distinguished logician, Rudolf

JL Carnap, develops in this book a

new method of semantical meaning

^analysis. After giving a detailed criti-

cal discussion of the traditional meth-

od, according to which any expression

of language (a word, a phrase, or a

sentence) is regarded as a name of one

unique entity (a thing, a property, a

class, a relation, a proposition, a fact,

etc.), Mr. Carnap concludes that the

various forms of this method of the

name-relation lead to numerous diffi-

culties and complications.

He proposes a new approach which he

calls the method of extension and in-

tension. The meaning of any expression

is analyzed into two meaning com-

ponents, the intension, which is ap-

prehended by the understanding of the

expression, and the extension, which is

determined by empirical investigation.

As one important application of this

new semantical method, Mr. Carnap

lays the foundation of a new system of

modal logic, that is, a theory of con-

cepts like necessity and contingency,

possibility and impossibility, which

philosophers and logicians will find

valuable in solving many puzzling]

problems.

MEANING AND NECESSITY

MEANING

AND NECESSITY

A Study in Semantics and Modal Logic

By RUDCJLF^C^RNAP

Professor of Philosophy in the University of Chicago

THE UNIVERSITY OF CHICAGO PRESS

CHICAGO ILLINOIS

THE UNIVERSITY OF CHICAGO PRESS, CHICAGO 37

Cambridge University Press, London, N.W. 1, England

W. J. Gage & Co., Limited, Toronto 2B, Canada

Copyright 1947 by the University of Chicago. All rights reserved.

Published 1947. Second Impression 1948. Composed and printed by

THE UNIVERSITY OF CHICAGO PRESS, Chicago, Illinois , U.S.A.

PREFACE

The main purpose of this book is the development of a new method for

the semantical analysis of meaning, that is, a new method for analyzing

and describing the meanings of linguistic expressions. This method, called

the method of extension and intension, is developed by modifying and ex-

tending certain customary concepts, especially those of class and property.

The method will be contrasted with various other semantical methods

used in traditional philosophy or by contemporary authors. These other

methods have one characteristic k^corfflHbi^Wrhey all regard an expression

in a language as a name of a concrete or abstract entity. In contradistinc-

tion, the method here proposed takes an expression, not as naming any-

thing, but as possessing an intension and an extension.

This book may be regarded as a third volume of the series which I have

called "Studies in Semantics", two volumes of which were published ear-

lier. However, the present book does not presuppose the knowledge of its

predecessors but is independent. The semantical terms used in the present

volume are fully explained in the text. The present method for defining the

L- terms (for example, 'L-true', meaning* logically true', ' analytic') differs

from the methods discussed in the earlier Introduction to Semantics. I now

think that the method used in this volume is more satisfactory for lan-

guages of a relatively simple structure.

After meaning analysis, the second main topic discussed in this book is

modal logic, that is, the theory of modalities, such as necessity, contin-

gency, possibility, impossibility, etc. Various systems of modal logic have

been proposed by various authors. It seems to me, however, that it is not

possible to construct a satisfactory system before the meanings of the

modalities are sufficiently clarified. I further believe that this clarification

can best be achieved by correlating each of the modal concepts with a cor-

responding semantical concept (for example, necessity with L-truth). It

will be seen that this method also leads to a clarification and elimination

of certain puzzles which logicians have encountered in connection with

modalities. In the Preface to the second volume of "Studies in Semantics,"

I announced my intention to publish, as the next volume, a book on

modal logic containing, among other things, syntactical and semantical

systems which combine modalities with quantification. The present book,

however, is not as yet the complete fulfilment of that promise: it contains

vi PREFACE

only analyses and discussions of modalities, preliminary to the construc-

tion of modal systems* The systems themselves are not given here. In an

article published elsewhere (see Bibliography), I have stated a calculus

and a semantical system combining modalities with quantification, and

have summarized some of the results concerning these systems. A more

comprehensive exhibition of results already found and those yet to be

found must be left for another time.

The investigations of modal logic which led to the methods developed

in this book were made in 1942, and the first version of this book was writ-

ten in 1943, during a leave of absence granted by the University of Chi-

cago and financed by the Rockefeller Foundation. To each of these insti-

tutions I wish to express my gratitude for their help. Professors Alonzo

Church and W. V. Quine reaorhe first version and discussed it with me in

an extensive correspondence. I am very grateful to both for the stimula-

tion and clarification derived from this discussion, and to Quine also for a

statement of his view and, in particular, of his reaction to my method of

modal logic. This statement is quoted in full and discussed in detail in the

penultimate section of this book. I am also indebted to Professors Carl G.

Hempel and J. C. C. McKinsey for some helpful comments. To Miss

Gertrude Jaeger I am grateful for expert help in the preparation of the

manuscript.

R. C.

CHICAGO

November 1946

CONTENTS

I. THE METHOD OF EXTENSION AND INTENSION i

1. Preliminary Explanations 2

2. L-Concepts 7

3. Equivalence and L-Equivalence 13

4. Classes and Properties 16

5. Extensions and Intensions 23

6. Extensions and Intensions of Sentences 25

7. Individual Descriptions 32

8. Frege's Method for Descriptions 35

9. Extensions and Intensions of Individual Expressions 39

10. Variables 42

1 1 . Extensional and Intensional Contexts 46

12. The Principles of In terchangeability 51

13. Sentences about Beliefs 53

14. Intensional Structure 56

15. Applications of the Concept of Intensional Structure 59

1 6. Lewis' Method of Meaning Analysis 64

II. L-DETERMINACY 69

17. L-Determinate Designators 69

1 8. The Problem of L-Determinacy of Individual Expressions .... 73

19. Definition of L-Determinacy of Individual Expressions 78

20. L-Determinacy of Predicators 81

21. Logical and Descriptive Signs 85

22. L-Determinate Intensions 88

23. Reduction of Extensions to Intensions 90

III. THE METHOD or THE NAME-RELATION 96

24. The Name-Relation 96

25. An Ambiguity in the Method of the Name-Relation 100

26. The Unnecessary Duplication of Names 106

27. Names of Classes in

28. Frege's Distinction between Nomina turn and Sense 118

29. Nomination and Sense: Extension and Intension 124

30. The Disadvantages of Frege's Method 129

31. The Antinomy of the Name-Relation 133

32. Solutions of the Antinomy 136

vii

viii CONTENTS

IV. ON METALANGUAGES FOR SEMANTICS 145

33. The Problem of a Reduction of the Entities 145

34. The Neutral Metalanguage M' 153

35. M' Is Not Poorer than M 157

36. Neutral Variables in M' 160

37. On the Formulation of Semantics in the Neutral Metalanguage M' , 161

38. On the Possibility of an Extensional Metalanguage for Semantics . 168

V. ON THE LOGIC OF MODALITIES 173

39. Logiqd Modalities 173

40. Modalities and Variables 177

41. Semantical Rules for the Modal System S 2 ' . . 182

42. Modalities in the Word Language 186

43. Modalities and Variables in the Word Language 191

44. Quine on Modalities 193

45. Conclusions 202

BIBLIOGRAPHY 205

INDEX 207

CHAPTER I

THE METHOD OF EXTENSION AND INTENSION

A method of semantical meaning analysis is developed in this chapter. It is

applied to those expressions of a semantical system S which we call designators;

