it which help to
determine the conditions in which it is true or false.
(Cf. Section 5.B, and 5.E.4.)
In general, the sentences which we use to make state-
ments or ask questions about things in the material world
contain some parts or features which determine the
particular things and properties talked about, and others
which do not limit the subject-matter in any way, but
merely help to determine the kind of thing which is being
said, or the way in which it is said. The latter are
the logical constants and comprise the logical form of
the statements. Thus, the sentences "All books are red",
"Not all books are red", and "Some books are red" say
different kinds of things though the subject-matter is
the same: they refer to the same things and properties.
Similarly, in each of the following pairs of sentences
the same kind of statement is made, but about different
topics:
(1) "All books are red" and "All horses are four-legged".
(2) "Fido is not black" and "Socrates is not triangular".1
___________
1.  "We could generalize this and show how logical
constants determine whether a statement, question or
command is expressed. Cf. 2.B.3)
    142

5.A.13. We see therefore, that the logical forms
of propositions correspond to common "structures" of
sentences (where the occurrence of logical words
other topic-neutral aspects counts as part of the
structure.) The logical form common to a pair of
propositions may be represented symbolically in the
usual way, by removing all non-logical words and
replacing them by symbols called "variables" which
indicate the kinds of words whose places they take,
and whether the same word occupies two or more places
in the sentence. These symbols may be described as
"sentence-matrices", such as "All P's are Q", or
"x is not P". Different kinds of letters may be used to
indicate positions of different kinds of non-logical
words, such as capitals for descriptive words or expres-
sions, and small letters for expressions referring to
particulars. Such a sentence-matrix represents
the structure common to a family of sentences obtained
by replacing the variable-letters by suitable non-logical
words. It therefore also represents the logical form
corresponding to that structure. (The symbol is not
itself the structure, any more than a blue-print is the
structure of the houses whose common structure it repre-
sents. The structure is a property of statements. It is
neither a symbol nor a physical object, and you will not
find it left behind when the things which are not the
structure are removed from a sentence, any more than the
structure of a house is left behind when the bricks and
other materials are removed. This point can cause con-
fusion and lead to talk about "unsaturated" entities,
which cannot exist on their own, etc.)

5.A.14. It is possible to study the logical properties
    143

and relations of propositions by classifying them
according to their logical forms (for reasons which will
become evident later on). Symbols are usually used
to represent those logical forms in the manner just des-
cribed, and systems of symbols represent sets of
propositions. In Appendix II, I shall try to show
briefly how a concentration on the geometrical properties
of such systems of symbolic representations can lead to
muddles about logic. Even where it does not lead to
muddles, this concentration on geometrical (or syntac-
tical)properties of sentences or symbols, cannot, unless
accompanied by a study of the functions of words and
symbols in determining the meanings of sentences, explain
anything. It can, at best, lead to description and
classification of logical properties of propositions
and inferences. I shall try to explain, leaving the classi-
fication to others.

5.A.15. To summarize. I have tried to show that the
distinguishing feature of logical constants is not that
they are governed by syntactical rules, but the fact that
their rules are topic-neutral. I have not yet described
in any detail the ways in which such words and constructions
contribute to the meanings of sentences which include them,
and this will be attempted in the following sections. I
hope eventually to provide an explanation of the fact
that some propositions are formal truths (i.e. true in
virtue of their logical form) and the fact that some
inferences are formally valid (i.e. valid in virtue of
their logical form), by showing how this comes about.
Once we understand what sorts of functions logical
words can have, we can see what is involved in using a
    144

logical word with one meaning rather than another, and
can apply criteria for identity of meanings of logical
words. This explains how it makes sense to say that
the English word "and" means the same as the German
word "und", and could take us one step further in the
programme of chapter two. (See 2.A.) But I shall
not go into this aspect in any detail, since ambiguity
of logical constants does not cause as much trouble in
connection with the analytic-synthetic distinction as
ambiguity of descriptive words (see 2.C.)

5.B.    Logical techniques

5.B.1.  So far I have explained in a vague sort of way
how to pick out those parts or aspects of sentences which
are purely logical, namely by seeing whether they are
topic-neutral. I have not yet said how they work, how
their occurrence in sentences contributes to the meanings
of statements which they express, but will do so now.
The explanation will be extended in the next section to
show how it is possible for a statement to be true in
virtue of its logical form. Later on, the account will
be generalized to show how it is possible for a stat-
ement to be analytic.
My descriptions of the functions of logical words will
have to be greatly oversimplified, and it will not be
possible to make more than a few qualifications near the
end, in 5.E.

5.B.2.  If, as pointed out in the previous section, it
will not do to say that the functions of logical words,
such as "or" and "not" are defined by the recursive rules
of a formal system, then how can we explain what their
    145

functions are? What are the topic-neutral rules which
enable them to occur significantly in sentences expressing
contingently true or false propositions? The answer,
as suggested in 5.A.2, seems to be that the rules help
to specify the conditions in which statements employing
those words are true, and conditions in which they are
false. Learning to use logical words and constructions
in sentences involves learning general principles for
recognizing conditions in which statements are true or
false. The rest of the chapter will simply be an
amplification of this statement.
(It is notorious that no matter how much one says
about what words mean, it is always possible for the
objection to be made that the account is either circular,
since it presupposes what it explains, or incomplete, since
it presupposes something else - one of the facts which
seems to have led to the doctrine of the "unsayable"
in Wittgenstein's "Tractatus". So the most that I can
hope to do is draw attention to certain features of what
we all know about our use of logical words, in the hope
that this will remove some puzzles. When I say "we
all know", I do not wish to imply that we can explicitly
formulate the knowledge. See Appendix III on "Implicit
Knowledge".)

5.B.3.  Frege, Russell and Wittgenstein each tried at
some stage to make use of the notion of a function (not
in the sense of a "role", but in the sense defined below)
to explain our use of logical and other words. I shall
use a slightly different notion, the notion of what I call
a "rogator", which will be explained presently. First
I shall say what is meant by a function. (The account
will be brief, as the notion is familiar.) Then I shall
    146

show why talking about such things seems to help, and
after that I shall offer my own variation on the theme.

5.B.3.a.    The notion of a "function" may be defined as
follows. A function is a rule, or a principle or a mapping
which correlates entities, called "arguments", with other
entities, called "values". More precisely, a function
correlates sets of arguments with values, one value to
each argument-set. A given function will have a
restricted domain of definition, so that not every set
of objects can be an argument-set with which the function
correlates a value. The class of argument-sets for
which a function has a value is called its "domain", or
"domain of definition". A function is defined, or set
up, by specifying a domain of definition, and by stip-
ulating either some generally applicable principle or
technique, or method of calculation, which enables the
value of the function to be discovered for every argument-
set of its domain, or simply by enumerating the arguments
and the values correlated with them. (A function "has"
or "yields" a value for a given argument-set, viz. the
one which it correlates with the set. The argument-set
"yields" a value for that function. The function may be
said to be "applied" to its arguments, or to argument-
sets, to yield its values.) Normally the value of any
function for an argument-set depends on the order of
the arguments in the set, and if we restrict ourselves
to functions whose domains contain only ordered sets with
the same number of elements, say n, then we can speak of
the function as "having n argument places" or as an
"n-ary function", and can speak of an argument as occurring
"in the i-th place" in some argument set. The number of
argument-places is usually indicated in a name or sign
    147

for the function by a string of so called "variable-
letters", one for each place. (Sometimes the letters
are called simply "variables".)

5.B.3.b.    If words or signs referring to arguments are
substituted for each of the variable-letters in the
sign for the function, and if the ordered set of objects
corresponding to the ordered set of argument-signs lies
in the domain of definition of the function, then the
new sign thus obtained is taken as referring to the value
of the function for that argument-set (or, more simply,
for those arguments). Thus, the sign "x + y" is a sign
for the arithmetical function, addition, and substitution
of the numerals "2" and "3" for the variables yields
the sign "2 + 3" which refers to the number which is
the value of the function for the set of numbers (2,3),
namely, the number 5. A set of names or signs for
arguments which form an argument-set, is called an
"argument-name-set", or, more shortly "name-set".

5.B.3.c.    When the domain of definition is restricted so
that some argument-places may be taken only by objects
of one kind, and some argument-places only by objects of
another kind, this may be indicated by a convention using
special kinds of variable-letters to indicate kinds of
arguments. (Cf. 5.A.13.)
Where the objects which are taken as values of a
function can all occur as members of the argument-sets
for which the function is defined, the function is
called an "operator," and its application may be
"re-iterated". (E.g. re-iterated application of addition
is symbolized thus: "x + (y + z)".) Where there is a
family of functions, such that the values of some may occur
    148

as the arguments of others, and the values of the latter
may be arguments for still others, etc., the functions
may be applied successively, as is customarily indicated
by such notations as "F(x, g(y, h(z,w)),u)" for the
successive application of the functions F(x,y,z), g(x,y)
and h(x,y). In this way more and more complex functions
may be built up by successive application. The rules or
techniques for finding values of such complex functions
are derived from or constructed out of the rules or
techniques of the component functions.

5.B.3.d.    An example of a kind of function which is often
an operator is what I shall call the "name-function"
of a function. If "F(x,y,z)" is the sign for a function,
then, as shown by the above remarks, there will always
be another function, called the "name-function" for
F(x,y,z), which takes as its argument-sets name-sets1
for the function F, and yields as its values the signs
obtained by substituting names of arguments for variables
in the sign for F. As the sign for the name-function,
I write "/F/"or "/F(x,y,z,)/", enclosing the name of the
function between strokes. So, in this case, if "a",
"b", "c" are names of arguments, then (a, b, c) is an
argument-set for F(x,y,z,), and ("a", "b", "c"), being
the corresponding name-set, is an argument-set for
/F(x,y,z)/, and its value, viz. /F("a", "b", "c")/, is the
name "F(a,b,c)". [There is a whole hierarchy of name-
functions, since to the function /F(x,y,z)/, there cor-
responds the name-function //F(x,y,z)//, taking for its
values such signs as "/F("a", "b", "c")/", which, as just

1.  See 5.B.3.b, above.
    149

indicated, is a name for the name "F(a,b,c)".] Mathe-
maticians and philosophers often confuse functions and
their corresponding name-functions, and so also arguments
and names of arguments, values and names of values. Some-
times this does not matter as the context makes clear
what is meant. But it does matter when attempts are
made to explain what propositions are in terms of the
notion of a function. (Note: the name-function is not
a sign for the name of a function. It is the sign for
a rule which is applied by substituting names of arguments
for variable-letters in the name of a function.)

5.B.4.  Now let us return, to the question: How do logical
constants contribute towards the meanings of statements
which employ them?
We may recall the fact, mentioned in 5.A.13, that
the logical form of a statement, i.e. the way in which
logical words and constructions occur in it, can be re-
presented symbolically by sentence-matrix in which the
non-logical words of the sentence expressing the state-
ment are replaced by variable-letters. Thus, starting
with the statement "Fido is a dog, and all dogs are four-
legged", we obtain the matrix: "x is a P, and all P's
are Q". We have here something strongly reminiscent
of the notation for functions, and this tempted Frege,
for example, to say that the original sentence must be
the name of a value of a function. He wished to say
that the thing named by the sentence (i.e. the value of
the function for Fido, etc., as arguments), was a truth-
value, the True or the False. This seemed odd, because
what the sentence was a name of, i.e. its truth-value,
    150

must depend on how things are in the world, whereas
what we understand by the sentence (or by a name
usually) does not depend on the facts, such as whether
Fido really is a dog. Frege, of course, was not faced
with this sort of difficulty, since he, unfortunately
for logic, was interested mainly in mathematical pro-
positions, whose truth-value does not depend on con-
tingent facts.

5.B.5.  One way out of the difficulty, would be to say
that the sentence is the name of a proposition, which is
the value of the function "x is a P, and all P's are Q"
for the arguments (Fido, dog, dog, four-legged), but
this would be of no use for our programme, since we
are concerned to explain what propositions are, and so
must not assume a knowledge of what they are.
I believe that we could regard a sentence as naming
a class of possible states of affairs (possible states
of the world). But I think there is a more illuminating
way of looking at things, which makes it easier to explain
how a statement may be true in virtue of its logical form,
or in virtue of what it means. We may allow that a
sentence corresponds to a truth value. But the meaning
of the sentence, what is understood by it, is a method
for discovering truth-values.
Meanwhile we may notice one thing. To every
sentence-matrix, no matter what function or other entity
it represents, there corresponds a name-function (see
5.B.3.d), which takes non-logical expressions as arguments,
such as "Fido", "Willie", "The animal under the table",
"dog", "horse", "four-legged", "hungry", etc., and yields
sentences as values. Thus, the function /"x is a P, and
all P's are Q"/ may take as a value the sentence "The
    151

animal under the table is a horse and all horses are
hungry". Sentences, therefore, may be regarded as
values of name-functions.

5.B.6.  Now in order to show how logical constants
contribute towards the meaning of a sentence,
I wish to introduce a new concept, the concept of a rogator,
which is something like the concept of a function, but
not quite. A function is a rule or principle which
yields a value for an argument-set, the value being
determined by the rule and the argument-set, whereas a
rogator is something which does not fully determine the
value but to which there corresponds a method or tech-
nique for discovering the value, which (i.e., the value)
may depend on contingent facts having nothing to do with
the rogator itself, or the principle on which it works.
A simple example of a rogator is the following,
R(x), which takes bottles for its arguments and the sun,
the moon or the earth for its values. In learning how
to find out the value of the rogator for any particular
argument, i.e. any particular bottle, one must learn to
apply the following technique:
Examine the bottle to see whether it is empty or
contains liquid, and, depending on whether it is
empty, less than half full of some liquid, or half
full or more, write down, respectively, "the sun",
"the moon" or "the earth". What has been written
down is then the name of the value of the rogator
R(x) for the bottle in question (at the time of
observation).
In this example, as in general, in order to know
what the value of a rogator is for a given argument, one
must know what the argument is (i.e. which object it is),
one must know the general technique for determining the
value, and one must know certain facts, or have performed
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experiments. The argument and technique alone do not
determine the value, for that depends also on what the
facts are, and the value of a rogator for given arguments
may change from time to time. (E.g. emptying a bottle
may change its value for R(x) from the earth to the sun.)
A time-dependent rogator can always be turned into one
which is not time-dependent by adding an argument-place
for a time, or time indicator. It is important to
notice that the technique for applying a rogator (for
determining its values) may be learnt by example, and
memorized, without the aid of an explicit description
of the way in which it is applied. (See Appendix III.)

5.B.6.  (note). Frege did not need to talk about
rogators since he was concerned with mathematics, in which
the values of functions are determined by general prin-
ciples, independently of empirical facts. Of course,
from a certain point of view, which takes account only
of the way things actually are in the world, and not of
what might have been the case, the notion of a rogator
collapses into that of a function. But one cannot
develop a complete theory of meaning without taking
into account possibilities as well as actual states of
affairs, since to understand a sentence is not merely
to know whether it is true or false. (See 2.C.5 and
4.B.6.) This is why "extensional" systems of logic
are of limited interest.

5.B.7.  I wish now to describe a more interesting kind
of rogator, illustrated by a game played with the aid of
arithmetical symbols, and in particular the symbols for
addition and multiplication, (x + y) and (x.y).
The game is played as follows. A machine, or some
    153

person (God) continually churns out little boxes, in each
of which is a slip of paper with a numeral, the name of
a positive or negative integer, such as "3", "?27",
"3862". (0 is taken to be a positive integer.) On
each box is written a letter or other sign, which is
described as its "name", e.g. "A", "B", etc., there
being no principle connecting the name on the box with
the numeral inside it.
The players make their "moves" in turn, by selecting
a name-function of some arithmetical function compounded
by successive application (see 5.B.3.c.) of addition and
multiplication [e.g. the function x.(x.(x.x)), or
(5.x + y2.z).(x + 3.w), and so on]. This name-function
is then applied to an argument-set consisting of an
ordered set of names of boxes. Thus if a player selects
the name-function /x + y.z/, and the names, "A", "B"
and "C", then he will make his move by reading out "A + B.C"
or "A plus B times C".
Each move is awarded a tick or a cross, as follows.
The boxes corresponding to the selected names are examined
and the value of the arithmetical function worked out
for the numbers referred to in those boxes as arguments.
Thus, if the numerals in the boxes named are found to be
"5", "?2" and "3", then the value of the function in the
case illustrated will be ?1, that is, 5 + (?2).3. A
tick is awarded if the value is positive, a cross if it
is negative.
The next player then makes his move, in the same
way, by selecting a function and set of names and "applying"
the function to the names, being awarded a tick or a cross,
depending on the results of examining the boxes so named
and calculating the value of the function. (The player
with most ticks is said to be "winning".)
    154

5.B.8.  This game provides us with a new kind of
rogator, which takes boxes or names of boxes as arguments
and ticks and crosses, or perhaps the words "tick" and
"cross", as values. Though derived from arithmetical
functions these rogators have different domains of
definition, and different domains of values, from arith-
metical functions, and they do not fully determine their
values for given sets of arguments (for that depends on
which numerals happen to be in which boxes).
Learning to play the game involves learning certain
techniques, such as the technique of calculating the
values of arithmetical functions for particular sets of
numerical arguments. But this is not all. One must
know how to decide whether a "move" is to be awarded a
tick or a cross, and this involves knowing how, given
the ordered set of names used in the move, and the
function employed, to select the appropriate boxes, look
at the numerals inside them, calculate the value of the
function, and then say "tick" or "cross", depending on
what comes out of the calculation.
We see therefore that a complicated technique must
be mastered by anyone who wishes to play the game. It
is a general or uniform technique, since new boxes are
continually being produced, with new names on them and
new numerals inside them, and one must know how to deal
with whatever turns up, and not just how to work with the
first twenty boxes which appear (e.g. by memorizing the
numerals inside them, and their values for a certain set
of functions). We can learn to apply such general
techniques quite easily, for example by watching others
and being given instruction in elementary arithmetic.
We need not, however, either hear, nor be able to for-
mulate, any explicit description of the techniques.
    155

Thus, knowing which rogator is involved in a move,
means knowing how to apply a general technique for
awarding a tick or a cross, given a set of names. We
may represent these rogators by means of symbols, such
as "P + Q.R", or "5.P2", etc., where "P" and "Q" etc.,
are variable-letters which indicate that argument-
places are to be filled by names of boxes, and the whole
symbol indicates which technique is to be applied for
working out the value of the rogator.
To each rogator there corresponds a name-function
(5.B.3.d.) very like that which corresponds to the
arithmetical function from which it is derived except
that one takes names of boxes (or names of names of
boxes) as arguments, and the other takes names of
numbers.

5.B.9.  It should be clear now what I am getting at.
Instead of regarding the symbols (sentence-matrices)
which represent the logical forms of propositions as
corresponding to functions (see 5.B.4), I shall regard
them as corresponding to rogators, which will be des-
cribed as "logical rogators". They may be represented
by such symbols as "x is P", "All P Q's are R", "x is
a Q and there are no R's which are S", etc. Corres-
ponding to them are also name-functions which take sentences
as values. (See end of 5.B.5.)

5.B.10. We can regard rogators as taking either things
or words which refer to them as arguments. For reasons
of convenience of exposition I shall describe logical
rogators as taking meaningful non-logical descriptive
words and referring expressions as arguments, their
    156

argument-places being represented by higher-case and
lower-case variable-letters, respectively. We could,
instead, talk about the things referred to as the
arguments. (A rogator, may, like a function, have a
restricted domain of definition. See 5.B.3.a and
5.B.3.c.) For the time being I shall take the words
"true" and "false" to be the values of logical rogators.
(But see 5.B.18, below.)
To each rogator there corresponds a general tech-
nique, which I shall describe as a "logical technique"
for determining its values, given an argument-set.
The technique involves looking at non-linguistic enti-
ties and then deciding to award the word "true" or the
word "false" for the "move" in language which is (or
would be) made by uttering the sentence obtained by
replacing variable-letters in the sign for a logical
rogator by suitable arguments for that rogator.

5.B.11. For example, for applying the logical rogator
"All P Q's are R", (derived from the logical form of
statements like "All red boxes are square"), we might
use the following technique. Given an argument-set of
three descriptive words, seek out the objects having the
properties referred to by the first two words, examine each
of them to see whether it has the property referred to
by the third descriptive word, writing down a tick if it
has, or otherwise a cross. When finished, look to see
if there is a cross amongst the things written down; if
not the value is "true" and otherwise "false".
In 5.A.3, we have already described the technique for
the use of the copula, in the rogator "x is P".
It is not essential that the techniques should be
described in these ways. There may be other techniques
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with the same effect, and there may be various ways of
describing the same technique. The important thing is
that there are techniques which can be learnt, and
which enable one, given a knowledge of the things (par-
ticulars or universals) referred to by non-logical words,
to examine "the facts" or "the way things are in the
world", and award truth-values to statements.

5.B.12. As before (see 5.B.8) generally applicable
techniques correspond to each logical rogator. For in
learning to use the logical form "x is a Q and there are
no Q's which are R", it is not enough to learn to deter-
mine the truth-value when one of the words "Tom", "Dick"
or "Harry" is taken as argument in the place of "x",
and the other arguments are "man" and "happy". Nor is
it enough to know how to find a truth-value for an
argument-set as things actually are. One must know how
to determine it for all suitable arguments in all possible
circumstances, otherwise one does not fully understand.
(Cf. 5.B.6.note.)
The techniques are, in fact, so general that it does
not matter to which particular material objects the
referring expressions correspond, nor to which properties
(or "improper" properties) the descriptive words correspond.
The techniques are topic-neutral. (See 5.A.3-4.)
(We shall see later on, in Section 5.E, that this must be
qualified.)

5.B.13. We are now almost in a position to say explicitly
what must be learnt when one learns to use logical words and
constructions. This requires a slight extension of
the notion of a logical rogator and the corresponding
name-function. (See 5.B.3.d.)
    158

So far we have considered only name-functions which
take sentences as values (see end of 5.B.5). But there
are name-functions which take referring expressions or
descriptive expressions as values. In addition, we must
allow not only descriptive expressions and referring
expressions as arguments, but also whole sentences.
Thus, the name-function /"the R of x"/ takes relation words
and referring expressions as arguments and yields
referring expressions as values, such as "the father of
Napoleon". The function /"P and Q"/ takes descriptive
expressions both as arguments and as values, as in "red
and round". The function /"? and ?"/ takes sentences
as arguments and yields sentences as values, such as
"Fido is a dog and all dogs are four-footed".
Since the value of a name-function may be a sentence
or a referring expression or a descriptive word, it may
occur as the argument of another name-function. Thus,
the function /"as P as x"/ may take as arguments "tall"
and the expression mentioned above, and yield as a
value the descriptive expression "as tall as the father
of Napoleon".
Thus, by successive application of name-functions
we can construct more and more complicated name-functions,
just as in our arithmetical game more and more complex
arithmetical functions could be constructed out of addition
and multiplication, by successive application. (See
5.B.3.c.) For example, we get the name-function /"x is
as Q as the R of z"/ by successive application of the
functions /"x is P"/, /"as Q as z"/ and /"the R of y"/.
Similarly, the logical form of a proposition may be regarded,
often, as constructed by successive application of logical
rogators to form a new, more complex rogator.
    159

5.B.14. We have already noticed that a name-function
which takes non-logical words as arguments and yields
sentences expressing statements as its values may be
thought of as corresponding to a logical rogator (a
technique for determining truth-values). In addition,
any of the name-functions described in 5.B.13 can be
thought of as corresponding to a logical rogator,
which takes as its arguments linguistic expressions of
various sorts, and as its values either a particular
object, or a property (proper or improper) or a truth-
value, depending on whether the values of the name-
function are referring expressions, descriptive expressions
or sentences. For example, the logical rogator
"the R of x" takes for its values particulars, such as
the father of Napoleon, which are referred to by the values
of the name-function /"the R of x"/. To each such
logical rogator there corresponds a technique which must
somehow be learnt for finding out which thing, property
or truth-value (etc.) is the value of the rogator, given
a set of arguments and their meanings. The technique,
as before, must be generally applicable. (See 5.B.12.)

5.B.15. Now the role of logical words and constructions
can be described explicitly: learning to use them
involves learning to apply name-functions to non-logical
words and expressions as arguments, obtaining referring
expressions, descriptive expressions and sentences as
values. (Formation-rules must be learnt.) Secondly,
it involves learning how the meanings of the resulting
combinations of words depend on the meanings of the
expressions taken as arguments, or, more specifically, if
the resulting expression is a referring one, one must know
how to tell which is the object to which it refers; if
    160

it is a descriptive expression, one must know to which
properties it refers (or, more generally, how to recog-
nize objects which it describes); if it is a full
sentence, one must know which logical techniques to
apply to the material objects, properties or other
non-linguistic entities referred to by the expressions
taken as arguments, in order to arrive at a truth-value.
(I.e. one must learn which logical rogators correspond
to which logical forms.)
The principles and techniques which must be picked
up if one is to use logical constants may be very com-
plicated and difficult to formulate explicitly.

5.B.16. I shall not try to describe in detail or
classify the various rules for the use of specific logical
constants (i.e. logical words and constructions) in all
contexts. That is the task of the formal logician, and
in any case it would be very complex since there is an
enormous variety of cases and many intricacies would have
to be taken account of, such as the fact that one and the
same English word can correspond to functions and rogators
taking various sorts of arguments and values. (E.g.
"or" as a function of descriptive expressions in "red or
round", and as a function of sentences in "That is red
or that is round"; "is" of identity and "is" as a copula.)
Moreover, the logical form of a proposition (or the
corresponding logical rogator) may not be fully determined
by its geometrical or syntactical form: other things,
such as the context of utterance, or the type of entity
referred to by one of the non-logical words may have to
be taken into account. (Compare, for example, "I want
that cake" and "I made that cake"; or "Fido is black",
and "The dog you heard is Fido".) In consequence, dif-
ficulties arise if we try to represent logical form in
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the usual way simply by removing non-logical words and
replacing them by means of variable-letters.1

5.B.17. I shall henceforth ignore the rules for the
individual logical constants, discussing only the
results of combining them with non-logical words and
expressions to form whole sentences. So I shall discuss
only the logical rogators or logical techniques which
correspond to complete sentence-matrices (5.A.13,
5.B.9-10). All we need notice in connection with
individual logical constants is that learning to use them
involves learning very general principles for dealing
with the non-logical words with which they may be com-
bined, or which are taken as arguments for logical
rogators. (See 5.B.12.) This generality, or topic-
neutrality helps to account for the fact that we can
cope with newly invented descriptive words (e.g. the
name for the colour of a new synthetic dye) without
formulating new rules for the use of logical constants in

1.  Some of these complexities are described briefly by
Holloway, in "Language and Intelligence" (see pp.144
to 152). He seems to think that pointing out these
irregularities demonstrates that signs in a calculus
do not work like words in a language (p.144). But
I have argued that there is a more fundamental
reason, namely, that signs in a calculus simply
cannot occur in true or false statements about
anything. (See circa 5.A.6, above.) What the
irregularities show, however, is that certain types
of calculus do not provide adequate symbolic representations
of forms of propositions, but this can
surely be remedied at the cost of a loss of elegance
by the choice of a calculus with suitably complex
formation rules, rules of derivation, etc. (There
would have to be different sorts of variables.) No
such remedy can, however, turn a calculus into a
language, for a language needs semantic rules as well
as syntactical ones. (See section 2.D.)
    162

sentences including them. It helps to explain how we
are able to construct sentences to deal with totally
new and unexpected situations. (However, see qual-
fications in 5.E.6,ff, below.) It explains how we
can successively apply name-functions to form more
and more complicated sentences, expressing more and
more complicated propositions.
These general principles for dealing with new
combinations of words, and for discovering truth-values
in new conditions, all have to be memorized in learning
to talk. (This does not mean that formulations of the
principles have to be memorized.) The fact that they
can be memorized is all that we need to explain the
possibility of learning and teaching the use of logical
words. (See quotation from Pap, in 5.A.2.) We cer-
tainly do not have to postulate the existence of any
"if-feelings" or other peculiar subjective entities
correlated with logical words.
(Though I do not plan to formulate principles for
the use of individual logical words, I have already
described explicitly the logical techniques which cor-
respond to certain logical forms of complete propositions,
such as "x is P" and "All P Q's are R", in 5.A.3 and
5.B.11.)

5.B.18. I have so far regarded the words "true" and
"false" as the values of logical rogators corresponding
to certain logical forms of statement, or sentence-
matrices. This, however, does not explain how we can
make statements to convey information, except by assert-
ing something like: "The sentence "The sun is shining"
corresponds to 'true' ". It is possible to modify my
account of logical rogators by regarding them as taking
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for their values, not truth-values, but sentences,
namely the sentences obtained by applying their name-
functions to argument-name-sets, in cases where the
value would be true, or slightly different sentences
including the word "not" where the value would be
"false". (Compare: in our arithmetical game, instead
of awarding ticks or crosses for "moves", the players
might simply write down whatever is read out
by a person in making a move in cases where it would
be awarded a tick. E.g. they write down "A + B.C"
when that move is awarded a tick. If it merits a cross
(i.e. if the value of the arithmetical function turns out
negative) they write down, instead, "(-l).(A + B.C)".
The reader may fill in details of how this works in the
game and what its point is.)
So in learning to talk we learn to utter sentences
themselves in cases where our investigations and appli-
cations of logical techniques would yield the value
"true". In cases where the value would be "false", we
learn to prefix the sentence with the words "It is not
the case that ...." or something similar, or, more
commonly, to utter another sentence which would correspond
to the value "true" (which, in many cases can be derived
from the original one by suitable insertion or removal
of the word "not"). Others, who have learnt to speak
the same language, then, if they hear us and trust us,
know what to expect when they look at the facts we have
observed in applying the logical techniques.1

1.  The rule might have been the other way round. We might
have learnt to play the game in such a way that utterance
of the original sentence took the place of utterance of the
word "false", and utterance of the amended sentence took
the place of the word "true", everything else being the
same. In that case, sentences in that language would
simply mean what their negations mean in ours. To utter
a sentence in that language would still, of course, be to
say what is the case, though we should take it to say what
is not the case, since we should not understand.
    164

5.B.19. We now see that whether a sentence S expresses
a true or a false proposition depends on three things:
(a) the logical form of S, which determines a
logical rogator or general technique for deter-
mining truth-values, or for determining when S
may be uttered and when not;
(b) the meanings of the non-logical words and
expressions taken as arguments (i.e. their sem-
antic correlations with things and properties);
and
(c) the way things are in the world, which is, in
general, discovered by carrying out observations
in the course of applying logical techniques.
(The fact that there is this third element shows that
there is something right in correspondence theories of
truth. Only by talking about rogators instead of functions
can we bring this out.)
These three elements are all illustrated by our
arithmetical game. For example, to three elements in
the claim to know that some statement is true correspond
three elements in the claim to know that a move in the
game deserves a tick, thus:
(a) Knowing which boxes correspond to which "names"
may be compared with knowing the meanings of
non-logical words.
(b) Knowing how to look into appropriate boxes and
apply the technique for deciding whether to
award a tick or a cross corresponds with knowing
the general principle or logical rogator
determined by the logical form of a proposition.
(c) Looking at numerals actually written out in the
boxes corresponding to names used in a move in
the game, is like observing the facts which
determine whether a sentence expresses a true
or a false proposition.
The analogy should not be taken too seriously. I
have tried to find a comparison which is close enough to
serve my purposes without being too close to be illuminat-
ing: very likely an impossible requirement!
    165

5.B.20. I have tried to describe the role of logical
constants in giving sentences their meanings by saying
that the way they occur in sentences determines which
logical rogators correspond to the propositions expressed
by those sentences. Thus they help to determine the
kind of technique which must be employed in discovering
whether a proposition happens to be true or not. So
they help to determine which sorts of facts can count
as verifying the proposition, or in which possible states
of affairs the proposition would be true. Logical con-
stants do not indicate which entities or kinds of enti-
ties a proposition is about (they are topic-neutral -
see 5.A), but they do indicate how things must be with
those entities if the proposition is true. (I have
ignored the role of logical constants in questions,
commands, and so on, but I think this could easily be
taken account of.)
It might be thought that this is an unnecessarily
long-winded and round-about way of describing how pro-
positions work. Certainly there are alternatives.
For example, we could regard sets of non-logical words
as determining functions and the logical words as the
arguments to which they are applied. (In general, there
is a symmetry between argument-sets and functions.) Or
we might try to avoid mentioning rogators and manage with
functions alone, as Frege and Wittgenstein and Russell did.
Notice, however, that my method eliminates many obscurities
in their accounts. Thus, there is no need to discuss
Frege's "unsaturated" entities of which nothing can be
predicated. Nor need we talk about "unsayable" facts
which merely "show" themselves, though, as will appear in
chapter seven, in connection with knowledge of necessity,
    166

this is only a temporary advantage. The price which we
have had to pay for explicitness is, of course, circularity.
But I hope the circle is so wide that this does not
matter.

5.B.21. This completes the account of the way in which
logical constants help to determine the truth-conditions
of propositions which they are used to express. (All
this shows, incidentally, that the employment of verbs
is not essential for the activity of statement-making.
Verbs have a special function which need not be described
here.) Now it remains to show how this sheds light on
the existence of logical properties of propositions or
logical relations between them. We must try to under-
stand, for example, how it is possible for propositions
to be true, or inferences to be valid, in virtue of their
logical form. Once again, the analogy of the arithmetical
game will be useful.

5.C.    Logical truth

5.C.1.  Logical properties of propositions and logical
relations between propositions can be shown to be due to
relations between the logical rogators involved in
construction. I shall first of all illustrate, using
the example of the arithmetical game (see 5.B.7.), the
general way in which rogators may be related and then
turn back to logical rogators and propositions.

5.C.2.  It will be recalled that "moves" are made in the
game by applying rogators, derived from arithmetical
functions, to names of boxes as arguments, ticks or crosses
being awarded according as the arithmetical functions have
    167

positive or negative values for the numerals in the
boxes named. The first thing to notice is that
relations of "entailment" may hold between moves in the
game.
Consider moves made with the two functions f(x,y)
and g(x,y), where the former is x2 + (?3).x.y and the
latter is x2 + (?3).x.y. + 27. The rogators derived
from these functions are f(P,Q) and g(P,Q), taking
names of boxes as arguments. To each rogator there
corresponds a general technique for working out its
value for any argument-set, the value being "tick"
or "cross" depending on which numerals are found in the
boxes referred to. (See 5.B.8). Now, since the
value of the function g(x,y) is always greater than the
value of f(x,y) by 27, for the same arguments, owing to
a relation between the techniques for calculating their
values, it follows that the former is positive whenever
the latter is. Hence, if any move made with f(x,y)
and names "A" and "B" is awarded a tick, then, in those
circumstances, a move made with g(x,y) and the same names
would also merit a tick.
Owing, therefore, to a relation between the tech-
niques for finding their values, the two rogators
f(P,Q) and g(P,Q) are themselves related so that the
value of the latter for a pair of names of boxes as
arguments must be "tick" whenever the value of the former
is, for the same arguments (in the same circumstances).
Of course, the value of f (P,Q) might be "cross" when the
value of g(P,Q), was "tick", but the converse could not
happen. We can tell, merely by examining the techniques
corresponding to the two rogators, without looking to see
what is in the boxes, that if the value of f is "tick",
then the value of g is also "tick". We may say of a
    168

pair of moves made with these rogators, such as
"A2 + (-3).A.B" and "A2 + (-3).A.B + 27" that the
former "entails" the latter.

5.C.3.  Now we must notice what happens when complex
rogators are constructed out of rogators between which
relations hold. Since the value of the arithmetical
function g is greater than the value of f for the same
arguments no matter what they are, it follows that the
arithmetical function g(x,y) + (-1).f(x,y) is positive
for all values of the arguments. Call this function
h(x,y). To it, as usual, there corresponds a rogator
h(P,Q), taking names of boxes as arguments, but with the
peculiarity that the value of the rogator for all argu-
ments is "tick" in all circumstances. The general
technique for discovering the value ensures that all
moves in the game made with h and any pair of names of
boxes must be awarded a tick, no matter what numerals
are in the boxes, and this may be discovered simply by
examining the technique for working out values of this
rogator.
Of course, this is not the only way in which a one-
valued rogator can be constructed. Other examples are
the rogators derived from the following arithmetical
functions:
x.x, x.y.x.y,   and x.x + y.x + y.y
each of which has positive values for all arguments.

5.C.4.  All this shows that although in general the value
of a rogator for a given set of arguments has to be
discovered by looking at the facts and applying the
general technique for determining its value, there are,
nevertheless, some cases where a complex rogator, com-
pounded out of simpler ones (see 5.B.3.c, 5.B.13) by
    169

successive applications, is a "freak" in that one can
discover its value merely by examining the technique
for working out its value, or by examining the tech-
niques of the simpler rogators out of which it is
compounded. Similarly, by examining the techniques for
working out the values of a pair of rogators, one may
discover that they stand in some "internal relation"
so that knowing the value of one of them may enable one
to work out the value of the other without consulting
the facts or actually applying its general technique.

5.C.5.  In some cases we can look at the way in which
the rogator determines its value independently of the
facts slightly differently. For example, in the
arithmetical game, the use of the function u.v + w.x + y.z
in a move may result in a tick or a cross being awarded,
depending on which names are used and which numerals are
in the boxes corresponding to those names. But if we
take the argument-set consisting of the names "A", "A",
"B", "A", "B", "B" then we obtain the move "A.A + B.A
+ B.B", which gets a tick no matter what is in the boxes,
since x2 + y.x + y2 is positive for all values of x and
y. This shows that in some cases we can look at the
value of a rogator as determined not only by properties
of the technique for working out its values, but also by the
"structure" of the argument-set. In these cases, by
examining the technique for determining values of the
rogator and the structure of the argument-set, we can
find a value which it must take for all argument-sets with
that structure, no matter what the facts are, although in
general the result of applying the techniques corres-
ponding to the rogators out of which the "freak" is con-
structed does depend on the facts, that is, on how things
    170

happen to be in the world. (See 5.B.6.)

5.C.6.  All this applies to logical rogators as well as
the ones which we have been discussing: logical roga-
tors may also generate "freaks". We saw (in section
5.B) that every sentence can be thought of as the
result of applying a logical name-function to a set of
non-logical words, the logical rogator corresponding to
that name function being what determines the conditions
in which the sentence so obtained expresses a true
proposition. As in the cases discussed above, it may
be possible, by examining the general techniques for
determining the value of a logical rogator, to discover
its values for all arguments, or for argument-sets with
certain structures, without consulting the facts at all.
So we can determine the truth-values of propositions
constructed with the aid of such rogators merely by
examining the logical techniques for discovering whether
those propositions are true or false, i.e. without
applying the techniques.
For example, the sentence "All red horses are red"
is a value of the logical name-function/" All P Q's
are R"/, for the argument set ("red", "horse", "red").
To it there corresponds a logical rogator and a technique
for determining truth-values. By examining that tech-
nique, and the structure of the argument-set, we can
discover the truth-value in question without actually
applying the technique (which would involve examining
all red horses to see whether they are red). We can
discover that the proposition expressed by the sentence
is true independently of the facts, and independently of
the actual meanings of the words in the argument-set (as
long as the structure is ("A", "B", "A")). We say that it is
    171

a "formal" truth, true in virtue of its logical form.