they include (declarative) sentences, individual expressions (i.e., individual

constants or individual descriptions) and predicators (i.e., predicate constants

or compound predicate expressions, including abstraction expressions) . We start

with the semantical concepts of truth and L-truth (logical truth) of sentences

( i, 2). It is seen from the definition of L-truth that it holds for a sentence if

its truth follows from the semantical rules alone without reference to (extra-

linguistic) facts (2). Two sentences are called (materially) equivalent if both

are true or both are not true. The use of this concept of equivalence is then

extended to designators other than sentences. Two individual expressions are

equivalent if they stand for the same individual. Two predicators (of degree

one) are equivalent if they hold for the same individuals. L-equivalence (logical

equivalence) is denned both for sentences and for other designators in such a

manner that it holds for two designators if and only if their equivalence follows

from the semantical rules alone. The concepts of equivalence and L-equivalence

in their extended use are fundamental for our method (3).

If two designators are equivalent, we say also that they have the same

extension. If they are, moreover, L-equivalent, we say that they have also the

same intension (5). Then we look around for entities which might be taken

as extensions or as intensions for the various kinds of designators. We find that

the following choices are in accord with the two identity conditions just stated.

We take as the extension of a predicator the class of those individuals to which

it applies and, as its intension, the property which it expresses; this is in ac-

cord with customary conceptions ( 4) . As the extension of a sentence we take

its truth- value (truth or falsity); as its intension, the proposition expressed by

it (6). Finally, the extension of an individual expression is the individual to

which it refers; its intension is a concept of a new kind expressed by it, which we

call an individual concept ( 7-9). These conceptions of extensions and inten-

sions are justified by their fruitfulness; further definitions and theorems apply

equally to extensions of all types or to intensions of all types.

A sentence is said to be extensional with respect to a designator occurring in

it if the extension of the sentence is a function of the extension of the designa-

tor, that is to say, if the replacement of the designator by an equivalent one

transforms the whole sentence into an equivalent one. A sentence is said to be

intensional with respect to a designator occurring hi it if it is not extensional

and if its intension is a function of the intension of the designator, that is to

say, if the replacement of this designator by an L-equivalent one transforms the

whole sentence into an L-equivalent one. A modal sentence (for example, 'it is

necessary that . . .') is intensional with respect to its subsentence ( n). A

psychological sentence like 'John believes that it is raining now* is neither ex-

tensional nor intensional with respect to its subsentence ( 13). The problem of

the semantical analysis of these belief-sentences is solved with the help of the

concept of intensional structure ( 14, 15).

2 I. THE METHOD OF EXTENSION AND INTENSION

1. Preliminary Explanations

This section contains explanations of a symbolic language system Si, which

will later serve as an object language for the illustrative application of the

semantical methods to be discussed in this book. Further, some semantical con-

cepts are explained for later use; they belong to the semantical metalanguage

M, which is a part of English. Among them are the concepts of truth Jalsity, and

(material) equivalence, applied to sentences. The term 'designator* is introduced

for all those expressions to which a semantical meaning analysis is applied, the

term will be used here especially for sentences, predicators (i.e., predicate ex-

pressions), and individual expressions.

The chief task of this book will be to find a suitable method for the

semantical analysis of meaning, that is, to find concepts suitable as tools

for this analysis. The concepts of the intension and the extension of an ex-

pression in language will be proposed for this purpose. They are anal-

ogous to the customary concepts of property and class but will be ap-

plied in a more general way to various types of expressions, including

sentences and individual expressions. The two concepts will be explained

and discussed in chapters i and ii.

The customary concept of name-relation and the distinction sometimes

made since Frege between the entity named by an expression and the

sense of the expression will be discussed in detail in chapter iii. The pair

of concepts, extension-intension, is in some respects similar to the pair of

Frege's concepts; but it will be shown that the latter pair has serious dis-

advantages which the former avoids. The chief disadvantage of the meth-

od applying the latter pair is that, in order to speak about, say, a property

and the corresponding class, two different expressions are used. The meth-

od of extension and intension needs only one expression to speak about

both the property and the class and, generally, one expression only to

speak about an intension and the corresponding extension.

In chapter iv, a metalanguage will be constructed which is neutral with

regard to extension and intension, in the sense that it speaks not about a

property and the corresponding class as two entities but, instead, about

one entity only; and analogously, in general, for any pair of an intension

and the corresponding extension. The possibility of this neutral language

shows that our distinction between extension and intension does not pre-

suppose a duplication of entities.

In chapter v, some questions concerning modal logic are discussed on

the basis of the method of extension and intension.

My interest was first directed toward the problems here discussed when

I was working on systems of modal logic and found it necessary to clarify

the concepts which will be discussed here under the terms of 'extension'

1. PRELIMINARY EXPLANATIONS 3

and 'intension' and related concepts which have to do with what is usual-

ly called the values of a variable. Further stimulation came from some

recent publications by Quine 1 and Church, 3 whose discussions are valu-

able contributions to a clarification of the concepts of naming and

meaning.

Before we start the discussion of the problems indicated, some explana-

tions will be given in this section concerning the object languages and the

metalanguage to be used. We shall take as object languages mostly sym-

bolic languages, chiefly three semantical language systems, S x , S 2 , and S 3 ,

and occasionally also the English word language. For the sake of brevity,

not all the rules of these symbolic systems will be given, but only those of

their features will be described which are relevant to our discussion. S r

will now be described; S 2 is an extension of it that will be explained later

( 41); S 3 will be described in 18.

The system 5 X contains the customary connectives of negation '~*

('not'), disjunction 'V' Cor'), conjunction '' ('and'), conditional (or

material implication) ' 3 ' ('if ... then ...'), and biconditional (or ma-

terial equivalence) ' =' ('if and only if). The only variables occurring are

individual variables '#', l y\ V, etc. For these variables the customary

universal and existential quantifiers are used: '(#)(..#..)' ('for every

x, : . x . .') and '(3#)(. . x . .)' ('there is an x such that . . * . .'). All

sentences in S x and the other systems are closed (that is, they do not con-

tain free variables). In addition to the two quantifiers, two other kinds of

operators occur: the iota-operator for individual descriptions ('(t#)

(..#..)', 'the one individual x such that . . x . .') and the lambda-

operator for abstraction expressions ('(X#)(. . x . .)', 'the property (or

class) of those x which are such that . . x . .') . If a sentence consists of an

abstraction expression followed by an individual constant, it says that the

individual has the property in question. Therefore, '(X#)(. . x . .)a'

means the same as ' . . a . .', that is, the sentence formed from '. . x . .'

by substituting 'a' for '#'. The rules of our system will permit the trans-

formation of '(X#)(. . x . .)a' into '. . a . .' and vice versa; these trans-

formations are called conversions.