5.C.7.  It is extremely important to notice, in all
these cases, that where the value of a rogator is
determined independently of the facts, this has to be
discovered by examining the technique, not by applying
it. But since one may have mastered a technique without
ever examining it (see appendix on "Implicit Knowledge"),
one may fail to notice that a rogator is a "freak" whose
values are determined independently of the way things
are, and go on as usual to find out its value by applying
the technique. (It should be recalled that the tech-
niques are generally applicable: they work for all
argument-sets, even those whose structure restricts the
possible outcome of applying the techniques. See 5.B.8,
5.B.12, 5.B.17.)
So, in the case of the game, the players may fail to
notice that a move such as "A.A + B.A + B.B" would merit
a tick no matter what numbers were found in the boxes
referred to, and go on in the usual way to look into the
boxes, calculate the values of the arithmetical functions,
and base their decision whether to award a tick or a cross
on the result of applying this technique. Similarly,
one may fail to notice that some proposition is true in
virtue of its logical form, and apply the usual logical
techniques for determining its truth-value by observation.
This is possible because the techniques are generally
applicable. Thus I can discover that "All the red horses
in this room are red" expresses a truth by examining the
red horses in the room (cf. 5.B.11), but there is no need
to, since I can see what the outcome would be merely by
thinking about the method which I should have to apply.
    172

(See 5.A.9 for another example.) The importance of
this will emerge in section 6.E on "Knowledge of
analytic truth".

5.C.8.  All this can be extended to explain the exis-
tence of logical relations between propositions, arising
from relations between their structures. For example,
we may learn that one proposition entails another in the
same sort of way as we found in 5.C.2. that one move in
the game could "entail" another. We may find, by
examining the logical techniques for discovering truth-
values of the two propositions, and the structures of
their argument-sets, that no matter how things are in
the world, if the outcome of applying one of these
techniques is "true", then so will the other be. E.g.
If "All black horses are hungry" expresses a true pro-
position, then so does "All big black horses are hungry".
In such a case, we may speak of "formal entailment".
The inference from one proposition to the other will be
"formally valid", or valid in virtue of its logical form.
(The logical form of an inference can be represented
by substituting variables for non-logical words, in much
the same way as the logical form of a proposition.
(Cf. 5.A.13).) Similarly, propositions may be formally
contradictory, or formally incompatible. As with formal
truth, such logical relations may pass unnoticed by
persons who apply logical techniques without examining
them. (We cannot give a general definition of "entails"
until after Chapter seven.)

5.C.9.  I wish to stress the (by now obvious) point that
these logical properties and relations of propositions
are not due merely to geometrical relations between symbols,
    173

but primarily to properties of and relations between
techniques which have to be learnt for doing things
with these symbols. Admittedly, in most languages
the rules for the use of symbols are probably so chosen
that to certain geometrical relations there correspond
relations between techniques (as implied by my remarks
in 5.B.14-15 about the connections between name-functions
and logical rogators). This is indispensable if there
are to be general principles for constructing more and
more complicated types of propositions out of a small set
of symbols without continually introducing new ad hoc
grammatical and logical rules of construction: to this
extent logic may be connected with syntax, though it
is never reducible to it. However, as remarked in
5.B.17(note), not all rules of formation of sentences are
quite like this, so the connections between logical
techniques and geometrical forms are not absolutely
indispensable and, in any case, it is not enough to
notice the connection between geometrical relations and
logical relations. Indeed, noticing this may blind
philosophers to the intermediary in virtue of which they
are connected, with unfortunate results, as I shall try
to show in Appendix II.
Part of the explanation of the tendency of philo-
sophers of logic to ignore these logical techniques, is
the fact that we can learn to use symbols and apply the
corresponding techniques, and sometimes even draw con-
sequences from the interconnections between these
techniques, without fully realizing what we are doing.
We need not even be aware of the existence of the tech-
niques. This is an illustration of a general point that
one may have knowledge which one cannot formulate, or
one may know that something is so without being quite
    174

aware of the reasons why they are so or how it is that
one knows this. One may claim, with perfect justi-
fication, to know that "If anything is red then it is
red" expresses a truth, and yet be completely inarti-
culate when there is any question of justifying the
claim. (This sort of thing is discussed in Appendix III,
and in 6.E.6.) I have been trying to make the missing
justification explicit, or at least to describe it in
general terms.

5.C.10. I have not, however, fully explained how we
can draw conclusions about the outcome of applying
certain techniques merely by examining those techniques,
without actually applying them. I have not explained
what goes on when one has the kind of logical insight
which is involved in perceiving that two logical tech-
niques are related in certain ways, or that a logical
technique has certain properties, apart from relating it
to the general way in which one may discover properties
of or connections between rogators.
Some would probably try to reduce logical insight
to a matter of seeing that a certain sequence of formulae
or symbols have certain syntactical properties, but this
would leave unexplained the kind of insight one has when
one sees that this is the case, which is a sort of mathe-
matical insight into the connections between geometrical
forms. In any case, we have already argued against
attempts to reduce logic to syntax. (See section 5.A
and Appendix II.) (For some reason it was only after
the discovery of (Gödel's famous incompleteness theorem
that some logicians began vaguely to appreciate this point.
Gödel expressed it as follows, in his contribution to
    175

"The Philosophy of Bertrand Russell", p.127-8: "It
has turned out that ? the solution of certain
arithmetical problems requires the use of assumptions
essentially transcending arithmetic i.e. the domain
of the kind of elementary indisputable evidence that
may be most fittingly compared with sense perception."
I do not wish to say that this transcends arithmetic:
I should rather say that it turns out that arithmetical
knowledge requires more than was once thought by some
logicians to be required. It transcends their concep-
tion of arithmetic.)

5.C.10.a.   It cannot be argued that when we have this
sort of insight or draw the sorts of conclusions under
discussion, by examining the structures of argument-sets
and the techniques for determining the values of logical
rogators, what goes on is that we consider statements
describing these techniques and structures and then
apply some formally valid procedure for inferring, via
formal entailments, that certain statements have certain
logical properties or stand in logical relations. That
would clearly be circular, since it is the nature of
formal validity that we are trying to explain by talking
about these properties of logical techniques. We cannot
without circularity explain this by assuming that their
having these properties is merely a formal consequence
of other facts. (Cf. 7.D.9,ff.)

5.C.l0.b.   It seems to me that what goes on when we have
this sort of insight, and in general when we discover
facts about the application of techniques by examining
them instead of applying them, is essentially the same
sort of thing as goes on when we discover necessary
    176

connections between, for example, geometrical structures
or properties, by examining those structures or pro-
perties and perhaps constructing informal proofs. The
difference lies in the degree of generality. (Logic
is topic-neutral: see section 5.A.) This sort of
thing will be discussed in more detail in the section
on "Informal proofs", in chapter seven.

5.D.    Some generalizations

5.D.1.  Let us leave aside questions about what goes
on when we examine logical and other techniques instead
of applying them, and consider how some of the remarks
of the previous section may be generalized, so as to
prepare the way for the discussion of propositions which
are analytic, or true by definition.
We have seen that although in general the value of a
rogator for any argument-set depends on how things are
in the world, nevertheless there are some "freak" cases
where the value can be discovered independently of
observing facts and applying the technique for that
rogator. In some cases we found that what determined the
value independently of facts was an interaction between
the general technique for discovering values, and the
"structure" of the argument-set. (See 5.C.5.) Let us
look a little more closely at this sort of case.
It is clear that when the logical form "All P things
are Q" is applied to the argument-set ("red", "red"), the
basic reason why the proposition expressed by the
sentence so obtained is true independently of what is the
case in the world, is not the fact that the two words in
the argument-set are identical, but that they have the
same meaning, that they refer to the same property.
    177

This is seen from the fact that if we define a new word,
"rot", say, to refer to the same property (proper or
"improper" property1) as "red", then applying the logical
form to the argument-set ("red", "rot") will yield the
statement "All red things are rot", which is true
independently of the facts for much the same reason as
the original one. So our description of the original
case was not sufficiently general. The fixed truth-
value of a statement like "All red things are red" is
not essentially due to the fact that it is obtained from
an argument-set with the structure ("P", "P"), but to the
fact that it is obtained from a set of two arguments stand-
ing in a certain relation (in our example the relation
is synonymy: the words refer to the same property).

5.D.2.  We see therefore, that when the value of a
rogator for certain arguments is determined independently
of the facts, this may be due to (a) the general technique
for discovering its values (e.g. the way the rogator
is constructed out of other rogators), (b) the structure
of the argument-set, and (c) relations between the
arguments. (The second is really a special case of the
third.) In this sort of case, the value of the rogator
is always the same for a set of arguments standing in the
appropriate relations, no matter what the arguments are,
and no matter how things are in the world. (As before
(see 5.C.7), a person may fail to notice this interaction
between a rogator and an argument-set, and work out the
value in the usual way by applying the technique as if
its outcome could depend on the facts.)

5.D.3.  Once again we can find an illustration in our
arithmetical game. (5.B.7.) If it is known that no

1.  3.B.5, 3.D.8.
    178

numeral occurs in more than one box in the game and
that every numeral will eventually occur in at least
one of the boxes, then we might, in the course of playing
the game, introduce new names for boxes, as follows.
If "B" is known to be the name of a box, then we say:
"Let 'A' be the name of whichever box contains the
numeral obtained by adding three to the number referred
to in ,'B' " We shall not, of course, know which box
is the one referred to by the name "A", but we do know
that whichever it turns out to be, the numeral in it will
be in the stated relation to the numeral in B, whatever that
may be. By considering this fact, and by examining the
technique for deciding whether moves are to be awarded
ticks or crosses, we can tell without looking into boxes
that the move "A + C.C + (?1).B + C" must be awarded a
tick since the value of the arithmetical function
x + y.y + (?1).z + w must be positive for all argument-
sets in which the second and fourth arguments are the
same, and the first argument is greater than the third.

5.D.4.  This example illustrates the same sort of thing
as may happen when we apply a logical form to descriptive
words whose meanings stand in some complicated relation
owing to the fact that they have been logically synthesized
in the manner described in section 3.B. Relations between
meanings of words in virtue of which the value of a rogator
is independent of the facts need not be as simple as in
the example of 5.D.1, where the relation was synonymy.
For example, if the word "U" refers to the property P,
and the word "V" refers to the combination of properties
Q and R, while the word "W" describes objects if and only
if they have the property P or do not have the property Q,
then the result of applying the name-function /"No F are G
    179

and not H"/ to the argument-set ("W", "V", "U") is the
sentence "No W are V and not U" which can be seen to
express a proposition whose truth is independent of
how things are in the world, owing to (a) facts about
the logical rogator corresponding to its logical form,
and (b) relations between the meanings of the words.
Since the proposition is not true merely in virtue
of its logical form (not all propositions with that
logical form are truths), but also in virtue of relations
between the meanings of some of the non-logical words,
the proposition is not a "formal" truth. (See 5.C.6.)
(We could alter customary philosophical usage and extend
the notion of the "logical form" of a proposition to
include such facts about the logical relations between
meanings of non-logical words, and this would be a good
thing insofar as it drew attention away from syntactical
properties of sentences, but I shall not do so.)

5.D.5.  We can now see that the fact that if words are
given meanings standing in certain relations then this
may have the consequence that some sentences in which they
occur express propositions whose truth-values are inde-
pendent of the way things are in the world, is just a
special case of a more general fact about rogators,
namely that relations between arguments in an argument-
set together with properties of the general technique for
discovering values of the rogator may in some cases
suffice to determine the value for that argument-set,
though in general it does not, since facts are relevant
too. This does not mean that the general technique
cannot be applied in order to discover the value in such
freak cases, but that it need not be. (Cf. 5.C.7.)
(We shall see later on that even an analytic proposition
    180

can be verified by empirical observations, though it
need not be.)

5.D.6.  It should be noted that when there are relations
between the meanings of descriptive words, although
this does not enable us to discover the truth-values of
all propositions which they may be used to express, there
will certainly be a great many whose truth-values are
determined. Thus, from the fact that the word "red"
refers to the same property as rot" we can infer not
only that "All red things are rot" expresses a true
proposition (See 5.D.1.), but also that each of the
following does: "Nothing is both rot and not red",
"If anything is not red then it is not rot", "A11 rot
and round things are red or the moon is made of green
cheese", etc. In addition, the following are false
independently of the facts: "Some red things are not
rot", "All round things are red and not rot and there is
at least one round thing", etc. (To any one relation
between the meanings of descriptive words, there cor-
responds a whole family of analytic propositions.
See 6.F.5.)
In all these cases we can discover the truth-value
in essentially the same way as before: namely by study-
ing the general logical techniques which would normally
be applied, in finding out the truth-values of statements
made with these logical words and constructions.
In addition, we could discover, by examining the
logical techniques and relations between meanings, that
certain relations such as entailment and incompatibility
hold between some propositions. (This is just an
extension of the remarks in 5.C.8.)
    181

5.E.    Conclusions and qualifications

5.E.1.  The time has come to summarize what has been
done in this chapter, and show how it fits into the
general programme of this thesis.
The aim of Part Two, which this concludes, was to
describe the general connection between meaning and
truth, in order to prepare the way for a description of
the connection between meaning and necessary truth, in
Part Three. (We have been mainly concerned with state-
ments containing only logical words and descriptive words
referring to universals, but many of the remarks of 5.B
apply also to statements in which particulars are
mentioned.)
The general connection can be summed up thus:
learning the meanings of words or sentences containing
them involves learning to recognize or pick out states
of affairs in which to utter such sentences is to make
true statements. I have tried to isolate out two
aspects of this learning process. first we have to
learn semantic correlations between non-logical words and
non-linguistic entities, and secondly we must learn the
use of logical words and constructions. (1) Semantic
correlations between descriptive words and universals
(observable properties and relations) were described
in chapters three and four. Learning these involves
learning to recognize the particular objects which may be
correctly described by such words. (2) Learning to
use logical words and constructions involves learning to
tell which logical rogator (which generally applicable
logical technique for determining truth-values) corres-
ponds to the way in which logical words and constructions
occur in a sentence. It is in virtue of this correspondence
    182

that the occurrence of such logical constants helps to
determine the conditions in which the proposition
expressed by the sentence is or would be true (or false).
All this showed how the truth-value of a proposition
expressed by a sentence containing logical words and
descriptive words depended on (a) the meanings of the
descriptive words, (b) the logical techniques corres-
ponding to the logical form and (c) the facts, i.e. how
things are in the world. (5.B.6, 5.B.19.) We saw
that this was just one instance of the general fact that
the value of a rogator depends on (a) the objects taken
as arguments (b) the general technique (or rule, or
principle) for discovering values, and (c) the way things
are in the world.
This completed the account of the general connection
between meaning and truth.

5.E.2.  Further investigation showed that the existence
of formal truths and formally valid inferences could be
explained in terms of properties of and relations between
rogators. (See 5.C.6, 5.C.8.) This eliminated the
need for making obscure, misleading or false remarks about
the connection between logic and syntax. (See section
5.A.) As remarked in the previous paragraph, the value
of a rogator depends, in general, on three things, but we
found some "freak" cases in which the third element dropped
out. In these cases, although the value could be dis-
covered by applying the general technique and investigating
facts or conducting experiments, nevertheless this was
not necessary, since the value could be determined a
priori.
We distinguished three types of freak case. (1) In
the most general case, the value depended on both of the
    183

first two factors (namely (a) and (b) above), and could
be discovered by examining the argument-set and the
technique for discovering values of the rogator. The
value was determined by relations between the arguments
together with properties of the general technique,
independently of the facts. It did not matter which
particular objects were taken as arguments: the value
was always the same, provided that they were related in
a certain way. (Section 5.D.) (2) A simpler type of
freak rogator was one whose value was the same for all
argument-sets with a certain structure. Here the value
could be discovered by examining the structure of the
argument-set and the general technique for finding values
of the rogator, independently of how things were in the
world, or which particular objects were taken as arguments.
(3) In the simplest sort of case, the value was completely
independent of which objects were taken as arguments, and
was fully determined by the general technique. Here both
the first and the third factors dropped out, leaving only
the second (b).
The second and third type of freak rogator sufficed
to explain the existence of propositions true in virtue
of their logical form, since such propositions corresponded
to logical rogators constructed in such a way that their
values for some or all arguments might be determined
independently of the facts. The first type will be used
to explain the more general fact that there are propos-
itions which are analytic, that is true in virtue of the
meanings of words, or true by definition.
A slight modification, taking account of relations
between rogators, due to relations between their general
techniques and their argument-sets, serves to account for
logical relations between propositions, such as entailment,
    184

incompatibility or logical equivalence.

5.E.3.  We see from all this that it is possible to give
an account of logically true propositions which arises
naturally out of a description of the general connection
between meaning and truth, covering contingent propositions
too. There is no need at all to explain away logical
truths as not being truths at all, or logically true pro-
positions as not being propositions at all, but rules or
conventions or expressions of acceptance of conventions,
etc. They are propositions, and their truth-values
may be discovered empirically by applying the general
logical techniques for discovering the truth-values of
contingent propositions expressed by sentences including
the same sorts of words and constructions. Their pec-
uliarity is only that their truth-values may also be
discovered in the other way which I have described.
(Cf. 6.E.1,ff.)

5.E.4.  It is probably obvious that what I have said is
closely related to Wittgenstein's explanation of logical
truth in "Tractatus Logico Philosophicus". (He too was
not content to classify and describe logical properties
of propositions and relations between them, but tried to
explain them.) His account, however, seems to me to
have involved some unnecessary obscurity, and was cer-
tainly not sufficiently general. I have tried to give
a more general account of logical form (of propositions
containing logical words, descriptive words correlated
with observable properties, and words referring to par-
ticular material objects). The logical form of a
proposition corresponds to the way in which the truth-
conditions of the proposition are related to the entities,
    185

such as material objects or properties, mentioned or
referred to in the proposition. Knowing the logical
form involves knowing in general how to tell whether
propositions with that logical form are true or false,
no matter what entities are referred to, and no matter
how things are in the world. (But see qualifications
below.)
This is what people are talking about when they
refer to the "real" logical form of a proposition, con-
trasting it with the "apparent" logical form suggested
by the verbal form of a sentence. Of course, every
intelligible sentence must fully determine the real
logical form (otherwise it could not be understood
correctly: we should not be able to recognize its
truth-conditions). It is only when instead of applying
the logical techniques we reflect on the logical form
that the form of the sentence can suggest anything mis-
leading to us. This, however, is a failing on our
part, due to our not thinking clearly about what, in a
way, we know quite well (see appendix on "Implicit
Knowledge"), and does not mean that there is anything
inaccurate or imprecise about the sentence. (E.g.
though we know quite well the difference between the
copula and "is" of identity we may get muddled when
talking about it.)

5.E.5.  It should be emphasized that I am; not talking
about a "perfect" language, or any one particular language.
I have been trying to bring out general facts about any
language which can be used for making true or false
statements about material things and their properties,
about the way things are in the publicly observable world.
Neither do I restrict my remarks to sentences in some
    186

special notation or "canonical form": what I say is
intended to apply to all sorts of statements using all
sorts of logical constructions, provided that it is
possible to think of the statements as built up out
of parts which have a general use in statements. (A
language in which there was just one sound, which had
to be learnt separately, corresponding to each statement,
would be very different from ours. There would be no
way of talking about possibilities, or of teaching the
meanings of false statements - and my remarks would
probably not apply in that case.)

5.E.6.  Despite its generality, my account of logical
form has involved a number of over-simplifications,
which must now be eliminated. The first oversimpli-
fication is concerned with presuppositions. I have
continually stressed the fact that to every rogator dis-
cussed so far there corresponds a generally applicable
technique for discovering its value for all permissible
argument-sets, which works in all possible circumstances
(Cf. 5.B.8 and 5.B.12). This must now be qualified,
for there may be some techniques which are applicable
only when certain conditions are satisfied. For example,
we described an arithmetical game (in 5.B.7) which
involved techniques for wording out the values of
rogators taking names of boxes as arguments. Those
techniques involved loosing into boxes and working out
the values of arithmetical functions taking the numerals
in the boxes as arguments. If, however, a box were
found to contain no numeral (e.g. there might be an
apple in it instead) or two different numerals, then the
technique could not be applied, and there was nothing in
the rules of the game to say how to deal with this case.
The rules (as I described them) did not say whether a
    187

move using the name of a box with an apple in it should
be awarded a tick or a cross or what. They merely took
it for granted that the question would not arise.

5.E.6.a.    Similarly, the applicability of logical
rogators presupposes the satisfaction of certain con-
ditions. Thus, the logical technique corresponding to
the logical form "All P Q's are R" as described in 5.B.11
presupposes that there are no objects which are borderline
cases for the descriptive words taken as arguments.
(Recall the various sorts of indefiniteness of meaning of
descriptive words described in chapter four.) If some
boxes turn up which are neither definitely scarlet, nor
definitely not scarlet, then that technique provides us
with no way of assigning a truth-value to the proposition
expressed by "All scarlet boxes are red". It should be
noticed that the technique does not even provide us with
a value for "All scarlet boxes are scarlet" in this case.
Of course, an examination of the logical technique and
the structure of the argument-set would, as described above,
lead us to say that the truth-value must come out to be
"true". But this presupposes that the technique yields
a value at all, that its applicability-conditions are
satisfied. (Compare the case where a player makes the
move "A.A + 3", and the box corresponding to "A" has
only an apple in it: he gets no tick although x2 + 3 is
always positive.)
So, had we been more precise and explicit, we should
continually have had to make qualifications of the form:
"? provided that the applicability-conditions of the
technique are satisfied". These were omitted in the
interests of clarity and simplicity (see 5.B.12.).
    188

5.E.6.b.    One kind of presupposition which has drawn
some attention is that if a non-logical word or expression
occupies an argument-place intended for an expression
which refers to some one entity of a certain kind, then
there is exactly one entity referred to and it is of
the correct sort. For example, the use of the logical
form "x is P" presupposes that the expression taking the
place of "x" refers to a particular object, and if there
is not exactly one to which it refers, then the tech-
nique (described in 5.A.3) for determining a truth-value
lacks application. Hence that technique does not pro-
vide us with a truth-value. (Section 2.D was concerned
to show that the same applies to descriptive expressions
substituted for "P".)

5.E.6.c.    To sum up: just as functions and rogators
may have restricted domains of definition, that is the
classes of argument-sets for which they yield values may
have certain limitations, so may there be restrictions on
the class of states of affairs in which the techniques
for determining their values can be applied. The domain
of applicability-conditions may be restricted. In many
cases, whether the technique yields a value or not, i.e.
whether one of its applicability-conditions obtains or
not, will depend on how things happen to be in the world
(e.g. on whether there happen to be any borderline cases
of instances of the colour scarlet, or whether there
happens to be no king of France). But there are probably
some cases where, from the way in which a rogator is
constructed, and from facts about the things taken as
arguments, one can discover without trying to apply the
technique that it cannot yield a value for those arguments.
That is to say, it may be impossible for the applicability-conditions
    189

of some complex rogator to be satisfied when
certain things are taken as arguments. A detailed
investigation of such things as limitations on domains
of definition and restrictions on applicability-con-
ditions of the techniques corresponding to logical rogators
would, I think, shed a great deal of light on the subject
of so-called "category mistakes", and, in particular,
show how they differ from straightforward contradictions.

5.E.7.  We see therefore that there are various ways
in which even a logically well-formed sentence may
fail to express a true proposition. The logical tech-
nique corresponding to it may yield the value "false", or
it may yield no value, for any of several different
reasons. This makes it look as if in some cases it is
correct to say that the proposition expressed by the
sentence is neither true nor false, or that no proposition
is expressed at all. But things are not quite as simple
as this, for, just as the semantic correlations between
descriptive words and properties may be indeterminate,
giving rise to difficult borderline cases, so may the
rules governing the use of logical forms and the prin-
ciples for deciding on truth-values be indeterminate,
giving rise to difficult borderline cases.

5.E.7.a.    For example, superimposed on the principle
for determining the value of a logical rogator for given
argument-sets, may be a general principle of the form:
"When the technique does not (definitely) yield the value
'true', then the value is 'false'." If this principle is
added to the original one (for example to the rules for
"is", "or" and "all" described in 5.A.3 and 5.A.11), then,
when the applicability-conditions of the original technique
    190

are not satisfied, this new general rule ensures that
the truth-value is false. But in that case trouble
arises from the fact that normally when the truth-value
of a proposition is "false", there is a proposition
derived from it by the insertion of the word "not" into
the sentence expressing it, namely, its negation, or one
of its contraries, which has the truth-value "true".
And usually the applicability-conditions for the tech-
nique corresponding to the new proposition are the same
as for the old one. Hence there is a conflict in cases
where applicability-conditions are not satisfied, between
what these rules lead to and what we should normally expect,
and there may be no rules which are definitely part of the
language to specify what is to be said in such cases.
This can be summed up by saying that there may be
more than sufficient rules for the use of logical forms,
which work in most cases but may come into conflict in
others, and then there may be no definite answer to the
question: "Is this proposition true or false?".
Various rules for the use of logical constants are
superimposed in an indeterminate way (Cf. 7.D.11, note.).
Failing to see that this is a case of indeterminateness
of linguistic rules, philosophers my argue in vain that
one or other answer or some third one is correct.
(See 6.D.4.) (Such controversies are not, of course,
completely useless, since they help us to see various ways
in which the principles governing the use of logical
operators can be made more definite. See Appendix IV.)

5.E.7.b.    Other kinds of indeterminateness in the principles
governing the use of logical forms arise out of the fact
that as a language develops, different ways may be found
for saying the same thing, and this may involve extending
    191

or changing the functions of logical and other words.
For example, instead of saying "All P things are Q",
we may learn to say "The class of things which are P is
included in the class of things which are Q", or "The
property P-ness is always accompanied by the property
Q-ness" or even "P-ness is possessed only by things which
have Q-ness". In this sort of way abstract substantives
referring to universals are allowed to enter into sen-
tences as if they had the same grammatical roles as
words referring to particulars. We learn to say things
like "Red is a colour", "Idleness is annoying", and these
sentences strongly resemble "Fido is a dog" and "My
table is brown", whose logical form is represented by
"x is P".
It looks therefore as if the domain of definition
of the logical rogator corresponding to this form has
been extended so as to include new argument-sets. We
cannot, however, simply say "let there be an extension",
for the technique for determining truth-values must be
extended too. This extension enables us to apply the
form "x is P" to argument-sets like ("my table", "brown")
and ("brown", "a colour") or ("brown", "attractive").
This may lead us to think that we have extended the
technique to cope with argument-sets like ("brown", "brown")
even when we have not done so. Again, there may be con-
flicts between rules, with nothing definite in the
language to settle them. We can, of course, extend the
technique to cope with this kind of argument-set if we
wish to do so, for we can give any form of words a use
if we wish, but we are tempted to think that we have
extended the technique and the domain of definition before
we have done so in fact, or that we are compelled to
extend it in a certain way, and this may lead us into
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difficulties such as Russell's paradox, and others.
(See what happens when people forget that division by
aero is not defined in arithmetic. We could extend
the notion of division to include division by zero,
bat if we do not wish our rules to lead into conflict
some other changes may have to be made. (Rules lead
to conflict when, for example, they enable a rogator to
have more than one value - e.g. both "true" and "false" -
for the same argument-set.))

5.E.7.c.    A similar sort of indeterminateness arises
when we use logical constants in talking about quite
new kinds of things, such as infinite sets, thinking
that their use is fully determined in these contexts
by the rules for their use in other contexts. People
may even disagree about the way in which the use is
determined in the new contexts, failing to notice that
this is a case of indeterminateness, where some new
convention must be adopted if the matter is to be
settled. (Cf. Section 4.0 and 7.D.10,ff.) So
philosophers of mathematics may disagree as to how the
logical constants are to be used in connection with
statements about infinite sets, without realizing that
there are alternative ways of using them, either of
which may be freely chosen. (Or it may not definitely
by the case or not the case that either can be freely
chosen: the rules of the language in question may not
definitely leave the matter quite undetermined.)

5.E.8.  These rather brief remarks give a rough indication
of some of the qualifications which must be made to all
my assertions about the general applicability of the
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techniques for determining truth-values, and about
the possibility of discovering values of rogators by
examining general techniques without applying them.
They also enable us to explain away many apparent
counter examples to the so-called "Laws of Logic", as
being cases where the applicability-conditions for
logical rogators are not satisfied, or where arguments
are taken from outside the domains of definition of
rogators. (Question: does this approach have any
advantages over the "ranges of significance" approach,
for a theory of types? Cf. 5.E.6.c.)
This concludes my account of the workings of
logical constants, and also my account of the general
connection between meaning and truth. We may now
proceed to discuss meaning and necessary truth, and in
particular to distinguish various ways in which relations
between arguments may determine the values of rogators.

















Part Three

MEANING AND NECESSARY TRUTH







_________________________________________________________
NOTE: This is part of A.Sloman's 1962 Oxford DPhil Thesis
     "Knowing and Understanding"
_________________________________________________________

    194

Chapter Six

ANALYTIC PROPOSITIONS

6.A.    Introduction

6.A.1.  The main stream in Part Three will be a con-
tinuation of the attempt to describe the various factors
which can determine or help to determine the truth-
value of a proposition. This will provide illustrations
for my explanation of the meanings of "analytic",
"necessary", "possible", and related words, which will
proceed at the same time. It will be shown that there
are several different ways in which a proposition may
be necessarily true, corresponding to a number of dif-
ferent ways in which its truth-value may be discovered.
In particular, it will be argued that, in the sense of
"analytic" which is to be defined in this chapter, not
all necessary truths are analytic. This is because
there are some properties which are necessarily connected,
although they can be completely identified independently
of each other. Hence their necessary connection is not
an identifying relation or a logical consequence of an
identifying relation. (What this means will be explained
presently.)

6.A.2. The first problem must be to get clear about the
meaning of "analytic". There are at least two ways in
which people can be unclear about the analytic-synthetic
distinction. The first is to be unclear as to what the
things are which it distinguishes. Thus, philosophers
are often confused about the sorts of things "propositions"
    195

or "statements" are, the entities to which they apply
the distinction. Even when they try to say explicitly
what it is that they are talking about, their usage often
conflicts with their explanations.
Sometimes it looks as if they are talking about
sentences, but a sentence considered simply as a
sequence of signs, or marks on paper, or sounds, cannot,
as such, either be analytic or fail to be analytic, any
more than it can be true or false. (Cf. 2.A.2, 2.D.3
(note), 5.A.8-9.) Before a sentence can be described
as true or false, or analytic or synthetic, it must be
thought of as a sequence of signs with a meaning or
linguistic function, and the meaning must be fairly
definitely specified. If a sentence is taken simply
to have its meaning in English, for example, then there
may be no answer to the question whether it is analytic
or not, owing to the ambiguities of the English language.
(An obvious example is provided by the sentence "All
mothers bore their children". More subtle examples were
discussed in section 2.C.) neither can the distinction
usefully be applied to formulae of a formal system,
since, as argued in the previous chapter (section 5.A)
and appendix II, a language is quite a different sort
of thing from a formal system. All this may seem
obvious, but, as will appear in a moment, it appears
not to have been noticed by some philosophers who try
to define "analytic". (Such as Carnap.)
We must apply the distinction to sentences only if
they are taken to have meanings, and we must know which
meanings they are taken to have. This involves knowing
what counts as "the same meaning". I have tried to show
how the properties referred to by descriptive words, and
the logical techniques corresponding to logical constants
    196

can be used to provide criteria for identity of meanings.
(Section 2.C, and chapters three and five.)

6.A.3.  The second kind of unclarity about the analytic-
synthetic distinction involves the way in which it is
applied. The word "analytic" is fairly common in
philosophical writings, and most philosophers have a
rough idea of what it means, but their usage is never-
theless bedevilled with confusions, obscurities, ambi-
guities and errors. As pointed out by Mario Bunge, in
Mind, April 1961 (p.239), the word is used with an
unrecognized, or at least unacknowledged multiplicity
of meanings.
For many philosophers (see for example p.21 of
Strawson's "Introduction to Logical Theory") the word is
apparently synonymous with "necessarily true", which is,
of course, a question-begging usage when the possibility
of synthetic necessary truths is being discussed.
Others offer "pragmatic" definitions, so that analyticity
admits of degrees. Quine's definition of "analytic"
as meaning "definitionally derivable from a logical
truth", was accepted with modification by Waismann in
his famous series of articles on the subject (in Analysis,
Dec. 1949, etc.: this will, be dealt with in detail below).
Some (following Frege?) turn the Quine-Waismann definition
around, and instead of talking about derivability from
logical truth by substitution from definitions, they talk
about logical derivability from definitions or "meaning-
postulates". (It is not usually noticed that these two
definitions are not equivalent.)
Sometimes followers of Carnap define the word
"analytic" in terms of "state-descriptions" and the rules
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of a so-called "language", L, which is admittedly precise,
but also quite useless, since it applies only to formulae
in a formal system, and not to statements in a language.
Such people are inclined to regard the analytic-
synthetic distinction as system-relative, so that
whether a proposition is analytic or not depends on the
"System" in which it occurs. (See, for example, Mario
Bunge, op.cit, p.239,ff.) I have never been quite sure
what these "systems" are supposed to be. My guess is
that philosophers who talk in this way are making the
mistake (see section 5. A and Appendix II) of confusing
formal systems and languages. Neither can I understand
why they regard propositions occurring in different
"systems" as the same proposition. Why not say that
they are different propositions, for then there will be
no need to regard the distinction as system-relative?
There appears to be some confusion as to whether they are
talking merely about sentences, which, admittedly, may
have different meanings in different languages, or about
propositions. (See 6.A.2.) Other philosophers talk
as if a proposition either is or is not analytic, no
relation to a system being regarded as relevant.
Often the word "analytic" is defined rather vaguely,
as "true in virtue of meanings", or "proposition which
cannot be denied without contradiction", or "proposition
which cannot intelligibly be denied". Sometimes it is
suggested or implied that we can decide to make a propo-
sition analytic, whereas others will allow only that we can
decide to let a sentence express a proposition which
already happens to be analytic, independently of our
choice. (Cf. 6.F.1, below.)
I shall not describe in detail, nor criticize most
of these accounts of the distinction. For detailed
    198

exposition and criticism the reader is referred to
"Semantics and Necessary Truth", by Arthur Pap. A few
critical remarks will be made by the way in some of the
discussion.

6.A.4.  Behind all this chaos and confusion there seems
to lie a fairly simple concept, familiar even in non-
philosophical contexts, for people often speak of some-
thing or other as being "true by definition", or use the
expression "by definition" to preface a remark in order
to indicate the sort of justification which they would
be prepared to offer for accepting it as true. All
my attempts to define the word "analytic" aim at trying
to clarify and make precise something like this ordinary
notion of a proposition which is true by definition.
The word does not correspond only to a technical distinction
invented for philosophical purposes.

6.A.5.  A small point to be cleared up is that I shall
use all three expressions "analytic", "analytically
true" and "analytically false" to describe statements or
propositions. The latter two expressions are unambi-
guous whereas the former is ambiguous. It may mean
either "analytically true", or "analytically true or
analytically false". This ambiguity is customary, and
should cause no confusion, as exactly what is meant will
be clear always from the context.

6.A.6.  It should be noted that the distinction between
analytic and synthetic propositions is not the same as
the distinction between propositions which are verbal
(or merely conventional!), and those which are non-verbal
    199

(or non-conventional, or independent of conventions of
any particular language). The verbal/non-verbal dis-
tinction works at a different level from the analytic/
synthetic distinction, as is shown by the fact that
philosophers seem to be trying to say something signi-
ficant when they say that all analytic propositions are
merely verbal, but the distinction is obscure. It is
very difficult to see what could be meant by saying
this sort of thing. I think that only Wittgenstein has
come close to being clear about it, but there will not
be space to discuss his view (in R.F.M.), as it can be
made intelligible only in the context of an account of
his general theory of meaning. Even if I manage to
demonstrate that some necessary truths are synthetic,
this will not settle the question whether all necessary
truths are verbal, or conventional. (See R.F.M. III.42).

6.A.7.  I shall now turn to a more detailed discussion
of some unsatisfactory accounts of the analytic-
synthetic distinction, in order to lead up to my own account.

6.B.    Some unsatisfactory accounts of the distinction

6.B.1.  In this section I shall describe a number of
attempts to explain what the analytic-synthetic distinction
is, picking on some of their weak points in order to
contrast them with my own definition later on. Its
purpose is purely introductory, and it should not be
taken too seriously, as it may be somewhat unfair to some
of the philosophers mentioned.
    200

6.B.2.  Kant's explanations of the distinction are not
very clear, though it is fairly easy to understand, at
least in a vague way, the sort of thing he is getting at.
For example, in A.6, B.11 ("Critique of Pure Reason")
he says:
"Either the predicate B belongs to the subject A,
as something which is (covertly) contained in this
concept A; or B lies outside the concept A,
although it does indeed stand in connection with
it. In the one case I entitle the judgement
analytic, in the other synthetic."
This is not very helpful, and seems to be too narrow a
definition for his purposes, especially as it applies
only to proposition in subject-predicate form (or
apparent subject-predicate form, such as "All A's are
B's"). (This, incidentally, illustrates the sort of
lack of clarity which can follow on too much concentra-
tion on "canonical forms" of propositions. Cf.
Appendix II. 11,ff.)
Kant's explanation is not made much clearer when we
are told, in A.716, B.744, that analytic knowledge is
obtained merely by meditating on concepts, or that
synthetic knowledge involves going beyond concepts in
an appeal to intuition (see A.721, B.749). (The notion
of an "appeal to intuition" will be clarified below.
Cf. 6.C.11, and sections 7.C and 7.D) I think that
what Kant was getting at in these passages will be
illustrated by my discussion of "identifying relations"
between meanings, below.
Most modern attempts to explain the distinction are
probably related to Kant's assertion that a judgement is
analytic if "its truth can always be adequately known in
accordance with the principle of contradiction" (A.151,
B.191).
    201

6.B.3.  Finding Kant's attempt to characterize the
distinction unsatisfactory, some philosophers have
tried to define the class of analytic propositions to
be those whose truth follows from the meanings of the
words occurring in them. This, however, is also
difficult to understand, as Waismann pointed out in
Analysis, Dec., 1949. He asks (p.27): "What can be
meant by saying that a statement follows from the very
meaning of its terms?"
The attempt to elucidate this by saying that what
is meant is a statement which follows from the definitions
of its terms, provokes another of Waismann's questions:
"If an analytic statement is characterized as one that
follows from mere definitions, why is it not itself
a definition? ? Why is it that what follows from
a definition is not, as one would expect, a definition,
but an analytic judgement?" (p.29.)
Quine also found it incomprehensible that definitions
should be available for founding truths. (See "Truth by
Convention" in Feigl & Sellars, p.259). However, he
allowed that they might be used to transform truths, and
this is echoed by Waismann: "Definitions are substitution
licences of a particular sort, ... and every substitution
licence can be re-written as an equivalence." (op.cit.
p.39).