Si contains descriptive constants (that is, nonlogical constants) of indi-

1 [Notes] (see Bibliography at the end of this book). Quine's views concerning the name-

relation (designation) will be discussed in chap, iii; and the conclusions which he draws from

them for the problem of quantification in modal sentences will be discussed in chap. v.

[Review C.] and [Review QJ. Church's conceptions will be discussed in chap, iii, in con-

nection with those of Frege. Church's contributions are more important than is indicated by

the form of their publication as reviews. It is to be hoped that he will soon find the opportu-

nity for presenting his conception in a more comprehensive and systematic form.

4 I. THE METHOD OF EXTENSION AND INTENSION

vidual and predicate types. The number of predicates in S r is supposed to

be finite, that of individual constants may be infinite. For some of these

constants, which we shall use in examples, we state here their meanings

by semantical rules which translate them into English.

1-1. Rules of designation for individual constants

V is a symbolic translation of 'Walter Scott',

'w' (the book) Waverley'.

1-2. Rules of designation for predicates

( H#' 'x is human (a human being)',

'RA#' 'x is a rational animal',

'Fx' ( x is (naturally) featherless',

'Bx' # is a biped',

' Axy y e x is an author of y\

The English words here used are supposed to be understood in such a way

that * human being' and ' rational animal' mean the same. Further, we

shall use 'a', 'b', V, as individual constants, and *P', 'Q', as predicator

constants (of level one and degree one) ; the interpretation of these signs

will be specified in each case, or left unspecified if not relevant for the

discussion.

In order to speak about any object language here the symbolic language

systems S x , etc. we need a metalanguage. We shall use as our metalan-

guage M a suitable part of the English language which contains transla-

tions of the sentences and other expressions of our object languages (for

example, the translations stated in i-i and 1-2), names (descriptions) of

those expressions, and special semantical terms. For the sake of simplicity,

we shall usually construct a name of an expression in the customary way

by including it in single quotation marks. In order to speak about expres-

sions in a general way, we often use *![',' 8/, etc -> for expressions of any

kind and**', '/, etc., for sentences, sometimes also blanks like '...',

'- -', etc., and blanks with a variable, e.g., '. . x . .', for an expression in

which that variable occurs freely. If a German letter occurs in an expres-

sion together with symbols of the object language, then the latter ones are

used autonymously, i.e., as names for themselves. 3 Thus, we may write in

M, for instance, 1 U% 33 a/; this is meant as referring to that expression

of the object language which consists of the expression H (whatever this

may be, e.g., 'Hs') followed by the sign ' ^\ followed by the expression

a/. (In symbolic formulas both in the object languages and in M, paren-

theses will often be omitted under the customary conditions.) The term

* See [Syntax], 42.

1. PRELIMINARY EXPLANATIONS 5

'sentence* will be used in the sense of 'declarative sentence'. The term

( sentential matrix' or, for short, 'mtfftix' will be used for expressions

which are either sentences or formed from sentences by replacing indi-

vidual constants with variables. (If a matrix contains any number of free

occurrences of n different variables, it is said to be of degree n; for ex-

ample, ' Axy V Px' is of degree two; the sentences are the matrices of de-

gree zero). A sentence consisting of a predicate of degree n followed by n

individual constants is called an atomic sentence (e.g., 'Pa', ' Abe').

A complete construction of the semantical system Si, which cannot be

given here, would consist in laying down the following kinds of rules:

(1) rules of formation, determining the admitted forms of sentences;

(2) rules of designation for the descriptive constants (e.g. i-i and 1-2);

(3) rules of truth, which we shall explain now; (4) rules of ranges, to be ex-

plained in the next section. Of the rules of truth we shall give here only

three examples, for atomic sentences (1-3), for ' V' (1-5), and for ' =' (1-6).

1-3. Rule of truth for the simplest atomic sentences. An atomic sentence

in S x consisting of a predicate followed by an individual constant is true

if and only if the individual to which the individual constant refers posses-

ses the property to which the predicate refers.

.This rule presupposes the rules of designation. It yields, together with

rules i-i and 1-2, the following result as an example:

1-4. The sentence 'Bs' is true if and only if Scott is a biped.

1-5. Rule of truth for ' V'. A sentence * V <S ; - is true in S x if and only if at

least one of the two components is true.

1-6. Rule of truth for ' = '. A sentence @< s @ y is true if and only if either

both components are true or both are not true.

There are some further rules of truth for the other connectives, cor-

responding to their truth-tables, and for the quantifiers; another example

of a rule of truth will be given in 3-3. The rules of truth together constitute

a recursive definition for l true in S r ', because they determine, in combina-

tion with the rules of designation, for every sentence in S x a sufficient and

necessary condition of its truth (as is given for 'Bs' in 1-4). Thereby they

give an interpretation for every sentence. Thus, for example, we learn from

the rules that the sentence 'Bs' says that (in other words, expresses the

proposition that) Scott is a biped. For the purposes of our discussion it is

not necessary to give the whole definition of truth. 4 It will suffice to pre-

< The first definition of the semantical concept of truth was given by Tarski [Wahrheitsbe-

griff]; I have given a slightly different form in [I], 7. For nontechnical discussions of the na-

ture of the semantical concept of truth see Tarski [Truth] and my [Remarks].

6 I. THE METHOD OF EXTENSION AND INTENSION

suppose that the term 'true' is defined in such a manner that it has its

customary meaning as applied to sentences. More specifically, we presup-

pose that a statement in M saying that a certain sentence in Si is true

means the same as the translation of this sentence; 5 for example, 'the

sentence 'Hs' is true in S x ' means the same as 'Walter Scott is human'.

On the basis of 'true', some further semantical terms are defined as fol-

lows, with respect to any semantical system S, e.g., Sj, etc.

1-7, Definition. @ is false (in 5) =DI ~ is true (in S).

Thus ' false' has here its ordinary meaning.

1-8. Definition. @ is equivalent to ,- (in 5) =DI @< ** @; is true (in 5).

This definition, together with the rule of truth for ' ==' (1-6), yields this

result:

1-9. Two sentences are equivalent if and only if both have the same truth-

value, that is to say, both are true or both are false.

It is to be noticed that the term 'equivalent' is here defined in such a

manner that it means merely agreement with respect to truth- value (truth

or falsity), a relation which is sometimes called 'material equivalence'.

The term is here not used, as in ordinary language, in the sense of agree-

ment in meaning, sometimes called 'logical equivalence'; for the latter

concept we shall later introduce the term 'L-equivalent' (2-3c).

I propose to use the term 'designator* for all those expressions to

which a semantical analysis of meaning is applied, the class of designators

thus being narrower or wider according to the method of analysis used.

[The word 'meaning' is here always understood in the sense of 'designa-

tive meaning', sometimes also called 'cognitive', 'theoretical', 'referen-

tial', or 'informative', as distinguished from other meaning components,

e.g., emotive or motivative meaning. Thus here we have to do only with

declarative sentences and their parts.] Our method takes as designators at

least sentences, predicators 6 (i.e., predicate expressions, in a wide sense,

s For detailed discussions of this characteristic of the semantical concept of truth, see

Tarski [Truth] and my [Remarks], 3.

6 Some terms with the ending '-tor* for kinds of expressions are customary, e.g., 'functor',

'operator*. The terms 'predicator* and 'designator* are formed in analogy to them. A still

wider use of the same ending might be taken into consideration with the aim of making the

terminology in the metalanguage somewhat more uniform. For this book, only the two terms

mentioned are adopted; but the following terms would seem to me quite suitable, too:

'descriptor* (for the customary 'description*), 'abstractor* (for 'abstraction expression*),

'connector* (for 'connective*). Other terms might seem more questionable, but perhaps still

worth consideration, e.g., 'individuator* (for 'individual expression*), 'propositor* or 'stator*

(for '(declarative) sentence*), 'conceptor* (for 'concept expression,* i.e., 'designator other than

sentence*). Morris, [Signs], uses a number of terms with '-tor* (or '-or*), among them some of

those mentioned here, for kinds of expressions or, more generally, of signs including non-

linguistic signs.

1 2. L-CONCEPTS 7

including class expressions), functors (i.e., expressions for functions in

the narrower sense, excluding proposition^! functions), and individual ex-

pressions; other types may be included if desired (e.g., connectives, both

extensional and modal ones). The term ' designator' is not meant to imply

that these expressions are names of some entities (the name-relation will

be discussed in 24), but merely that they have, so to speak, an inde-

pendent meaning, at least independent to some degree. Only (declarative)

sentences have a (designative) meaning in the strictest sense, a meaning of

the highest degree of independence. All other expressions derive what

meaning they have from the way in which they contribute to the meaning

of the sentences in which they occur. One might perhaps distinguish

in a vague way different degrees of independence of this derivative

meaning. Thus, for instance, I should attribute a very low degree to

' (', somewhat more independence to* V', still more to ' + ' (in an arithmet-

ical language), still more to'H' ('human') andV ('Scott'); I should not

know which of the last two to rank higher. This order of rank is, of course,

highly subjective. And where to make the cut between expressions with no

or little independence of meaning ('syncategorematic' in traditional

terminology) and those with a high degree of independence, to be taken as

designators, seems more or less a matter of convention. If a metalanguage

is decided upon, then it seems convenient to take as designators at least

the expressions of all those types, but not necessarily only those, for which

there are variables in the metalanguage (compare [I], 12, and references

to Quine, below, at the beginning of 10).

2. L-Concepts

By the explication of a familiar but vague concept we mean its replacement

by a new exact concept; the former is called explicandum, the latter explicatum.

The concept of L-truth is here defined as an explicatum for what philosophers

call logical or necessary or analytic truth. The definition leads to the result that

a sentence in a semantical system is L-true if and only if the semantical rules

of the system suffice for establishing its truth. The concepts of L-falsity, L-

implication, and L-equivalence are defined as explicata for logical falsity, logical

implication or entailment, and mutual logical implication, respectively. A sen-

tence is called L-determinate if it is either L-true or L-false; otherwise it is called

L-indeterminate or factual. The latter concept is an explicatum for what Kant

called synthetic judgments. A sentence is called F-true if it is true but not L-

true; F- truth is an explicatum for what is known as factual or synthetic or

contingent truth. The concepts of F-falsity, F-implication, and F-equivalence

are defined analogously.

The task of making more exact a vague or not quite exact concept used

in everyday life or in an earlier stage of scientific or logical development,

8 I. THE METHOD OP EXTENSION AND INTENSION

or rather of replacing it by a newly constructed, more exact concept, be-

longs among the most important tasks of logical analysis and logical con-

struction. We call this the task of explicating, or of giving an explication

for, the earlier concept; this earlier concept, or sometimes the term used

for it, is called the explicandum; and the new concept, or its term,

is called an explicatum of the old one. 7 Thus, for instance, Frege and,

later, Russell took as explicandum the term 'two' in the not quite exact

meaning in which it is used in everyday life and in applied mathematics;

they proposed as an explicatum for it an exactly defined concept, namely,

the class of pair-classes (see below the remark on (i) in 27); other

logicians have proposed other explicata for the same explicandum. Many

concepts now defined in semantics are meant as explicata for concepts

earlier used in everyday language or in logic. For instance, the semantical

concept of truth has as its explicandum the concept of truth as used in

everyday language (if applied to declarative sentences) and in all of

traditional and modern logic. Further, the various interpretations of de-

scriptions by Frege, Russell, and others, which will be discussed in 7

and 8, may be regarded as so many different explications for phrases of

the form 4 the so-and-so'; each of these explications consists in laying down

rules for the use of corresponding expressions in language systems to be

constructed. The interpretation which we shall adopt following a sug-

gestion by Frege ( 8, Method Illb) deviates deliberately from the mean-

ing of descriptions in the ordinary language. Generally speaking, it is not

required that an explicatum have, as nearly as possible, the same meaning

as the explicandum; it should, however, correspond to the explicandum in

such a way that it can be used instead of the latter.

The L-terms CL-true J , etc.) which we shall now introduce are likewise

intended as explicata for customary, but not quite exact, concepts.

'L-true* is meant as an explicatum for what Leibniz called necessary

truth and Kant analytic truth. We shall indicate here briefly how this

and the other L-terms can be defined. In the further discussions of this

book, however, we shall not make use of the technical details of the fol-

lowing definitions but only of the fact that 'L-true' is defined in such a

way that the requirement stated in the subsequent convention 2-1 is ful-

filled. This is in accord with the purpose of this book, which is intended

not so much to carry out exact analyses of exactly constructed systems

as to state informally some considerations aimed at the discovery of

concepts and methods suitable for semantical analysis.

7 What is meant here by 'explicandum' and 'explicatum' seems similar to what Langford

means by 'analysandum' and 'analysans'; see below, n. 42, p. 63.

1 L-CONCEPTS 9

We shall introduce the L-concepts with the help of the concepts of

state-description and range. Some ideas of Wittgenstein 8 were the start-

ing-point for the development of this method. 9

A class of sentences in S x which contains for every atomic sentence

either this sentence or its negation, but not both, and no other sentences,

is called a state-description in S,, because it obviously gives a complete

description of a possible state of the universe of individuals with respect

to all properties and relations expressed by predicates of the system. Thus

the state-descriptions represent Leibniz' possible worlds or Wittgenstein's

possible states of affairs.