6.B.4.  Having noticed that definitions could be thought
of, as rules or licences permitting substitutions of
synonymous expressions without change of truth-value,
Quine, and later Waismann, decided to define "analytic
proposition" to mean "logical truth definitionally
abbreviated" ("Truth by Convention", p.251), thereby
removing the difficulty of explaining how a definition
    202

could make true a proposition which was not itself a
definition. This presupposes the notion of a "logical
truth", which seems to be the notion of a proposition
which is true in virtue of its logical form, or, in
my terminology (see 5.A.9 and section 5.6), a "formal
truth". Waismann's version of the definition of
"analytic" was as follows: "A statement is analytic if
it can, by means of mere definitions, be transformed
into a truth of logic." (op.cit. p.31.)
For example, on this view, "All bachelors are
unmarried men" is analytic, since, by definitional sub-
stitutions, it can be transformed into "All unmarried
men are unmarried men", which is a formal truth. Any
proposition which cannot in this way be transformed
into a formal truth would, according to this definition,
be synthetic, not analytic. Pap, on p.5 of "Semantics
and Necessary Truth", seems to indicate his acceptance
of this definition, when he writes: "One may be inclined
to characterize as synthetic, necessary statements whose
descriptive terms occur essentially yet cannot be
eliminated through analysis." (I.e. analysis of
meanings, on the basis of which one can replace defined
symbols with their definientia.)

6. B.5. Now, there does seem to be a close connection
between propositions which are analytic and propositions
which are true in virtue of their logical form, as will
appear later on when it is shown that formal truths are
just a particular kind of analytic truth. But the con-
nection cannot be that the class of analytic propositions
is defined as suggested above in terms of derivability
from formal truths, for this definition does not seem to
be wide enough. This is because there are propositions
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which are true in virtue of partial definitions, of the
sorts discussed above (in section 4.C), and these pro-
positions cannot be derived from formal truths by means
of synonymy substitutions.
The examples discussed there were all concerned
with incompatibility relations between colour words
(such as "red" and "orange"), but similar remarks might
be made about relations between certain sound-concepts.
Consider, for example, the expressions which refer to
the kind of feature of a sound which we call its "timbre",
such as " flute-timbre" or "the sound of a bassoon".
It seems that the meanings of these expressions might be
taught ostensively, in such a way as to leave them
indeterminate in some respects so that the questions:
"Can a sound have two timbres at the same time?" and
the question: "Can a sound be the sound of a flute and
the sound of a bassoon at the same time?" would not have
definite answers, (For sources of indeterminateness
see chapter four, section A.)
Consider the sound produced by the loudspeaker when
a (monophonic) recording of a duet for flute and bassoon
is being played on a gramophone: here it is possible
clearly to "hear" both instruments in the one sound
coming out of the loudspeaker. But is it really one
sound, or is it two sounds? If it is one sound, does
it have two different timbres at the same time? That
is, does it have the flute-timbre and the bassoon-timbre
or does it have no timbre at all? Or does it have a
third timbre different from that of the sound produced
by either a solo flute or a solo bassoon? I believe
that as far as the English language is concerned, there
is no definite answer one way or the other to these questions,
At any rate if this sort of case is not produced during
    204
the process of teaching someone to use the word "timbre",
then there may be nothing in what the pupil understands
by the word to settle these questions: the meaning which
he associates with the word does not determine "in
advance" what he should say about this sort of pheno-
menon. See (2.D.2, 3.C.4, etc.)
In that case, each of the questions could be
settled one way or another by the adoption of a linguistic
convention for the use of the words describing sounds,
which might have the consequence that certain statements
were "true by definition" though not definitionally
derivable from formal truths.

6.B.5.a.    We might, for example, adopt the convention
that the sound in question was to be described as "one
sound with two timbres". Or we might adopt a rule
to the effect that "flute-timbre" and "bassoon-timbre"
were to be incompatible descriptions, which would rule
out the possibility of describing the sound of the
duet as a sound with both timbres. (It would not,
however, tell us whether the sound was to be described
as having either of the two timbres alone, or as having
no timbre at all, etc. The rule leaves each of the
individual concepts as indefinite as it was without the
rule: see 4.0.3-4.)
Such an incompatibility convention is an arbitrarily
chosen linguistic rule which helps to remove certain kinds
of indeterminateness of meaning (4.C.4). It serves as
a partial definition of the word "timbre", say. It does
not define any expression as being synonymous with any
other, it sets up no synonymy relations, but it does have
the consequence that sentences like "No sound has a
    205

flute-timbre and a bassoon-timbre at the same time"
express true propositions, owing to the incompatibility
between the two descriptive expressions.
We have therefore found a statement which is true
in virtue of the fact that certain words have certain
meaning or are governed by certain linguistic rules,
but which is not derivable from a formal truth by
substitution of synonyms. (Notice, incidentally, that
this partial definition does not rule out any kind of
experience as impossible: it does not make the exper-
ience of hearing the duet impossible, but merely rules
out the possibility of describing it in a certain way.
This should be borne in mind when the incompatibility
of colours is under discussion. Compare 3.B.4.d.)

6.B.5.b.    Another sort of proposition which is, in an
obvious sense, true by definition though not derivable
from formal truths by substitution of synonyms is pro-
vided by an ostensively-defined relational expression
with an added verbal rule. The ostensive teaching of
an expression like "to the left of" might show that
expressions of the form: "X is to the left of Y" are
applicable to a whole range of cases, including pairs
of objects at various distances apart, without specifying
the use in connection with just one object. The
indefiniteness might then be removed by the adoption of
an arbitrary convention, giving one or other answer to
questions like "Can an object be to the left of itself?"
or "In the expression 'X is to the left of Y', can 'X'
and 'Y' refer to the same thing?" For example, it might
be decided that the expression was to be irreflexive, in
which case "X is to the left of X" would express a false
    206

proposition no matter what referring expression took
the place of "X", and the statement "Nothing is to the
left of itself" would be true by definition.
Once more, we have an example of an analytic pro-
position which is not definitionally derivable from a
formal truth, since the linguistic convention in virtue
of which it is true does not generate any synonymy-
relations: it is, as before, a partial definition.

6.B.6.  In these examples of propositions which are
true by definition without being definitionally abbre-
viated logical truths, descriptive terms, such as
("flute-timbre", "to the left of", etc.) occur essentially
yet cannot be eliminated through analysis (see end of
6.B.4). Since they are obvious candidates for the title
of "analytic" propositions, the Quine-Waismann definition
of "analytic" in terms of derivability from formal truth
cannot be wide enough.
These examples show that Quine was wrong when he
wrote (in "truth by Convention", p. 258) that " ...
definitions are available only for transforming truths,
not for founding them." We seem to have discovered
propositions whose truth is founded in linguistic con-
ventions. At any rate, they are not merely derived from
some other truths by some kind of transformation.

6.B.7.  However, even if we were wrong about these examples,
the question would arise: What is it for a statement to be
true in virtue of its logical form? Surely only that the
statement is true in virtue of the meanings or functions
of the logical words and constructions employed in it.
But to say that formally true statements are true in virtue
of the meanings of logical constants surely cannot mean
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that they are true in virtue of being derivable from
formal truths by substitution of synonyms: this would
be circular, or lead to a vicious infinite regress.
So there must be some other sense in which a proposition's
truth may follow from the fact that its words are
governed by certain rules, than the one suggested by
Quine and Waismann. If there is this other way in
which a proposition may be true in virtue of meanings or
linguistic conventions, why should it be restricted to
formal truths, why should it not also explain the sense
in which other analytic propositions are true by definition?
We must try, therefore, to find a wider definition of
"analytic" than the Quine-Waismann definition, which
avoids this last objection, but before doing so let us
see what Frege had to say.

6.B.8.  Frege, as we shall see, did not limit the role
of definitions to that of "substitution licences".
According to him, the question whether a judgement is
analytic or not, is a question not about the content
of the judgement, but about "the justification for making
the judgement" (Grundlagen, p.3). The judgement
that some proposition is analytic is not concerned with
its being true, or with what it means, but is "a judge-
ment about the ultimate ground upon which rests the
justification for holding it to be true."
Now notice how Frege goes on. In order to discover
whether a proposition is analytic or not, we have to find
the proof of the proposition and then follow it right
back to the primitive truths on which it is based. "If,
in carrying out this process, we come only on general
logical laws and on definitions, then the truth is an
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analytic one, bearing in mind that we must take account
also of all propositions upon which the admissibility
of any of the definitions depend." ("Grundl." p.4.)
(We need not worry about the fact that different
persons may justify a proposition in different ways:
what Frege clearly means is that a proposition is
analytic if there is some justification resting ulti-
mately only on general logical laws and definitions.
As will be shown later on, an analytic proposition may
be justified empirically too. So Frege ought not really
to talk about "the ultimate justification" or "the
ultimate ground", as if there could be only one.)

6.B.8.a.    This may seem clear at first, but it becomes
mysterious as soon as we try to find out what Frege means
by a "definition" or how he thinks a proof can rest on
definitions.
From what is said in "Grundlagen", and in his essays
"On the Foundations of Geometry" (Phil. Rev. I960), it is
clear that he thinks (or did at least once think) of a
definition as some kind of proposition, which first of
all "lays down the meaning of a symbol" and then "trans-
forms itself into a judgement ... (which) ... no longer
introduces the object, (but) is exactly on a level with
other assertions made about it." ("Grundl." p.78.)
Admittedly, he says (Phil.Rev. p.4.): "Although
definitions which have been made into (sic) statements
formally play the part of basic propositions, they are
not really such", but it is apparent from the context that
he does not mean to deny that they are propositions on a
level with other propositions in the proof, but only that
they are basic propositions, i.e. statements of general
logical laws or axioms, and that they "extend our knowledge"
    209

(Phil.Rev. p.5.)
He also regards a definition as a means of "deter-
mining a reference of a word or symbol" (Phil.Rev. p.4)
or as something which "lays down the meaning of a
symbol" (Grundl, p.78.). How can a definition both
lay down meanings and serve as a proposition "on a
level with other propositions"?

6.B.8.b.    In order to understand what lies behind all
this, we must remember that Frege required a rigorous
proof to satisfy certain conditions. "All propositions
used without proof should be expressly mentioned as such,
so that we can see distinctly what the whole construction
rests upon" and "all the methods of inference used must
be specified in advance. Otherwise it is impossible to
ensure satisfying the first demand." (See "Translations",
p.137.) But the principles of inference specified "in
advance" by Frege permit inferences to be drawn only in
cases where the premises and conclusion stand in a certain
kind of formal or logical relationship (having nothing to
do with their content), which requires that the premises
should be propositions "exactly on a level with" the
conclusions.
So if a definition is to serve as a premise in such
an inference, then it must have the same general form as
the other kinds of propositions which may serve as pre-
misses. Now we can see why Frege requires his definitions
to lead a double life: first they must somehow or other
specify that certain words or symbol have certain meanings,
and secondly, in order to serve as premisses for inferences,
they must be propositions "exactly on a level with other
assertions" ("Grundl", p.78). Thus we find Frege talking
about definitions as propositions which first do one thing,
    210

and then "transform themselves" into something else.
(Mario Bunge, in Mind, 1961, p.140-141, manifests the
same confusion in talking about "linguistic conventions
taking the form of propositions, not of proposals".)

6.B.8.c.    But how can definitions lead this kind of
double life? How can anything which works like an
ordinary proposition do what a definition is supposed
to do, namely lay down the meaning of a symbol? How
can a definition first assign a meaning to a symbol and
thereupon "transform itself" into a proposition?
Frege seems not to have realized that in order to
define a word one must mention it, or somehow indicate
that one is making a statement about words and their
meanings, and that such a statement is not "on a level
with" other statements which use those words.
One might try to defend him by saying that a pro-
position using a proper name can say what its reference
is, as in "The number five is the number of fingers on
a normal human hand". This certainly tells us which
number is the one referred to by "the number five", but
it does not tell us the sense of the expression, since
it leaves open the question whether it is a matter of
definition or a matter of fact that the number of fingers
on the normal human hand is what is referred to by "the
number five". Similarly, a proposition using a descriptive
word can tell us something about the extension
of that word, but it does not tell us the meaning of the
word. Thus "A square is a rectangle in which the
adjacent sides are equal" tells us something about the
extension of "square" if we happen to know the meanings
of the other words, it tells us which objects happen to be
described by the word, but it does not say why it describes
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them. (Cf. 2.C.8.) The statement does not say what
the meaning of "square" is, it merely describes a pro-
perty which can be found in squares, leaving open the
possibility that this is an accident. Frege, however,
regards this statement as a definition (see p.145 of
"Translations").

6.B.8.d.    Of course, if I say "That vase on the top
shelf is turquoise in colour", the person I am talking
to may guess that I mean to tell him what the word
"turquoise" means, and he may guess the meaning correctly.
But I have not told him the meaning, for what I say
leaves open the possibility that "turquoise in colour"
means the same as "on the top shelf"! "Tomatoes are
red in colour" could be used to teach the meaning of
"red": does that make it a definition? An explicit
definition must not merely be something which enables
a meaning to be guessed: it must say what the meaning
is. It is not enough to state a fact which happens
(though this is not asserted) to be true by definition.
So in order to state a definition of "gleen" I must say
something like "The word 'gleen' means the same as
'glossy and green'" or "By definition of', gleen', a thing
is gleen if and only if it is glossy and green", and not
merely "All gleen things are glossy and green and all
glossy and green things are gleen".
Thus Frege's device (in "Grundgesetze") of adding a
vertical stroke to a formula to indicate that it is a
definition is not enough, for it leaves unsettled exactly
which symbol is being defined and exactly what is being
said about it: its only effect is to assert that a
formula expresses a statement which is true in virtue of
some definition, but it does not say which word is defined
    212

by it nor what the definition is.

6.B.9.  I have been trying to force Frege into a dilemma
of the following sort. When he talks of definitions as
propositions which may occur in a completely explicit
proof without themselves being proved, then either he
means to refer to definitions proper, that is statements
which are explicitly about words and their meanings, or
he wishes to refer to statements (in the "material mode")
which do not mention words or meanings but are never-
theless true in virtue of the fact that certain words
have certain meanings.
In the former case, some kind of explanation is
required of how the definitions can be used as premises
for logical inferences to propositions which use the words
defined.
In the latter case, Frege is already making use of
the notion of an analytic proposition when he talks about
"definitions", and so he has not begun to explain how
to tell that a proposition is analytic in the first place;
he has, at most, shown us how we can tell that a pro-
position is analytic if we already know that other
propositions are analytic, namely by seeing if it can be
derived from them using purely logical (formal) truths
and formally valid inferences.
In either case he has left unanswered Waismann's
Questions: "Why is it that what follows from a definition
is not, as one would expect, a definition, but an analytic
judgement?" (See 6.B.3.) He has not shown us how to
get from explicit definitions to statements which are true
in virtue of those definitions.

6.B.10. To sum up there is a gap to be bridged between
    213

statements which are about meanings of words and state-
ments which are true in virtue of meanings. Frege
eliminated the gap by failing to distinguish two different
kinds of statement properly, and talking instead about
one thing which could do two kinds of jobs. Waismann
and Quine, on the other hand, tried to eliminate the
gap by giving up the idea of inferring true propositions
from definitions. Instead they regarded definitions as
substitution licences, which permit analytic statements
to be derived from formal truths. But we have seen that
this is not sufficiently general, for it does not take
account of statements true in virtue of partial defin-
itions. (6.B.5,ff.)
It is clear that what is needed here is a new
explanation of the notion of a definition or linguistic
convention, and a new description of principles according
to which from definitions or statements about meanings
one can infer that certain sentences express true pro-
positions. In short: the notion of a "definition"
must be clarified, and the notion of a logically valid
inference must be generalized.

6.B.11. It is at this stage that we must turn back to
what was said in chapter five, especially sections C, D
and E (5-E.1 - 5.E.4). We have seen that although the
value of a rogator for an argument-set will usually
depend on how things happen to be in the world, which
may affect the outcome of applying the technique for
discovering values, nevertheless, there are some "freak"
cases where the value is determined independently of the
facts and may be discovered by examining the argument-set
and general technique, without applying the technique.
This showed us how it is possible to discover that some
    214

sentences express true propositions merely by examining
the non-logical words occurring in them (taking their
meanings into account, of course) and the logical
techniques for discovering the truth-values of the
propositions expressed by those sentences.
Thus, by examining the argument-set ("red", "rot")
and the logical technique corresponding to the logical
form "All P things are Q", we could see that the pro-
position expressed by "All red things are rot" must be
true, without actually applying the technique and
examining all red things. (See 5.D.1. The word "rot"
was defined to refer to the same property as "red".)
So we have already seen that it is possible to
infer from facts about the meanings of words and the
functions of logical constants that a sentence expresses
a true proposition. When that happens, we know that
we have the right to assert that proposition, since in
general we know that we can utter a sentence when the
logical rogator corresponding to its logical form yields
the value "true" (see 5.B.18). Thus we have an answer
to Waismann's questions quoted in 6.B.3: "What can be
meant by saying that a statement follows from the very
meaning of its terms?" and "Why is it that what follows
from a definition is not, as one would expect, a
definition, but an analytic judgement?" The answer is
simply that there are ways of drawing conclusions from
the fact that certain descriptive words and logical
constants have certain meanings or functions, such as the
conclusion that a sentence including them expresses a
true proposition.

6.B.12. This shows that Waismann's conception of a
logically valid inference was too narrow, for he failed
    215

to see that the inference from a truth about meanings of
words to a proposition using those words can, in a sense,
be logically valid even though it is not formally valid,
like the inference from "All roses attract bees" to "All
red roses attract bees", which is valid in virtue of its
logical form. It should not surprise us that an infer-
ence from a statement about words to a statement using
those words should be logically valid. After all, the
inference from (1) "The sentence 'Plato was precocious'
expresses a true proposition" to (2) "Plato was
precocious", is surely logically valid?
I describe such inferences as logically valid since
they do not depend on special facts about the subject-
matter referred to by the words and sentences mentioned.
They depend on very general facts about the logical
techniques for determining truth-values and the con-
ditions in which it is appropriate to utter a statement.
All the essential features of the examples of propositions
whose truth-values could be determined independently
of the facts (in 5.C and 5.D), were topic-neutral features.
This is why I call an inference from the fact that a
proposition possesses those features to the fact that
the proposition is a true logical inference.

6.B.12.a.   We might say that we have discovered logical
theorems which could be formulated in some such manner as
the following:
If a word "P" refers to the same property as the word
"R", "P" "Q" and "R" being descriptive words, and
if the relevant logical constants are used as
described in 5.B.11 then the sentence "All P Q's
are R" expresses a true proposition.
This statement is certainly not a formal truth, like
"If all red things are rot, then all red boxes are rot",
    216
for it is not true in virtue of its logical form in the
same way. (Content is relevant as well as structure in
deciding that the theorem is true:  metalinguistic words
and expressions occur essentially.  Its truth has to
be established by considering the things it is about,
and in particular by investigating the logical tech-
niques corresponding to the logical constants mentioned.)
Such logical theorems may be described as "non-
formal truths of logic". They state the facts which
justify the (non-formal) inferences which we make in
deciding that analytic propositions are true. It should
be noted that although such truths and inferences are
not formally valid, this in no way implies that they
are lacking in rigour, though we must remember that,
as pointed out at the end of chapter five, they may pre-
suppose that certain conditions are satisfied. There
will be a more general discussion of non-formal proof
in chapter seven. (Though these non-formal truths of
logic are often appealed to implicitly, and sometimes
stated explicitly, logicians appear not to have taken
them into account when explaining what they mean by
talking about "logical truths" or "logically true pro-
positions". Almost always they seem to think they are
talking only about propositions which are true in virtue
of their logical form, the formal truths described in
5.C.)

6.B.13. So much for the (non-formal) logical theorems
and logical inferences which are implicitly employed when
we discover that some sentence is true in virtue of what
it means. However, we are not yet quite ready to offer
a definition of "analytic", for we must first turn our
attention to facts about meanings, in order to clarify
    217

the notion of a "definition". We shall then be able
to follow Frege in defining analytic propositions to
be those whose truth-values can be discovered merely
by examining facts about the meanings of words used
to express them, and making inferences justified on
general logical grounds (of a topic-neutral kind). For
this we require the concept of an "identifying fact
about meanings".

6.C.    Identifying relations between meanings

6.C.1.  The discussion of chapter five brought out three
factors which, in general, determine the truth-value
corresponding to a sentence (at any time), namely (a)
the meanings of the non-logical words, (b) the logical
techniques corresponding to the logical form and (c) the
way things happen to be in the world. (See 5.E.1.)
In some "freak" cases, we found that the third factor
dropped out as ineffective (though one might fail to
notice this and take the facts into account in the usual
way in finding out the truth-value). In these freak
cases, the truth-value of the proposition expressed by a
sentence could be discovered by examining (i) the
logical techniques corresponding to its logical form
(ii) the "structure" of the argument-set (set of des-
criptive words) to which the logical form was applied and
(iii) relations between the arguments, or, more speci-
fically, relations between the non-logical words. Thus,
from facts about the meanings of the words we were able
to infer that combining words in certain ways produced
sentences expressing true propositions.
We must now show how it is possible to pick out a
    218

class of facts about the meanings  of words which cor-
respond to definitions in virtue of which analytic
propositions may be true. Let us describe them as
"identifying" facts about meanings. We may also talk
about identifying relations between meanings, or
analytic relations between meanings. Statements about
meanings which do not state identifying facts about
meanings state non-identifying facts, or describe
synthetic relations between meanings. A definition is
then a statement of an identifying fact about the
meanings of words. What does all this mean?

6.C.2.  It was argued in 6.B.8.c-d that a statement can
be a definition only if it mentions words. But not
every statement which mentions words is a definition,
even if from the fact that it is true we can infer that
some sentence using those words expresses a truth, For
example, it may be the case that
(1) the class of objects with the property referred
to by the word "red" and the class of objects
with the property referred to by the word "mat"
are mutually exclusive,
in which case we can infer (logically, but not formally)
that
(2) the sentence "No red things are mat" expresses
a true proposition.
From the relation between the words in the argument-set
("red", "mat") described in (1), we can infer, by consider-
ing the appropriate logical techniques for determining
truth-values, that applying the logical form "No P things
are Q" to that argument-set yields a true proposition
(as described in 5.D). But the relation in question may
hold simply because of the contingent fact that nothing
which is red happens to have a surface with a mat texture.
The statement (1) is not a definition, for one could fully
    219

understand the words it mentions without knowing that
they stand in the relation it describes. In order to
know that they stand in that relation one must not only
know the meanings of the words, but also have carried
out some empirical observation of the class of red
things or the class of objects with mat surfaces.
So (2), though inferred from a statement about words
is not inferred from a definition, or from the statement
of an identifying fact about meanings.

6.C.3.  When is a statement about words a definition?
How do we discover whether a statement states an identi-
fying fact about meanings? What must a person ask
himself when he asks whether a statement defines the
meanings which he associates with the words which it
mentions?
The answer seems to be suggested by the example
just mentioned. The fact (if it is a fact) that the
words "red" and "mat" have mutually exclusive extensions
is not an identifying fact about their meanings because
it is possible fully to specify what they mean, by
indicating the properties to which they refer, without
mentioning the fact or anything which logically implies
it. One could successfully draw a person's attention
to either property in a object without mentioning or
getting him to think about or attend to the other property
in any way. The relation of incompatibility is not an
identifying relation because to assert that the relation
holds is not essential to a full specification of the
meanings of the words. Their having the meanings which
they do have is not even partly constituted by their
being incompatible descriptions, as it would be if there
    220

were an n-rule to the effect that they could not both
describe the same object at the same time.
In short to discover whether a statement identifies
meanings one must ask "Is it possible to know exactly
what I mean by these words without knowing the fact
stated by this statement or any other logically equi-
valent to it? (This last clause is included because
there is no need for us to say that words have different
meanings when their definitions are logically equivalent:
our criteria for identity of meaning need not be quite
as sharp as that - see sections 2.A, 2.C - though for
some purposes it might be necessary to discriminate
between different forms of definitions )

6.C.4.  An identifying statement about the meanings with
which words are used states something which must be known
if one is to know what those meanings are, in the sense
of knowing how to use the words with those meanings.
One need, not, however, know that the statement is true.
For the statement may employ metalinguistic concepts
without which it cannot be understood, though one can
perfectly well use the words it mentions without having
them. I can use a word to refer to a property, and yet
not understand the expression "refer to a property". I
may know that a word, such as "gleen" refers to a com-
bination of two properties, without knowing that the words
"glossy" and "green" refer to those two properties
separately. Nevertheless, the statement that "gleen"
refers to the same property as "glossy and green" states
an identifying fact about the meaning of "gleen", since
it correctly describes the way "gleen" is used by anyone
who uses it correctly, who knows what it means. In such
cases I say that the identifying fact about the meanings
    221

is one which must be known at least implicitly by
anyone who knows the meanings of the words in question.
(For more detailed remarks on "implicit" knowledge, see
Appendix III.)
     What is required for the implicit knowledge to
become explicit may be the acquisition of new meta-
linguistic concepts, or the acquisition of a new voc-
abulary, or any of the other sorts of things described
in Appendix III (see III.5). In these cases one is
not learning the meanings of the words, but merely
learning to say what, in a way, one already knows about
the meanings of the words, since one knows how to use
them. One does not have to carry out observations of
the objects described by those words, nor examine the
properties referred to by the words in order to discover
new aspects (Cf. 7.D.) That is to say, for making
implicit knowledge of meanings explicit, neither
experience nor insight is required. If, in addition
to knowing the meanings of words, one must have some
experience or insight, in order to see that a statement
is true, then it does not state an identifying fact about
meanings. (Cf. 6.C.10, below.)

6.C.5.  If a statement states an identifying fact about
the meanings of certain words, then this means that
unless that fact is (at least implicitly) known, the full
meanings of the words will not be known. But this
leaves open the possibility that part of their meanings
may be known. For example, in section 4.C it was shown
that a partial definition might be adopted according to
which the two hue-words "red" and "orange" were to be
incompatible descriptions. A person who had been taught
    222

their meanings ostensively, without being told about
this incompatibility convention, would know something
about their meanings, for he would be able to decide
correctly in most cases whether objects were describable
by these words, but he would not fully understand the
words as used by persons who followed the incompatibility
rule. He would not know that certain descriptions of
borderline cases were excluded, such as "both red and
orange.
So in some cases one may know part of what a person
means by a word without knowing all the identifying facts
about the meaning: but then the partial meaning is
likely to be less determinate than the full meaning.

6.C.6.  Some identifying facts about meanings are
"purely verbal" and some are not. For example, suppose
the word "V" to be semantically correlated with the
property P and the word "W" with the combination of pro-
perties P and Q (cf. 3.B.3). Then either of these
correlations may be set up without the other. For
example, a person may use the word "V" to refer to P,
while another person uses the word "W" to refer to the
combination of P and Q, though neither of them has a word
synonymous with that used by the other. Nevertheless,
the statement that the word "W" refers to a combination
of the property referred to by "V" with another property
states an identifying fact about the meanings of the
words: the relation it describes is an identifying
relation between their meanings. On the other hand, if
two words, such as "red" and "orange" are related by an
incompatibility rule of the sort described in 4.C.2, then
it is impossible to say exactly what either of the words
    223

means without mentioning the other word, unlike the
previous case, where one can fully explain the meaning
of either "W" or "V" without mentioning the other.
In the one case we have a purely verbal identifying
relation, which holds merely in virtue of a rule relating
two words, whereas in the other case we have an identi-
fying relation which is not purely verbal because it
holds primarily on account of rules correlating words
and properties.1

6.C.7.  If an identifying relation holds between
entities, then at least one of them is not capable of
existing on its own. Certainly the hue redness (i.e.
the observable property) may exist on its own (since one
may be fully acquainted with it without being acquainted
with the hue orangeness). But if the words "red" and
"orange" are related by a defining incompatibility rule,

1.  Here we see one of the things which may be meant by
the distinction between "real" definitions and
"nominal" definitions. The rule relating "red" and
"orange" is purely nominal, whereas the definition
of "W" in terms of "V" and some other word would be
"real" since it would be a correct definition in
virtue of the prior correlation of these words with
properties. There is a sense in which one of the
definitions is quite arbitrary, since there is
nothing to justify the statement that the words are
related by an incompatibility rule, except the fact
that unless they were so related they would not be
the same words (or at any rate they would have
different meanings), whereas the other is non-arbitrary,
since the statement of the relation
between words is justified by the fact that those
words are correlated with certain properties, though,
of course, this correlation is itself as arbitrary
as any linguistic convention. But this is a
digression.
    224

then the properties to which they refer are "improper"
properties (cf. 2.D.6, 3.B.5), not objects of experience,
and neither could exist alone. If the property referred
to by the word "gleen" is the combination of the pro-
perties referred to by "glossy" and "green", then the
former ("improper") property cannot exist unless the
latter two do, though either of the latter two may, of
course, exist independently of the other.
In general, when several entities stand in some
identifying relation, at least one of them is a thing
which could not exist unless the others did: for other-
wise it would not stand in the relation in question, and
so it would not be what it is. This brings out the fact
that an identifying fact about meanings may be essential
only to the full specification of the meaning of only
one of the words mentioned. But if a statement states
an identifying fact about meanings then there must be at
least one word whose meaning could not be fully specified
without it.

6.C.8.  As we have seen, the statement (1) of 6.C.2. does
not state an identifying fact about the meanings of the
words "red" and "mat". There are some kinds of state-
ments which can only state identifying facts about
meanings, if they are true. Examples are statements of
the form:
(1) The word "U" refers to the same property as "V".
(2) The word "U" refers to the combination of the pro-
perties referred to by "V" and "W".
(3) "U" means the same as "V".
On the other hand, the following may either be logical
consequences of identifying facts, or they may describe
synthetic (i.e. non-identifying) relations between the
    225

meanings of words.
(4) The words "U" and "V" have the same extension.
(5) The extension of the word "U" is the intersection
of the extensions of "V" and "W".
(6) The property referred to by "U" is possessed by
all objects which have the property referred to
by "V".
In general, if the meaning of a word is logically syn-
thesized out of properties (see section 3.B), then the
relation which holds between the meaning of this word
and words referring to the properties from which its
meaning is synthesized must be an identifying relation,
since one cannot know its meaning without knowing (at
least implicitly) that this relation holds. On the
other hand, a statement about the classes of objects
correctly describable by certain words may fail to state
an identifying relation between the meanings of those
words. (Cf. 6.C.2.)

6.C.9.  In addition, it should be noted that it is
possible for words whose meanings are non-logically
synthesized to stand in identifying relations. For
example, if the word "red" is governed by a p-rule of
the sort described in 3.D.2, and if the words "red-inf"
and "red-ult" refer to the two specific shades taken as
boundaries, then the statement "The word "red" refers to
shades of colours lying between the shades referred to
by 'red-inf' and 'red-ult' " states an identifying fact
about the meaning of "red" and the relation it describes
between the three words mentioned is an identifying
relation between their meanings, if they are used in the way specified.
Similarly, the statement "The word 'tetralateral
    226

refers to the property of being bounded by four plane
surfaces" states an identifying fact about the meaning
of the word "tetralateral", introduced in 2.C.8.
However, the difference between these statements
of identifying facts, or identifying relations between
meanings, and those discussed previously is, as pointed
out in 3.D.10, that the relations between words which
hold in virtue of non-logical syntheses are not logical
relations, that is to say, they are not relations which
are describable in quite general topic-neutral terms,
but are relations which can hold only between meanings
of words referring to special kinds of properties.
Similarly, we may say that these identifying facts are not
facts which can be described in purely logical, or
topic-neutral terms. This will be important when we
come to discuss ways in which one can infer that sen-
tences express true propositions from the fact that
certain words occurring in them are identifyingly related.

6.C.10. We can now use the concept of an "identifying
relation between meanings" to complete the definition of
"analytic" which was begun at the end of section 6.B.
We define "analytic" so that a statement S, obtained
by applying a logical form F to an argument-set A (i.e.
an ordered set of non-logical descriptive words) is
analytic if it is possible to determine the truth-value
of that statement merely by examining some or all of the
following: (i) the logical technique corresponding to F,
(ii) the "structure" of the argument-set A and (iii)
identifying relations between the meanings of the words
in A, provided that only purely logical ( i.e. topic-neutral)
considerations are relevant in the inference.
    227

Thus, if a statement S is analytic, then any other
statement with the same logical form will also be
analytic, if its non-logical words stand in the same
sort of identifying relation as the non-logical words
of S, no matter what the topic with which they are
concerned (e.g. no matter which properties they refer
to: this is the force of the underlined part of the
definition).
For example, the statement "All gleen things are
glossy and green" can be seen to be true by considering
the general logical technique corresponding to the
logical form "All P things are Q and R", by and taking
note of the fact that the words in the argument-set
("gleen" glossy "green") stand in the following
identifying relation: the property referred to by the
first is the combination of the properties referred to
by the other two. Since all one needs to know is that
the words stand in this relation, without knowing what
sorts of properties they refer to, or what sort of topic
they are concerned with (i.e. only purely logical
topic-neutral considerations are relevant), one can
conclude that any other statement of the same logical
form whose non-logical words stand in the same identi-
fying relation is true, and therefore analytically true,
no matter what those non-logical words mean. (See also
the example in 5.D.1.)
The proviso that only general logical considerations
and identifying relations between meanings can be rele-
vant rules out the cases where the truth-value of a
statement has to be discovered either by applying the
technique corresponding to its logical form and carrying
out empirical enquiries concerning the things referred
to by the non-logical words, or by examining any such
    228

things as the properties referred to by these words:
the latter would not be a topic-neutral enquiry since
it would presuppose an acquaintance with these
properties. (Cf. 6.C.4.)

6.C.11. This account of the analytic-synthetic dis-
tinction will become clearer later on, when analytic
propositions are contrasted with synthetic propositions
which are necessarily true. Meanwhile it may be noted
that much of what I have said could be construed as an
attempt to clarify some of Kant's remarks about the
distinction.
For example, when he considers the possibility that
"... the predicate B belongs to the subject A as some-
thing which is (covertly) contained in this concept A"
(see 6.B.2.), it seems that he is considering just one
of the kinds of facts which I should describe as an
identifying fact about the meanings of words. This is
in the same spirit as his remarks that analytic knowledge
is obtained merely by meditating on concepts.
When he says that synthetic judgement involves going
beyond concepts in an appeal to intuition, he seems to
have in mind the same sort of thing as I have when I rule
that some statements about meanings do not state identi-
fying facts since in order to know that they are true
it is not sufficient merely to know the meanings of the words
mentioned: in addition one must examine the things
referred to, either by carrying out empirical investi-
gations, or by examining properties to discover that they
are synthetically related (see chapter seven). Appeals
to "intuition" are also ruled out by the condition that
knowledge of the truth of analytic propositions must be
based only on logical considerations, which do not presuppose
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acquaintance with any special kind of property or object
(cf. 3.B.10, 7.C.2, 7.D.3,ff). Kant's account simply
happens to be less general than mine, and is not as
detailed, since it does not explain how it is possible
for a statement to be analytic. (Cf. Sections 5.C & 5.D.)
In addition, I think that my definition of "analytic"
brings out what people are getting at when they describe
some statement as "true by definition", or say such things
as "If that's what you mean by so and so then you must
admit that ...." or "You cannot really believe that,
unless you mean it in an unusual sense", etc.

6.C.12. It should be noted that in my definition of
"analytic", the phrase occurred ".. if it is possible
to determine the truth-value ..." The point of this is
the fact that one may fail to notice that a statement
is analytic, and then discover whether it is true or not
in the usual way, by carrying out empirical enquiries.
I shall explain the significance of this presently.
But first we must take note of the complexities
in our ordinary use of words which were described in
chapter four.

6.D.    Indefiniteness of meaning

6.D.1.  It should not be thought that every time anyone
makes a statement that statement is either analytic or
synthetic, even if it falls within the class of statements
we have selected for discussion (in 1.C.2). For there
may be some fact about the words used to express the
statement which is not clearly an identifying
    230

fact about the meanings of those words. If it is
possible to infer from this fact that the statement is
true, then it may not be clearly analytic or clearly
not analytic, since it is not clear whether the truth
of the statement can be logically inferred from an
identifying fact or not.
The reason why a fact about words may be neither
definitely an identifying fact nor definitely not an
identifying fact is simply that words may have meanings
which are indeterminate in any of the ways described in
chapter four. This indeterminateness has the effect
that sharp criteria for identity of meanings cannot be
applied. As pointed out in section 2.C and section 4.B,
this does not matter much for normal purposes, but it
does matter when philosophers are discussing the
analytic-synthetic distinction, and say such thins as
"A particular speaker on a particular occasion who uttered
'All phosphorus melts at 44°C' without knowing whether he
was following a rule which made this analytic or not, would
be cheating" (Pears, in mind, 1950, p.204). To use a
word with an indeterminate meaning is to cheat only when
one implies that one means something perfectly definite
by it, that one would know in all cases what would count
as settling questions about the truth-value of a statement
using the word, but one need not imply this when one uses
a word.

6.D.2.  For example, if the word "U" is correlated by an
indeterminate d-rule with a range of properties with
indeterminate boundaries (4.A., ff. ), and the word "V"
refers to one of the borderline properties, then the
statement "Possession of the property referred to by 'V'
    231

is a sufficient condition for being correctly describable
by the word "U' " is neither definitely an identifying
statement about the meaning of "U" nor definitely not
an identifying statement about the meaning of "U", even
if it is generally believed that all things which are
V are also U, for some such reason as that the property
referred to by V happens always to accompany some other
property which is taken as a sufficient condition for
being describable by "U". (Far more complex and
interesting cases are possible.) In such a case, the
statement "All things which are V are U" is neither
definitely analytic nor definitely synthetic, even if
it is definitely true. In this case the proposition
would cease to be definitely true if someone produced
an object which had the property referred to by "V"
without definitely having anything else which sufficed
to ensure correct describability by "U".
More interesting cases, from the point of view of
this essay, concern necessarily connected properties.
If what was said in 2.C.8 and following paragraphs is
correct, than the property of being bounded by three
straight sides is different from the property of being
rectilinear and having three vertices. But if we ask
whether the English word "triangle" refers to one of
these two properties, or to the other, or to both con-
junctively, or to both disjunctively, then there will
surely be no answer, for since these two properties are
always found together it makes no difference for practical
purposes (e.g. when a man orders a table with a tri-
angular top) to which of them or which combination of
them the word refers. This helps to account for our
use of the expression "can be defined ..." in such contexts
as "The word 'triangle' can be defined as referring to the
    232

property of being bounded by three straight lines"
(Cf. 2.C.10, where it was shown how this could lead to
a question-begging argument.) The word can be defined
in this way for normal purposes, since a word so defined
will, for normal purposes, do exactly the same job as
the ordinary English word "triangle": loose criteria
for identity of meanings are all we need. Since it
is not definitely the case that the word has to be
defined this way, or that it has to be defined as
referring to the property of having three angles,
we cannot say definitely that the statement "All triangles
are bounded by three sides" is analytic, as understood
in English, or that it is definitely synthetic.