It is easily possible to lay down semantical rules which determine for

every sentence in S x whether or not it holds in a given state-description.

That a sentence holds in a state-description means, in nontechnical terms,

that it would be true if the state-description (that is, all sentences belong-

ing to it) were true. A few examples will suffice to show the nature of these

rules: (i) an atomic sentence holds in a given state-description if and only

if it belongs to it; (2) ~ @ holds in a given state-description if and only

if @i does not hold in it; (3) @ V @, holds in a state-description if and

only if either @< holds in it or @,- or both; (4) @ t s @ y holds in a state-

description if and only if either both <S* and @, or neither of them hold

in it; (5) a universal sentence (e.g., '(#)(?#)') holds in a state-descrip-

tion if and only if all substitution instances of its scope ('Pa', 'Pb', Tc',

etc.) hold in it. Iota-operators and lambda-operators can be eliminated

(for the former, this will be shown later, see 8-2; for the latter, see the

explanation of conversion in i). Therefore, it is sufficient to lay down a

rule to the effect that any sentence containing an operator of one of these

kinds holds in the same state-descriptions as the sentence resulting from

the elimination of the operator.

The class of all those state-descriptions in which a given sentence <5,-

holds is called the range of @. All the rules together, of which we have just

given five examples, determine the range of any sentence in S r ; therefore,

they are called rules of ranges. By determining the ranges, they give, to-

gether with the rules of designation for the predicates and the individual

constants (e.g., i-i and 1-2), an interpretation for all sentences in S x , since

8 [Tractatus]; see also [I], p. 107.

' The method which I shall use here is similar to, but simpler than, the one I have de-

scribed in [I], 19, as procedure E. The simpler form is possible here because S t contains

atomic sentences for all atomic propositions. The procedure to be used here seems to me the

most convenient among those known at present for the semantical construction of a system

of deductive logic; I have used it, furthermore, for modal logic in [Modalities] and for induc-

tive logic, that is, the theory of logical probability or degree of confirmation in [Inductive].

io I. THE METHOD OF EXTENSION AND INTENSION

to know the meaning of a sentence is to know in which of the possible

cases it would be true and in which not, as Wittgenstein has pointed

out.

The connection between these concepts and that of truth is as follows:

There is one and only one state-description which describes the actual

state of the universe; it is that which contains all true atomic sentences

and the negations of those which are false. Hence it contains only true

sentences; therefore, we call it the true state-description. A sentence of

any form is true if and only if it holds in the true state-description. These

are only incidental remarks for explanatory purposes; the definition of

L-truth will not make use of the concept of truth.

The L-concepts now to be defined are meant as explicata for certain

concepts which have long been used by philosophers without being defined

in a satisfactory way. Our concept of L-truth is, as mentioned above, in-

tended as an explicatum for the familiar but vague concept of logical or

necessary or analytic truth as explicandum. This explicandum has some-

times been characterized as truth based on purely logical reasons, on

meaning alone, independent of the contingency of facts. Now the mean-

ing of a sentence, its interpretation, is determined by the semantical rules

(the rules of designation and the rules of ranges in the method explained

above). Therefore, it seems well in accord with the traditional concept

which we take as explicandum, if we require of any explicatum that it

fulfil the following condition:

2-1. Convention. A sentence @< is L-true in a semantical system 5 if

and only if @< is true in 5 in such a way that its truth can be established

on the basis of the semantical rules of the system S alone, without any

reference to (extra-linguistic) facts.

This is not yet a definition of L-truth. It is an informal formulation of

a condition which any proposed definition of L-truth must fulfil in order

to be adequate as an explication for our explicandum. Thus this conven-

tion has merely an explanatory and heuristic function.

How shall we define L-truth so as to fulfil the requirement 2-1? A way

is suggested by Leibniz' conception that a necessary truth must hold in all

possible worlds. Since our state-descriptions represent the possible worlds,

this means that a sentence is logically true if it holds in all state-descrip-

tions. This leads to the following definition:

2-2, Definition. A sentence <S< is L-true (in S x ) =Df @* holds in every

state-description (in S z ).

2. L-CONCEPTS ti

The following consideration shows that the concept of L-truth thus

defined is in accord with the convention 2-1 and hence is an adequate

explicatum for logical truth. If < holds in every state-description, then

the semantical rules of ranges suffice for establishing this result. [For ex*

ample, we see from the rules of ranges mentioned above that 'Pa' holds

in certain state-descriptions, that '~Pa' holds in all the other state-

descriptions, and that therefore the disjunction 'PaV~Pa' holds in

every state-description.] Therefore, the semantical rules establish also the

truth of <S because, if @< holds in every state-description, then it holds

also in the true state-description and hence is itself true. If, on the other

hand, @< does not hold in every state-description, then there is at least

one state-description in which @i does not hold. If this state-description

were the true one, @ would be false. Whether this state-description is

true or not depends upon the facts of the universe. Therefore, in this

case, even if @ t is true, it is not possible to establish its truth without

reference to facts.

L-falsity is meant as an explicatum for logical or necessary falsity or

self-contradiction. L-implication is meant as explicatum for logical impli-

cation or entailment. L-equivalence is intended as explicatum for mutual

logical implication or entailment. The definitions are as follows:

2-3. Definitions

a. @ t - is L-false in (S r ) =Df ~ @ is L-true.

b. @ L-implies @/ (in Sj) =Df the sentence @< D @y is L-true.

c. @t is L-equivalent to @y (in S x ) =Df the sentence <S @, is

L-true.

d. is L-determinate (in S x ) = Df is either L-true or L-false.

The following results follow easily from these definitions, together with

2-2:

2-4. @< is L-false if and only if @ does not hold in any state-description.

2-6. @< L-implies @y if and only if @, holds in every state-description

in which @ holds.

2-6. @ is L-equivalent to @/ if and only if @< and / hold in the same

state-descriptions.

The condition for L-falsity stated in 2-4 means, in effect, that @< can-

not possibly be true. The condition for L-implication in 2-5 means that

it is not possible for @ to be true and for @y to be false. The condition

for L-equivalence in 2-6 means that it is impossible for one of the two

sentences to be true and the other false. Thus these results show that

L-falsity, L-implication, and L-equivalence as defined by 2-3a, b, c, may

12 I. THE METHOD OF EXTENSION AND INTENSION

indeed be regarded as adequate explicata for the explicanda mentioned

earlier.

*

We have seen that our concept of L-truth fulfils our earlier conven-

tion 2-1. Therefore, according to the definition 2-3d, a sentence is L-de-

terminate if and only if the semantical rules, independently of facts, suf-

fice for establishing its truth-value, that is, either its truth or its falsity.