6.D.3.  When a word or sentence is used with an
indefinite meaning, it is often possible to make the
meaning more definite in one way or another by adopting
an additional linguistic convention. This was illus-
trated by the n-rule correlating the words "RED" and
"redange" in 3.B.4, ff. Sometimes one way of making
meanings more definite makes a statement analytic,
while another way of making the meanings of the same
words more definite makes the statement synthetic. In
such a case the original statement is neither analytic
nor synthetic: its meaning is too indeterminate for the
distinction to apply. When a sentence does not definitely
express this or that proposition, we cannot always ask
whether the proposition which it expresses is analytic
or synthetic.
Normally there are several different ways in which
the meaning of an ordinary sentence could be made more
determinate: for example, if one of the words in it is
correlated with a range of properties whose boundaries
    233

are indeterminate, then there must be many different
possible determinate boundaries. To each way of making
a concept or meaning more determinate there corresponds
what may be described as a (relatively) "sharply iden-
tified" concept or meaning, since sharp criteria for
identity can be used for distinguishing these new
concepts. (Cf. 2.C, and 3.C.9-10). All of these
different sharply identified meanings may be thought of
as somehow "superimposed" in the old meaning, to produce
the indeterminateness, (as if several different but
similar faces were superimposed to produce a blurred
photograph). (In 3.E.2, it was remarked that several
different concepts of the sorts distinguished in chapter
three were superimposed, in our ordinary concept "red".
Cf. 7.D.11.note.)
As pointed out in 4.B.7, this indeterminateness can
occur at different levels in a language and manifest
itself in different ways.

6.D.4.  This sort of thing is sometimes ignored by
philosophers when they are engaged in "conceptual
analysis". For example, they may argue over the question
whether it is part of the meaning of "daughter" that a good
daughter behaves in a certain way, or whether it is part
of the meaning of "table" that the word describes objects
which are used for certain purposes. One side will
argue that it is part of the meaning, and another will
argue that it is not, that it is just a fact which is
generally taken for granted in one way or another, and
both may fail to consider the possibility that as far as
ordinary use goes the concept may be too indeterminate for
either side to be correct, or perhaps as some people
understand the word it has one sort of meaning, and as
    234

others understand it it has the other. In such a case,
the argument can only be pointless. (But see 5.E.7.a.)

6.D.5.  It should not be assumed that the analytic-
synthetic distinction can never be applied to ordinary
statements using words which have indefinite meanings.
If two areas have fuzzy boundaries it may not be clear
whether they overlap or not, but this doesn't rule out
the possibility that a circle with fuzzy boundaries may
be completely inside another, or completely outside
another, or that they may definitely overlap. Similarly
a concept may stand in a definite relation to some other
concept even though each of them is indeterminate in
some way. For example, no matter how indefinite the
concept "horse" may be it is certain that the statement
"Every horse chews everything it eats at least five times"
is synthetic, whether it is true or false. Some ordinary
statements may be definitely synthetic despite the
indefiniteness of some of the concepts employed in them.
If all statements in some language were to be either
analytic or synthetic then even the most subtle ambi-
guities would have to be eliminated (which need not be
done for ordinary purposes). But as soon as the most
flagrant ambiguities have been eliminated it is possible
for some statements to be definitely analytic or definitely
synthetic.

6.D.6.  All this shows that there is no need to say that
there is no such distinction as the distinction between
analytic and synthetic statements just because some
statements fail to fall on one side or the other of the
distinction. (It is often difficult to tell whether an
    235

utterance is meant as a question or as a statement:
should we therefore abandon the distinction between
questions and statements? Indeterminateness can make
a statement neither definitely true nor definitely
false, where borderline cases turn up: should we say
that there is no distinction between true and false
statements?) Certainly we cannot assume that the dis-
tinction can be unambiguously applied to everything
which can be described as a statement: if that was Kant's
assumption, then it is an assumption he should not have
had. We could try to make the distinction cover all
cases by making it a "pragmatic" distinction and talking
about degrees of analyticity, as suggested by Pap in
"Semantics and Necessary Truth" (p.352, etc.), but that
would be of interest only in answering empirical questions
about how statements as understood by certain persons
ought to be classified, and would certainly not be
relevant to the problems discussed by Kant (and in the
next chapter of this thesis). The alternative of
adopting a "system-relative" distinction (suggested, for
example, by M. Bunge in Mind April, 1961) is quite
unintelligible to me unless it is simply an obscure and
confused version of what I have said, namely that we
must be clear as to what meanings (definite or indefinite)
are associated with words before we can ask whether the
statements they express are analytic or synthetic.

6.D.7.  To sum up: the fact that words may be used
with indeterminate meanings has the consequence that,
as understood in some language, or by some person, or
group of persons, a sentence may express a statement
which is neither definitely analytic nor definitely not
    236

analytic. This is because the relations which hold
between the meanings of some of the words occurring
in such sentences may not be definitely identifying
relations, nor definitely not identifying relations.
This possibility is sometimes ignored by those who believe
that there is a distinction. But others who are aware
of the possibility go to the opposite extreme and ignore
the fact that the distinction can be applied in a clear
way in some cases. My description of the way logical
and descriptive words work has the advantage of being
able to take account of or explain both of these facts,
namely the fact that the distinction can be applied in
some cases, and the fact that it cannot be applied in
all.

6.E.    Knowledge of analytic truth

6.E.1.  We are now able to see the resemblances and
differences between analytic propositions and synthetic
ones. Both kinds of proposition are expressed by
sentences built up out of descriptive words and logical
words (and constructions) which have a perfectly general
application in statements about the world. So in both
cases it is possible to discover truth-values by taking
account of meanings of non-logical words and applying
logical techniques corresponding to the logical forms of
propositions, i.e. by investigating facts. The difference
is that in the case of analytic propositions there is
another way of discovering truth-values, namely by
examining the meanings of non-logical words and the logical
techniques which have to be applied in discovering the
truth-value in the other way.
    237

This shows that the notion that "the way we get to
know the truth of necessary propositions is by inspecting
them" is not quite as misleading as Malcolm suggested
in Mind, 1940 (p.192). For it is one way in which we
can get to know the truth of some kinds of propositions.
But it is not the only way in which one can come to
know that they are true. For, as repeatedly pointed
out, analytic propositions are the same sorts of things
as all other propositions, insofar as they are expressed
by sentences which refer to non-linguistic entities, and
whose logical form determines what sorts of observations
count as verifying them. That is to say: analytic
propositions merely form a subclass of the class of
propositions which can be verified empirically. (Cf.
5.A.9, 5.C.7, 5.D.5, 5.E.3.)

6.E.2.  For example, a person may fail to notice that
the proposition "All gleen things are glossy" is analytic
("gleen" as I have defined it means "glossy and green"),
and look to set whether it is true or not by examining all
the things which are gleen to see whether they are glossy
or not. (Cf. .B.11). We know that he must find that
they all are, but he may simply fail to notice that one
of the steps in recognizing that something is gleen is
recognizing that it is glossy, and so he may fail to see
that his search is superfluous. Perhaps he divides the
investigation into two stages: he first examines all
gleen things and finds that they are ellipsoidal in shape,
and then he examines all ellipsoidal things and finds
that they are glossy. In either case, on the basis of
what he has observed, of what he knows about the meanings
of "gleen", "glossy", etc., and of what he knows about
    238

the general technique for verifying statements of the
form "All P's are Q's" he is justified in asserting:
"All gleen things are glossy".
(This shows, incidentally, that although only an
analytic proposition can be formally entailed by an
analytic proposition, nevertheless, any kind of proposi-
tion, whether analytic or not, can entail an analytic
proposition. For example the proposition (1) "All
bachelors are happy men and all happy men are unmarried",
which is certainly synthetic, and may perhaps be false,
formally entails the proposition (2) "All bachelors are
unmarried," which is analytic. Some people - such as
P. Long in Mind, April 1961, pp. 190-191 especially -
find this sort of fact surprising. But the fact that
a synthetic proposition can entail an analytic proposition
is no more surprising than the fact that a true pro-
position may be entailed by a false one. Even a purely
formal truth of the form "All P are P" may be entailed
by a false contingent proposition of the form "All P are
Q and all Q are R and all R are P".)
This fact that analytic propositions have in common
with synthetic propositions the possibility of being
verified empirically helps to show how wrong people are
when they say that analytic propositions are not really
propositions but rules, or when they say that empirical
facts are irrelevant to their truth-values. (They are
not irrelevant but ineffective.)

6.E.3.  When knowledge of an analytic truth is based on
empirical enquiries, all three elements in the justification
of the proposition are involved (see 5.E.1). But we have
seen that the third element (observation of the facts)
need not be involved in a justification for asserting
    239

the proposition. This, however, does not mean that the
other two elements can, on their own, provide a justi-
fication in the same way as they did when accompanied by
the third. For when all three are involved, the
logical technique corresponding to the form of the
proposition has to be applied, and the technique can
only be applied when facts are observed. When we dis-
pense with empirical observation, we no longer apply our
knowledge of meanings and logical techniques: instead
we study them, which is quite a different matter.
(Similarly, in showing that when numbers standing in
certain relations are taken as arguments the value of
some arithmetical function must be positive, no matter
what those numbers are, we do not apply the calculating
techniques for determining the value of the function:
we study them. They can only be applied when actual
numbers are taken as arguments. This step to a higher
level is concealed in normal mathematical procedure owing
to the technical devices used for proving general theorems
with the aid of variables, which helps to give the mis-
leading impression that techniques are being applied
(e.g. to entities of a special kind known as "variable
numbers"), owing to a misleading formal analogy. In
proving the algebraic theorem normally stated thus
"(a + b)2 = a2 + 2ab+b2" we do not add or multiply:
we study the general effects of adding and multiplying
in certain ways.)
This is important, because it explains why people
may fail to see that a proposition is analytic even though
they understand it perfectly well. For they may under-
stand it perfectly in the sense of knowing the meanings
or functions of all the words and constructions involved
    240

and knowing how to tell whether statements expressed
by sentences using them are true or false, and yet fail
to notice the aspects of these meanings or functions in
virtue of which some of these statements are analytic.
(Cf. 5.C.7.) People may be quite good at applying a
technique automatically without being able to think
clearly about it. It may never even occur to them to
study it. (See appendix on "Implicit Knowledge".)
Having failed to see that a statement is analytic, one
may also fail to notice that it in true.
By describing several different ways in which one
may come, mistakenly, to think a proposition false when
it is analytically true, I shall now try to bring out
the inadequacy of the definition of an analytic proposition
as a proposition which cannot be intelligibly denied, or
which one must know to be true if one knows what it means,
or which is such that when a person denies it this is a
sufficient justification for saying that he does not
understand the meanings of the words used to express it.

6.E.4.  First of all, we may get out of the way a whole
series of cases which are rather puzzling from a certain
point of view, but need not be discussed here: the
cases where a person seriously denies something simply
because he is temporarily muddled, or confused, or absent-
minded. We must simply take these for granted as possi-
bilities which can explain completely why a person denies
or affirms anything at all, whether it is true or false
or analytic or synthetic.
Secondly, we must notice that a person who has failed
to see that a proposition is analytically true may then
deny it on account of mistakes of the same sort as could
lead him to deny a true synthetic proposition which he has
    241

tried to verify by making empirical enquiries or obser-
vations. For example, he may have asked someone and been
given the wrong answer. Or he may have misunderstood
one or more of the individual words or constructions, or
failed to take in the structure of the sentence expressing
the proposition, so that he thinks it expresses some
other proposition than the one which it does express, in
which case he may take certain facts as falsifying it
when they do not really do so. Failing to take in the
structure of the sentence properly is a different matter
from failing to understand one or more of the descriptive
or logical words. One may know perfectly well what the
elements of a sentence are, and how they work, but simply
be mistaken as to the way in which the sentence is built
out of them, or mistaken as to the way in which, the
verifying technique corresponding to its logical form
is in this case determined by the rules for the individual
logical words and constructions (see circa 5.B.15).
So there are many different ways in which one can
get the wrong idea, and not all can be described simply
as "failing to understand the meanings of the words".
(Notice, incidentally, that instead of getting the wrong
idea one may, as a result of imperfectly grasping the
structure of a sentence, not have any clear idea at all
of what it means, even, in some cases, without realizing
that one does not have a clear idea.)
In addition to all these factors, which may account
for a person's making the wrong enquiries, or drawing
the wrong conclusions from what he observes, there is
also the possibility of mistakes of observation.
For example, someone who wishes to find out whether all gleen
things are glossy may try to collect all gleen things
    242

together and, having done so, look them over and mis-
takenly think he sees one which is not glossy. Or he
may make a mistake which has nothing to do with special
features of the proposition in question, such as
establishing the two premises "All bachelors are happy
men" and "Not all happy men are unmarried" and mistakenly
concluding that "Not all bachelors are unmarried".
Obviously still more complex mistakes may explain a
person's denying some proposition which is analytic.

6.E.5.  But all this presupposes the possibility of
failing to notice that the proposition is analytic.
What sorts of things can account for this failure? Once
again, there are many distinct possibilities.
A person who knows perfectly well how to tell whether
any particular object is or is not correctly described by
some word may fail to notice what he is doing in doing
this, and so fail to observe that there is some identi-
fying fact about the meaning of that word: he knows the
meaning of the word in a practical way, since he can use
it, he can apply his knowledge, but he is not explicitly
aware of its connections with other words. In the same
way one may be quite good at counting, and be able to
decide which numeral follows any given numeral, and yet
be unable to formulate the general principle on which one
constructs the new numerals. (E.g. one may never have
thought about it.) Failing to notice the relations
between the meanings of words one may fail to see a
consequence of the fact that they stand in these relations.

6.E.5.a.    In a similar way one may fail to notice some
general feature of the logical technique corresponding to
    243

the logical form of a proposition, despite the fact that
one can apply it in particular cases in deciding whether
statements are true or not. Such a person fails to take
explicit note of the fact that in deciding that the
proposition "All gleen things are glossy" is true he
examines each thing which has the combination of pro-
perties referred to by "gleen" to see whether it also
has the property referred to by "glossy". Failing to
notice the general procedure corresponding to the logical
form of the proposition, he fails to notice facts about
the application of that procedure which determine its
outcome.

6.E.5.b.    Alternatively, one may be perfectly well aware
of the relations between the non-logical words in a
sentence, and be able to describe, in a general way, the
logical technique corresponding to the way logical
constants occur in that sentence, and yet fail to notice
how all this applies to the particular case in question,
for some reason. (It is possible to fail to notice some-
thing which is well within one's field of view.)

6.E.5.c.    Each of these, and perhaps other possibilities,
may explain a person's failing to notice that some pro-
position is true independently of the facts, even though
in a clear sense he understands the proposition, since
he understands its parts, he knows how they are put
together, and he knows what counts as the propositions
being true. We could say that in such a case he doesn't
fully understand it unless he notices that empirical
enquiries are unnecessary, but then are adopting a
new terminology, giving a new sense to the notion of "full
understanding", and to say that a person who sincerely
    244

denies an analytic proposition cannot fully understand
it is correct and not misleading only if the new ter-
minology has been made clear, and even then it is not
very informative, since it slurs over the differences
between possible explanations of a failure to notice
the truth of an analytic proposition.

6.E.6.  All this may suggest that in order to notice
that a proposition is true by definition, that it has
to be true on account of its meaning, one has to be an
expert logician who can formulate facts about meanings
and logical constants and draw conclusions from them, or
perhaps formulate non-formal truths of logic of the sort
described in 6.B.12.a. But this is not so, for one may
know these facts implicitly (without being able to formulate
them) and see, perhaps in a "dim" sort of way, what
they imply. One may know something, and be fully jus-
tified in asserting it; and yet be unable to say what
justifies the assertion.
This can be illustrated by much of the ordinary
person's knowledge of arithmetical facts. It is fairly
easy for someone who knows what a recursive definition is
to say explicitly how the series of numerals used in
counting is generated (for example either in Arabic
notation or in English words). But a person who knows
perfectly well how to go on producing new numerals when
counting may be quite unable to formulate the general
principle which he is following, and may even be unable
to recognize a correct formulation suggested by someone
else. Nevertheless, he knows the principle, since he
can apply it and distinguish incorrect from correct moves
in accordance with it. We may say that he knows it
    245

"implicitly" (Cf. Appendix III.) Now this implicit
knowledge may give him the right to assert with con-
fidence some general statement about numerals such as
numerals ending in "6" are always closer to numerals
ending in "3" than they are to numerals ending in "O".
("Closer" = "separated by fewer numerals".) In par-
ticular, it may justify his asserting that between "0"
and "100" there is no numeral ending in "6" which is
closer to one ending in "O" than to any ending in "3".
He is justified by his knowledge of the general principle
for constructing numerals, properties of which he can
see more or less clearly: so there is no need for him
to justify the assertion by writing out the sequence
for "O" to "100", though he could do this.
Similarly a person who says "It is true and has to
be true that all gleen things are glossy" may be per-
fectly justified in making this claim, on account of
his implicit knowledge of the techniques for working
out the truth-values of such propositions. He may be
quite inarticulate about the reasons why it "has to be
true", yet what he says is correct, and he is justified
in saying it. He need not have seen a logician's proof
(or anyone else's).

6.E.7.  We see therefore, that a person may see, cor-
rectly, that a proposition is analytic, that on account
of what it means it has to be true no matter now things
happen to be in the world, without fully understanding
the reasons why it is true, or why it would he true in
all possible states of affairs. He does not fully
understand what makes it true, because he has not noticed
that it is true on account of a general feature which the
proposition shares with other propositions expressed by
    246

sentences whose meanings are related in the same way.
Most philosophers have, of course, hitherto been in
this position, which is why they have not been able
accurately to characterize the class of analytic pro-
positions, or propositions true by definition. Their
misunderstanding is shown, for example, when they say
that the difference between statements which are true in
virtue of their logical form (such as "All bachelors are
bachelors") and analytic statements which are not formal
truths (like "All bachelors are unmarried") is that in
the sentences expressing the latter, some non-logical
words occur "essentially". This is mistaken, since it
is not in virtue of any special property of the words
"bachelor", "unmarried" that the sentence "All bachelors
are unmarried" expresses an analytic proposition, any
more than it is an essential property of the word "red"
which accounts for the truth of the proposition "All red
things are red". All that is essential is that the
words occupying their positions in the sentences should
have meanings which stand in certain identifying relations.
That is to say, an analytic proposition is not true in
virtue of the fact that the non-logical words have the
special meanings which they do have, but in virtue of
the fact that those meanings, whatever they may be, stand
in certain relations, and other propositions, including
words with quite different meanings, may be true for the
same reason. One need not know the meaning of a sentence
in order to know that it expresses an analytic proposition,
one need only know certain facts which must be known
implicitly by anyone who knows the meaning. (Though if
one knows only these facts one will not, of course, know
which proposition the sentence expresses.)
This shows how the class of formal truths is merely
    247

a subclass of the class of analytic truths. (Compare
5.D.1.)

6.E.8.  One thing that is clearly brought out by all
these examples, which is sometimes overlooked, is that
there is a difference between knowing that a proposition
is true and knowing that it is analytic. This was
pointed out as long ago as 4.B.5, in connection with the
more general fact that there is a difference between
knowing that the propositions expressed by certain sen-
tences are true, and knowing what those sentences mean.
(This does not contradict the thesis that knowledge of
meanings can often be explained as knowledge of the
general way in which the various words and constructions
used in sentences contribute towards determining the
conditions in which the propositions expressed are true,
or false.) Knowledge that an analytic proposition is
true can be based on observations in the same way as
knowledge of the truth of any empirical proposition,
whereas the knowledge that it is analytic has to be based
on a priori considerations of facts about meanings and the
techniques for determining truth-values. (Cf. 5.C.7,
6.E.3.)

6.E.9.  It is possible to get into a muddle by failing to
distinguish knowledge that some proposition is analytic
from knowledge that a sentence expresses an analytic
proposition as understood by certain people. The latter
presupposes knowledge of empirical facts about the way
the people concerned use words, the former does not. We
may put this by saying that knowledge that some proposition
is analytic is knowledge that if certain words are used
with certain kinds of meanings, then the proposition which
    248

they express when combined in a certain way is analytic.
In order to discover this one need not know anything
about the way in which anybody actually uses words, though
one will not be able to report one's discovery unless one
knows which language is understood by the people to whom
one is reporting it. Of course, one will not be able
to think about such matters without having learnt a
language, and one learns a language by seeing and hearing
what happens when others speak it (and trying to speak
it oneself), which involves picking up some empirical
knowledge. But that is irrelevant. I believe Malcolm
must have been muddled about all this when he wrote (in
Mind 1940) that a child learns necessary truths by
observation of the way people talk (p. 193) and that we
answer questions about entailments "by finding out certain
empirical facts about the way we use words" (p.195).
What one learns by this sort of observation is how people
understand sentences. The discovery that the propositions
expressed by those sentences are analytic, or stand in
relations of entailment, require a further step which
is quite different from empirical observation. (Pointing
to the way people use the words "triangle", "line", etc.,
is not relevant in the proof of a theorem about triangles.
Neither would it be relevant to establishing an empirical
fact about triangles.)

6.E.10. To sum up: I have tried to bring out various
factors involved in the justification of the claim to
know that some proposition is true when that proposition
is analytic. One way of bringing them out is to describe
ways in which one may fail to see that such a proposition
is true. This also reveals the superficiality of some
commonly accepted definitions of "analytic".
    249

One of the most important points to be stressed
is that analytic propositions can be verified empirically
like other propositions using the same empirical con-
cepts. (This suggests that the empirical-nonempirical
distinction should be applied to ways of knowing, not
to kinds of proposition.) The importance of this is
that it shows in what sense they are propositions,
capable of being true or false.

6.F.    Concluding remarks

6.F.1.  The contents of this chapter may now be summarized.
I have tried to show how it is possible for propositions
to be analytic, by showing how the truth-value of the
proposition expressed by some sentence may be determined
by identifying relations between the meanings of non-
logical words in that sentence together with facts about
the logical technique corresponding to its logical form.
This is equivalent to saying that there is something
which is essential to the proposition's being the one
which it is, from which it follows logically (but not
formally: cf. 6.B.12) that the proposition has the
truth-value which it does have. It could not have any
other truth-value unless the descriptive words expressing
it had different meanings or the logical words and con-
structions involved in it had different functions, in
which case it would have been a different pro-
position (since we are using strict criteria for identity of
meanings and propositions - see section 2.C.). It
follows that anyone who talks about "declaring a proposition
to be analytic" is muddled or using loose
criteria for identity, since one cannot simply declare
    250

a proposition to be analytic without risk of changing
the proposition. The most one can do is declare that
words are to be used in a certain way, which may have
the consequence that a certain combination of these
words expresses an analytic proposition, but that
proposition is analytic whether those words are used
to express it or not. (Cf. Waismann in "Analytic-
Synthetic II", p.25. See 6.A.3, above.)

6.F.2.  The identifying facts about meanings, which
must be known if it is to be known what those meanings
are, are what correspond to the notion of a "definition"
in the crude account of an analytic proposition as one
whose truth follows logically from definitions. The
fact that metalinguistic concepts are required for
stating these identifying facts, shows why Frege was
wrong to think of definitions as occurring on "the same
level" as other propositions. (Cf. 6.B.8.a,ff.) The
fact that we can, in a non-formal (but perfectly
rigorous) way, draw logical conclusions from facts about
meanings, shows how Quine and Waismann were wrong in
denying that "definitions" could be used to found truths,
and in asserting that they were merely substitution
licences for transforming truths. (6.B.3-4,ff.)

6.F.3.  We found, in section 6.B, that the Quine-Waismann
definition of "analytic proposition" as "proposition
derived by substitution for synonyms in formal truths"
was not sufficiently wide. The question now arises
whether propositions which are analytic in their sense
are also analytic in the sense defined in 6.C.10: is our
new definition simply wider than the old one, or is it
completely different?
    251

We can show that propositions which are analytic
in the old sense are also analytic in the new sense, as
follows. A formal truth is one kind of analytic pro-
position, for its truth-value is determined in a purely
logical (topic-neutral) way by (i) its logical form and
(ii) the structure of the argument-set of non-logical
words to which that form is applied. But the example
of 5.D.1 shows clearly that replacing a word or expres-
sion by another word or expression referring to the
same property or properties, in some sentence, cannot alter
in any essential way the factors which determine the
truth-value of the proposition expressed by that sentence,
since the actual shapes of the words used in the sentence
do not matter: it is what they refer to or what their
functions are that matters. Hence, if general logical
considerations about logical form and meaning suffice
for a determination of the truth-value of a proposition
P, and if P' is derived from it by substituting synonyms
for some of the non-logical words used to express it,
then P' can be shown to be true by general logical con-
siderations of the same sort as before, taking into
account also some new identifying facts about meanings.
Hence, if P is analytic, then so is P', and if P is a
formal truth then P' is analytic, since formal truths
are analytic.
This shows that the Quine-Waismann definition is
included under our definition, which simply happens to
be more general, since it allows us to take account of
identifying relations other than synonymy, as shown in
6.B.

6.F.4.  We have just proved a logical "theorem" by
    252

non-formal considerations, namely, the theorem that
analyticity is preserved by synonymy substitutions.
We could prove, in a more general way, that analyticity
is preserved by any substitution of one descriptive
expression for another which is such that those two
expressions are logically equivalent, in the sense that
identifying facts about meanings, together with purely
logical considerations, can show that if anything is
correctly described by one of these expressions then it
is also correctly described by the other, no matter how
things happen to be in the world.
In addition, we can prove that if a proposition is
analytic, then any other proposition formally entailed
by it (5.C.8) is also analytic.
For example, suppose that by general logical cons-
iderations we are able to deduce from some identifying
fact about the meanings of "A", "B" and "C" that the
logical rogator corresponding to "All P things are Q and
all Q things are R" takes the value "true" for the argument-
set ("A", "B", "C"), in which case the proposition expressed
by the sentence "All A things are B and all B things are
C" must be analytic. In addition, we know that this
proposition formally entails the one expressed by "All
A things are C", since we can discover, from a consider-
ation of the logical techniques corresponding to their
logical forms, that the latter must be true whenever the
former is, no matter how things happen to be in the world.
(See 5.C.8). Now add these considerations to the previous
ones which showed that the former sentence expressed an
analytic truth, and we find that we have a way of showing,
by consideration of identifying facts about meanings and
general logical principles, that the latter sentence
    253

expresses a proposition which must be true independently
of what is the case in the world.
This example should show that a general theorem
could be proved to the effect that analyticity, like
truth, is preserved by formal entailment. (The reader
who wishes to test his grasp of my definition of
"analytic", "formal", etc., may try to write out the
general proof in detail.) This is a theorem of non-
formal logic. (I.e. it is not just a formula in a
formal system derivable from axioms using rules of
inference, nor is it a proposition about what is derivable
in formal systems.)

6.F.5.  The fact that any analytic proposition formally
entails many other propositions implies that if there
is one proposition which is true in virtue of some
identifying fact about the meanings of non-logical words,
then there will be many others which are true in virtue of
the same fact (though different logical considerations
may be required in order to establish their analyticity).
We are reminded, therefore, of the fact pointed out in
5.D.6, namely that to any identifying fact about meanings
of non-logical words, there corresponds a whole family of
analytic propositions whose truth it helps to guarantee.
This fact is sometimes made use of when people try to
define "analytic". Hare's definition, in "The language
of Morals" (pp. 41-42) went as follows, for example:
"A sentence is analytic if, and only if, either (1)
the fact that a person dissents from it is a
sufficient criterion for saying that he has mis-
understood the speaker's meaning or (2) it is
entailed by some sentence which is analytic in
sense (1)."
We have shown how (2) is not part of the definition of
    254

"analytic", in our wide sense, but is a theorem about
analytic propositions. The importance of this is that
we do not need to say that some propositions are analytic
in one sense and others in another: the class of analytic
propositions is homogeneous on our definition.

6.F.6.  The homogeneity of the class of analytic pro-
positions is due to the fact that definitions or state-
ments of identifying facts about meanings must be thought
of as being on a different level from propositions using
the words they mention. So the "definitions" in virtue
of which analytic propositions are true are not themselves
included in the class of analytic propositions. There
are not some analytic propositions, or propositions
true by definition, which are themselves definitions,
or "registers" of linguistic conventions, or direct
expressions of linguistic conventions, or linguistic
"proposals" (see end of 6.B.8.b), while the remainder
are analytic only in virtue of being logical consequences
of these propositions. Instead we must say that no
analytic proposition is the definition in virtue of which
it is true, and all analytic propositions can be shown
to be true in the same way, namely directly from a con-
sideration of meanings and logical (topic-neutral) facts,
without the mediation of other analytic propositions.
This homogeneity relieves us of the task of dis-
covering which of the analytic propositions are the
definitions and which are merely entailed by these
"definitions", a task which could be quite embarrassing.
In addition, this homogeneity enables my definition to
escape the objection to the Quine-Waismann definition of
"analytic" which I raised in 6.B.7.
    255

6.F.7.  For reasons given in 1.C.2, and Appendix I,
I have deliberately restricted the discussion of the
analytic-synthetic distinction to a small class of
propositions, namely those which are universal in form
(i.e. mention no particular objects) and include only
fairly simple descriptive words in addition to logical
constants. The main reason for this restriction is
that any attempt to define " analytic" straight off for
all kinds of propositions seems to lead to muddle and
confusion. However, now that we have taken the first
steps in the clarification of the notion, the question
arises whether it may not be generalized.
The first and most obvious generalization would
be to take account of relational expressions, which have
not been mentioned since 6.B.5.b, where it was pointed
out that the irreflexiveness of the relation "to the
left of" might be due to a purely linguistic convention,
in which case "Nothing is to the left of itself" must
express an analytic proposition. The extension of the
notion of an identifying fact about meanings to take
account of relational expressions, and the extension of
the notion of a logical rogator to allow relational
expressions as arguments, can be easily accomplished
I shall say no more about this.

6.F.8.  Secondly, one might try to extend the notion of
an analytic proposition to include those using proper
names and other singular referring expressions, by
extending the notion of an identifying fact to include
facts about the meanings of proper names and other
referring expressions. For example, it might be said to
be an identifying fact about the meaning of the word
"Socrates" that the thing referred to by the word is a
    256

human being and a man. This assumes that we can talk
about knowing the meaning of a referring expression in
a sense which is different from knowing which thing is
its referent (for I might very well know which material
object was called "Socrates" without having the faintest
idea whether that object was a human being: for example
I might not have the concept of a human being).
Whether or not it is possible to extend the notion
of an analytic proposition to include propositions like
"Socrates is human", "Tom's bachelor uncle is unmarried",
etc., need not concern us now, since we are mainly
interested in necessary connections between universals.
(See Appendix I. For an attempt - not very successful
in my view - to show that proper names may occur in
analytic statements of the form "x is P", see Searle's
D.Phil. thesis.)

6.F.9.  The next possible generalization concerns words
or expressions for which there are "appropriateness-
conditions" instead of just truth-conditions. (See
2.D.9.) In general, rules laying down appropriateness-
conditions for utterances (such as "Alas!") are concerned
not with observable states of affairs which make pro-
positions true, but with such things as the purposes
which may be served by utterances, the contexts of
utterance, the things which are to be expected of the
person producing the utterance if he is not to be said
to have been deceitful or changed his mind, and so on.
There is an enormous variety of cases, and it is
not to be expected that we can deal with them all at
once, except in a very vague way, as follows: it may
be possible to extend the notion of an identifying fact
    257

about meanings to include identifying facts about the
functions of words or expressions governed wholly or
partly by appropriateness-rules. For example, it may
be that the conditions in which it is appropriate to
say "Ouch!" are identifyingly related to the conditions
in which it is appropriate or true to say "Something
just hurt me". Perhaps there are identifying relations
between the conditions in which it is appropriate to
utter sentences expressing statements and the con-
ditions in which it is appropriate to utter the words
"I believe that ....." followed by such sentences.
(Thus identifying relations between appropriateness
conditions may help to account for so-called "pragmatic"
implications, and also the notion of "logical oddness".
See 2.B.9) Perhaps there are identifying relations
between the conditions in which expressions of moral
judgements are appropriate and the conditions in which
statements to the effect that one has decided to do
something are appropriate, though we must be prepared to
find that such relations are extremely complicated and
difficult to describe, if there are any (we are prepared
for complicated identifying relations by the example of
3.B.5, and comments thereon).

6.F.10. In general, we may talk about an identifying
fact about the meanings or functions of words (or perhaps
an identifying fact about concepts), wherever there is
some fact which must be known if the meanings or functions
of those words (or expressions, or constructions) are to
be known. (Cf. 6.C.3-4.) But it is doubtful whether
this extension of the notion of an identifying fact to
include facts about appropriateness-conditions, or, more
    258

generally, meanings or functions of words, always leads
to an extension of the notion of an analytic proposition, or
analytic utterance. For the connection between analy-
ticity and identifying facts was explained in terms of
the ways in which truth-values of propositions might
be determined, and the whole point about appropriateness-
conditions, for example, is that they need not be con-
cerned with truth. For this reason, I regard with sus-
picion the use of the terms "analytic" and "synthetic" in
connection with imperatives, or moral judgements, or
aesthetic judgements, unless it is made clear that a
special new terminology is being used. I should prefer
to talk about analytic and synthetic connections between
meaning or functions.

6.F.11. This concludes my account of analytic truth.
I have tried to show in what way analytic statements are
true, and why they are true independently of facts, that
is, independently of how things happen to be in the
world. It should be clear that their being true, or
even necessarily true, does not rule out any states of
affairs as actual or even as possible states of affairs,
since their being true is fully determined by matters
which have nothing to do with observable states of
affairs, though this may be concealed by the fact that
they can be verified by observation in the usual way.
In addition, they have the appearance of saying something,
they have a meaning, they seem to state facts. ("It is
a fact that all bachelors are unmarried, just as it is a
fact that pieces of wood fall to the ground when dropped,
only the former fact couldn't have been otherwise.")
However, their having a meaning comes only to this: they
are constructed out of words and expressions which have
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meanings and can occur in statements which are not
analytic. To say of any statement that it states a
fact can be very misleading. Certainly if it is true
it is true in virtue of some fact, which may be des-
cribed as the fact which it states, but if it is false
we have to talk about "possible facts". We might
try saying that the fact stated is the one which if it
actually existed, would make the proposition true: but
in general there are indefinitely many different
possible states of affairs in which a proposition would
be true (e.g. the truth of "Tibs is on the mat" leaves
open the possibility of many different arrangements of
the cat and the mat). Is the fact stated by the pro-
position a collection of all these possible states of
affairs, or is it something common to them all, or what?
Until questions of this sort have been answered, it is
not at all clear what the significance is of the assertion
that even analytic statements state facts, unless it
simply, means that carrying out the normal procedure for
discovering their truth-values by making empirical
observations will always yield the result "true". But
that should not surprise us, since we selected this as
the characteristic property of analytic propositions.
What we must now ask is whether there is any other
way in which a proposition can be true in all possible
states of affairs, or any other way in which the truth-
value of a proposition can be discovered independently
of applying the normal logical techniques. Are there
any other ways than those which we have described, in
which the truth-value of a proposition may be due to facts
about the meanings of the words and constructions used to
express it? Is there any other way in which a proposition
can be a necessary truth than by being analytic?
The meanings of these questions will be clarified, and
answers suggested, in the next chapter.
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Chapter Seven
KINDS OF NECESSARY TRUTH
Introduction
In chapter six I explained what is meant by saying
that a proposition is analytic, and showed how it is
possible to know that such a proposition is true inde-
pendently of any observation of facts. The features of
an analytic proposition in virtue of which it is true
ensure that it would be true in all possible states of
affairs, so we can say that it could not possibly be
false, that it must be true, that it is necessarily true,
and so on. All these truth-guaranteeing features are
topic-neutral and can be described in purely logical terms,
such as that the proposition is made up of certain logical
words in a certain order, with non-logical words whose
meanings stand in certain identifying relations. This
chapter will be concerned with the question whether there
is any other way in which a proposition can be necessarily
true.
In order to give this question a clear sense I must
explain what is meant by "necessary", that is, give an
account of the way in which the necessary-contingent
distinction is to be applied. I shall start off by
talking about the meaning of "possible". The next
section will attempt to explain the meaning of "necessary".
The rest of the chapter will be concerned to describe and
distinguish kinds of necessary truths, and ways in which
a proposition may be known to be true independently of
observation of contingent facts.

_________________________________________________________
(Transcription checked and corrected by Luc Beaudoin 15 May 2016)
NOTE: This is part of A.Sloman's 1962 Oxford DPhil Thesis
     "Knowing and Understanding"
NOTE (24/06/2016): When this chapter was written I knew nothing about
programming and Artificial Intelligence.
In retrospect, much of the discussion of procedures for applying concepts is
directly relevant to the problems of
designing human-like intelligent machines. References to "morons" can be
interpreted as references to computer models.
_________________________________________________________

    261

(Throughout the chapter it must be remembered that
this thesis is written from the point of view described
in section 1.B.)

7.A.    Possibility

7.A.1.  We have reached the stage at which it is not
enough to have only a rough intuitive grasp of the
necessary-contingent distinction. If we are to make
any further progress with the problem of synthetic
necessary truth we must try to see clearly exactly what
this distinction, or family of distinctions, comes to.
It is often pointed out that there is a close
connection between the notion of necessity and the notion
of possibility. A statement is necessarily true if it
would be true in all possible states of affairs, or if
it is not possible that it should be false. This is
sometimes put by saying that necessity is definable in
terms of possibility and negation. I do not think the
connection is quite as simple as some logicians would
have us believe (See 7.B.1, 7.B,10). There certainly
is a close connection between the two notions, however,
so I shall try, in this section, to explain how we can
understand talk about possibility, or about "what might
have been the case".
In order to do this, I shall make use of the very
general facts which, in chapter two, I argued to be
presupposed by statements about the meanings of words in
English and similar languages. (See section 2.B,
especially 2.B.6.) These are facts such as that our
sentences describe states of affairs which can be thought
of as made up of material objects possessing observable
properties and standing in observable relations. More
specifically, I shall rely on some of the arguments in
    262

2.D to the effect that in this conceptual scheme
universals (properties and relations) are not essentially
tied to the particular objects which happen to instan-
tiate them. [Cf. 2.D.5,ff., 3.C.3,ff.)

7.A.2.  It is worth noting that the notions of necessity
and possibility are not merely technical notions invented
by philosophers, for we are all able to use the follow-
ing words and expressions; "necessary", "necessarily",
"possible", "impossible", "must", "had to happen",
"couldn't have happened", "cause", "if so and so had
happened", "if only I had done so and so ?", etc.
Think of the words of the popular song: "That1a the
way it's got to be"!
Despite their familiarity, these notions are
puzzling because they are "non-empirical" in a strange
way. We can point to what is the case, but we cannot
point to what isn't the case and might have been.
Worse still, we cannot point to what is not the case
and could not have been. At any rate, we cannot produce
examples to be looked at, in the way in which we can
produce or point to actual observable states of affairs,
or events. How then do we learn to understand these
kinds of expressions in the first place? The clue seems
to be provided by a fact pointed out in 2.C.6, namely
that in order to decide that something or other is
possible, we have to consider properties, or, more
generally, properties and relations and ask whether they
are connected or not. (There are really many different
kinds of necessity and possibility: I shall not discuss
them all.)

7.A.3.  Let us consider some examples. The piece of
    263

paper in front of me is not blue and square, but it
might have been, or at any rate there might have been a
blue and square piece of paper in front of me. The
piece of paper which is in front of me is white and
oblong, but it might have been different. There is
a cardboard box on my table; it has a lid which is
neither white nor oblong, though it might have been both.
There is no paper on the floor near my chair, but there
might have been, and it might have been either white and
oblong or blue and square (or it might have had other
shapes and colours).
What lies behind all this, is simply the fact, to
which I have already drawn attention,1 that universals
are not essentially tied to those particular objects
which happen to instantiate them. Universals are not
extensional entities, they exist independently of the
classes of objects which actually possess them. As
remarked previously, one can have a property in mind,
think about it, attend to it, recall it, associate a
word with it, talk about it, etc., without thinking
about any actual particular object which has that property.
Neither the property of being blue and square nor the
property of being white and oblong is essentially tied
to the particular material objects which actually have
them. Nor are they essentially tied to the times and
places at which, as a matter of fact, they can observed.
When we see the properties of objects, or the relations
in which they stand, we can see that they are not the
sorts of things which have to occur where they do occur.

1.  (2.B.6, 2.D.5, 3.C.5, etc.)
    264

7.A.4.  This possibility of recurrence is, after all,
what makes us describe properties and relations as
universals, and contract them with particulars. We
can isolate out three aspects of their universality.
First of all there is actual recurrence. The
whiteness of the piece of paper on which I am typing
is a property which it shares with many other objects
existing at the same time and at different times.
Secondly, other objects exist which, although they are
not instances of the property, might have been. The
box on table is not white, but it might have been.
Thirdly, there might have existed objects which do not
in fact exist (there might have been a piece of paper on
the floor next to my chair), and if they had existed,
then they might have had these properties. If there
had been a piece of paper on the floor next to my chair,
it might have been white.
Some philosophers would explain the universality of
properties in terms of the first of these three aspects,
namely actual recurrence, but this will not do, for there
are probably properties, such as very complicated shapes,
which are, as a matter of fact, instantiated by exactly
one object, or possibly by no objects at all.

7.A.5.  Universals can recur, even when they do not in
fact do so. Now how do we know this?
When I look at an object and pay attention to one of
its properties which does not have any other instances,
how do I tell that that property is the sort of thing
which could occur elsewhere, even though it does not.
Is it simply a generalization from experience? Is it
because I have seen many objects which share properties
that I come to believe that this specific property at
    265

which I am looking is also the sort of thing which could
be shared by several objects? If it were an empirical
generalization, then I should have to leave open the
possibility of an empirical refutation, or at least
counter-evidence, but there does not seem to be any such
possibility. Apart from the fact that I do not know what
sort of experience would count as a refutation or as
counter-evidence, the suggestion seems to be nonsensical
because the sort of doubt which is appropriate to an
empirical generalization does not seem to be appropriate
here. (Indeed, the possibility of recurrence of universals
is presupposed by any empirical generalization.)
It seems that when I look at the shape or the colour
of an object, I can see then and there that what I am
looking at is the sort of thing which can recur, since
there is nothing about it which ties it essentially to
this object, or this place and time.

7.A.6.  When I look at a colour and see that it is the
sort of thing which could occur in other objects at other
places and times, I do this by abstracting from the
particular circumstances of its occurrence, such as the
fact that it is possessed by this piece of paper here and
now, is being looked at by this person, can be found to
be two feet away from that particular table, and so on.
I believe that this sort of abstraction is often
confused with another kind, namely abstraction from
specific features, for example in Kant's remark (in
C.P.R, A.713, B.741):
"The single figure which we draw is empirical, and
yet it serves to express the concept without impairing
its universality ? for in this empirical intuition
we consider only the act whereby we construct the
concept, and abstract from the many determinations
    266

(for instance, the magnitude of the sides and of the
angles), which are quite indifferent as not altering
the concept 'triangle'."
I think Kant confuses two things which are very often
confused, but which must be very carefully distinguished,
namely universality and generality, both of which may be
involved in one universal (or concept) "expressed" by a
particular material object. Universality is common in
the same way to all properties, but some properties are
more general (or less specific) than others. The
universality of a property consists in the possibility
of its occurring in other objects than those which
actually instantiate it. The generality of a property
(or concept) consists in the possibility of its occurring
in several different objects in different forms. I do
not know whether there is a sense in which properties
can be thought of as general or non-specific in any
absolute way, but there is certainly a relative general-
specific distinction. The property of being a triangle
is more general, or less specific, than the property of
being an isosceles triangle. A specific shade of red
is more specific, or lass general, than the hue, redness.
(Cf. 3.A.1.)
In order to perceive the (relative) generality of a
property we have to abstract from the specific features
of an object which has that property. In Kant's example,
we have to abstract from the specific ratios between the
sides of a triangle and the specific sizes of the angles,
its specific orientation, and so on, in order to perceive
the generality of the property of being triangular. But
this sort of abstraction is not what concerns us at
present: we are interested not in abstraction from
specific features, but in abstraction from particular
    267

circumstances: that is what occurs when we see a
property as the sort of tiling which can recur, whether
it actually does so or not, that is, as a universal.
It is presupposed by the other kind of abstraction, for
only if it makes sense to talk of a property as being
possessed by other objects does it make sense to talk
of other objects as possessing other determinate
(specific) forms of this property.

7.A.7.  It should be clear that I am not trying to define
the notion of possibility or "what might have been" in
terms of what can be conceived or imagined: I am not
saying "P is possible" means "P can be imagined".
Imaginability is not a criterion for possibility nor
vice versa. There may be things which are possible
though no human being can imagine them, either owing
to lack of experience, or owing to complexity. There
may be colours which have never yet been seen and cannot
be imagined at present, or shapes too complicated to be
taken in. Worse still, people have imagined or conceived
of things which later turned out to be impossible.
For example, one has to be a rather sophisticated
mathematician to be unable to imagine trisecting an angle
with ruler and compasses, and in the sense in which most
of us can imagine that, it is surely possible for someone
who is still more unsophisticated than we are to imagine
seeing a round square. (Someone might draw a straight
line and say that it was a picture of a round square seen
from the edge!) So it will certainly not do merely to
say that what is possible is what can be imagined, or that
what is necessary is what could not be conceived to be
otherwise: it will not do to offer this as a definition.
    268

What I am saying is rather this: look at what goes
on when you imagine what it would be like for something
to be the case, and then you will see more clearly, from
a philosophical point of view, what it is to describe
a state of affairs as "possible". The important thing
is that the various properties (and relations) which we
can see in the world need not be arranged as they are,
in the instances which they happen to have, and we acknow-
ledge and make use of this fact when we imagine non-
existent states of affairs, or when we talk about them
or write stories about them or wish for them, or draw
pictures of then. In short, that which makes imagin-
ability possible in some cases, is what explains how
states of affairs my be possible though not actual,
namely, the loose connection between universals and
their actual instances.

7.A.8.  All this may be used to explain the notion of the
set of "truth-conditions" of a proposition. We have
seen (of. 5.E.1) that in general whether the proposition
expressed by a sentence S is true or not depends on three
things:
(a) the non-logical words in S and their meanings,
(b) the logical form, and corresponding logical
technique
(c) facts, or the way things happen to be in the world.
This shows that when a logical form, and a set of non-
logical words are combined to form a sentence expressing
a proposition, the linguistic roles of the logical constants,
and the semantic correlations governing the non-logical
words together determine a set of possible states of
affairs in which to utter the sentence would be to make a
true statement. These are the "conditions" in which
    269

applying the logical technique for determining its
truth-value would yield the result "true". There is
not usually just one truth-condition, as pointed out in
6.F.11. Every statement ignores some aspects of the
states of affairs in which it would be true. Variations
in these "irrelevant" aspects help to increase the size
of the class of possible states of affairs in which such
a statement would be true. Whether it is true or not
depends on whether one of these possible states of affairs
actually obtains, i.e. is a fact. (This can be gener-
alized. If R is any rogator, and T is one of the values
which it can take, and if A is an argument-set for
that rogator, then R and A together determine a class of
possible states of affairs, or T-conditions, namely those
in which applying the technique for determining the value
of R for A would yield the result T. "Rogator" was
defined in 5.B.6.)

7.A.9.  By taking note of the fact that universals can
recur, that is by abstracting from the particular cir-
cumstances in which we see shapes, colours, and other
properties, we are able to learn such things as that this
book might have been the colour of that one, there might
have been a box on the floor the same shape as the one on
the table, there might have been pennies in my right-hand
pocket instead of in my left-hand pocket. It should
be stressed that there is nothing mysterious in this:
apprehending the universality of a shape or colour or
other property, does not involve making use of "inner-eyes"
or other occult faculties: it is just a matter of using
ordinary intelligence and ordinary eyes and imagination.
We thereby take note of very general facts but for which
our language, thought, and experience could not have been
    270

the sorts of things which they are. (See chapter two,
section B.)

7.A.10. All this shows that there is a non-linguistic
kind of possibility. By this I mean merely that when
we talk about possibilities we are not talking about
combinations of words which are permitted by the rules
of some language. Contrast this with Schlick's remark
(Feigl and Sellers, p.l54): "I call a fact or process
'logically possible' if it can be described, i.e. if
the sentence which is supposed to describe it obeys the
roles of grammar we have stipulated for our language."
The class of possible states of affairs is much
more complex and numerous than the class of sentences
formulable in any language. Sentences are discrete
and individually describable, and, at any one time,
either finite in number or able to be arranged in a
fairly simple sequence, unlike possible states of affairs,
which shade continuously into one another in many
different dimensions. (Austin: "Fact is richer than
diction.")

7.A.11. Further, it should be noted that the concept of
possibility cannot be reduced to that of logical possi-
bility or analytic possibility. To say that a proposi-
tion is not a formal or analytic falsehood is to say that
one cannot show it to be false merely by considering the
meanings of words and the logical techniques of veri-
fication corresponding to its logical form. This simply
means that observation of the facts may be relevant to
determining its truth-value. It does not imply that
any state of affairs is a possible one, or that there is
    271

a non-empty class of possible states of affairs corres-
ponding to it as truth-conditions. For the question
whether, for some other reason, the truth-value would
come out as "false" in all possible states of affairs
is still left open. At any rate, some argument is
required to show that it is not left open: and that
shows that there are different concepts of possibility
here.
To sum up, knowing what possibility is, is not a
matter of knowing the laws of logic and seeing which
descriptions of possible states of affairs do not
contradict them, neither is it a matter of knowing which
combinations of words are permitted by the rules of
some language. It is a matter of knowing that the
world is made up of material things and their properties
and relations, and knowing that these properties and
relations are not essentially tied to those material
things which actually instantiate them, that they need
not occur in the arrangements in which they do occur, or
at the places and times at which they do occur. Other
factors might have been taken into account, such as
loose ties between particulars and the actual places
and times at which they exist. (This table is here now,
but it might have been next door.) These factors will
be ignored, not being relevant to our main problem.

(Note. This conception of possibility can be used to
solve some philosophical problems. For example, it makes
puzzles about the identity of indiscernibles disappear.
If "indiscernible" means "could not possibly be different
in some respect", then the principle that indiscernibles
are identical is true. If "indiscernible" means "is
not actually different in some respect", then the
principle is false. A sphere in an otherwise empty
universe will have two halves, despite its symmetry,
because one of them could be a different colour from the
    272

other, even if it is not. If the principle were
correct in its second sense, then, although an unsym-
metrical object could exist alone in the universe, it
could not be gradually transformed into a sphere, for
on becoming a sphere it would consist of two (or
Indefinitely many - if we take account of sectors)
parts which have all their properties and relations in
common. What would happen to it then? It is clear
that the principle is absurd.)

7.B.    Necessity

7.B.1.  The concept of possibility has been shown to have
an application on account of the loose tie between uni-
versals and actual instances.
But understanding talk about possibilities is not
enough for an understanding talk about necessity. For
that, one must know what is meant by "the range of all
possibilities", or "what is not possible". The use of
negation or the word "all" has to be defined afresh in
these modal contexts, and corresponding to different
ways in which its use is explained there may be different
kinds of ranges of possibility, different Kinds of
necessity, different kinds of impossibility. We must
therefore proceed with caution.

7.B.2.  We have seen that the concept of "analytic"
possibility is not very substantial (7.A.11). Similarly,
the kind of necessary truth which we have found in
analytic propositions does not seem to affect the range
of all possible states of affairs, since the necessary
truth of such a proposition is merely a matter of its
having certain general features which ensure that it
comes out as true, no matter what particular things
or kinds of things it is about, and no matter how things
    273

happen to be in the world. So its necessary truth is
not due to anything at all specific which has to be the
case in all possible states of affairs. Hence its
being necessarily true imposes no special restrictions
on what may be the case: it does not seem to limit the
range of all possible states of affairs in any way.
Let us try to find a more substantial and more
general concept of necessity by following up what Kant
said in the "Critique of Pure Reason" (B.4),f namely that
a statement 'is necessarily true If
"it is thought with strict universality, that is in
such a manner that no exception is allowed as
possible, it is not derived from experience."
(The last clause, "not derived from experience", will be
ignored for the time being. See Appendix VI.)

7.B.3.  What is strict universality? How can no
exception be allowed as possible?
Suppose the following were true: (1) "All triangles
are red". It would then be a universal truth
with no exceptions, but it would not be strictly universal,
since it is clear that triangles do not have to be red:
even if all triangles happen to be red, I can see, just
by looking at a triangle that although it is not green
it might have been, while still a triangle. Although
there are no exceptions, nevertheless exceptions are allowed
as possible. By contrast, the proposition (2) "All
squares have exactly four angles" is not only a universal
truth without exceptions, but it is strictly universal,
since there could not be any exceptions: even if there
were any other squares than the ones which there actually
are, they would all have exactly four angles. (2) is a
necessary truth, whereas (1) would not be necessary, even
if it were true.
    274

7.B.4.  This can be expressed more generally If it is
tied up with some of the remarks in 7.A about possibility
and properties. First of all, let us consider pro-
positions of the form "All P's are, Q's", where "p" and
"Q" are descriptive words referring to observable pro-
perties. Now recall that the universality of a property
has three aspects (7.A.4). Firstly, the property may
occur in several different actual particular objects.
Secondly, it might have been Instantiated in some of those
particular objects which are not in fact instances.
Finally, there might have been objects, which if they had
existed, might have had the property. We may therefore
say that the property referred to by "P" is necessarily
connected with the property referred to by "Q", and the
proposition "All P's are Q's" is necessarily true, if,
and only if, all the objects in the first class, namely
all those which actually have the property referred to
by "P", also have the property referred to by "Q"; all
the objects in the second class, that is all those which
might have had the property P, would, if they had had
it, also have had the property Q; and, finally, all the
objects in the third class, namely those which might
have existed though they do not, would, if they had
existed and had the property P, also have had the property
Q. In short, there are three sorts of potential counter-
examples to a proposition of the form in question, namely
those objects which have the property P, those which do
not, but might have had it, and those which, if they had
existed, might have had it; and to say that no exceptions
are allowed as possible, is to say that none of these
objects, if it had (existed and) had the property P would
have been without the property Q.
    275

7.B.4.a.    We can generalize this further if we recall
that only propositions are being discussed which are
universal in form and mention no particular objects
(see Appendix I). For a sentence made up only of
logical constants and descriptive words referring to
properties is true if and only if certain relations hold
between the classes of objects possessing certain pro-
perties,relations such as inclusion, or mutual exclusion.
Such a proposition is necessarily true, then, if all
the classes of objects with the specified properties do
in fact stand in the specified relations, and, in
addition, they would do so even if other objects had the
properties in question than the ones which actually do
so, even if there were other objects in existence than
the ones which there actually are. In such a case not
only are there no exceptions, but, in addition, no
exceptions are possible.
This could be generalized a stage further to include
propositions referring not only to properties, but also
to relations, such as "two feet away from", "brighter
than", "inside", and so on, but I shall leave that to
the reader.
It should be noted how this definition differs from
the definition of an analytic propositions here we make
no mention of "Identifying relations" between meanings,
nor do we restrict the sources of necessity to topic-
neutral features of propositions. Thus, it is so far
an open question whether all necessary truths are analytic.

7.B.5.  All this may suffice for a definition of
"necessity", but it is not very helpful, since it dose
not explain how it comes about that any statement is
necessarily true, or how we can ever tell that it is.
    276

What is missing is an explanation of how we can tell
whether a counter-factual conditional statement asserting
that no exceptions are allowed as possible is true, that
is, how we can tell what would have been the case if
certain objects had had certain properties. How do I
tell that if the piece of paper on which I am now typing
had been square then it would have had four angles?
How do I tell that if there had been a tetralateral block
of wood on my table then it would also have been tetra-
hedral? (Cf. 2.C.8.) It should not be assumed that
simply because I know how to tell that something or
other might have been the case, I know how to tell what
else would then have been the case, or that I know how
to tell what would be the case in all possible states
of affairs. (See 7.B.1.)
There seems to be so complicated a range of possible
worlds and possible states of affairs that it is hard
to see how anything at all could be excluded from the
range. There might be worlds in which space had five
dimensions, or only two. There might have been a world
in which there were only sounds, and no space or spatial
objects or spatial properties (see Strawson's "Individuals",
chapter two). There might have been worlds in which
properties and relations existed which were quite unlike
anything we can imagine. Or might there?
It seems clear that there is a tangled and complicated
question here, which is not really relevant to such prob-
lems as concern us, for example, the problem whether it
is both necessary and synthetic that two properties which
actually do exist always occur together (such as the
property of being four-sided and the property of having
four angles). The source of the trouble is that there
    277

are different concepts of necessity, and different kinds
of ranges of possibilities.

7.B.6.  But our definition of "necessary truth" was
restricted in such a way that we need not take account
of all these complexities, for it is concerned only with
classes of objects possessing properties which actually
do exist in our world. We therefore have no need to
talk about all possible worlds, since we can limit
ourselves to talking about all possible states or configu-
rations of this world, where "this world" describes a
world in which the same observable properties and relations
exist as exist in our world. (It should be recalled
that the existence of universals need not involve actual
existence of instances. See section 2.D.) Thus, since
we are talking only about states of this world, we need
not consider worlds without space and time, or five-
dimensional worlds. (Compare what Kant says about his
Copernican Revolution in the Preface to the second
edition of C.P.R. B.xvi-xvii, etc.) (see note at end
of this section.)

7.B.7.  Now we may return to an explanation of how it is
possible to tell that a statement is necessarily true.
Once again we shall make use of general facts about
universals, that is, observable properties and relations.
It has already been pointed out (7.A.4,ff) that a
property exhibited by an object is the sort of thing which
can recur. Now we must notice further that one object
may possess more than one property at the same time. A
material object may be both red and round. It may be
cubical and transparent. It may be cigar-shaped, glossy
and green.
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When two or more properties are exhibited by an
object, we may be able to see that some of them have no
connection with the others. For example, the fact that
a box is cube-shaped has nothing to do with the fact that
it is red. Not only could the property of being cube-
shaped occur in other objects, in addition it could
occur in other objects without the colour which acco-
mpanies it in this one. Even if neither of these pro-
perties  did occur without the other (which, of course,
is not the case), we could still nevertheless see that
there might have been an object which was cubical without
being red, or red without being cubical. one need not
have seen either of the properties actually exhibited
without the other in order to see that they are capable
of occurring separately, any more than one must have seen
the shape or the colour in another object at another
time or place in order to see that it can have other
instances. (Cf. 7.A.5.) All we need is our eyes and
intelligence, and the knowledge of what it is to be
cube-shaped and of what it is to be red, and then one can
see that it is possible to recognize either property in
an object without its mattering whether the other is
there or not.
Similarly, where there are two properties which we
have never, as a matter of fact, seen in the same object
at the same time, we may be able to tell that there could
be an object with both of them. I have never, as far as
I know, seen an object which is both cigar-shaped and
blue, but there is nothing in either property, insofar as
it is an observable property, which excludes the presence
of the other. I know what it would be like to recognize
both properties in one and the same object.
    279

Thus, even if the two statements 'All cubes are red"
and "Nothing cigar-shaped is blue" are true, that is have
no exceptions, nevertheless they do allow exceptions as
possible. We can see, by examining the properties
concerned, that they are not necessarily connected.
It is by contrast with this sort of case that I shall
explain how statements can be necessarily true.

7.B.8.  We have added a refinement to our concept of
possibility by taking note of the fact that not only
are universals not essentially tied to their actual
instances, So that they can be instantiated in other
places and times than they are in fact, but, in addition,
they are not essentially tied to one another, so that
they can occur in different combinations from those in
which they in fact occur. Universals are unfettered
by their instances, and also, sometimes, by one another.
Not always, however, and limitations on this second sort
of freedom generate the kind of necessity which is of
interest to us. There may be something in the constitution
of a property which ties it to another property,
or which prevents its occurring with another property.
If so, this may have the consequence that a statement
using words which refer to.: those properties is not only
true, but necessarily true, since no exceptions are allowed
as possible. (Exceptions would be objects in which these
tied properties occurred separately, or in which incompatible
properties occurred together.)
If there are any such relations between properties
which are not identifying relations, then they will
provide us with a new class of relations between descriptive
words referring to properties so related, and here, as in
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the case of analytic propositions, the relations between
descriptive words, together with features of the logical
technique for discovering truth-values of a statement,
may determine the outcome of applying the technique in
any possible state of affairs (cf. 6.C.1.). So if
there is some way of knowing that the properties
referred to by words stand in such relations, then we
may be able to determine the outcome of an empirical
investigation to discover the truth-value of a proposi-
tion, without actually making that investigation. If
this is so, then we shall have found a new type of
illustration of the fact pointed out in 5.E.1 and
5.E.2, that although in general the value of a rogator
depends on (a) the arguments, (b) the technique for
discovering values and (c) the facts, nevertheless there
are cases where without knowing any facts (i.e. without
having any empirical knowledge of how things happen to
be in the world) we can discover the value by taking
note of relations between the arguments and examining
the general technique for determining values. We shall
have found a way of telling, without knowing which
particular objects there are in the world, nor what
properties and relations they instantiate, that none of
them is an exception to what is asserted by some statement.
That is, we shall have found a new kind of a priori
knowledge of the truth of a statement. (See end of 7.B.2.)

7.B.9.  If there are these connections between properties,
and if we can know that they exist, for example by
examining the properties in question, then this will
explain how we can be in a position to assert such state-
ments as "If this had been square then it would have had
    281

four angles", or "If this had been turquoise, then it
would not have been scarlet". Thus, by talking about
properties, and their ties with or independence of one
another and their instances, we are able to explain
some uses of the words "necessary" and "possible", and
counter-factual conditional statements.
To sum up: since properties are not tied to their
actual instances we can talk about what might have been
the case in the world, and since they may be tied to one
another (this includes incompatibility), we can talk about
what would have been the case if so and so had been the
case. Hence we can talk about statements to which no
exceptions are possible, that is, statements which are
necessarily true.

7.B.10. All this should show that the concept of
necessity is far more complicated than the concept of
possibility. (See 7.A.1, 7.B.1.) Only the latter is
required if we are to be able to use our ordinary language
to describe new situations, to ask questions about unknown
facts, to understand false statements. We need only
understand that the range of things which might have been
the case is wider than the range of things which are the
case. The concept of necessity is required when we grow
more sophisticated, when we wish to do more than simply
describe what we see or ask questions about what is to be
seen in the world. It comes in when we wish to draw
inferences, when we wish to know about the properties of
all things of a certain kind without examining them all,
when we do mathematics or philosophy, or try to explain
what "makes" things happen as they do, or when we ask
whether happenings are avoidable or not. It comes in
also when we try to justify the assertion of a counterfactual
    282

conditional statement, about what would have happened
if something or other had been the case. In order to
understand talk about possibility, one need only see
that states of affairs are possible which are not actual,
whereas in order to understand talk about necessity, one
must, in addition to understanding talk about possibility,
also see the reasons why the range of possibilities is
limited in certain ways. The former requires only a
perception of the loose tie between all universals and
their actual instances (by abstraction from particular
circumstances), the latter requires perception of the
strong ties between some universals and other universals.

(7.B.note.  It should not be forgotten that in all this
we are talking only about the kind of necessity which
arises out of limitations on the possible states of
this world, in which objects have properties and stand
in relations only of the sorts which are capable of
having instances in our world. There may, however,
be other kinds of necessity, other kinds of limitations
on what may be the case in the observable world.
(Compare 7.B.6.)

For example, there may be limitations on the range
of possible states of affairs - or possible worlds -
which can be talked about in a language using a distinction
between subjects and predicates. Or there may be
limitations on what can be the case in states of affairs
which are observable by beings with senses of any kind.
(E.g. a sense which enables them to perceive magnetic
and electrical properties directly.) Perhaps there is
some other kind of necessity, called "natural necessity"
by Kant, which is operative when types of events or
states of affairs stand in causal connections.

Kant talked also about a kind of necessity which
involves particular objects, such as the necessity in the
synthesis of an experience of a particular object (cor-
responding to the "form" of the object), but this sort
of thing need not concern us. We have decided to ignore
statements mentioning particulars - see Appendix I - and
in any case the ascription of necessity to such statements
    283

can usually be explained in terms of their being instances
of some universal statement which is necessarily true,
as when we say "Tom's bachelor uncle must be unmarried".

There is no space here to discuss a sufficiently
wide concept of necessity to allow us to take account of
all these cases and such questions as whether it is
necessary that space is three-dimensional. It is not
clear to me that there is a perfectly general and
absolute concept of necessity. For example: if a
statement is necessarily true, then it is not obvious
that it makes sense to ask whether it is necessarily
necessarily true. See end of Appendix I. There may
be only a relative concept, operating at different levels,
each level being characterized by the type of thing
which can count as the reason why a statement is necessarily
true. At the level which concerns us, the
reason must be that there is a perceptible connection
between observable properties or relations.)

7.C.    Synthetic necessary connections

7.C.1.  This chapter has so far shown, by drawing atten-
tion to certain features of the conceptual framework which
we presuppose in using descriptive words and referring
expressions (of. 2.B.4-6), how we can understand talk
about possibility, and, in a vague way, what is meant by
saying that some statements are necessarily true. A
statement of the sort under discussion (using only des-
criptive words and logical constants) is necessarily true
if there are connections between the properties referred
to by the descriptive words, which ensure that no particular
object could be a counter-example to the statement,
since certain combinations of those properties in
one object are ruled out by the connections between them.
Now we must ask whether all such connections between
properties are identifying: connections (see 6.C) or
whether some non-identifying or synthetic connections
    284

between properties can ensure the necessary truth of a
statement.

7.C.2.  Let us be clear about what we must look for.
if knowing the meanings of words (sharply identified
meanings, that is, see 6.D.3 and section 2.C), suffices,
on its own or together with purely logical (topic-neutral)
considerations, to justify the claim to know
that properties are related in some way, then that is
an identifying relation, not a synthetic relation.
For relations between properties to be synthetic, know-
ledge of them must require something more than the know-
ledge of which properties they are, and the "something
more" must not be purely logical. But what more could
there be? Is there some way of examining properties
themselves (a non-logical enquiry, since it presupposes
actual acquaintance with a special kind of subject-
matter) in order to discover that there is a connection
between them, a connection which need not be known in
order to know which properties they are? We must now
investigate some examples, and see whether this sort of
insight is possible. If any such insight is possible,
it will explain Kant's talk about "appeals to intuition".
(See 6.B.2, 6.C.11.)

7.C.3.  The most interesting examples come from geometry,
though there are many other kinds which cannot be described
here for reasons of space. (More examples will
be found in Appendix V. See also 2.C.8, 3.C.10 and
3.D.10).
In 2.C.8. we defined the words "tetralateral" and
"tetrahedral", the former referring to the geometrical
property of being bounded by four plane faces, the latter
    285

to the property of being bounded by plane faces and
having four vertices. I argued that the two words refer
to two different properties which can be identified
independently of each other, since one can notice either
property, attend to it, think about it, or talk about
it, without being aware of the existence of the other.
So in order to know that they are inseparable it is
not enough to know which properties they are: in
addition one must carry out some sort of construction,
either in imagination or with sheets of cardboard, or
with diagrams, or somehow examine the two properties,
in order to be sure that all possible ways of putting
four plane surfaces together to bound a closed space
must result in there being exactly four vertices, and
that no other number of plane surfaces can yield exactly
four vertices. This examination presupposes acquaintance
with a special kind of property, and cannot take a
topic-neutral form. It does not, therefore, involve
drawing conclusions in a purely logical way, so cannot
account for knowledge of an analytic truth, according to
the definition of 6.C.10.
I call such a construction, carried out for the
purpose of enabling oneself or someone else to perceive
the connection between two or more properties (or
relations), an "informal proof". (For more detailed
remarks see next section.)

7.C.4.  It seems, therefore, that since an informal proof
of a non-logical kind is required, in addition to a
specification of the meanings of "tetrahedral" or "tetra-
lateral", for a justification of the assertion (1) "All
tetralateral objects are tetrahedral", this must be a
    286

synthetic statement. Its justification is quite
different from the justification of (2) "All gleen things
are glossy", which proceeds by specifying that the word
"gleen" refers to a combination of the property referred
to by "glossy" with another property (that referred to
by "green", say), and then taking account of purely
logical properties of the technique for verifying
statements of the form "All P things are Q". There is
no identifying relation between the meanings of "tetra-
lateral" and "tetrahedral", from which a logical proof
of (1) could proceed.

7.C.5.  There are many more examples of this sort of
connection between properties, some more problematic than
others. Here are a few. (In most cases "improper"
or synthesized properties are involved.)
(a) The property of being bounded by three straight
lines is necessarily connected with the property
of being a plane figure with three vertices.

(b) In 3.D.3 and 3.D.4 we described two different
"procedures" for picking out triangular shapes, of
which the first involved memorizing one triangular
shape and picking out others on account of their
deformability into it, while the second involved
pointing at sides in turn and reciting "Bing bang
bong". Here are two synthesized properties which
seem to be necessarily connected. Can the connection
be shown, by purely logical considerations,
to follow from identifying relations?

(c) No closed space is bounded by three planes. Is
the incompatibility between the property of being a
closed space and the property of being bounded by
three plane surfaces analytic?

(d) If a cube is inside a sphere and a piece of wire
is inside the cube, then the wire is inside the
sphere. Is the transitivity of the relation
"inside" due to some identifying fact, or can one
know which relation it is without being aware of its
transitivity, or anything which logically entails
its transitivity?
    287

(e) Any pattern made up of regularly spaced rows
of regularly spaced dots is also made up of a
sequence of regularly spaced columns of dots.
...................
....................
.....................
......................
.......................
........................
.........................
..........................
It also consists of an array of diagonal rows of
dots. (Diagonal rows inclined at various angles
may be seen in the array.) All these several
aspects of one pattern seem to be necessarily
connected: the presence of some of them can be
seen to entail the presence of others Are these
identifying connections between the aspects? Are
they purely logical consequences of identifying
connections?

(f) Consider Kant's example; no left-handed helix
may be superimposed on a right-handed helix.

(g) Consider Wittgenstein's example (R.F.M., Part I,
50.): any rectangle can be divided into two parall-
elograms and two triangles (by a pair of parallel
straight lines passing through opposite corners, and
a third parallel line between them). Is this due
to some identifying fact about the meaning of
"rectangle"?


                


(h) Any object with the property of having a shape
which occupies the space common to three cylinders
equal in diameter whose axes pass through one point
at right angles to one another, has also the property
of being bounded by twelve equal foursided faces,
each of which is part of a cylindrical surface
(This is the shape obtained by pushing a hollow
cylindrical cutter through a potato three times in
mutually perpendicular directions in a symmetrical
way.)
    288

(i) If the side of one square is the diagonal of
another, then the former can be divided into pieces
which, on being rearranged, form two squares con-
gruent with the latter.

                

In all these cases try, seriously, to say what the
linguistic conventions are on which we must be relying
implicitly when we perceive these necessary connections.
(See Wittgenstein's "Remarks on the Foundations of
Mathematics" for a serious attempt to meet this challenge.
Cf. 7.10,ff., below.)

7.C.6.  To all this the following objection may be made:
"Of course there is no simple identifying relation between
the meanings of the words 'tetralateral' and 'tetrahedral',
and between the other words used in your examples, but
this does not mean that the necessary connection between
them is not a logical consequence of identifying relations
between meanings. For the meanings of the words must
be explained in terms of more fundamental geometrical
words, such as 'plane', 'line', 'point', intersection',
etc., and the meanings of these words stand in identifying
relations, from which the connections to which you have
drawn attention can be deduced by purely logical con-
siderations." The objector will thereupon produce some
axiomatic system of geometry, in which his "more funda-
mental" words occur as primitive or undefined terms,
which he will use to define the words which interest us,
and then, triumphantly, he will deduce from the axioms
of his system, together with his definitions, using only
logically valid inferences, that such statements as
"All tetrahedral objects are tetralateral" are theorems.
    289

But this is not enough. He must show first of all
that his definitions do not simply take as identifying
relations, relations which can be regarded as synthetic
necessary connections. That is to say, it is not
enough for him to show that words like "tetrahedral"
can be defined as he suggests, he must also show that
they have to be so defined, that it is impossible to
understand them in some other way (e.g. by associating them
with immediately recognizable properties), otherwise he
will be arguing in the manner criticized in 2.C.10 and
3.C.10 (i.e. trading on ambiguities, and using loose
criteria for identity of meanings.) In short, he must
show that his definitions are definitions of the words they
purport to define. Secondly, he must show that his
"axioms" are in some sense true by definition, that they
state or are logical consequences of identifying facts about
the meanings of the primitive terms, and that they are
not themselves statements which are necessarily true and
synthetic. We could, of course, adopt additional verbal
rules of the sort described in section 4.C to make the
sentences expressing his axioms into expressions of
analytic propositions, but he must show that only if
such rules are adopted can the words in these sentences
be understood as referring to those geometrical features
to which they do refer. Once again: it is not enough
for the objector to show that words can be defined in
such a way as to make certain sentences express analytic
propositions. He must show that as ordinarily understood
they have to be so defined, or at least that unless they
are so defined they cannot refer to properties which are
necessarily connected.
    290

7.C.7.  It is far from obvious that there must be
identifying relations of the sorts which could corres-
pond to the axioms of an axiomatic system of geometry.
(See remarks about superfluous "links between descriptive
expressions", in 2.D.3 and 2.D.4.)
After all, we are not concerned with an abstract
system of lifeless symbols having no empirical use,
but with words which describe properties or aspects of
physical objects which we can perceive and learn to
recognize. (See 3.A.2) These words occur in ordinary
everyday statements, such as "Here's a table with a
square top", or "These three edges of this block of
wood meet in a point", expressing contingent propositions
which may be true or false. But (as pointed out in
2.D.2 note, and again in section 5.A), no system of
axioms can suffice to give words meanings of this sort,
for, in addition, semantic rules are required, correlating
the words with non-linguistic entities such as
observable properties. If we must have such semantic
correlations, is it not conceivable that they may, on
their own, suffice to give words their meanings, and
determine their use and their relations to one another,
without the aid of any further "axioms" or "linguistic
conventions"? If so, it is surely an open question,
requiring further investigation, whether all such rela-
tions are either identifying relations between meanings
or logical consequences thereof. As remarked in (3.A.4.
(of. 3.D.9), in order to be able to use a descriptive word
one must be able to recognize some universal immediately
so it is an open question whether some of these immediately
recognized properties or features stand in relations
with others, of a kind which must be discovered by
examining them: why should the only things we can see
    291

using our eyes and intelligence be facts about
particulars?

7.C.7 (note).   It is very common nowadays to think that
any geometrical proof must start from axioms which are
all arbitrarily selected, serving as expressions of
linguistic conventions of some kind, specifying the
meanings of the geometrical words involved in them.
the reason why people think this is that different
systems of axioms may all be internally consistent,
as in usually pointed out in connection with systems
which do not include Euclid's parallel axiom. Consider
Hempel's assertion (in Feigl and Sellars, p.243):
"The fact that these different types of geometry
have been developed in modern mathematics shows
clearly that mathematics cannot be said to assert
the truth of any particular set of geometrical pos-
tulates; all that pure mathematics is interested
in, and all that it can establish, is the deductive
consequences ....." (Compare Russell's definition
of pure mathematics at the beginning of "The
Principles of Mathematics." 2nd. Ed, p.3.)
All this, however, presupposes that internal consistency
and deductive consequences are all that interest us, but
it need not be all, if geometrical theorems are intended
to state facts about observable geometrical properties.
In that case, the axioms include words which refer to
non-linguistic entities, and may be true or false, as
well as consistent or inconsistent with one another.
They are then not definitions, since the words in them
are given their meanings independently, by correlations
with different properties. It is not an accident that
the kind of geometrical proof which involves not logical
deductions from axioms, but the construction of diagrams,
with construction-lines and sides and angles marked as
    292

equal, etc., occurs in a branch of what mathematicians
call "pure" mathematics. Some philosophers (unlike
Frege and Kant) give the impression that they are quite
unaware of this, as is shown by the quotation from
Hempel and many remarks which I have heard in discussions.
Perhaps it is wrong to think that by examining
geometrical concepts or properties we can "discover" that
the parallel axiom is true, but that is a very special
case, since it concerns infinitely long lines (and
therefore not ordinary observable properties), and it does
not follow that other axioms are also merely matters of
convention. For example, the "theorem" that every rec-
tangle is divisible into two triangles and two paralle-
lograms is on quite a different footing from the assertion
that two parallel lines never intersect. it is not a
mere defining postulate, and neither is it a contingent
fact. (See 7.E for more on this.)

7.C.8.  The argument so far may be summarized thus: at
some point in the explanation of the meanings of des-
criptive words we must point to objects of experience
with which they are correlated (i.e. we have to appeal
to what is "given in intuition"). But then it is an
open question whether these non-linguistic entities
(properties) are so related as to ensure that some of
the statements using words which refer to them are
necessarily true, or whether all such relations must be
identifying relations between meanings. This is not a
question which can be settled merely by pointing to a
set of axioms or linguistic conventions which could set
up identifying relations and make statements analytic,
for to say that they can do the job of making statements
necessarily true is not to say that they are indispensible.
    293

And to say that anything else which does the job must
give the words the same meanings anyway, is to base a
question-begging argument on the use of loose criteria
for identity of meanings. (2.C.10, 3.C.10.)
I am trying to show that some very superficial
and slipshod thinking lies behind many denials of the
existence of synthetic necessary truths.

7.C.8. note.    We are not interested in the question
whether some statement in some actual language is or is not
analytic, or whether the relation between certain sets
of words in some actual language is an identifying
relation. (Cf. 6.E.9.) This sort of question is of
little philosophical interest and has to be based on an
empirical enquiry in order to discover exactly what
people mean by the words they use, and the discussion
of chapter four and section 6.D shows clearly that there
may be no definite answer to such a question, or there
may be answers which can be summarized only in a statis-
tical form. (Cf. 4.B.7.) We are concerned only with
the question whether certain sorts of statements have
to be analytic, or whether it is possible to give their
words meanings which are identified independently of one
another, and then discover, by examining the properties
referred to, that the outcome of applying the logical
techniques for determining the truth-values of such
statements can yield only the result "true". Even if
statements referring to such properties are analytic in
English, or in some axiomatic geometrical system, owing
to the fact that auxiliary rules have been adopted,
setting up identifying relations between meanings, that
does not prove anything, for the rules may be superfluous.
    294

Failure to appreciate this point can cause people to
argue at cross-purposes, for example over the question
whether it is analytic that nothing can be red and green
all over at the same time.