This suggests the following definition, 2-7, as an explication for what Kant

called synthetic judgments. The subsequent result, 2-8, which follows

from the definition, shows that the concept defined is indeed adequate as

an explicatum.

2-7. Definition. @ is L-indeterminate or factual (in S,) =Df @ is not

L-determinate.

2-8. A sentence is factual if and only if there is at least one state-descrip-

tion in which it holds and at least one in which it does not hold.

The concept of F-truth to be defined by 2-ga is meant as an explicatum

for what is usually called factual or synthetic or contingent truth in con-

tradistinction to logical or necessary truth. The concepts defined by

2-9b, c, d, are meant as explicata in an analogous way. The adequacy of

these F-concepts as explicata follows from the adequacy of the L-concepts.

2-9. Definitions

a. @ is F-tnte (in Si) =Df <5t is true but not L-true.

b. ,- is F-false (in Sj) = D f ~ @. is F-true.

c. <5< F-implies y (in S t ) = D f @ 3 @y is F-true.

d. <5, is -equivalent to @>y (in Si) =DI @< s y is F-true.

The following are simple consequences of these and the earlier defini-

tions:

2-10. @t is F-false if and only if @, is false but not L-false,

2-11. @i is F-equivalent to / if and only if @ is equivalent but not

L-equivalent to @/.

As an example of F-truth, consider the sentence *Bs'. We found earlier

with the help of a rule of truth and rules of designation, that 'Bs' is true

if and only if Scott is a biped (1-4). This result does not tell us whether

'Bs' is true or not; it merely states a sufficient and necessary condition

for the truth of the sentence l Bs'. This is all we can learn about *Bs' from

the semantical rules alone. If we want to determine the truth- value of

'Bs', we have to go beyond the mere semantical analysis to the observa-

tion of facts. We see from 1-4 which facts are relevant: we must look at

the thing Walter Scott and see whether it is a biped. Observation shows

3. EQUIVALENCE AND L-EQUIVALENCE 13

that this is the case. Therefore, ' Bs' is true. Since the semantical rules do

not suffice for establishing its truth, it is not L-true; hence it is F-true.

3. Equivalence and L-Equivalence

The symbol ' = ', customarily used between sentences, is used here also

between designators of other kinds, especially between predicators and between

individual expressions. T s= Q' is to mean the same as* (x) (Px = Q#)'. 'a ss b*

is used, instead of the customary 'a = b j , as an identity sentence, saying that

a is the same individual as b. Then the concepts of equivalence and L-equiva-

lence, previously applied to sentences only, are defined for designators of any

kind; these two concepts are fundamental in our method. Two designators are

said to be equivalent if the s -sentence connecting them is true; they are said

to be L-equivalent if this sentence is L-true. It follows that *P J and *Q* are

equivalent if they hold for the same individuals. And 'a' and ( b' are equivalent

if a is the same individual as b.

We have defined the terms ' equivalent ' and 'L-equivalent' so far only

for sentences (r-8 and 2-3c). Now we shall extend their use so as to make

them applicable to all kinds of designators, especially also to predicators

and individual expressions. Extended in this way, the two concepts will

become the fundamental concepts in the method of semantical analysis

to be proposed here.

We begin by extending the use of the symbol ' ='. It is customary as a

connective between sentences. We shall use it in our systems between

two designators of any kind, but only if both designators are of the same

type. This use is introduced by the following rules of abbreviation. If the

extended use of ' ==' is taken as primitive, then suitable rules of ranges are

to be laid down which lead to the same results (for example, the result

that ( P 25 Q' has the same range as, and hence is L-equivalent to,

' (x)(Px s= Q#)'). The reasons for choosing just these interpretations for

' ~= ' with the various kinds of designators will soon become apparent.

The first rule introduces ' =' between predicators:

3-1. Abbreviation

a. Let 31* and 21, be two predicators of the same degree n in S x .

2li s a/ for Or)(# 2 ) . . GO

b. Hence for degree one:

We shall use in S x the connective ' * also between predicators, but, for

the sake of a convenient notation, in a way different from the use of ' s='

just introduced. The resulting expression (e.g., *PQO is here taken as a

predicator, not as a sentence as in the case of ' = ' (e.g., C P as Q'). We de-

fine it for degree one:

14 I. THE METHOD OF EXTENSION AND INTENSION

3-2. Abbreviation. Let 31, and Sly be two predicators of degree one in

S x . ^.Sl y for(X^)[Sl^M.

Thus, for example, 'F*B' is short for ( (\x)[Fx9^x]\ and hence is an ex-

pression for the property of being a featherless biped.

Furthermore, we introduce ' SB* as a primitive sign of identity of indi-

viduals instead of the customary l =* \ For this purpose we lay down the

following rule:

3-3. Rule of truth. If SI, is an individual expression in Si for the individual

x and Sly for y, then Sl s Sly is true if and only if x is the same individual

asy.

[If S is an extensional system containing, in distinction to S x , a predi-

cator variable '/', then we can achieve the same result as 3-3 by defining

2l = Sly, in a way similar to Russell's, as short for (/)[/(Sl<) = /(Sly)].]

If a system S contains, in distinction to S x , functors also, then ( = ' can

be defined for them in a way similar to the above definition for predica-

tors. The method may be indicated briefly by stating the definition for the

simplest type, namely, functors for singulary functions from individuals

to individuals; the definitions for other types are analogous. This defini-

tion will not be used in our further discussions.

3-4. Abbreviation. For functors 21* and Sly in S:

[Note that here on the right-hand side the sign ' =' stands, not between

sentential matrices, as in 3-ib, but between full expressions of functors,

which are for this type individual expressions.]

Now we shall define 'equivalent', 'L-equivalent', and ' F-equivalent' in

a general way for all kinds of designators.

3-5. Definitions. Let Sit and Sly be two designators of the same type in S x .

a. Slf is equivalent to Sly in (S x ) =Df the sentence SI, = Sly is true (in Si).

b. SI, is L-equivalent to Sly (in S x ) =DI Sl = Sly is L-true (in S x ).

c. 21, is F -equivalent to Sly (in S x ) = D f SI* = Sly is F-true (in S x ).

Now let us see what the concepts just defined mean for the various

kinds of designators. We begin with predicators. Let *P' and *Q' be two

predicators of degree one in S x . According to 3-sa, they are equivalent if

and only if 'P == Q J is true, hence, according to 3-ib, if and only if

' (x)[Px s Qx]' is true, hence if 'P' holds for the same individuals as 'Q'.

The result is analogous for two predicators of any degree n, say 'R' and

*R' J . They are equivalent, according to 3-$a and 3-ia, if and only if

3. EQUIVALENCE AND L-EQUIVALENCE 15

. . (# n )[R#j #n m R'ffi #J' is true, hence if the two predicators

hold for the same sequences (of length ri) of individuals.