7.C.9.  All this should at least show that the question
whether some necessary truths are synthetic, on account
of being true in virtue of relations between universals
which are neither identifying relations nor logical
consequences of identifying relations, is an open question.
It has to be settled by a closer investigation of what
goes on when one examines a pair of properties, such as
the property of being tetralateral (bounded by four plane
surfaces) and the property of being tetrahedral (having
four vertices), and discovers, possibly with the aid of
an informal proof, that there is some unbreakable con-
nection between those properties. That is, we must look
at what goes on when a person discovers, perhaps with the
aid of an informal proof, that, owing to the relation
between some properties (relations such as entailment
or incompatibility, a sentence expresses a universal
proposition which not only has no exceptions in fact, but
which allows no exceptions as possible. (See 7.B.2,
7.B.4.ff.)

7.D.    Informal proofs

7.D.1.  So far, I have tried to show that just as we can
see (using our eyes and ordinary intelligence, cf. 7.B.7)
that the redness of a round and red object is able to
occur elsewhere without the roundness, even if as a
matter of fact it does not, so can we see that some
    295

properties are unable to free themselves from certain
others, with which they are always found, or unable to
cohabit with some with which they are never found. For
example, I have argued that by examining the appropriate
properties and discovering their relations we can detect
the necessary truth of such statements as "All tetra-
hedral objects are tetralateral" or "No closed spaces are
bounded by three plane surfaces". This shows that not
only are we able to see that the actual state of affairs
in this world1 is not the only possible one, i.e. by seeing
that universals are not essentially tied to actual
particular instances nor to one another, but, in addition,
we can see that there are some limitations on the ways
in which these universals can be instantiated, some
limits to what may be found in a possible state of the
world. (This can be used to explain Kant's distinction
between "form" and "content", in some contexts, and also,
since it is concerned only with connections between
properties and relations "tangible" to the senses, why
he talked about "the form of sensible intuition" See
C.P.R. A.20, B.34,ff; A.45, B.62; B.457 n.)
Section 7.C was concerned to establish that the
necessary truths discovered in this way are not analytic,
since first of all their necessity is due to non-
identifying relations between properties, and, secondly,
in order to become aware of them, one requires some kind
of insight which is not purely logical, since it pre-
supposes acquaintance with a specific kind of subject
matter and is therefore not topic-neutral. For the
first step I relied on arguments very like those of
__________
1.  (7.B.6.)
    296

3.C.9 and 3.C.5-7, to show that the properties are
independently identifiable, and for the second I relied
on the fact that the insight into the connections
between properties always requires some kind of exam-
ination of those properties. In this section I wish
to say a little more about what goes on when one
examines properties, by talking about informal proofs.
(7.C.3.)

7.D.2.  What happens when I construct an informal proof
to enable someone (possibly myself) to see that proper-
ties are related in some way? There is a very great
variety of cases. For example:
(a) I might enable someone to see that nothing can be
both circular and square by drawing a circle on trans-
parent paper and getting him to try to draw a square on
which it can be superimposed, in the hope that he will
perceive the incompatibility of the two properties.
I might point to a curved bit of the circle and a straight
bit of a square and say: "This sort of thing can never
fit onto that sort of thing".
(b) To show someone that if a triangle has two equal
sides then it has two equal angles, I may point out that
if it is picked up, turned over, and laid down in its
former position with the sides interchanged, it must
exactly fit the position it occupied previously since
each of the two equal sides lies where the other was, and
the angle between them does not change by being reversed.
Hence each of the two angles which have changed places
fits exactly on the position occupied formerly by the
other, so the angles are equal.
(c) To show someone that if a figure is bounded by four
plane surfaces then it must have four vertices, I may
hold up sheets of cardboard and show that the only way
to get four of them to enclose a space is first to form
an angle with two of them, then to form a "corner" or
pyramid without base, by adding a third, then to complete
the pyramid by adding the fourth. He can then count the
number of vertices, or corners. (This also helps to show
why three planes cannot enclose a space.)
    297

(d) To enable a person to see that if anything has both
the property of being a "kite" (four-sided figure with
a diagonal axis of symmetry) and the property of being
a rectangle, then its shape is square, I may draw a
kite and show that a pair of adjacent sides must be
equal if it is symmetrical about a diagonal, and remind
him that a rectangle with a pair of adjacent sides
equal must be square.
(e) To enable a person to see that any rectangle has
the property of being divisible into two triangles and
two parallelograms, I may simply draw a rectangle, and
then draw three parallel lines obliquely across it so
that each of the two outer ones passes through one of
a pair of opposite corners. (See 7.C.5, example (g).)
Owing to the enormous variety of cases, I shall be
able to make only a very few rather vague and general
remarks. (See Wittgenstein's "Remarks on the
Foundations of Mathematics", for a detailed discussion
of many more examples.)

7.D.3.  First of all, I claim that each of these proofs
is perfectly rigorous, and having once seen and under-
stood it I am perfectly justified in asserting and
believing the general statement which it is supposed
to prove, such as "Nothing is both circular and square",
or "Every rectangular figure is divisible into two
triangles and two parallelograms". (The claim that
such a proof is valid is a mathematical claim, not a
philosophical one, since it is to be tested mathematically
by trying to construct counter-examples: more on this
presently.)
What happens when I see such a proof as a proof, when
I see, as a result of going through the proof, that two
properties are connected, and a universal statement
necessarily true? The answer seems to be that I pay
attention to a property, and notice that although it can
    298

be abstracted from the particular circumstances in which
it is instantiated (see 7.A.6), so that it could occur
elsewhere, and be recognized, even if it does not
actually do so, nevertheless, it cannot be abstracted
from the fact that it occurs in an object which has
some other property (or from which some other property
is absent). In particular, the construction of the
proof may show me how, once I have found any other object
which has the first property, I can repeat the method of
construction of this proof in order to demonstrate that
the other object has (or has not, as the case may be)
the other property. The proof gives me a general prin-
ciple for going from one property or aspect of an object
to another, thereby showing me the reason why no excep-
tions to the proved general statement are "allowed as
possible" (7.B.2,ff. 7.B.ff.) In Wittgenstein's
terminology: the proof serves as the "picture of an
experiment" (see R.F.M. I.36.) It may be better to say:
the proof serves as a picture of a proof.

7.D.4.  Perhaps we can see more clearly what goes on by
distinguishing token-proofs from type-proofs. A token-
proof is the particular event or set of marks on paper
etc., spatio-temporally located, observed by you or me.
The type-proof is a new universal, a property common to
all token-proofs which use the same method of proof.
The function of the token-proof is to exhibit the common
property, the type-proof (a pattern); and to have grasped
the proof, to have seen "how it goes", is to have seen
this new universal. Now, the essential thing about
the new property is that it is made up of various parts
connected together (compare: the shape of a cube is a
property made up of various parts connected together,
    299

such as the several faces - see section 3.D on non-
logical synthesis). We may think of it as a "bridge-
property" which connects one or more of its parts with
others. Thus, when I start with a rectangle, draw
three parallel lines, and end with a figure divided
up into two parallelograms and two triangles, I have
exhibited a bridge-property which starts from the pro-
perty of being rectangular and goes to the property of
being divided up in a certain way. This bridge-property
is a temporal property, like the tune common to two
occurrences of sequences of sounds (cf. 3.A.5.): it has
to be exhibited by an "enduring particular". What
the token-proof shows me, when it shows me that the
property P is connected with the property Q, is that any
object with the property P is capable of being used in a
token-proof of the same type, since P is the starting
point of a bridge-property which leads to Q. Thus the
proof makes evident the connection between two properties
by exhibiting them as parts of a new property. The
token-proof shows how P and Q both "fit" into the type-
proof.
This reveals a relation between words which are
semantically correlated with those properties. In
virtue of this relation some propositions using those
words are necessarily true. (Cf. 5.E.2, 6.C.1.).

7 D.5.  But how do we discover the connection between
the initial property and the bridge-property, or the
connection between the bridge-property and the final
property? How do we see that any object with the property
P must be capable of being used in a proof of this type?
This is a crucial question. Consider a particular
    300

instance: how does the proof that every rectangle is
capable of being divided into two triangles and two
parallelograms show me that every rectangle is capable
of being the starting point of such a proof? Do we
need another bridge-property here, starting from the
property P and ending in the former bridge-property?
Obviously not. Then why not? The answer seems to be
simply that a proof must start somewhere, and wherever
it starts there must be something which is taken as not
needing proof, namely that the first steps of the proof
are possible. The reason why this needs no proof is
that it may be discovered simply by inspecting the
original property. Just by inspecting the property of
being a rectangle I can see that if anything has the pro-
perty then a line traversing it obliquely may be drawn
through one of its corners. A person who cannot see
even this will not follow the proof in question.
In other words, the account of the function of a
proof in terms of type-proofs, or bridge-properties,
is incomplete, since it leaves out the essential fact
that at every stage of the proof something just has to
be seen by examining a universal, namely that the next
stage may proceed from there: this must be something
which requires no proof. The whole point of a proof is
to bring out a connection which is not evident. Where
a connection is evident no proof is required, and this
kind of connection which displays itself must be found
at every "step" in a successful proof.

7.D.6.  By pointing to a particular object I may draw
someone's attention to some property or other universal
instantiated in that object, but I cannot force him to
see it. Similarly, by drawing his attention to a
    301

property or pair of properties I may succeed in drawing
his attention to a connection between those properties
in virtue of which one cannot occur without the other,
or in virtue of which they are incompatible, but I cannot
force him to see it. In some cases I can help him
too see it by constructing a proof, by drawing his
attention to a new property, a bridge-property of which
the other two are somehow parts or constituents. But
I cannot force him to see the bridge-property (I cannot
force him to see what the type-proof is so that he could
recognize it again in another instance: I cannot force him
to see how the proof goes), and I cannot force him to see
how it reveals a connection between the two properties in
virtue of their connection with it.
This is very vague, and ignores differences between
different kinds of proof. Perhaps it will be made
a little clearer by the replies to some objections.

7.D.7.  The first objection is that all my talk about
"seeing" properties and connections between properties
is far too psychological to serve as an account of what
a proof is.
It is important to be clear about the sense in which the
account is psychological and the sense in which it is not.
Certainly the fact that someone takes a proof as valid is not
what makes it valid (cf. 7.A.7, where a similar objection was
raised to my account of possibility). But no account of what
a proof is can avoid using psychological concepts such as
"belief", "certainty", "understanding", "conviction"; for
what is a proof supposed to do? It certainly cannot make
the proposition true which it is supposed to prove. It
cannot make it necessarily true either. The necessary
truth of "Nothing is both round and square" in no way
    302

depends on the fact that anyone has ever proved it
Perhaps its necessary truth might be said to depend on
the possibility of constructing a proof. But what
makes this a possibility is a connection, or set of
connections, between properties - which is precisely
what is shown by the proof. The existence of the proof
(token-proof) does not bring the connections between
properties into existence, for the proof depends upon
them for its own existence, and they are capable of
ensuring the necessity of the proved proposition without
the intermediary of the proof.
The proof neither makes the proposition true, nor
makes it necessarily true. Rather, it brings out the
reason why it is necessarily true. "Bringing out" can
only mean "making evident to someone or other", for the
reason is there, doing its job faithfully, whether any
proof is constructed or not. So the role of proof is
to enable someone to see that a statement is true, or
necessarily true, and it follows that psychological
concepts must be employed in a description of what proof
does and how it does it.

7.D.8.  The error in the objection is to assume that
because psychological terms are used to explain what a
proof is, the validity of proofs is a psychological
matter. But this is not so. (Necessity is not defined
in terms of inconceivability, but both are explained
together. See 7.A.7.)
A person cannot simply turn up and say that he knows
that it is possible for a round square to exist because
he has seen and been convinced by a proof of its possi-
bility. If he has seen a proof, and followed it, then
    303

he has become acquainted with a new universal (the type-
proof: 7.D.4), and, as pointed out in 7.A, a universal
is the sort of thing which can recur, so he must be able
to reproduce the proof and point out its relevant
features to us. He cannot get by with the remark that
he remembers how the proof goes, and can imagine it, but
cannot produce it for us, for what can be imagined by
him proves nothing unless it is the sort of thing which
could be drawn on paper, or otherwise concretely recon-
structed and subjected to scrutiny. Neither is it
enough for him to draw a straight line and say that it
is a picture of a round square seen from the end, for he
must explain in virtue of what this can serve as a picture
of a round square, i.e. how it exhibits the roundness and
the squareness of the thing it is meant to represent.

7.D.8.a.    Of course, a person may produce a perfectly
valid proof which, for the time being, no one else can
follow, on account of its complexity. But this does
not make its validity a subjective matter, any more than
the possession by an object of a complex property (e.g.
a shape, or other structure) is a subjective matter in
cases where only one person happens to be able to "take
in" the property. We all know, at least in a vague way,
what would count as an objective refutation of the pro-
position alleged to be necessarily true, for if we
understand the proposition then we are able to recognize
counter-examples, should they turn up. (This must be
modified to take account of existence-proofs, or proofs
of possibility.)
Even if the proposition happens to be necessarily
true, but not validly proven, we know what would count
    304

as a public demonstration of the invalidity of the proof,
for there must be something about the proof in virtue of
which it is supposed to establish the connection in
question (i.e. there must be a type of which it is a token),
and it could be shown to be invalid by a construction
based on the same principles (a token of the same type)
which leads to a proposition which has demonstrable
counter-examples.
(A full account would describe and classify various
ways in which one may fail to see the validity of a
valid proof, or come to think an invalid proof is valid.
E.g. one may think one has seen a property which one has
not seen. One may have seen a bridge-property, but not
one which does quite what it is taken to do, as when one
notices a connection which works in most cases without
seeing that there is a special class of counter-examples.)

7.D.9.  Next it may be objected that what I say is just
wrong, since what really goes on in a proof is that, in
a "dim" way, we are shown how a formalized proof would go,
starting from identifying relations between concepts and
drawing purely logical conclusions, without any need for
such things as "showing" the connections between proper-
ties. Since it looks as if I am not doing any such thing
when I use an informal proof to enable someone to see that
all tetralateral objects are tetrahedral, it is up to the
objector to say what the formalized proof is that I am
unwittingly presenting, e.g. by showing me the identifying
relations or definitions from which he thinks I am drawing
purely logical conclusions, whereupon the arguments of
7.C.6 will come into play.
Secondly, the objector will find himself in difficulties
    305

as soon as we ask how the conclusion follows from those
statements of identifying facts which he claims to be
implicit premises in the proof. For, as pointed out in
5.C.9 and 5.C.10, etc., in order to see that some
inference is logically valid, it is necessary to per-
ceive properties of general logical techniques correspond-
ing to the logical forms of propositions, or connections
between such techniques, and this is just the sort of
insight into the connections between universals which is
provided by the informal proofs whose existence the
objector wishes to deny. (The difference is only that
logical techniques are topic-neutral, whereas we are
discussing Informal proofs concerned with special kinds
of observable properties.)

7.D.9.a.    This last point is important, because one of the
strong motivating forces behind the desire to establish
that all necessary, truths are analytic, or that all
apriori knowledge is knowledge of analytic truth, is the
desire to eliminate the need to talk about special kinds
of "insight" into the relations between universals. It
is apparently thought that if all necessary truth and
apriori knowledge can be shown to be derived by purely
logical considerations from definitions then that need
will be eliminated, but the remarks of the previous para-
graph, and 5.0.9-10, show that this merely shifts the
problem.
The amazing thing is that some philosophers thought
this problem could he avoided by explaining all logical
connections and all perception of logical connections in
terms of formal systems and derivability of theorems from
specified axioms according to rules of derivation specified
in advance. It is amazing for two reasons, firstly because
    306

it is hard to see how anyone was ever able to think that
merely talking about rules for manipulating symbols
could explain logical properties and relations of
statements (see appendix II), and secondly because even
if talk about proofs in formal systems did explain
logical connections and our knowledge of them, this would
be at the cost of reducing logic to geometry, and there
would remain the problem of explaining what sort of
insight was involved in perceiving that strings of
symbols stood in certain geometrical (or syntactical)
relations to others. For, after all, the formal logician
is not trying to establish the merely contingent fact
that he can here and now derive (or has here and now
derived) this particular set of marks from that particular
set of marks (all tokens) while trying to follow certain
rules: he wishes to show that a relation holds between
types of marks, or, in other words, that geometrical
properties (or patterns) stand in a connection of the
kind which we have been discussing and he thinks he can
explain away.

7.D.9.b.    Sometimes it is argued that this question about
the justification for asserting that a formula has a
formal property need not arise (e.g. the property of
being a theorem) since the rules of the formal system are
so devised that even a moron, or a machine, could be
instructed to check a proof to see that it went in accord
with the rules. But this misses the point, for it relies
on the very fact to which attention was drawn above
(7.D.5), namely that each step in a proof must depend on
connections between (e.g.) properties which are so evident
as to need no proof.
If I "prove" that the statement "If a triangle is
    307
inside a circle, and a dot is inside the triangle, then the
dot is inside the circle" is necessarily true, by drawing
a diagram, namely a circle surrounding a triangle with a
dot in the middle, or if a moron comes upon the diagram
and utters the statement, then there is no more doubt
about mistakes here than when a moron asserts that some
pattern of formulae satisfies a recursive definition
of "proof-sequence".


                

 In any case, pointing out that a moron can apply some
test for picking out certain sequences of sentences does
not answer the question why sequences of sentences picked
out in this way are valid proofs. Are morons supposed
to be able to tell that any sequence of statements, no
matter how complicated, which satisfies some recursive
test constitutes a valid proof?

7.D.9.c.    The assertion that an informal proof is really
a formal proof in disguise does not seem to be of any
help at all. It would be truer to say that a formal
proof is an informal proof in disguise. The formulae
in a formal proof represent propositions, or the logical
forms of propositions, and serve the purpose of drawing
our attention to logical relations between those pro-
positions, in an indirect way. For the symbols are so
chosen, that geometrical relations between them represent
relations between the logical techniques corresponding
to the logical forms of propositions (see 5.C.9). So,
when we look at the logician's symbols, our attention is
drawn (half consciously) to the relations between these
logical techniques for discovering truth-values, and we
are thereby enabled to see the relations between truth-
values of propositions (i.e. relations between the outcomes
of applying the logical techniques). It is not essential
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to this process that the symbols which draw our attention
to logical relations between propositions should con-
stitute a "proof-sequence" in some formal system.
(Herein lies the answer to Wittgenstein's puzzle in
R.F.M. part II sections 38 and 43: a Russellian proof
is cogent only insofar as it has geometrical cogency,
yet one may "accept" it without ever noticing the
geometrical application. This is because the geometrical
application is a consequence of the contingent fact that
the rules of our language correlate logical forms of
propositions, or logical techniques of verification, with
geometrical forms of sentences in a uniform way.)
A fully explicit logical proof would draw attention
directly to logical techniques and their interrelations,
in much the same way as the informal proofs under discussion
draw attention to connections between geometrical
properties. Such an informal logical proof would help
someone to perceive logical properties of or relations
between propositions, which is what a formal proof does
only indirectly and implicitly. (Compare Appendix II.)

7.D.10. We have now dealt with two objections, first
that the account of informal proof was too psychological,
and secondly that there is a formal proof, proceeding
from definitions or identifying facts about meanings,
underlying every informal proof. The first objection
was met by asking what a proof is supposed to do, the
second by showing that only an informal proof can do this,
even if it is purely logical. But it is still open to
someone who wishes to deny the existence of synthetic
necessary truths to admit that informal proofs are possible,
while asserting that they can only start from identifying
relations between meanings or properties and proceed
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logically, so that at every stage only topic-neutral
considerations are relevant. Such a proof could only
demonstrate the truth of an analytic proposition,
according to the definition of 6.C.10. This assertion
might be based on the argument that only an identifying
relation between meanings can guarantee that there will
not be an exception or counter-example to the proposition
proved.
How can I be sure, simply because I have seen one
rectangular figure divided up into a pair of triangles
and a pair of parallelograms, that there will never be
a rectangle which cannot be divided up in this way?
How can I be sure that I shall never see a figure which
has both the property of being round and the property
of being square? The suggestion is that I can be sure
only if I adopt a linguistic convention ruling out the
possibility of describing any unexpected object as a
counter-example to the proved proposition. That this
suggestion has a point is shown by the history of
mathematics. For it has happened more than once that
a proof has been accepted as valid, and then later shown
to be invalid. (Dr I. Lakatos has investigated such
cases.) Many of Wittgenstein's remarks in R.F.M. seem
to be directed towards showing that the possibility of
an unexpected counter-example can never be eliminated
except by a convention relating the meanings of words,
in something like the way in which the "purely verbal"
rules described in section 4.C were able to rule out the
possibility of a counter-example to statements like "No
red thing is orange" (see 4.C.3-4). (Similar, though
more vague, arguments were used by von Wright on p.38
of "Logical Problem of induction', 2nd Ed.)
    310

7.D.10.a.   Suppose, for example, that a (token) proof
connecting a property P with a property Q purports to
show that any object with P can serve as the starting
point for another (token) proof of the same type. If
so, the object must also provide an instance of the
"bridge-property" starting with P and ending with Q,
so it must be an instance of Q, and thus no counter-
example to "All P things are Q" is possible. But
suppose the proof does not work: an object turns up
which has the property P, but from which the bridge-
property cannot start, that is, an object which has P
but cannot enter into a token-proof of the required type
(the construction cannot be carried out). Then the
only way to save the theorem proved is to adopt a new
linguistic convention. E.g. we may say that being
the starting point for the bridge-property is one of the
defining criteria for having the property P, and so the
new object does not really have P and is not really a
counter-example. Now the argument we are considering
claims that even before any counter-example has turned
up, the only way we can guarantee that none will do so
is by adopting this sort of new defining criterion for
having some property, and what the informal proof does
is show us how the new criterion works by displaying
the bridge-property which we thereafter take to be
identifyingly related to the property P. Similarly, we
may have to take the connection between the bridge-
property and the property Q as an identifying fact about
the meaning of the word which formerly referred to Q, in
order to ensure that no counter-example may turn up to
break the link at the other end. (This applies only to
one, relatively simple, kind of proof, of course.) By
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adopting these new rules we have given a more determinate
meaning to the words expressing the theorem which was
meant to be proved, and we have also set up identifying
relations between their a meanings from which it follows
logically (see 6.B.11,ff.) that the sentence expressing
the theorem in question must correspond to the truth-
value "true". In Wittgenstein's terminology: "in the
proof I have won through to a decision" (R.F.M. Part II,
27.)

7.D.10.b.   Now it is very likely that this sort of thing
happens sometimes in mathematics: we may think that we
have completely identified some complicated geometrical
or arithmetical property when in fact it is indeterminate
in some respects (see section 4.A), and borderline cases
could occur to provide potential counter-examples to
some theorem about that property. (Cf. end of 7.B.4.)
Then the construction used in a proof of that theorem
may show us a new way of defining the meanings of the
words used to express the theorem so as to rule out these
counter-examples. But it is important to notice the
differences between borderline cases which produce the
following reactions (a) "I had not thought of that
possibility, so I was wrong after all", (b) "I had not
considered that possibility, but it does not matter as
it is not the kind of thing that I was talking about" and
(c) "I had not considered that possibility, and now I
don't know whether to say it is the kind of thing I meant
to refer to or not." Case (a) is an admission that the
proof was invalid and the theorem wrong after all. (b)
rejects the borderline case as not providing a counter-
example. Only (c) leaves room for a new decision to
    312

adopt a convention as to how to describe the borderline
case in order to save the theorem. The thesis that
every proof is covertly a logical proof from identifying
relations between concepts which we implicitly accept
in accepting the proof requires that every geometrical
concept be indeterminate in such a way that there is
room for this sort of decision.
It is not possible to go into the question whether
every concept is indeterminate in this way without
embarking on a general enquiry concerning meaning and
universals, and all the topics raised by Wittgenstein in
his discussion of the notion of "following a rule" in
"Philosophical Investigations". I shall say only that
the fact that in some cases mathematicians have failed to
see the possibility of counter-examples to propositions
which they believed to have been proved, does not in the
least convince me that no non-logical informal proof is
secure, and neither does the fact that in some cases a
property or a proof may be so complex that it cannot be
"surveyed" properly and has some indeterminate aspects:
there are other cases where properties are sufficiently
simple for their connections to be quite perspicuous,
leaving no room for any doubt that something will go
wrong. I am perfectly certain that if anyone brings me
an object which is alleged to be bounded by four plane
surfaces, and not to have four vertices, then it will
turn out that either the four planes do not bound the
object, or there are not exactly four of them, or they
are not planes, or he has miscounted the vertices, or ....
(Why should I specify in advance all the mistakes which
could possibly be made?)

7.D.10.c.   I conclude that there is little reason to doubt
    313

that there are some connections between properties of
such a kind as to prevent their occurring in certain
combinations in particular objects, that these connec-
tions need not be identifying connections, and that
they may either be quite evident, or sometimes made
evident by an informal proof, which enables us therefore
to see that some proposition states a necessary truth.
This does not deny that there are cases where indeter-
minateness of meaning makes it necessary to adopt purely
verbal rules (4.C) to rule out the possibility of
counter-examples, neither does it imply that we are
infallible and can never wrongly think we see connections
between complicated properties.
It is worth noting that the difference between a
doubtful borderline case of an instance of a property,
and other objects which clearly are or clearly are not
instances of it cannot be merely a numerical difference
between them. There must be a difference in kind between
borderline and non-borderline cases, they must exhibit
different properties (e.g. an object with a borderline
shade of red looks different from one which is bright
scarlet). So even where a new verbal rule is required
to ensure that borderline cases do not provide counter-
examples to theorems proved by an informal proof, the
verbal rule has to be applied only in some kinds of
situations, involving objects which differ in certain
respects from those which are not borderline cases. In
other sorts of instances of the properties referred to,
the connections between the properties are as shown in
the informal proof, so no verbal rule is required to
ensure that no counter-example to the theorem can arise
amongst them, that is amongst objects which do not have
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the properties peculiar to the borderline cases. This
shows that even if it is true that some verbal rule is
always required to make it certain that no unexpected
borderline cases can provide counter-examples to a
theorem proved in the manner under discussion; that is,
even if every necessary truth has an analytic aspect,
nevertheless there is a synthetic aspect, brought out
by the informal proof, which shows a necessary connec-
tion in at least a limited range of cases. ("No rec-
tangle can look just like this one and fail to be
divisible into two triangles and two parallelograms, and
I do not need to adopt any convention to ensure that,
for it is evident to anyone who examines the shape of
this rectangle".)

7.D.11. Now our persistent objector may argue that even
if it cannot be shown that what goes on in an informal
proof is always implicit logical deduction from implicitly
acknowledged identifying relations between meanings or
properties, nevertheless it remains for me to demonstrate
that the "connection" revealed by such proofs are not
breakable, that they are necessary connections, allowing
no exceptional particular objects as possible. How do
we know that the constant conjunction of these two
immediately perceptible features, or their failure to
occur together, as the case may be, is not just a con-
tingent fact? Certainly it is up to me to show that the
propositions in question are necessarily true, but I do
not do this by means of a philosophical argument, I do so
by means of the proof which we are discussing!
If the objector cannot see for himself by examining
properties (either in particular instances, or in his
imagination if he is well acquainted with them), or by
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going through an informal proof, that nothing can be both
round and square, that every tetralateral object is
tetrahedral and no exceptions are possible, if he cannot
see the necessary connections between these independently
identifiable properties, then he cannot be forced to
see them, as already pointed out, and he must forever
remain in doubt as to whether, perhaps in the depths of
darkest Africa, there lies hidden somewhere a slab of
some hitherto unknown material, whose boundary can be
seen to be at once both square and circular, or perhaps
a little pyramid, completely enclosed by exactly four
flat sides, but with only three vertices, or five.
The objector can surely not expect me to convince
him by offering a proof which starts from definitions, and
then draws purely logical inferences, for the whole point
of this section and the last is to show that only by
altering the meaning of the statement proved can one
replace a non-logical informal proof by a logical one.
(To show that the statements in question are necessarily
true is not a philosophical task.)

7.D.11.note.    All this must be qualified by the remark
that our ordinary geometrical concepts ("round", "square",
"straight", "flat", etc.) are extremely complicated in a
way which makes it very difficult to describe
what goes on when a normal person is confronted with an
informal proof of the sort used in school geometry. I
am referring to the fact that where we have one word,
such as "ellipse", there are usually very many concepts
superimposed in its meaning, in a quite indeterminate
way, and this is not noticed, owing to our use of loose
criteria for identity of meaning, which cannot distinguish
these different superimposed concepts. Consider each
of the various definitions of the notion of an ellipse
which might be given by a mathematician, and blur its edges
a little. Add the semantic correlation between the word
and the visual property or range of visual properties
which we associate with it. Load all these meanings onto
one word in an indeterminate way: and then try to explain
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what goes on in the mind of someone who uses the word
in this overdetermined fashion when he sees a proof of
a theorem about ellipses! (For other examples of
superimposed concepts see 3.E.2 and 6.D.3. Compare
also 5.E.7.a. I believe that our ordinary arithmetical
concepts are indeterminate and overdetermined in a
similar fashion, which is why different philosophies of
mathematics have all been able to claim some plausibility:
logicism, formalism, intuitionism, empiricist theories.
Each has picked out one aspect of the truth, while
making the mistake of claiming it to be the whole truth.
There is no time now to show how a unifying theory could
be developed.)

7.D.12. Some of the things said by Kant about synthetic
apriori knowledge are explained by this discussion, in
particular that it requires an "appeal to intuition".
This is firstly because without the intuition (acquain-
tance with properties, etc.) one cannot know which
concepts are involved or that there are any empirical
concepts involved which can be applied to observable
objects; and secondly an appeal to intuition is required
since without it one cannot come to see how the concepts
(or properties) are connected. (See, for example,
C.P.R., A.239-40, B.298,ff; 308-9; A.716,B.744).
The fact that looking at a diagram (real or imagined) can
play an essential part in perceiving the connections
between properties shows that in doing so one is not
merely drawing logical inferences whose validity depends
on topic-neutral principles. (This might also be put by
saying that the type-proof, or bridge-property, mentioned
in 7.D.4, is not logically synthesized out of the properties
whose connection it is supposed to reveal: the synthesis
in the proof - i.e. the way in which it is constructed -
is non-logical, for reasons of the sort given in 3.D.6,ff.)
I think my account of informal proof helps also to
    317

explain why Frege believed that one could have synthetic
apriori knowledge concerning geometry. (See sections 14
and 90 of "Foundations of Arithmetic".) It also explains
some of the talk of Intuitionists about "mental constructions"
(see Heyting "Intuitionism", e.g. p.6,ff),
though my account would have to be modified to take
account of proofs of theorems about the properties of
infinite sets, since these are not perceptible properties.

7.D.13. Whether this brief and highly condensed sketch
is correct or not, one thing should be clear: informal
proofs certainly do something, and what they do is
different from what is done by a proof of an analytic
proposition starting from identifying facts about meanings
and proceeding logically. This shows that if the "proved"
statements are necessarily true in both cases, then it is
very likely that there are at least two kinds of necessity
worth distinguishing. My suggestion is that the way to
distinguish them is to notice that in both cases the
propositions exemplify the notion of a "freak" case of a
rogator whose value happens to be determined by relations
between its arguments together with facts about the tech-
nique for working out its values, the difference being that
in the one case the arguments are identifyingly related,
and all arguments standing in the same relation must yield
the same truth-value, whereas in the other case the
relations between the arguments are not identifying, and
they are not relations of a sort which could hold between
any sorts of entities at all (they are not topic-neutral).
However, it is open to anyone who does not like talking
about synthetic necessary truths or synthetic apriori
knowledge to reject my terminology and say that in both
cases the propositions are analytic since their truth is
    318

determined, however indirectly, by what they mean. But
then "true in virtue of meanings" seems to be synonymous
with "necessarily true" in this terminology, and the
assertion that all necessary truths are analytic says
nothing and says it in a redundant terminology which
fails to take account of distinctions which are of some
interest.

7.D.14. It should not be thought that the assertion
that there are synthetic necessary truths has any great
metaphysical significance, or that it justifies any claim
to have perceived with the inner light of reason, or any
other mysterious faculty, moral or theological truths, or
truths of a transcendent nature. (See 7.D.8). So far,
the assertion has been justified only by a discussion of
ways of perceiving connections between simple empirical
statements (i.e. between the techniques corresponding to
logical rogators). If it can be extended to cover
other cases, such as the principle of causality, then
this has to be shown by detailed investigation.
I claim only to have given an informal proof of the
existence of synthetic necessary truths of a simple and
uninteresting kind, or at least to have shown that there
is a distinction to be made between different sorts of
necessary truth. But the topic is difficult and complex,
and I have been unable to do much more than provide an
introduction to it by showing how its problems are related
to and can arise out of general considerations about thought
and language and experience. [I am very dissatisfied
with the discussion of this section, though I believe it to
be a first step in the right direction. I have included
it for the sake of completeness.]
    319

7.E.    Additional remarks

7.E.1.  This chapter may now be summarized. Chapter
six had explained how the truth of a statement may be a purely
logical consequence of identifying facts about
the meanings of the words used to express it. Such a
statement would be true in any possible state of the
world because its truth-value is determined independently
of how the world happens to be. In this chapter I have
tried to explain why it makes sense to contrast the way
the world happens to be with ways it might have been, so
as to give a fairly clear sense to the question: Are
there any statements which would be true in all possible
states of the world, besides those described in chapter
six? That is: are there any non-analytic necessary
truths? I was able to give a sense to this question,
by making use of some of the very general facts about
our language which were pointed out in chapter two,
especially 2.B and 2.D, namely the fact that we use a
conceptual scheme with provision for a distinction between
universals (observable properties and relations) and
particulars, and the fact that universals are not essen-
tially tied to actual particulars. The question then
became: Are there any limitations on the distribution
of universals to be found in any actual or possible state
of the world, apart from purely logical limitations,
which are in no way concerned with anything special about
specific properties but are topic-neutral? [This was
the fundamental question, but in order to take account
of "Improper" or "synthesized" universals (see chapter
three), we asked the question in the following form:
Are there any connections between universals (i.e. res-
trictions on the ways in which they may be instantiated)
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which are not due simply to (a) identifying facts about
those universals and (b) purely logical or topic-neutral
restrictions?]

7.E.2.  I tried to answer this question by drawing
attention to observable connections between observable
properties, where (a) the properties can be identified
independently of each other and (b) logical considerations
alone do not account for the connections between them,
since the properties themselves must be examined for the
connection to be perceived. Thus, a slightly more
general account was available of the way in which relations
between the arguments of a logical rogator might help to
determine its value, than the account given in chapter
six. In short, we saw that a sentence may express a
proposition which would be true in all possible states
of the world, though it is not analytic. The reason
why no exceptions to such a synthetic necessary truth are
possible is that exceptions would have to be objects in
which properties were combined in ways which are excluded
by the connections between those properties.

7.E.3.  This also explained why one might have the right
to make such statements as "If this had been P then it would
also have been Q" or "If this had been P then it would not
have been R" The connections between properties which
make some statements necessarily true, also give a sense
to counterfactual conditional statements, by giving us
the right to assert some of them as true.

(Note: We could generalize this slightly to explain
the concept of entailment. The proposition p entails
the proposition q if something ensures that if p were true
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then q would be. this can be put more precisely:
The proposition p entails the proposition q if and only
if there is some relation R satisfying the following
conditions:
1)  The relation holds between p and q.
2)  The relation holds between some propositions which
are neither necessarily true nor necessarily false.
3)  if the relation holds between two propositions Ř
and ?, then this ensures that in any possible state
of the world in which Ř would be true ? would also
be true.
(The relation may be purely formal - i.e. it may be
a topic-neutral relation, or it may be concerned with the
content of the two propositions.)
I suspect that our ordinary expressions of the form
"If Ř then ?" are more like assertions of entailment than
like assertions of material implication, though probably
much more complex than either, as can be seen by examining
the sorts of things which are normally regarded as
justifying such assertions. In consequence, it is not
obvious that what I said in chapter five about logical
forms of propositions, and the logical rogators which
correspond to them, applies without modification to
conditional statements.)

7.E.4.  The discussion of informal proofs was intended
to explain how we can become aware of connections between
properties of the sorts which ensure the necessary truth
of some synthetic propositions. It also provided a
partial answer to the question raised in section 5.C
about the manner in which one can become aware of the
relations between logical techniques in virtue of which
propositions whose logical forms corresponded to those
techniques might have logical properties or stand in
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logical relations. The answer was very vague, namely that
perceiving connections between logical techniques is the
same sort of thing as perceiving connections between
(say) geometrical properties.
It is clear that there is a lot more work to be
done on the subject of informal proof, as I have talked
only about some very simple cases, and left many questions
unanswered.

7.E.5.  For example, there is a puzzling fact which I
have hardly mentioned, except in 6.E.6-7, and without
an explanation of which it is impossible to give a
complete account of the way in which informal proofs
work, or the way in which we normally come to have know-
ledge of necessary truth, namely the fact that a person
may be justified in claiming or believing something,
and for the right reasons, without his being able to see
clearly or say clearly what the reasons are.
This is exemplified by a layman's assertion of an
analytic proposition which he correctly justifies by
saying that it is "true by definition", even though he
may be quite unable to explain what it is for a proposition
to be true by definition. Similarly, he may have seen
an informal proof, and so be quite justified in asserting
the proposition which it proves, saying "It must be so",
and yet be quite incapable of saying how the proof proves
the proposition. This is connected with some of the
remarks in the appendix on "Implicit Knowledge".
It is pretty certain that if ever a philosopher does
manage to give a clear and accurate account of how
informal proofs work and why we are justified in asserting
the propositions which such proofs are taken to justify,
we shall not be able to retort to him that we knew it all
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before, just as the person who cannot see that some
statement is true until he has studied a proof cannot
claim to have known it all before, even though the proof
does lead him on from things which he did know before.
Perhaps there is an analogy here between what happens
when a mathematician convinces us of the truth of some
surprising theorem by drawing construction-lines and
what happens when a philosopher solves some kinds of
problem: the philosopher draws "construction-lines" of
a different sort in order to bring out connections between
concepts, such as the connection between the concept of
a diagram used in a geometrical proof and the concept of
necessity. (Was I drawing philosophical "construction-
lines" when I talked about rogators in chapter five, in
order to give an account of logical form and explain the
connection between formal properties of sentences and
logical properties of propositions?) This suggests
that if a mathematical proof can enable one to see that
some synthetic proposition is necessarily true, then
perhaps philosophical investigations may also reveal
synthetic necessary truths. This is something which
requires detailed investigation. (For some remarks on
philosophical analysis, see Appendix IV.)

7.E.6.  Another subject which requires investigation is
the relation between our ordinary empirical concepts of
shape and colour, and the idealized concepts which, at
times, it may have appeared that I was discussing. (See
the disclaimers in 3.A.2. and 7.C.7.) Idealized concepts
are somehow extrapolated from our ordinary concepts.
Examples are the concept of an "absolutely specific"
shade of red, or an "absolutely specific" triangular shape,
and the concept of a "perfect" geometrical shape, such as
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the shape of a perfect cube. Philosophers sometimes
suggest that there is no connection between these
idealized concepts and our ordinary empirical ones.
At any rate they are usually unclear as to how they are
derived from our empirical concepts, perhaps because they
fail to see that two sorts of idealization are involved.