To give an example, let us assume the following as a biological fact:

3-6. Assumption. All human beings are featherless bipeds and vice versa.

Then the following holds:

3-7. The sentence '(x)[H.x = (F*B)*]' is true (in S x ), but not L-true,

hence F-true.

According to 3-ib, the sentence just mentioned can be abbreviated by

<H = F*B'. Hence, 3-5 yields:

3-8. The predicators ' H' and ' F *B ' are equivalent (in Si) , but not L-equiv-

alent, hence F-equivalent.

On the other hand, the truth of the sentence ( (x)[Hx s RA#p can be

established without referring to facts by merely using the semantical rules

of S x , especially 1-2 (see the remark following this rule) and the truth

rules for the universal quantifier and for ' s*. Therefore:

3-9. ( (x)[Hx s RA*]' is L-true.

According to 3-ib, the sentence just mentioned can be abbreviated by

3-10. 'H = RA' is L-true.

Hence, 3~5b yields:

3-11. The predicators 'H' and *RA' are L-equivalent (in SJ.

Now let us apply our definitions to individual expressions. The follow-

ing result is obtained from 3-3 and 3~sa:

3-12. Individual expressions are equivalent if and only if they are expres-

sions for the same individual.

Examples for L-equivalence and F-equivalence of individual expres-

sions will be given later ( 9).

A consideration of these results for predicators and individual expres-

sions shows the following: If 'P' and *Q' are equivalent predicators, then

'Pa' and 'Qa' are either both true or both false and hence, in any case,

equivalent; the same holds for 'Pb' and 'Qb', etc. Furthermore, if 'a' and

'b' are equivalent, then *Pa' and 'Pb' are either both true or both false

and hence, in any case, equivalent; the same holds for *Qa' and *Qb', etc.

An analogous result for functors follows from rules like 3-4. It can be

shown that the following two theorems hold generally for our systems Si,

S 3) and S 3 , and likewise for any similar systems, including those contain-

16 I. THE METHOD OF EXTENSION AND INTENSION

ing functors, provided that definitions analogous to those given above are

laid down.

3-13. If two designator signs are equivalent, then any two sentences of

simplest form (in S x : atomic form) which are alike except for the occur-

rence of the two designator signs are likewise equivalent.

3-14* If two designators (which may be compound expressions) are L-

equivalent, then any two sentences (of any form whatever) which are

alike except for the occurrence of the two designators are likewise L-

equivalent.

These two results show that our choice of the interpretation for the ex-

tended uses of ' 55 ' and of the definition for the extended use of the terms

1 equivalent' and 'L-equivalent' was not arbitrary. In fact, the choice was

made with the intention of reaching these results. In particular, the first

result 3-13, in its application to individual expressions, may be regarded

as supplying a justification for the use of ' ==' as a sign of identity, which

might at first perhaps appear strange.

On the basis of equivalence and L-equivalence for designators we define

the following two concepts:

3-15. Definitions. Let 31; be a designator (in SJ.

a. The equivalence class of 3l =Df the class of those expressions (in

S x ) which are equivalent to 31*.

b. The L-equivalence class of 2l =Df the class of those expressions

(in Si) which are L-equivalent to 31 .

It is easily seen that 31* itself belongs to both classes, that the L-equiva-

lence class is a subclass of the equivalence class, and that both classes con-

tain only designators of the same type as 8l<.

4. Classes and Properties

It is customary to regard two classes, say those corresponding to the predica-

tors T' and 'Q', as identical if they have the same elements, in other words,

if 'P' and *Q* are equivalent. We regard the two properties P and Q as identical

if T' and 'Q' are, moreover, L-equivalent. By the intension of the predicator

T' we mean the property P; by its extension we mean the corresponding class.

It follows that two predicators have the same extension if they are equivalent,

and the same intension if they are L-equivalent. The term 'property' is to be

understood in an objective, physical sense, not in a subjective, mental sense;

the same holds for terms like 'concept', 'intension', etc. The use of these and

related terms does not involve a hypostatization.

In analyzing the meaning of an adjective, e.g., 'human', or a cor-

responding predicator in a symbolic language, e.g., 'H', it is customary

4. CLASSES AND PROPERTIES 17

to speak of two entities on the one hand, the property of being human

or, as we shall write for short, the property Human; on the other hand,

the class of human beings, or the class Human. 10

The metalanguage M must contain certain translations of the sen-

tences of the object languages to be dealt with in M. The translation can

often be formulated in different ways. Take as an example an atomic sen-

tence in S,, say 'Hs'. Its simple, straightforward translation into M is as

follows, according to our rules of designation for 'H' and V (1-2 and i-i) :

4-1. ' Scott is human'.

There are two other translations of 'Hs' which in a sense are more explicit

by using the terms ' property' or 'class' but which have the same logical

content as 4-1 :

4-2. 'Scott has the property Human'.

4-3. 'Scott belongs to (is an element of) the class Human'.

As another example, take the sentence '(#)[H# D B#]'. Here, likewise,

there is a direct translation (4-4) and two more explicit ones with 'prop-

erty' (4-5) or ' class' (4-6) :

4-4. 'For every x, if x is human, then x is a biped'.

4-6. 'The property Human implies (materially) the property Biped'.

4-6. 'The class Human is a subclass of the class Biped'.

In these examples the terms 'property' and 'class' seem unnecessary,

since there are forms which avoid those terms (4-1 and 4-4). Thus the im-

portant question may be raised as to whether semantics could not do en-

tirely without those terms. However, we shall first accept them, so to

speak, uncritically, endeavoring merely to make their customary use more

exact and consistent. Later only shall we come back to the question men-

tioned; it will then be shown how the apparent multiplicity of entities

which seems to be introduced by the admission of these and other terms

can be reduced ( 33 f.). Thus our present acceptance of the two more

explicit forms of translation is merely an introduction of two ways of

speaking; it does by no means imply the recognition of two separate kinds

of entities properties, on the one hand; classes, on the other.

10 Since a brief formulation seems desirable and since phrases of the form 'the property

human' and 'the class human* are contrary to English grammar and sometimes even am-

biguous, I have used in earlier publications (see [I], p. 237) double quotation marks, e.g.,

'the property "human" '. However, this use of quotation marks differs from their normal

use. Therefore, I prefer now the method of capitalizing; I shall use it not only in connection

with 'property* and 'class* but likewise with other words designating kinds of entities, e.g.,

'relation', 'function', 'concept', 'individual', 'individual concept', and the like. In connection

with nouns instead of adjectives I often use also the customary form with 'of, e.g., I write

either 'the concept of equivalence' or 'the concept Equivalence'.

i8 I. THE METHOD OF EXTENSION AND INTENSION

The above examples seem to show a certain parallelism between the two

modes of speech, the one in terms of 'property' and the other in terms of

' class'. However, there is one fundamental difference, leaving aside the

inessential, merely idiomatic difference that in the one case the connecting

phrase is 'has' or 'possesses', while in the other it is 'belongs to' or 'is an

element of. The fundamental difference is in the condition of identity.