7.E.6.a.    First there is the idealization towards perfect
specificity, which explains our use of expressions like
"exactly the same shade of colour as" or "exactly the
same shape as". We see pairs of objects which are more
and more alike in some respect, and then extrapolate to
the limit, on the assumption that it makes sense to do
so, even though we are not able to discriminate pro-
perties finely enough to base the notion of "exact likeness"
directly in experience. This kind of exact likeness is
supposed to be transitive, unlike perceptual likeness.
So the absolutely specific shade of colour (for example)
of my table is a property common to all objects exactly
like it in respect of colour. (There can be no bor-
derline cases.) Perhaps some argument can be given to
justify the assumption that it makes sense to talk like
this. I do not know.

7.E.6.b.    The second sort of idealization is quite
different, and helps to explain such concepts as the
concept of a "perfectly straight" line, or a "perfectly
smooth" curve, or a "perfectly plane" surface, or a
"perfectly perpendicular" pair of lines. It may also
be connected with such notions as a "perfectly pure"
shade of red, or a "perfectly pure tone", or a "perfect"
musical octave. For the purposes of this sort of
idealizations it is first of all assumed that it makes
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sense to speak of absolutely specific properties (shapes,
colours, tones, etc.) and then use is made of the fact
that in some cases the properties can be arranged in a
series, apparently tending towards a limit. Thus one
line looks straighter (smoother, more nearly circular,
etc. than another, and a third looks straighter (etc.)
than the first, and so on: so we extrapolate and assume
that it makes sense to talk about the perfectly straight
(smooth, circular, etc.) line which lies at the end of
the series and is straighter (smoother, etc.) than the
others. A similar process may account for the concept
of an infinitely long line, or the concept of
parallel lines, or the concept of infinity in arithmetic
or set theory. Similarly, one colour looks a purer blue
than another, and so on, so we assume that there could
be a perfectly pure shade of blue. In some cases there
may be more than one route by which the limit is
approached.

7.E.6.C.    It is taken for granted that such methods of
extrapolation fully define the "perfect" concepts which
they generate, and that different methods of extra-
polation may define the same limit. And it used to be
thought that facts about these perfect concepts could be
discovered with the aid of old-style Euclidean proofs.
But it is more likely that although the method of generation
of such idealized concepts fully determines some things
about them (thus, the relation "inside" applied to
perfect squares, triangles, circles, etc., is transitive),
nevertheless in order fully to define them it may be
necessary arbitrarily to stipulate that certain relations
hold between them, or that certain statements about them
are true (such as Euclid's parallel axiom). Since such
    326

a stipulation is arbitrary (there is nothing in virtue
of which it is "correct"), we could adopt alternative
"axioms" and complete the definitions in another way.
This is the tiny grain of truth which lies behind current
opinions of the sort which I criticized in 7.C.7(note).

7.E.6.d.    It is also sometimes not noticed that the
process of idealization does not remove all empirical
elements from these "perfect" concepts. Hence it is
assumed that geometrical proofs which are concerned with
them have nothing to do with objects of experience. This
is why philosophers sometimes talk as if a perfectly
sharp distinction can be made between "pure mathematics"
and "applied mathematics" the latter being regarded as
an empirical science, perhaps a branch of physics.
There is no space here to explain in detail why this is
muddled.

7.E.7.  A failure to understand the nature of these
"perfect" mathematical concepts, or to see the difference
between those "axioms" which served the purpose of com-
pleting the definitions of concepts and the "theorems"
whose truth in no way depended on arbitrary stipulations
of identifying conventions, left people unable to cope
with the shock of the discovery of alternative internally
consistent axiom-systems for geometry. The notion of a
proof as something which served to establish the truth
of a theorem was therefore undermined, and philosophers
tried to salvage what was left by treating proofs as
nothing more than methods of deducing consequences from
arbitrary hypotheses or postulates. This at least seemed
to be secure: for, by means of formalized systems of logic
one could at least give fool-proof criteria for the
    327

validity of a proof. Criticisms of this conception of
proof have been made elsewhere (in 5.C.10,ff, 7.D.9,ff and
Appendix II). It seems not to have been realized that
such a conception severs the concept of "proof" completely
from the concept of "truth". It seems not to have
been realized that if proofs are intended to serve the
purposes described in section 7.D, namely, to enable
people to perceive the truth of propositions, to bring
out the reasons why propositions are true, then the search
for a fool-proof criterion of validity is futile: for,
no matter what criterion is adopted, questions remain
about the justification for accepting proofs which
satisfy that criterion, and the justification for the
statement that any particular type of proof satisfied
the criterion. If a justification is offered, then its
validity cannot be constituted by satisfaction of the
criteria in question - that would be circular. The
only way to avoid a circle is to give up talking about
criteria of validity, and either follow Wittgenstein
in his talk about conventions (in R.F.M.) or try to
explain how we can simply see necessary connections
between properties and other universals by examining
them, perhaps with the aid of informal proofs. (Are
these really distinct alternatives?)

7.E.8.  Finally, the reader is reminded that although an
informal proof enables one to discover that a proposition
like "All tetralaterals are tetrahedrals" is true without
discovering how things happen to be in the world (i.e.
without looking to see which particular objects exist
where, and what properties they have, etc.), nevertheless
it is possible to verify such a proposition empirically,
    328

just as (cf. section 6.E) it is possible to verify an
analytic proposition empirically. Thus, one might
carry out a survey of all objects bounded by four plane
sides in order to discover whether they also possess
the property of having four vertices. Such an empirical
justification for the assertion of the proposition is
adequate, despite the fact that it is unnecessary.

================================================================
NOTE: This is part of A.Sloman's 1962 Oxford DPhil Thesis
     "Knowing and Understanding"
================================================================

                                                        329

Chapter Eight
Concluding Summary

This thesis will be concluded with a brief summary,
which may be supplemented by section C of chapter one.
(see especially 1.C.2, concerning the limitations on the
discussion.)

Meaning and truth.
The description of the general connection between
meaning and truth (between understanding and knowing) began
with some general remarks about the presuppositions of talk
about meanings and propositions, and criteria for identity
of meanings, preparing the way for much of what followed.
We saw that descriptive words have their meanings in
virtue of semantic correlations with combinations of ob-
servable properties (or relations), which one must learn
to recognize in learning to use the words. A system for
classifying such words on the basis of the ways in which
they are correlated with universals was described, which
enabled us to give an account of many hidden complexities
in the meanings of simple-looking adjectives and common
nouns.
The role of logical words in sentences was described
by developing some ideas of Frege and Wittgenstein. It
is possible to regard the logical form of a proposition as
corresponding to a rogator, which takes descriptive words
and expressions as arguments and takes as values the words
"true" and "false": to each logical rogator there corres-
ponds a "logical technique" for determining the value given
the meanings of the non-logical words taken as arguments,

                                                        330

the outcome of which generally depends on how things happen
to be in the world. In learning the use of logical words
and constructions, we learn how their occurrence in sen-
tences determines the logical form of the propositions
expressed, by determining which logical techniques or
which rogators correspond to those sentences.
Thus, the semantic correlations between descriptive
words and universals, together with the correlations between
"logical forms" of sentences and logical techniques, deter-
mine the conditions in which sentences containing descriptive
and logical words express true or false propositions. This
is how meanings of statements are determined by the meanings
or functions of individual words.
(This inquiry was not without by-products. We found
reason to reject the reduction of logic to syntax. We
were able to clarify the difference between "presuppositions"
of a statement and its "implications", by talking about the
conditions in which logical techniques are applicable.
This, and the notion of the "domain of definition" of a
rogator, looked like a suitable basis for a doctrine of "types"
and "category rules" slightly more general and less arbitrary-
looking than theories based on "ranges of significance"
of predicates. See end of Chapter Five.)

Meaning and necessary truth.

Making use of some of the early general remarks about
meanings and propositions, especially the remarks about
conceptual schemes, we analysed some aspects of the con-
cepts of "possibility" and "necessity" by drawing attention
to general and fundamental facts of experience, but for
which our language and thought could not be as they are,
such as the fact that universals are not essentially tied

                                                        331

to their actual particular instances. Necessity was
explained in terms of connections between universals
which limit the possible ways in which they might have
occurred in other instances than those which actually
possess them. Such connections between universals may
also explain our use of subjunctive conditional state-
ments in some contexts.

The descriptions of the connection between meaning
and necessary truth (between understanding and knowing
apriori)followed on naturally from the earlier description
of the general connection between meaning and truth.
Though the value of a logical rogator for a set of argu-
ments normally depends on how things are in the world,
and has to be discovered by applying the appropriate tech-
nique, nevertheless there are "freak" cases where the
truth-value may be discovered by examining the technique
and relations between the meanings of non-logical words
taken as arguments. (Though even here it may be dis-
covered also in the normal way, by applying the technique.)
Relations between the meanings of descriptive words, which
may help to determine the truth-value of a statement in
all possible states of the world, may either be identifying
relations, corresponding to definitions or partial
definitions, or non-identifying relations, corresponding
to connections between universals (observable properties
and relations). Thus there are two sorts of propositions
which are necessarily true, namely those which are analytic
and those which are synthetic. The discovery of the
relations which may make a synthetic proposition
necessarily true is made by examining observable pro-
perties or relations, possibly with the aid of an informal
proof.

                                                        332
(It is assumed throughout that the statements under
discussion do have truth-values, that the applicability-
conditions for rogators are satisfied. This may not
always be discoverable apriori. See 5.E.6, ff.)

All this showed that there were four types of true
or false statements using only descriptive and logical words.
1)  Formal truths and falsehoods, whose truth-values are
determined by their logical form alone.
2)  Analytic, but non-formal, truths or falsehoods, whose
truth-values are determined by both their logical
form and identifying relations between meanings of
some non-logical words.
3)  Synthetic necessary truths and falsehoods, whose
truth-values are determined by the factors mentioned
so far, together with synthetic or non-identifying
relations between the meanings of some of the non-
logical words.
4.  Synthetic contingent statements, whose truth-values
depend on their logical form, on the meanings of their
non-logical words, and on how things happen to be in
the world (i.e. on which particular objects have which
properties, etc.).

In order to know the truth-value of the first kind, it
is enough to know how logical constants (topic-neutral
words and constructions) work, and perceive properties of
the corresponding logical techniques. Of the other words
one need know nothing except that they are descriptive
words referring to properties.
For knowledge of the truth-value of the second kind,
something must, in addition, be known about the descriptive
words, such as that some of them are used as abbreviations
for other expressions, or that the meanings stand in cer-
tain identifying relations. What the meanings are need
not be known.

                                                        333
    Knowledge of the truth-value of the third kind of
proposition requires, in addition to the factors so far
mentioned, a complete understanding of at least some of
the descriptive words. One must know which properties
are referred to, in order to be able to examine them and
discover the connections between them.
    Finally, not only is complete understanding required
for knowledge of the truth-value of propositions of the
fourth kind, but also an empirical enquiry to find out
how things stand with the particular objects which have
(or do not have) the properties referred to. Here know-
ledge of meanings and logical techniques is applied, where
in the other cases it was only examined.

    The discussion of section 2.C showed that the failure
of many philosophers to see all this could be explained
not only by their confused understanding of the terms
"synthetic", "necessary", etc., but also by their unwitting
use of loose and fluctuating criteria for identity of
meanings. They have failed to use Kant's "engraver's
needle", partly on account of not having noticed that a
theory of universals (properties and relations) need not
rely on the oversimplified "one-one" model. (Cf. 2.D.6-7,
3.B.5, 4.B.1, etc.).
    This concludes my answer to the main question raised
in section 1.1  Many subsidiary questions have been
raised which could not be answered in the limited space
available - some of these are dealt with briefly in the
appendices. I claim to have shown that Kant was justified
in describing some kind of knowledge as both synthetic
and a priori,(1) and, which is perhaps more important, to

________________

 (1). See Appendix VI.


                                                        329

have revealed some relations between very general con-
cepts, such as "property", "meaning", "truth", "proof",
"possibility" and "necessity".

====================================================
This is part of A.Sloman's 1962 Oxford DPhil Thesis
     "Knowing and Understanding"
====================================================



APPENDICES

  I. Singular referring expressions          335
 II. Confusions of formal logicians          340
III. Implicit knowledge                      357
 IV. Philosophical analysis                  372
  V. Further examples                        381
 VI. Apriori knowledge                       386


    335

Appendix I

SINGULAR REFERRING EXPRESSIONS

The discussion in the thesis has hardly been concerned
at all with singular referring expressions that is,
proper names, definite descriptions and other words or
pronouns which, in one way or another, refer to actual
particular objects. The main reason is that in order
to illustrate the existence of synthetic necessary truths
it is enough to discuss sentences containing words which
do not refer to anything besides universals.
There is another reason why it is most helpful to
define the analytic-synthetic and necessary-contingent
distinctions in such a way as not to apply to propositions
which mention particulars, namely to avoid difficulties
with such propositions as the following:
i) The Queen of England is a woman.
    ii) I am speaking now.
    iii) John's uncle is a man.
    iv) He is a male.
    v) Socrates is human. (Searle, D.Phil. thesis, pp. 132-7.)

The difficulty is that in each case there are reasons
for regarding the proposition as analytic and reasons
for regarding it as synthetic. The reason for regarding
each of then as analytic is that its negation looks self-
contradictory in some sense. "Queen" surely means "female
monarch" So the Queen of England could not fail to be a
woman. (Etc.) The reason for regarding each of them
as synthetic is that it presupposes the existence of some
particular, which it mentions, and so one cannot discover
    336

that it is true merely by investigating the meanings of
the words and the logical form: one must know that the
particular object exists, as the result of some empirical
enquiry.
The same sort of difficulty arises when we consider
inferences to propositions mentioning particulars. For
example, from (a) "If anyone is an uncle then he is a man"
we may apparently infer (b) "If Tom is an uncle then he is
a man", and the inference looks as if it is logically
valid: certainly it is normally regarded as valid But
then, if (a) is analytic, why should (b) not be analytic
too, since anything logically implied by an analytic proposition
is itself analytic? One way out would be to
say that the inference is not purely logical, since (b)
presupposes something which (a) does not, namely that
there is a person referred to by the word "Tom", and only
one, and this cannot be guaranteed by logic
Now consider the inference from (b) to (c) "John's
uncle, Tom, is a man" Again, the inference looks as if
it is logically valid, and is normally treated as if it
were, but, as before, the conclusion presupposes something
which is not presupposed by the premiss, namely that there
is exactly one person who is referred to by "John", that
he has exactly one uncle, and that that uncle is named "Tom".
Inferences of this sort are normally regarded as valid
because we take their presuppositions for granted, and when
this can be done logic suffices to ensure that the con-
clusion will be true if the premiss is. Nevertheless,
the fact that there is the additional presupposition in
each case seems to provide a good reason for saying that
the conclusion does not state a necessary truth, and is
therefore not analytic.
    337

Whether we describe statements like (b) or (c) as
"analytic" or not does not matter very much. What is
usually of philosophical interest is whether statements
of the form of (a) are analytic, or necessarily true.
In other cases we can talk about the connections between
the meanings of words as being analytic, or, in the
terminology of chapter six, identifying relations, without
asserting that the propositions which they express are
analytic.
There is a further reason for withholding the necessary-
contingent distinction from statements mentioning par-
ticular objects. Necessity is defined, in chapter seven,
in terms of what would be the case in all possible states
of affairs; a statement is necessarily true if no
exceptions are "allowed as possible", Thus, in order
to ask whether the statement "This round box is not square"
was necessarily true, we should have to ask whether it would
be true in all possible states of affairs, or whether one
of them would provide an exception. But the normal
criteria for identity of material objects simply do not
extend far enough to enable us to decide, in all possible
states of affairs, which box, if any, was the same box as the
one referred to in the statement in question. Hence we
cannot decide in every possible state of affairs whether
anything is "this box", and so cannot decide whether the
statement would be true in all possible states of affairs.
(Indeed, in some possible states of affairs there would
be nothing which was "this round box".)
The criteria for identity work up to a point, but only
if we try to identify objects in different possible states
of affairs which are not too different: there must be a
    338

certain amount in common between them. This explains
why we can intelligibly say: "This box might have been
red", "If that wall had been painted yesterday it would
have been dry by now", and so on. We can talk like this
because we do not imagine other things to be very different,
which is why criteria for identity work. But they
could break down. Suppose the house had been bombed
twenty years ago, and the wall rebuilt as before. Would
it still have been this wall? And if it had been built
of a different material? Or with a different thickness?
Or in a slightly different position? ...
Of course, we can, if we wish, treat singular state-
ments such as "This round box is not square" as necessary
truths, simply because they are substitution-instance of
universal statements which are necessarily true, This
is the fact which underlies our use of such idioms as
"Tom's uncle has to be a man", "If the box is square then
it must be rectangular". More commonly, perhaps, the
phrase "not necessarily" is used to deny the existence
of a universal statement which is necessarily true of
which some statement would be a substitution instance:
"Her aunt is not necessarily a spinster". But I shall
not apply the necessary-contingent distinction to state-
ments mentioning particulars, for the reasons already
given. This does not mean that singular statements are
contingent, but that the question simply will not arise.
It is of no interest for our purposes, as all interesting
questions about necessity can be reduced to questions about
necessary connections between universals. (See 7.B.)

It should be noted that I have defined "necessary"
in such a way that it cannot always be applied to pro-
    339

Positions which mention universals. Thus, the statement
"Roundness is incompatible with squareness" is neither
necessary nor contingent according to my definition,
since it is not clear what could count as an exception
to it, unless it means "Nothing is both round and square"
The definition of "necessary" in chapter seven is concerned
only with what would be the case in all possible states
of this world (any world in which the same universals
exist as in our world). We can then ask whether in
some possible state of affairs two properties would exist
in the same object, and if the answer is in the negative,
then the properties are incompatible. But there is no
further intelligible question whether in some possible
state of affairs the answer would be positive! Hence
there is no clear sense to the question whether the
properties would be compatible in some possible state of
affairs. We may be able to invent some sense for the
question whether there might have been a world in which
these properties were not incompatible, but it would
certainly not be a question about all possible states
of this world. This illustrates the general fact that
there is no question of truths being necessarily necessary
or only contingently necessary, in my terminology
    340

Appendix II

CONFUSIONS OF FORMAL LOGICIANS

II.1.   In chapter five I tried to illustrate a way of
studying logic by explaining the logical properties of
propositions and relations between propositions. In
this appendix I wish to show how people may be led astray
as a result of a concentration on methods of classifying
and describing propositions according to their logical
properties and relations to one another. I shall
exaggerate some features of this "formal" approach to
logic, in the interests of brevity and clarity: it should
not be regarded as an accurate historical survey. There
is probably no philosopher who has consistently made all
the mistakes which I shall describe.

II.2.   Many philosophers have thought that logic consists
in the study of propositions which are true in virtue of
their logical form and inferences which are valid in
virtue of their logical form. (See, for example, p. 10
of Russell's "The Principles of Mathematics".) They have
not often been quite clear about the distinguishing char-
acteristics of logically true propositions and logically
valid inferences, but they do seem to mean to refer to what
I call "formal truths" and "formally valid inferences"
(See 5.C.6, 5.C.8.)
The first step in such a study seems to be to describe
and classify propositions according to their logical
properties, or, more commonly, to classify inferences.
As shown in chapter five, a convenient way of classifying
them according to their logical form is to replace all non-
    341

logical words in the sentences concerned by variable-
letters. For example, "All P's are Q's" represents the
logical form of "All horses are animals", and the logical
form of the inference from "All stallions are horses and
all horses are animals" to "All stallions are animals"
is represented by "All P's are Q's and all Q's are R's =
ergo = All P's are R's", or some such symbol. In this
way all propositions or inferences with the same logical
form may be represented by the same symbol, and the
geometrical or typographical properties of the symbol
are supposed to show or represent the logical properties
and relations of the propositions and inferences represented
by the symbols. (Instead of such symbols, we could, of
course, use such words as "disjunctive", "conditional",
"universal affirmative", etc., to describe logical forms
for purposes of classification.)

II.3.   For a long time logicians apparently did little
more than give these various forms (or at least the ones
they had bothered to symbolize) names, and list those
which corresponded to logically valid inferences, or
logically true statements, and gradually they came to
think they could forget about the propositions and in-
ferences with which they had started and concentrate
entirely on the symbols representing their logical forms.
They even went so far as to mistake symbols for propositions
with dire consequences, as will be seen.

II.4.   Eventually came the discovery that a system of
symbols representing logical forms could be regarded as
something like an algebraic system, as follows. A few
of the (simpler) symbols representing logically true
    342

propositions could be taken as "axioms", and a few, or
perhaps even only one, of the symbols representing logically
valid inferences could be interpreted as expressing
a rule for deriving new symbols from given ones, in such
a way that the class of "theorems", that is symbols
derivable from the axioms by successive applications of
the rule(s) of derivation, constituted the class of
symbols representing propositions true in virtue of their
logical form.
The discovery of such "formal systems" amounted to the
discovery of a recursive method of characterizing a class
of symbols which represent propositions true in virtue of
their logical form: for the axioms and rules of derivation
provide a recursive definition for the predicate "is a
theorem".

II.5.   Extensive mathematical investigations were carried
out of the various ways of characterizing such recursive
systems and comparisons were made between different methods
of defining the same class of "theorems", and between
quite different systems, many entertaining mathematical
results being obtained.
Unfortunately, some people mistook this mathematical
study for a philosophical one: thus, to show that some
symbol was a theorem in a formal system was thought of as
a sufficient explanation of the fact that propositions
with the logical form represented by that symbol were
true. For example, Waismann, in his article "Analytic-
Synthetic" in Analysis, December 1949 (pp. 31, 33, 36),
implied that correspondence with a theorem of 'Principia
Mathematica' or some other text-book of logic was a
necessary and sufficient condition for being a logical truth.
    343

Sometimes the construction of derivations of theorems
in such a system was thought of as constituting a proof
of the propositions whose forms were represented by those
theorems. Logicians apparently failed to notice that any
class of symbols can be represented by a suitably chosen
formal system (though in some cases there may be no rules
of derivation and all theorems may have to be taken as
axioms), and so they thought that there was something
significant about the fact that logical truths could be
represented by such a system. (There is something significant
namely that our rules for the use of logical
words enable more and more complex sentences to be built
up using those words, their meanings being determined by
the way they are constructed. But by this time philo-
sophers had forgotten about meanings, having been
mesmerized by symbols.)

II.6.   The trouble was that the possibility of repre-
senting properties of propositions or relations between
propositions by symbols in a formal system, together with
the geometrical resemblance between such symbols and
sentences in a language, led philosophers to think of
such a formal system as a kind of language, or even as
constituting a part of our own language. At any rate,
those parts of symbols which looked like or represented
our logical words, such as "and", "or", "not", etc., were
identified with our logical words. A formal system was
somehow thought of as providing a framework for our lan-
guage in which non-logical words or concepts could be
embedded. (This attitude is clearly expressed in the
writings of Carnap.) Sometimes it was hinted that insofar
as any language fell short of this ideal it crust be deficient.
    344

II.7.   Thus, a language came to be thought of as made up
of a set of logical words, whose functions were fully
defined by the rules of some formal system, together with
extralogical words governed by rules of some other kind.
But, since the rules of a formal system are concerned only
with the symbols in that system and their geometrical
relationships, since they mention nothing extralinguistic,
it looked as if the rules for the use of logical words must
themselves mention no non-linguistic entities. Thus it was
claimed that the rules were purely syntactical. Similarly,
since the property of being a theorem in the system was a
recursively defined syntactical property of symbols, and
since theorems represented logically true propositions,
it was claimed that logical properties were purely syn-
tactical properties of sentences. Thus it was thought
that logic could be reduced to syntax. I have already
argued against this in section 5.A: if logical constants
were governed by purely syntactical rules, then they could
never have any essential function in sentences expressing
contingent truths or falsehoods.

II. 8.  If formal systems provide frameworks for languages,
and if formal systems have axioms and rules of inference,
then surely languages must have them too? So languages
came to be talked of as "systems" with axioms and rules
of inference of their own. (I have argued that these
concepts do not intelligibly apply to real languages, in
"Rules of Inference or Suppressed Premisses?" which should
appear in Mind soon.) Since different formal systems may
have different sets of axioms and rules, and even different
classes of theorems, it seemed that languages too might
have different systems of logic, and so people quite
    345

happily talked shout "the logic" of a given language, or
talked of logical truth in some way relative to a
system (See M. Bunge, in Mind, 1961.)

II.9.   The discovery of formal systems and the possi-
bility of giving recursive characterizations of the class
of symbols representing propositions in the system (by
"formation rules") in addition to the possibility of
recursively characterizing the class of "theorems",
sparked off a number of reductive programmes. It was found
that the set of logical constants employed in a formal sys-
tem could be decreased in size without effectively dimini-
shing the number of theorems since all theorems contai-
ning other logical constants than those chosen as primitive
could be reintroduced merely by giving "definitions" for
the eliminated constants in terms of the constants taken
as primitive and substituting "synonymous" symbols.
Thus the class of theorems and the class of logical
constants were "reduced" to subclasses by definitional
elimination, and it was thought that a similar procedure
could be employed for reducing the concepts and theorems
of arithmetic to those of logic. Since logic was thought
of by some as merely a matter of syntax and since it was
thought that arithmetic could be reduced to logic, it
seemed that arithmetic should be thought of as a syntacti-
cal science concerned with the manipulation of symbols in
a formal system. Frege brought powerful arguments to
bear against such views, but it is not quite clear whether
he realized that they could be used against Formalist
philosophies of logic as effectively as against Formalist
philosophies of mathematics

II.10.  One of the very strong motivations behind
the search for formal systems was the desire to find
some absolutely rigorous and explicit method of proof:
    346

hardheaded philosophers did not like to talk about "self-
evidence". The discovery of formal systems made it look
as if the notion of logical proof or justification could
be "reduced" to formal derivation from a fixed set of
axioms and definitions by means of predetermined rules
of inference. The production of proof-sequences and
proofs of theorems in formal logic came to be thought of
as the paradigm of rigorous argument. Only deduction in
a logical calculus could be regarded as strictly valid
reasoning.

II.10.a.    Eventually, some thought that the only way of
proving anything was by producing a sequence of statements
starting from axioms and definitions and proceeding
according to the rules of inference which were supposed
to be among the rules of our language. (See II.8, above.)
No notice was taken of the fact that there must be some
other way of being justified in accepting something as
logically true, or as logically following from something
else: the question "What right have we to accept the
axioms or to follow the rules of inference? How do we
know that we shall not be led to affirm false statements?"
was not given a proper examination. Thus people tried to
"analyse" the actual processes of reasoning which we follow
when we are not involved in logical investigations by
looking for "suppressed premises" and "rules of inference".
They regarded informal proofs, such as the proofs employing
diagrams which are used in geometry, as somehow inferior or
inadequate: such proofs had to be replaced by something
formalized. (Cf. 7.D.ff.)
The logical conclusion of this line of thought is that
unless one has worked through a proof of some formal system
    347

such as "Principia Mathematical one is really not fully
justified in believing that three plus two equals five,
or in believing that if all wooden boxes are red, and not
all wooden boxes are round, then not all red boxes are
round, or in believing that nothing can be round and square
at the same time. But people are justified in believing
these and other things, and completely justified, without
having gone through formalized proofs, and this shows once
again that there must be some other kind of justification
than that given by a formal proof. (I have tried to
describe this other justification in section 7.D on informal
proofs.) But perhaps, if we have this other kind of
justification, then formalized proofs may be superfluous?

II.10.b.    Even if proof-sequences gave some sort of justifi-
cation for acceptance of statements corresponding to the
formulae terminating them, this justification could not
be complete. The point of asserting an arithmetical or
logically true proposition cannot simply be to announce
that the sentence expressing it is derivable in a formal
system from other symbols according to fixed rules. For
then asserting such a proposition would be analogous to
displaying a chess-board with the pieces arranged in a
certain way in order to announce that they are in a position
obtainable from the "starting position" via moves in accord
with the rules of the game. If asserting the proposition
has some further point, then a justification is required
for assuming that mere derivability in some system
guarantees that it is true, if asserted with this further
point: a justification is required for regarding the
axioms as true and for regarding the rules of derivation
as truth-preserving. (It is forgotten that we knew how
    348

to select logical truths and logically valid inferences
before we constructed formal systems of symbols repre-
senting them.)
All that a formalized proof can do, is "show" that a
sentence has a certain geometrical or syntactical structure,
or that its structure is related to other structures in some
way. But this is not enough for logic: in addition one
must know that when sentences have certain structures, or
when structures are related in certain ways, then the pro-
positions expressed by those sentences have certain logical
properties or stand in logical relations, and one can learn
this only by taking account of the roles in the language
of various aspects or parts of sentences, not by looking
at formal systems.

II.10.c.    I do not wish to imply that the construction of
a formalized proof can never serve any useful purpose.
It can be used to demonstrate that certain syntactical
relations (i.e. geometrical relations) hold between sen-
tences expressing certain propositions. That is to say,
a formalized proof rigorously, but informally, proves a
theorem of combinatorial geometry,
A formalized proof can demonstrate that if logical
relations correspond to certain syntactical relations,
then those logical relations hold between certain pro-
positions (It does not show that logical relations
correspond to these syntactical relations, nor does it
show why they do.) In addition, just as formulae in a
calculus can be thought of as providing a useful symbolic
representation or "map" of propositions in a language
(of, II.2 and II.4), so can proof-sequences, which pick
out certain sorts of "routes" along such a map, usefully
    349

represent patterns of formally valid arguments, for pur-
poses of classification or summary, for example. A
formal proof can be a useful guide when one is trying to
understand an argument to see whether it is valid or not,
but the proof does not make understanding superfluous.
I am not trying to show that such proofs are quite
useless so much as to draw attention to some mistaken
views as to the purposes which they can serve, and to
show how they are connected with other mistakes which
arise out of a concentration on the symbols which represent
forms of logical truth or logically valid inferences. The
argument has been aptly summarized by Frege; (in "Trans-
lations" p. 201):
"Apparently we are being tacitly referred to our
knowledge of meaningful arithmetic. But if we have
a knowledge of meaningful arithmetic, we have no
need of formal arithmetic."
Replace the word "arithmetic" by "logic" in this state-
ment, and it will serve as a summary of all my remarks.

II.10.d.    All this seems to show that there is something
fundamentally misguided in the attempt to produce absolutely
explicit proofs, except perhaps as an exercise in a branch
of mathematics - combinatorial geometry. It shows that
there is something wrong with the Leibnizian dream of an
ideal language which somehow has its meaning written on
its face, so that one can settle questions about truth
and falsity by mechanical manipulations alone. It is
misguided because, no matter how much is written on the
face of a symbol, there will always be something left out:
an explanation of what the "writing" means, a description
of its function in the language. Semantics cannot be
reduced to syntax.
    350

(This does not mean, of course, that there is no such
thing as a rigorous proof. It merely means that one kind
of analysis of rigour is wrong. See chapter seven,
section D.)

II.11.  One of the sources of an oversimplified view of
logic (logic = syntax) is the selection of a class of
"canonical" forms for study. It is obvious that if all
possible symbols corresponding to the logical forms of
propositions and inferences were constructed in the usual
manner by replacing non-logical words in sentences by
variable-letters, then many more different sorts of symbols
would be obtained than have ever been encompassed within
the class of symbolic forms discussed in any one text-book
of logic. For example, the following are not usually
listed separately by logicians:
(1) All A's are B's.
(2) Every A is a B.
(3) If a thing is an A then it is a B.
(4) Only B's are A's.
Instead, they represent the whole lot by one symbol, such
as
(5) Whatever x may be, if x is an A then x is a B,
which is then described as a "canonical" form. (This is
similar to the old mistake of thinking that logicians need
consider only propositions in subject-predicate form.)

II.11.a.    Normally the selection of canonical forms is
done as a matter of course, following a philosophical
tradition originating with some philosopher's (understandably)
limited survey. But sometimes an attempt is made to
justify the failure to discuss statements or inferences
    351

not in canonical form. Several different sorts of
reasons may be offered.
i) It is obviously more convenient to classify only a
small class of logical forms than to take all varieties
into account.
ii) The omission of forms like (1) - (4) above may be
defended by the assertion that any proposition with one
of those forms "obviously" means the same as a proposition
with the canonical form (5), and may therefore be replaced
by it without any loss of generality. (See for example,
Quine's remark in "word and Object", p. 228: "... Such
a canonical idiom can be abstracted and then adhered to
in the statement of one's scientific theory. The doctrine
is that all traits of reality worthy of the name can be
set down in an idiom of this austere form if in any idiom.")
It may be added that statements in canonical form are clear
and precise, whereas other statements are vague, ambiguous,
or unclear. Moreover, if anyone wishes to use one of the
other forms with a clear meaning he may do so by redefining
it in terms of the symbols and constructions employed in
the canonical forms. (Cf. "Word and Object", p. 188.)
iii) Finally, it seems to be thought, sometimes, that
certain canonical forms are most suitable for representing
logical properties of propositions. For, if written in
these forms the sentences may more effectively "show"
their logical properties and relations. (" ... the inner
connection becomes obvious ...": Wittgenstein, "Tractatus"
5.1311.)

II.11.b.    There is certainly no reason why, if we find it
convenient, and If only some facts of logic interest us, we
should not select only a subclass of the whole class of
    352

logical forms for purposes of study and systematic
representation in symbols. But the choice of such
canonical forms must always be, at least to some extent,
arbitrary, depending on such subjective factors as what
interests us, or what we find "obvious" and therefore
not worth recording in our symbolism.
Why should we regard the following as different forms,
whose logical equivalence is worth recording,
(5) Whatever x may be, if x is an A then x is a B,
(6) There is no x such that x is an A and x is not a B,
while the equivalence between (5) and, for example,
(4) Only B's are A's,
is regarded as "trivial" or "obvious", or "merely
linguistic", or "merely a matter of meaning"?
Surely the equivalence between (5) and (4) is as much
in need of study and explanation as the equivalence between
(5) and (6)? What should we say if someone turned up who
claimed to find the latter "trivial, obvious, and merely a
matter of meaning" while the former equivalence was
tremendously important for him? If some proposition
of the form (5) turns out to be true in virtue of its
logical form (e.g. if the same predicate is substituted
for both "A" and "B") then why should this be any more
interesting or important than the fact that the corres-
ponding proposition of the form (4) is a logical truth?

II.l1.c.    What I am driving at is this: if any explanation
of the logical truth of, or logical relations between,
propositions makes use of the fact that they are expressed
by sentences in canonical form, then the explanation points
to something inessential, for logical properties and
relations of the same kind are found amongst propositions
    353

expressed by other sentences. If there is some notation
which renders certain logical connections perspicuous, a
symbolism in which the logical properties and relations of
propositions "show" themselves any more obviously than
they do in other notations, then this is merely an inter-
esting fact about that notation and its effect on us, and
does not reveal any general truth about the logical pro-
perties of propositions.
After all, propositions expressed in this notation do
not as obviously "show" their logical connections with
propositions expressed in other notations, nor the connections
between those other propositions. This is because the
function of a sign is not generally shown by that sign,
though if we know the function, then, in some cases, we
may more easily be able to see the consequences of their
having these functions than in other cases.

II.11.d.    Whether propositions are expressed by sentences
in canonical form or not, their logical properties and
relations are due to the fact that they are built up in
certain ways with logical words and constructions which
have been given functions of the sort described in section
5.B. The connection between logical properties and
relations of propositions and geometrical or syntactical
properties and relations of sentences is a consequence
of the fact that the logical constants have the functions
which they do have in determining the conditions in which
propositions are true or false. It may help someone to
see that a proposition has certain logical properties or
stands in certain logical relations to other propositions
by "rewriting" it in canonical form, but pointing to the
geometrical features of the new sentence does not explain
    354

the logical properties of the proposition expressed by
the old one: that has to be done by talking about their
functions.

II.11.e.    It is a failure to see this that sometimes leads
philosophers to talk about the "real" logical form of a
proposition as opposed to its apparent logical form, shown
by the grammatical form of a sentence. But the logical
form of a proposition is the way in which its truth-
conditions are determined by the meanings of the non-
logical words, and that must be quite correctly shown
by the sentence itself, for otherwise we could not under-
stand it properly, and we do understand our ordinary
sentences whether they are in canonical form or not.
The reason why we find some types of sentences misleading
is that we are philosophers who have swallowed a short-sighted
traditional philosophical doctrine and fail to see a counter-
example to that doctrine for what it is. (Or if we are
not philosophers, then we find certain forms misleading only
because we fail to think clearly and allow ourselves to be
convinced that because an analogy or comparison works in
some cases it must work in all.)

II.11.f.    Finally, in connection with canonical forms it
should be noted that the tendency to regard formalized
proofs as having some kind of exalted status is quite
analogous to the tendency to regard some forms of pro-
positions as somehow "superior" from the logical point of
view. We think of these special kinds of proofs, or
deductions, as having a "canonical form" in which logical
relations are most efficiently demonstrated. Having
noticed that these proofs convince us, we fail to ask why
    355

they should do so, or why they alone should do so.

II.12.  I have tried, in this Appendix, to carry out a
very brief survey of some of the mistakes and confusions
which arise when philosophers restrict their attention to
the forms of sentences and neglect the functions of words
and constructions. Partial description is mistaken for
complete explanation, largely because a formal system,
which is a device for representing certain features of
propositions, is thought of as containing propositions,
owing to the physical resemblance between its formulae and
sentences in a language. The study of methods of
representing facts of logic, and classifying them, leads
to a mathematical study of various methods of recursively
defining a class of combinations of symbols, and this
study, which is really a branch of Geometry, is mistaken
for a philosophical study of logic or truth or inference.
The concepts invented for the purposes of such mathematical
studies are mistakenly assumed to have some philosophical
application: geometrical concepts referring to shapes
of symbols and their interrelations are employed by philoso-
phers when they should be talking about the functions of
symbols and their interrelations. (Analogous mistakes
are sometimes made by physicists, when they assume that
concepts which apply only to mathematical models also
have application to the reality which these models are
supposed to represent. But such mistakes are less fre-
quent because there is, fortunately, no physical resemblance
between the symbols used by mathematicians and the things
which physicists take them to represent.)
I think that I have been drawing attention to some of
    356

the facts which led Wittgenstein to complain:
'Mathematical logic' has completely deformed the
thinking of mathematicians and of philosophers, by
setting up a superficial interpretation of the forms
of our everyday language as an analysis of the
structures of facts. Of course, in this it has
only continued to build on the Aristotelian logic.
("Remarks on the Foundations of Mathematics",
IV-48.)
The discussion of this appendix and chapter five
(especially sections 5.A and 5.B) may be construed as
an attempt to sort out the geometrical from the philo-
sophical questions.
    357

Appendix III

IMPLICIT KNOWLEDGE

III.1.  Throughout the thesis I have been making remarks
about things which must be known by persons who use words
to make statements. But I have often qualified them by
saying that such knowledge need not be explicit. In
this Appendix I wish to describe some examples of what
I call "implicit" knowledge and explain why it is possible
to talk about knowledge in such cases. I shall not be
able to deal with the subject thoroughly or systematically,
and will content myself with a few disorganized remarks.
It is important to clarify the notion of implicit know-
ledge if we are to be clear about philosophical analysis
and the nature of analytic propositions.

III.2.  First I shall give a list of examples of the sort
of thing I mean to talk about.
(a) In his article "Philosophical Discoveries" (in
Mind, April 1960) Hare talked about some persons who all
know how to do a certain kind of dance but are unable to
be sure about the correct description of the way the dance
goes until they actually try to do it, and he compares
this with knowing what a word or expression means without
being able to say what it means or how it is used
(b) Another example is provided by a person who
wishes to mention the fact that he has recently seen
someone, but cannot for the moment, recall his name.
He may say: "Of course I know it - it's on the tip of
my tongue."
(c) I know a tune very well, and can recognize it as
    358

soon as I hear it, but try as I will, I cannot, for the
moment, sing it or even imagine how it goes. (But if you
sing the first two bars, I may be able to carry on from
there.)
(d) I know a tune and can recognize it on hearing it, but
if someone writes it out I may not be sure whether he has
written it out correctly, until I play what he has written
on the piano.
(e) I am familiar with a face, or a style of painting
or musical composition, yet quite unable to say how I
recognize it. I cannot say what it is about the face,
or style, in virtue of which I recognize it and distinguish
it from others. Even when confronted with the face, or
an example of the style, I may be unable to describe the
distinguishing characteristics.
(f) A person who can type very easily, even when
blindfold, may find it very difficult to describe from
memory the relative positions of the keys on the type-
writer.
(g) A person tries to describe everything in a room
he has just left, and is sure he has left out nothing.
Then someone asks: "Was there a carpet?" He replies:
"How silly of me! Of course there was a carpet, and I
knew that very well. I don't know why I didn't think
of it."
(h) We can all count, and can tell, given any numeral
written out in English or in Arabic notation, which is the
next one in the series. But most persons who can do this
cannot give a general formulation of the principle for
going from one to the next, despite their ability to apply
the principle. (Cf. 6.E.5, 6.E.6.) Even if someone else
offers a formulation, they may not be able to think clearly
    359

enough to tell whether it is correct or not.

III.2.a.    This should be compared with some of the following
facts mentioned in the thesis.
1) I asserted that talk about meanings presupposes
the existence of criteria for identity of meanings, at
all levels, In section 2.B; then, in 2.B(note), I allowed
that people who talk about meanings need not explicitly
know which criteria they are relying on.
2) In chapters three and four I described various
kinds of correlations between words and properties which
explain how we use descriptive words. But one need not
be able to formulate explicitly the principle on which one
decides whether to call objects "horses" or not. One may
use a word according to a complicated procedure, and yet
not know in an explicit way what that procedure is. (Cf.
3.D.9.) One may use a word according to several different
rules superimposed in an indeterminate way, without
realizing this until (e.g.) one starts thinking about
difficult borderline cases. (Cf. 3.E.2, 7.D.11. note, and 4.B.2.)
3) In 5.A.3 and 5.A.11 I described techniques which
we have to learn to use for discovering whether statements
using the words "is", "or" and "all" are true or not.
A person must know what the technique is in order to
understand sentences using the words: but he need not
know in an explicit way, for he may be unable to dis-
tinguish between correct and incorrect formulations of
the rules for the words. The techniques may be learnt
by example and memorized, without any explicit description
ever being formulated by pupil or teacher. (5.A.6,
5.B.8, 5.C.9.)
    360

III.3. Each of these examples is puzzling. In each case
we want to say that there is something a person knows all
the time, or really knows, despite his inability to give a
correct answer to a question about it. He knows what he
is doing, that something is the case, how to do something,
what something is, etc., and yet, without deliberately
deceiving, gives the impression of not knowing. What do
we mean by saying that he really knows? What explains his
inability to answer correctly in these cases?
I believe that the answer to these questions is given
by the fact that there are a great many different tests
for knowing any one thing, and passing any one of them
counts as a sufficient justification for the claim to know,
or the assertion that someone knows, provided that there is
no reason to think that success in the test can be explained
as an accident or some kind of lucky guess.
The fact that I can type correctly without looking
justifies my claim to know the relative positions of the
keys on a typewriter, despite the mistakes in my attempted
description of their positions. I know where my pen is
because, as soon as I need it, I go straight to the right
place, despite the fact that if someone had asked me where
it was I might not have been able to answer correctly. I
know how the features on a face are arranged because I can
recognize the face and distinguish it from other faces
despite my inability to describe the peculiarities in
virtue of which I recognize it. I know what the tech-
nique is for deciding whether a statement of the form
"x is P" is true or not, despite my inability to formulate
the technique, since, when confronted with such statements,
and told the meanings of the non-logical words, I am able
to decide whether they are true or not.
    361

III.4.  There are many different tests for a person's
knowing any one thing, and his passing any one of them counts
as strong evidence that he knows. But there is no one of
them that he must pass in order to show that he knows:
when he fails one of the tests this need not count as
strong evidence for his not knowing, since it may often
be assumed that there is an explanation for his not passing
the test. (E.g for his giving the wrong answer or not
being able to think of the right answer.) Some of these
explanations of an apparent lack of knowledge are the
following:
First of all there is a whole family of cases which
need not be discussed in detail. A person may quite sincere-
ly give a wrong answer despite his knowledge of the correct
one, simply because of a slip of the tongue, or on account
of his being absentminded, or preoccupied with something
else, or because he has misheard the question, or because
giving the correct answer requires concentration and he
has a headache. All these are cases of temporary muddle
or confusion, and can usually be detected by asking the
person again, in a suitable tone of voice!

III.4.a.    Next there is the relatively uninteresting case
where the person is unable to express his knowledge In
words simply because he does not know any words which
could express it adequately, either because he hasn't
learnt any or because he cannot think of them at the
moment. A person who knows the difference between the
sound of a clarinet and the sound of a flute may simply
not think of saying that the former has more upper harmonics,
or that the latter is "purer" or "more naive", or "less
reedy".
Connected with this is the case where a person is
    362

not able to give the correct answer to a question simply
because he has not thought of that answer as a possible
one. But as soon as it is suggested he recognizes it as
correct, and if several are suggested he can pick out the
correct one from among them. His passing this teat (per-
haps in addition to his displaying his knowledge in his
behaviour) shows that he "really knows", despite his failing
the more difficult test of having to think up the right
answer for himself. (Compare: in philosophy the diffi-
culty often lies in thinking up the correct answer, not
in seeing that it is correct, once stated.) (Compare
example (g) in III.2.)

III.4.b.    In some cases, a person may be unsure whether a
description is correct or not, not because he has not
noticed the possibility, but because it describes from
a new point of view. I may know how to recognize a tune
by its sound, and yet be unsure about recognizing it when
written down, despite the fact that I can read music.
(See example (d).) I may, in addition, be unable to
recognize it when played backwards or upside down. I
may recognize the feel of something (e.g. a familiar
chair) and yet be unable to distinguish it from others
by the way it looks: and for this reason I may be unsure
about the correct description of the way it looks. I
believe this may apply to one's knowledge of how to do
something, such as a dance, or touch-typing. (See examples
(a) and (f), of III.2.) A person who is able to tell
whether he is doing a dance correctly or not (he knows
which movements he is supposed to make, and he knows which
movements he is making), may be quite unable to be sure
whether other persons are doing it correctly when he watches
them. Similarly, if he is given a description of the
    363

dance from the point of view of a person watching its
performance he may not be sure whether it is correct or
not. He may find out by making the movements and looking
to see whether they fit the suggested description or not.
In that case he is observing himself in two ways at once,
or, more accurately, he knows in two different ways what
he is doing.

III.4.c.    a slightly different case is the following: a
person may know how something is done In the sense that he
can do it in an unthinking way when he has to, and yet when
he stops and thinks about it he may not be sure. For
example, an experienced flautist may be able to play a des-
cending chromatic scale at high speed on the flute, but
if his fingers are placed in the position for one note and
he is asked to Indicate which must be moved up and which
must be moved down for the next note down the scale, he
may have to pause and think for some time, especially if
he is not actually holding a flute. He may find it even
more difficult to say which fingers must move if he is not
first permitted to move them. This fact, that he cannot
be sure about the individual steps of a routine which he knows
how to apply quickly and unthinkingly, may help to explain
his difficulty in telling a pupil which fingers to move
and when, in addition to the factor already pointed out,
namely that he may be able to recognize the correct move-
ments from the point of view of an agent while being unsure
about them from the point of view of an external observer.

III.4.d.    Yet another possible explanation of a failure
to give the correct answer is the fact that one may be
able to apply a technique which is very complicated,
    364

insofar as it has several stages, or insofar as exactly
how it goes at any stage may depend on other things. For
despite one's ability always to go on from one stage to
the next correctly, even where how one goes on depends on
what the technique is being applied to, one may not be able
to think about all the stages, or to recall all the possible
variants, when one is trying to describe the technique in
the abstract. For example: a chemist who has been trained
to identify samples of some substance, which can occur in
several varieties, for which the tests are slightly differ-
ent, may make a mistake in describing all the tests,
despite the fact that he never errs in performing them.
Similarly, we may find it difficult simply to sit back
and describe all the observable factors which we have
learnt to take into account in deciding whether an object
is a horse or not (or in deciding whether a person has an
intention to do something or not), despite our ability to
make the decision when necessary.
In the previous case (III.4.c.) a person who could
recognize correct applications of a complete routine might
be in doubt about individual stages of the routine. In
this new case a person who is sure about any one of the
stages when asked may be in doubt if asked simply to
describe them all. The complexity of the routine explains
his failure.

III.5.  I said that in each of these cases a person may
be described as knowing something or other despite the
fact that he fails some test for knowing. The reason
why we do not take the failure as a criterion for his not
knowing, is that we are able to explain the failure in
some other way than by saying that he doesn't know. This
    365

is supported by the fact that in most cases he may be able
to pass the test later on without having gone through a
process of acquiring the relevant knowledge in between.
For example, the person who gives the wrong answer
because he is absent-minded simply has to think again.
He need not learn again.
When I have a person's name "on the tip of my tongue",
and correctly pick it out from among several suggestions
offered to me, all I needed was to hear the name to bring
it back to mind: in such a case I do not learn that that
is his name.
The person who cannot describe the way a dance goes
until he does the dance does not thereby learn how the
dance goes. What he learns is how to describe the dance
from a new point of view.

The person who cannot describe something with which
he is perfectly familiar because he must first learn the
appropriate vocabulary is not thereby acquiring the know-
ledge which he is able later on to express in his new
vocabulary. Learning the name of a colour is not the
same thing as learning that that is the colour of my
table, even though it may enable me for the first time
to say correctly what the colour of my table is.
To summarize: many different sorts of things may
enable a person to know in an explicit way something which
he previously knew only implicitly. We say in such cases
that he nevertheless "knew" previously because the process
in which he learns to express his knowledge or to pass one
of the tests which he previously failed, is not the same
sort of process as is required for acquiring the knowledge.
(He does not make the relevant empirical observations, or
    366

examine properties to discover their inter-relationships.)
He is all the time potentially able to pass the test, and
the evidence for this, apart from his later success, is
the fact that he was previously able to pass some other
test which displayed his knowledge (cf. III.3.). (I
shall not discuss the question how passing one test counts
as evidence of ability to pass others.)

III.6.  All this may help to explain why I was able to
describe some of the things which people know when they
know how to talk, without fear of being contradicted by
the fact that my descriptions would probably come as news
to many people who know how to talk! These people know
how to talk, but they do not know explicitly that their
descriptive words have the meanings which they do have in
virtue of being correlated with observable properties as
described in chapters three and four, and neither do they
know explicitly that to the logical form of a proposition
there corresponds a technique for determining truth-values
for sets of non-logical words by examining the way things
happen to be in the world. Their knowledge of all this
is implicit, and to say that they know it all is to say
that it accurately describes what they are doing when
they decide whether to use some word to describe an object,
or whether a sentence expresses a true proposition, or
whether two persons understand some sentence in the same
way. Their not knowing explicitly may be explained by
factors of the sorts described already.
But there is one sort of factor which can be very
important, and which I have not yet discussed; I shall
do so now.
    367

III.7.  A possible explanation of a person's not knowing
that in doing A he is doing B, C, D ... may be the fact
that he does not have the concepts which would enable
him to think about his activity in this way. More speci-
fically, he may not have the metalinguistic concepts which
would enable him to think about and describe the way he
uses words. For example, the child who can use the word
"cat", but does not have the concept "word", can hardly
be expected to know explicitly how it uses the word "cat".
Similarly, the explanation of a person's inability to say
how he uses the word "horse", for example, may not only be
that the procedure for picking out horses is too complex
(see III.4.d and also 3.B.5 and 4.A.6), but in addition that
he does not have the concepts "meaning", "property",
"semantic correlation", "disjunctive range", etc.: he
does not have the concepts which I have used in my des-
cription of what people learn when they learn to talk.
(Similarly, philosophers have hitherto been unable to give
a correct explicit account of analytic propositions, despite
their implicit knowledge of what it is for a proposition
to be true by definition, on account of not having something
like the concept of a "rogator" [see section 5.B], or so
it seems to me.) A person may acquire the concepts which
enable him to know explicitly that he uses his logical
words or descriptive words in a certain way, without
actually being taught that he uses them in that way. He
thereby learns to say what the words mean, but he does not
learn what they mean - he knew that all along, since he
could use and understand them.

III.7.a.    Now we can see how to cope with the difficulty
mentioned in 6.C.4. I defined "identifying fact about
the meaning of words" to mean "fact which must be known
    368

by anyone who knows the meanings of those words". The
difficulty was that someone might learn to use the word
"gleen" to refer to the combination of the properties
which are referred to by the English words "glossy" and
"green", without knowing explicitly that the property
referred to by "gleen" was the combination of the properties
referred to by "glossy" and "green". He might not know
the meanings of the words "glossy" and "green", and he might
not have the metalinguistic concepts which would enable
him to understand a statement about the meaning of desc-
riptive words. Still, he knows the fact in question
implicitly, since he makes use of it in employing the word
"gleen". He decides whether to describe an object as
"gleen" or not by looking to see whether it has the two
properties in question. If he decides in any other way
he does not understand the word "gleen" as I have described
it: if he does not know, even implicitly, that it is
correlated with those two properties, then he does not
know its meaning. (This is still not quite clear: a
complete discussion would require an investigation and
comparison of the following expressions: "Knowing what the
word 'W' means", "Knowing how to use the word", "Knowing
the meaning of the word", "Knowing that the word 'W' means
...", "Being able to understand the word 'W' " and so on.)

III.7.b.    It is important to distinguish the acquisition
of new metalinguistic concepts, which enable one to express
one's knowledge of facts about the meanings of words, from
the acquisition of other sorts of concepts, which enable
one to discover new facts about the meanings of words,
facts which one did not know previously even though one
understood those words. Consider, for example, the
concept of a "starlike" figure, introduced in 3.D.5. A
    369

starlike figure is one which is bounded by straight lines
meeting alternately in reflex and acute angles. Now a
person may be able to use the word "square" to refer to
the usual recognizable property, without having the con-
cept "starlike". I may teach him the new concept by
giving a definition, or showing him examples, without
mentioning squares at all. Having acquired the new
concept he may then notice, for the first time, that no
square is starlike. He discovers a new fact about the
property referred to by the word "square", and thereby
learns that the words "square" and "starlike" are in-
compatible descriptions. But he does not thereby learn
an identifying fact about the meaning of the words, for
in order to acquire this knowledge it was not enough for
him to acquire new metalinguistic concepts.
Consider another examples I may know how to use the
expression "Daisy-daisy" as the name of a tune, and be able
to recognize the tune on hearing it, without knowing, even
implicitly, that the first and second intervals of the
tune are thirds (first a descending minor third, then a
descending major third), or that the first three notes
form a major chord, on account of not having the concept
of a musical interval, or a major chord. Suppose someone
teaches me to pick out musical intervals and name them,
and then one day I hear the tune I knew previously, and
notice immediately that the first two intervals are both
thirds. Have I acquired explicit knowledge of something
I knew previously in an implicit way? Surely not. It
seems much more reasonable to say that I have discovered
a new aspect of the tune; I had not previously noticed
the possibility of looking at a tune as a sequence of
musical intervals, and I in no way made use of the
    370

possibility, implicitly or otherwise, for I could only
have done so if I had had the appropriate concepts. (How
can a person who is unable to tell whether two intervals
are the same or not ever make use of the sameness of two
intervals? It should not be forgotten that this is a
phenomenological essay: I am not interested in what would
be given as a causal explanation of how he recognizes the
tune. See section 1.B.)

III.7.c.    This difference between acquiring metalinguistic
concepts which enable one to say explicitly how one had
previously been using words, and acquiring other sorts of
concepts which enable one to discover new facts about the
meanings of one's words, or about the properties to which
they refer, is one of the factors which lies behind the
distinction between an identifying (or analytic) fact
about meanings and a non-identifying (or synthetic) fact
about meanings, of which so much use was made in chapter
seven. This is what justifies talk about synthetic
necessary truths (whose necessity has to be discovered by
examining properties, perhaps with the aid of informal
proofs - see 7.D), and shows that the term "synthetic
a priori" is not just an old label with little explanatory
force, as averred by Hare, on p. 145 and p. 153 of
"Philosophical Discoveries".

III.8.  To sum up: there are many kinds of things which
one may know implicitly without being able to express the
knowledge in words, or to answer questions about it.
This inability may be explained in any one of a number of
different ways. It may also be removed in a number of
different ways, none of which involves actually acquiring
    371

the knowledge in question: they all involve merely
learning to express the knowledge in a new way. We say,
in such cases, that one knows, despite the inability to
express the knowledge, because one is able to use it, in
applying a technique, in carrying out a routine, in making
allowance for facts, etc.
I have tried to distinguish cases where implicit
knowledge of how words are used is made explicit, from
cases where a new discovery is made, where something new
is discovered about previously familiar meanings, namely
a connection with other meanings or some other previously
unnoticed aspect.

(Some more remarks about implicit knowledge will be found
in Appendix IV on "Philosophical Analysis". See also
remark about "implicit justification" in 7.E.5.)
    372

Appendix IV

PHILOSOPHICAL ANALYSIS

IV.1.   It was suggested in 1.A.5 that clarification of
the analytic-synthetic and necessary-contingent dis-
tinctions might help to solve problems about the nature
of philosophical analysis, and perhaps lead to methodo-
logical advances. Not much has been said on this in the
thesis, and in this appendix I shall make a few vague
remarks, in the hope of suggesting lines for more detailed
investigation.

IV.2.   Philosophical analysis, which may also be described
as "conceptual analysis", is essentially the search for
identifying relations between the meanings or functions
of words or sentences or types of linguistic constructions.
(See section 6.C, and 6.F.7, ff.) This is what seems
to lie behind such questions as:
(a) Can one be pleased without being pleased at or
about anything?
(b) Is there something queer in the assertion "I
have definitely decided to go, but I am sure I
shall not"?
(c) Is it part of the meaning of 'table' that tables
are used in certain ways, or have certain functions,
or is it just an additional fact about tables?
(d) When a person asserts that something is the case,
does he imply that he believes that it is the case?
In answering questions like these one is presumably drawing
attention to connections between concepts or features of
concepts, and this means drawing attention to identifying
facts about the meanings or functions or uses of words
or other expressions. From the discussion of section 2.B
    373

and 6.F.7,ff., it should be clear that there are many
different "levels" at which meanings or functions can be
related, any of which can explain the existence of impli-
cations between utterances, or the queerness or oddness
of certain utterances (self-contradiction, or analytic
falsity, is just one sort of case). It would seem to
be important to devise some principle for systematically
classifying such identifying relations, if philosophical
analysis is not to look like the piecemeal collection of
linguistic oddities.

IV.3.   For example, there are connections between the
describability-conditions of descriptive words, between
the techniques of verification corresponding to logical
forms of sentences, between the purposes served by the
utterance of statements, between the preconditions (exist-
ence of social habits or institutions or empirical
regularities in our physical environment) for the efficacy
of certain sorts of utterances. In all these cases there
may be Identifying relations at one level, or relations
between one level and another. Thus, If one of the
conventions of a language is that the utterance of a
statement of the form "I intend to do X" primarily serves
the purpose of giving people the assurance that X will be
done, then identifying relations between appropriateness-
conditions for this sort of utterance and truth-conditions
for the utterance of statements of the form "I believe
that I shall not do X" may generate a kind of queerness
manifested in the utterance of "I intend to do X but I
believe that I shall not do it". The queerness is to
be accounted for, not by conducting empirical observations
of what people can believe and intend, but by examining
    374

the knowledge of meanings which we normally apply when
we talk, just as we account for the necessary falsity of
analytically false statements. (See section 6.E.)
The difference is merely that in the latter case we are
concerned only with truth-conditions.
I shall not now try to describe a system for classi-
fying such identifying relations between meanings and
functions and the consequences which they can have.
(It may turn out that we must also allow for the
possibility of non-identifying relations between meanings,
analogous to those relations between universals which were
described in sections 7.C and 7.D.)

IV.4.   Now it would certainly be of some interest to see
what sort of system could be used for classifying various
sorts of connections between meanings, and the consequences
of such connections. This would amount first of all to an
extension of the Aristotelian classification of forms of
inference, of far greater significance than the mere ex-
tension to take account of more varieties of logically
valid inference (the much-vaunted achievement of modem
symbolic logic), and secondly to an extension of my
explanation (in chapters five and six) of the existence
of logical relations and properties of propositions. But
this interest in principles of classification can certainly
not explain why philosophers should debate with such great
interest questions of the kind illustrated in IV.2.
(System-building is, after all, supposed to be out of
fashion.) Why are they interested in finding out whether
it is actually part of the meaning of "table" that tables
have certain functions, and whether the English words "red"
and "green" are actually analytically incompatible, instead
of merely noting that there are these possible linguistic
    375

conventions, and that adopting them would have certain
consequences?

IV.5.   Perhaps some philosophers are simply interested in
empirical questions about how people use words, and these
questions are not entirely trivial, despite the fact that
we may already know how the words are used. For our know-
ledge may be implicit, and, as pointed out in the previous
appendix, there may be some difficulty in expressing such
knowledge explicitly, especially if superficial analogies
lead us mistakenly to expect that the answers will be of
a certain kind. (Cf. App. II.11.e, above.) But the
best way to serve this sort of interest in empirical
questions is to carry out empirical surveys (e.g. using
statistical methods), and it would be important to allow
for the possibility that in general there will be no
definite answers to such questions, since how a word is
understood may vary from person to person, and even an
individual may understand it In an indefinite way. (See
the discussion of indeterminateness in chapter four, and
also 3.E.2, 5.E.7.a, 6.D.3, 7.D.11.(note).)

IV.6.   However, most of those who indulge in conceptual
analysis are not inclined to use the methods of popularity
pollsters, and this is not merely a matter of laziness:
they are not really interested in how people actually talk,
though their words often seem to belie this. What else
are they trying to do then? What sorts of non-empirical
questions can they be trying to answer?
One kind of non-empirical question is the question
whether a philosophical distinction is vacuous, or whether
a philosophical system of classification works in this way
    376

or that, or not at all. Thus, a philosopher (Kant) who
has begun to describe a system of classification of the
sort envisaged above, might try to illustrate its appli-
cation by assigning particular examples to their place in
it, and his opponents might dispute that his principles
of classification have been correctly applied in these parti-
cular cases. He says the statement S has certain features
in virtue of which it satisfies the conditions which he
has laid down for being synthetic. They say that it does
not. But now there comes a confusion between the question
whether S has those features as it is in fact understood,
the question whether having those features entails satis-
faction of the conditions for being synthetic and the
question whether it is possible for any statement to
satisfy those conditions. Compare the following case.
A mathematician tries to prove that any geometrical figure
which has the property P also has the property Q, and he
does so by drawing a diagram with construction-lines, etc.
(See section 7.D).) Now if the case is sufficiently com-
plex there may be a debate as to whether the figure which
he has drawn actually has the property P, or whether it
actually has the property Q, or whether the construction-
lines as he has drawn them actually serve the purpose they
are meant to serve. These (semi-empirical) questions
about the particular diagram may then be confused with
other questions about properties, such as whether they
are necessarily connected, or whether it is possible for
objects to possess them at all, etc.
Thus, what really underlies an interest in an
apparently empirical question about how words are actually
used is an interest in a non-empirical question as to
whether and how a system of classifying possible ways of
using words may be applied. But the failure to distinguish
    377

the empirical from the non-empirical questions may lead
philosophers into endless disputes about "What we really
mean" - endless because what we really mean is too in-
determinate for either side of the dispute to be correct.
(This seems to me to be clearly illustrated by disputes
as to whether "I know I am in pain" is odd, disputes as
to the connection between expressions of intentions and
predictions of one's future actions, and disputes as to
whether it is part of the meaning of "good" that certain
types of men are good men or whether it is part of the
meaning of "good" that believing certain types of men
to be good men is connected with being inclined to behave
in certain ways. In the last case, not only are empirical
and non-empirical disputes confused, but, in addition,
practical disputes about how we ought to use the word
"good" are mixed in too.)
The existence of this sort of confusion is what led
me, in 2.C.10 and 7.C.8, to stress the fact that even if
it is established that in English the statement "All
triangles have three sides" is analytic, this does not close
the question whether it is possible to use the word "triangle"
to refer to the property of having three angles in such a
way as to make the sentence "All triangles have three sides"
express a synthetic necessary truth. We were not concerned
with the question whether some statement in English is actually
synthetic, but with the question whether it is possible
for certain sorts of sentences to express synthetic
propositions.

IV.7.   We have so far found that conceptual analysis can
serve the following purposes: 1) It can provide empirical
reports on linguistic usage, by making our implicit know-
ledge of how we talk explicit, though this purpose might
    378

be better served by conducting statistical surveys.
2) It can be a disguised account of the workings of some
system of classification of kinds of relations between
meanings and functions of linguistic items, and their
consequences. 3) The discussion of the way in which
particular words are actually used may serve to illustrate
or clarify or provide counter-examples to general state-
ments about systems of classification (or about possible
ways of using words) in much the same way as the par-
ticular diagram used in an informal proof can enable one to
see the truth of some general statement about geometrical
properties, or provide a counter-example.

IV.8.   However, this may make it look as if philosophical
analysis is concerned only with language and theories
about language, but that is not so. For a conceptual
analysis, in drawing attention to previously unnoticed
facts about the ways in which we actually use words, serves
also to draw attention to a possible way of classifying
the things referred to or described by these words: types
of material objects, states of mind, kinds of behaviour,
etc. It may draw our attention to a system of classi-
fication with which we are all familiar in one way, since
we employ it all the time, though in another way it has
general features of which we are unaware, for the sorts
of reasons described in the previous appendix. ("So
that's what I've been doing all the time - I'd never have
guessed" may be the expression of having made a philoso-
phical discovery!)

IV.8.a. what interests the philosopher, however, is not
so much the fact that we do classify things in this way
or in that way, as the fact that it is possible to
    379

classify them in one way or another. (This may be
important in dispelling philosophical prejudice as to
what it means for consciousness to exist, for example.)
The interest in possible ways of classifying things
need not be fed only by analysis of concepts which we
actually employ: the scientist or philosopher may draw our
attention to new possible ways of classifying things, per-
haps by teaching us to use new concepts.
(A very interesting and difficult question, which
underlies much of Wittgenstein's discussion in "Philo-
sophical Investigations" is the question whether and to
what extent the possibility of adopting certain systems
of classification depends on what is actually the case
in the world. In a more specific form, this becomes
the question whether I have been right in saying that
the existence of observable properties does not depend
on which particular objects actually have those properties.
(See 2.D and 7.A.) Would it make sense to talk of the
size or shape of objects if everything were constantly
changing in size and shape? Colours? Etc.)

IV.8.b. The wish to understand the world, in a philo-
sophical way, is, at least partly, constituted by the wish
to know how things in the world may be classified. (Can
we divide the world up into material entities and mental
entities, or are there only material entitles, some of
which have certain observable properties while others
look and act differently? Are there electrons and other
subatomic particles out of agglomerations of which the
other things in the world are formed, or must the place
for electrons in our scheme of things be explained in terms
of ways of classifying observable macroscopic phenomena
    380

such as flashes on fluorescent screens and clicks in
geiger counters?) The wish to know how things may be
classified may be satisfied by finding out the answers
to questions of the form "Can we say that ...?" "If
such and such had been the case, might we have said that
...?" "Can the word so and so be applied in such and
such a sort of case without changing its meaning?" Indeed,
this may be the only way of making satisfactory progress.
But the fact that these questions are explicitly about
words disguises the fact that they are implicitly about
things. This is why the criticism that so-called
"linguistic philosophy" is just a dilletantish enquiry
into empirical facts about linguistic conventions, merely
misses the point. (This misunderstanding is excusable,
however, since its practitioners often miss the point
themselves. This, like the rejection of the possibility
of synthetic apriori knowledge, is one of the manifestations
of the neurotic fear of doing anything resembling old-style
metaphysics.)

IV.9.   To summarize: in addition to the purposes listed
in IV.7, above, philosophical analysis can also, in an
indirect way, help to provide answers to very general
questions about the world of our experience, which,
according to the Pocket Oxford Dictionary, is the business
of philosophy.
    381

Appendix V

FURTHER EXAMPLES

V.1.    In section 7.C the notion of a non-identifying
relation between the meanings of words was illustrated by
means of geometrical examples. Geometrical properties
are not the only ones which may stand in synthetic
relations, but they give the assertion of the existence
of synthetic necessary truths its strongest support.
Some additional problematic examples will be mentioned
now, but not discussed in any detail.

V.2.    Our first examples involve colour-concepts. Dis-
cussions of the relations between colours or between colour-
words can be very confused if the distinctions made in
chapter three are not taken seriously. Thus, we have
seen that one and the same colour-word "red" may have
different sorts of meanings depending on whether it is an
f-word directly correlated with a hue (see 3.A.1), a d-word
disjunctively correlated with a range of specific shades
of colour (see 3.B.2) a p-word correlated with a range of
specific shades picked out by some procedure (see 3.D.2),
or a word correlated with the disposition of normal persons
to say "red" (see 3.D.2.note). The words given their
meanings in these various ways may have the same extension.
So our first set of problematic examples come from questions
of the form: is it analytic that everything which is red
in one of these senses is red in another of these senses?
(This is a question which cannot be asked in ordinary
English, since the word "red" as ordinarily understood
has neither one of these meanings nor another, and our
    382

ordinary vocabulary does not make provision for distinguishing
these several senses easily. See 3.E.2.)

V.2.a.  Now suppose that the word "scarlet" is an f-word,
referring to just one specific shade (a shade of red).
Then we can ask whether the sentence "All scarlet things
are red" expresses an analytic proposition. Owing to the
indeterminateness of the meanings with which words are
normally understood, this question probably has no answer
(see section 6.D). By distinguishing various possible
(sharply identified) meanings of "red" we can understand
the question in such a way that it has an answer. Thus,
if "red" is a d-word disjunctively correlated with a range
of specific shades, of which the shade referred to by
"scarlet" is one, then the sentence in question expresses
an analytic proposition.
On the other hand, if "red" is an f-word simply
correlated with the hue redness (3.A.1), then the meaning
of "red" and the meaning of "scarlet" can probably be
identified independently of each other: one could learn
to recognize the specific shade without being able to
see the hue, and one could learn to recognize the hue
without ever having seen the specific shade. It follows
that if there is a necessary connection between the hue
and the shade this must be discovered by examining the
two properties, in which case the proposition that every-
thing which has the shade in question has the hue in
question must be synthetic and necessarily true.

V.2.b.  People sometimes say that it is merely analytic
that nothing can be two colours at the same time. Cer-
tainly, we could adopt n-rules to ensure the incompatibility
    383

of colour words (see section 4.C), and it may be the case
that we do. But perhaps we do not need to: perhaps
there are ways of understanding colour-words so that the
connections between them are synthetic. Exactly what
sort of relation holds between a pair of colour words will,
of course, depend on the sorts of meanings they have.
Thus, if "red" and "yellow" are both d-words dis-
junctively correlated with ranges of specific shades of
colour (with no overlap between the ranges), then the
question of the incompatibility of "red" and "yellow" is
logically equivalent to the question of the incompatibility
of different specific shades of colour.
On the other hand, if "red" and "yellow" are both
f-words, directly correlated with hues, then their in-
compatibility depends not only on the impossibility of
finding an object with two different specific shades of
colour, but also on the impossibility of finding an
object which has a shade of colour which is simultaneously
a shade of red and a shade of yellow, that is an object
which has two hues at once, though only one shade of colour.
If one is a d-word correlated with a range of
specific shades of colour, and the other is an f-word
correlated with a hue, then the incompatibility will
depend on the impossibility (for example) of finding an
object with one of the specific shades correlated with
"red" and the hue correlated with "yellow". Is it anal-
ytic that nothing which is scarlet in shade can be yellow
in hue?
If both words are p-words, correlated with properties
by means of procedures, of the sort described in section
3.D, then the relation between them may be still more
complex and problematic.
    384

V.3.    Examples are also forthcoming when we consider
properties of sounds.
What sort of fact is it that if two musical intervals
with the characteristic sounds of a fifth and a fourth are
added together, then the outer interval will have the
characteristic sound of an octave?
What sort of fact is it that only if the three notes
of a chord are separated by two of the following intervals:
a third, a minor third, and a fourth, can they form a
triad with the characteristic sound of a major chord?
Can all these musical properties be identified indepen-
dently, or can they be identified only by specifying their
relations? Or are the relations contingent?

V.4.    Another example: no surface is both glossy and
mat at the same time. Analytic or synthetic? Necessary
or contingent?

V.5.    There seem also to be relations between mechanical
properties, which are really the same sorts of things
as the geometrical properties already discussed, except
that motion comes in too.
For example, if a rod remains straight and its mid-
point remains fixed while one end moves down, then the
other end moves up.
If two gear wheels are meshed, their shapes and
distance apart remaining constant, then if one of them
turns about its axis, and neither penetrates the other,
then the second one will move too. (This is the sort
of thing which enables us to predict what will happen
when one part of a machine starts moving, if none of the
parts bends or disintegrates or penetrates the others.)
Is this analytic?
    385

V.6.    I believe that more examples can be found by consider-
ing relations between numbers. First of all it should be
noted that among the several different concepts superimposed
in our ordinary arithmetical concepts are "perceptible"
numerical concepts. For it is possible to learn to recog-
nize the number of objects in small discrete collections
just by looking, and without counting or otherwise cor-
relating the objects with anything else. Similarly, one
might learn, by being shown examples, to recognize simple
operations performed on such sets, such as addition and
subtraction, nothing being allowed to come into or out of
existence or merge with anything else during such an operation.
Then, by examining these observable numerical properties and
operations, perhaps with the aid of informal proofs of the
sorts described in section 7.D, we my be able to see conn-
ections which justify the assertion of such statements as
"A two-set added to another two-set with which it is disjoint
yields a four-set". "A five-set can be divided into a two-
set and a three-set."
It is arguable that such statements, if understood in
one way, are both synthetic and necessarily true. But I
shall not go into the argument.

V.7.    All these examples rely on the fact that there are pro-
perties which are independently identifiable. The sort of
thing that is meant by saying that such properties can be
identified Independently of one another was illustrated by
the discussion in section 3.C of the way in which properties
can explain our use of descriptive words.
In addition, the examples rely on the fact that in order
to perceive the necessary truth of the statements which are
alleged to be both synthetic and necessarily true, it is
necessary to be acquainted with the specific kinds of pro-
perties referred to: purely logical, or topic-neutral,
enquiries will not suffice.
    386

Appendix VI

APRIORI KNOWLEDGE

VI.1.   I think the arguments of chapter seven show that
the common tendency to confuse the terms "analytic" and
"necessary" ought to be resisted. Similarly it is
tempting to confuse the terms "necessary" and "apriori".
I shall now try to show briefly that this is undesirable.

VI.2.   Kant asserted that one of the marks of apriori kno-
wledge was necessity, and there is something in this. If
a statement is contingently true, then it would be false
in some possible state of affairs, so there are no reasons
why it should be true which can be known without discovering
which possible state of the world is the actual
one. Hence the truth of a contingently true statement
has to be ascertained by empirical observation of the facts
to make sure that there are no counter-instances. Thus
no statement which is not a necessary truth can be known
apriori to be true, from which it follows logically that
if a statement can be known apriori to be true then it is
necessarily true.
But the converse does not follow. There may be state-
ments which are necessarily true which are not known to be
true at all, let alone known apriori. Some necessarily
true statements may be known to be true only on the basis
of empirical enquiry: a person who fails to realize that
every object bounded by four plane faces must have four
vertices may establish the truth of "Every object bounded
by four plane faces has four vertices" by carrying out
a survey of objects bounded by four plane faces and count-
ing their vertices. Indeed, it seems likely that there
    387

are some truths which cannot be known apriori by any human
being despite their necessity, owing to the complexity of
the connections between universals in virtue of which they
are necessarily true. Perhaps there are kinds of conn-
ections between properties which simply cannot be dis-
covered by examining those properties: causal connections
may be like this, in which case there are necessary truths
which cannot be known a priori at all. At any rate, it
is clear that the concepts "necessary" and "known apriori"
are distinct, though there is a connection between them.

VI.3.   This helps to show that the term "apriori" should
not be applied to statements or truths: it applies to
kinds of knowledge or ways of knowing. I have not defined
the expressions "apriori" and "empirical", and it is
probable that their use In philosophical discussion is even
more confused than the use of the terms "analytic" and
"necessary". One way in which the aprior-empirical dis-
tinction may be applied is as follows. Where the truth-
value of a statement depends on whether or not certain
particular objects have certain properties or stand in
certain relations, it is, in general, necessary to ascertain
the truth-value by carrying out observations to see which
particular objects exist, and which relations and properties
they instantiate. This is an empirical way of acquiring
knowledge: one observes particular contingent facts.
However, as shown in the thesis, there are some cases
where it is not necessary to carry out such an investi-
gation, since its outcome can be discovered merely by
examining the properties and relations concerned and per-
ceiving connections between them. This is an apriori way
of acquiring knowledge: one does not discover by empirical
observation how things happen to be in the world which
    388

might have been otherwise (see section 7.A). This is
the distinction which I have had in mind whenever I used
the term "apriori", namely the distinction between kno-
wledge obtained by observation of particular facts and
knowledge obtained without observation of particular facts.

VI.4.   It is sometimes suggested that apriori knowledge,
and knowledge of necessary truth, is not derived from or
dependent on experience. But it is not at all clear what
this means, unless it is an obscure version of what I have
just said.
For there are two ways in which the examples of apriori
knowledge of necessary truth mentioned above are derived
from or at least dependent on experience. First of all,
each of the necessarily true statements which I have dis-
cussed employs descriptive words referring to observable
properties, and, in general, in order to have the requisite
concepts and understand the statements It is necessary to
have had some experience of objects with these properties,
or properties similar to them. At any rate, this is how
we do in fact usually acquire such concepts. Secondly,
experience comes in when one perceives the connections between
such properties. For example, one may have the required
insight while looking at a particular diagram which ex-
hibits the properties in question. So in this sense apriori
knowledge may be based on particular experiences (rather than
on detached and lofty exertions of "Pure Reason"). But
it is still not empirical in the sense defined above.
Whether there is knowledge of a sort which is completely
independent of all experience in some sense is dubious:
what could it be knowledge of?
                                                                        389


====================================================
This is part of A.Sloman's 1962 Oxford DPhil Thesis
     "Knowing and Understanding"
====================================================


BIBLIOGRAPHY

(Abbreviations used in the thesis are placed between square brackets.)

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