Classes are usually taken as identical if they have the same elements.

Thus, for example, on the basis of our earlier assumption (3-6), the class

Human has the same elements as the class Featherless Biped. Therefore :

4-7. The class Human is the same as the class Featherless Biped.

Under what conditions properties are usually regarded as identical is less

clear. It seems natural, and sufficiently in agreement with the vague

customary usage, to regard properties as identical if it can be shown by

logical means alone, without reference to facts, that whatever has the one

property has the other and vice versa; in other words, if the equivalence

sentence is not only true but L-true. Thus with respect to our earlier ex-

amples (3-7 and 3-9) the following holds:

4-8. The property Human is not the same as the property Featherless

Biped.

4-9. The property Human is the same as the property Rational Animal.

It is easily seen, on the basis of our definitions in the preceding section

(3-ib and 3-$a, b) that the identity conditions stated above can be formu-

lated in the following way with respect to predicators (of degree one) :

4-10. Classes are identical if and only if predicators for them are equiva-

lent.

4-11. Properties are identical if and only if predicators for them are

L-equivalent.

Now we shall introduce the terms 'extension* and 'intension* with

respect to predicators. If two predicators apply to the same individuals

in other words, if they are equivalent it is sometimes said that they are

coextensive or that they have the same extension (in one of the various

customary uses of this term). The use of 'intension' varies still more than

that of 'extension'. It seems in agreement with at least one of the custom-

ary usages to speak of the same intension in the case of L-equivalence.

Thus we lay down the following two conventions:

4-12. Two predicators have the same extension if and only if they

are equivalent.

4. CLASSES AND PROPERTIES 19

4-13. Two predicators have the same intension if and only if they

are L-equivalent.

These conventions determine only the use of the phrases ' have the same

extension' and 'have the same intension'. For many purposes this is suf-

ficient. If, however, we wish to go further and to speak of something as the

extension of a given predicator, and of something else as its intension,

then these conventions do not suffice; but they help us by narrowing the

choice of suitable entities. The first convention means that we may take

as extensions of predicators only something which equivalent predicators

have in common. According to 4-10, this condition is fulfilled by the cor-

responding classes. The second convention means that we may take as in-

tensions of predicators only something which L-equivalent predicators

have in common. According to 4-11, this condition is fulfilled by the cor-

responding properties. This suggests the following conception of the ex-

tension and the intension of predicators:

4-14. The extension of a predicator (of degree one) is the corresponding

class.

4-16. The intension of a predicator (of degree one) is the corresponding

property.

This seems sufficiently in agreement with customary usage. If this is ap-

plied to the predicator 'H' in S x , we obtain:

4-16. The extension of 'H' is the class Human.

4-17. The intension of 'IT is the property Human.

Both results hold also for the predicator '(\#)(H#)', which is L-equiva-

lent to 'H' in Sx.

It is obvious that there are many other ways for choosing entities as ex-

tensions and intensions of predicators (of degree one) so as to satisfy our

conventions (4-12 and 4-13). One alternative is as follows: It is possible

to take as the extension of a predicator its equivalence class (s-isa) and

as its intension its L-equi valence class (3-isb). This conception seems less

natural than the one we have chosen (4-14, 4-15), because it leads to

linguistic instead of to extra-linguistic entities. On the other hand, this

conception of intensions has the advantage that it is possible in an exten-

sional metalanguage; this will be explained later. (Compare definitions by

Russell and Quine mentioned below, at the end of 33.)

It may perhaps be useful, in order to avoid misunderstandings, to add

some informal remarks concerning the use in this book of the term

'Property*. This term will be used as synonymous with words like 'qual-

ity', 'character', 'characteristic', and the like in their ordinary use. It

20 I. THE METHOD OF EXTENSION AND INTENSION

is to be understood in a very wide sense, including whatever can be said

meaningfully, no matter whether truly or falsely, about any individual.

The term is used here not only for qualitative properties in the narrower

sense (for example, the properties Blue, Hot, Hard, and the like) but also

for quantitative properties (for example, the property Weighing Five

Pounds), for relational properties (e.g., the property Uncle Of Some-

body), for spatiotemporal properties (e.g., the property North Of

Chicago), and others. It is important to note what is not meant here by the

term 'property'. First, it does not refer to linguistic expressions; to the

symbol *H' and the corresponding word 'human' we apply the term

'predicator', not 'property'; by a property we mean rather what is ex-

pressed by a predicator (of degree one). Second, the properties of things

are not meant as something mental, say images or sense-data, but as

something physical that the things have, a side or aspect or component

or character of the things. If an observer sees that this table is red, then

the table has the character Red and the observer has the corresponding

character Red-Seeing. By the property Red we mean the first, not the

second; we mean that physical character of the thing which the physicist

explains as a certain disposition to selective reflection, not that psycho-

logical character of the observer which the physiologist explains as a

certain disposition to a specific reaction by the sensory part of the nervous

system.

Suppose we understand some predicators in a given language; that is to

say, we know which properties they express. Suppose, further, that we

have experienced each of these properties; that is to say, we have, for each

of them, found some things which, according to our observations, have

that property. We can form compound predicators out of the given predi-

cators with the help of logical particles. Then we understand a compound

predicator because its meaning is determined by the meanings of the com-

ponent predicators and the logical structure of the compound expression.

It is important to notice that our understanding of a compound predica-

tor is no longer dependent upon observations of any things to which it

applies, that is, any things which have the complex property expressed

by it.

In order to construct examples, suppose that the system S x contains

not only the predicator *H' for the property Human, but also the predica-

tor "F for the property Twenty Feet High. Then we can, for example,

form the following compound predicators (provided we permit the use of

'~ ' and ' V' in predicators in analogy to the use of ' ' introduced by 3-2) :

'~H' expresses the property Non-Human, 'H V T' the property Human

4. CLASSES AND PROPERTIES 21

Or Twenty Feet High, and 'BUT the property Human And Twenty Feet

High. We know things which exemplify the first of these three properties,

and likewise some for the second. But we have never seen any things that

exemplify the predicator ' H*T ; , and there are presumably no things of this

kind in the world. Nevertheless, 'H*T' is not meaningless. Since it is a

well-formed predicator (of degree one), it expresses a property, although

this property does not apply anywhere. We shall say of both the predica-

tor and the property that they are empty. One can understand ' H*T' just

as clearly as the other compound predicators; and one may indeed under-

stand this or any other compound predicator before he knows whether

and, if so, where it is exemplified. The understanding of a compound

predicator is based upon the understanding of the component predica-

tors. Exemplification in experience is required only for primary predica-

tors, with the help of which the others are interpreted.

Now consider the predicator 'H* ^H\ No factual knowledge is needed

for recognizing that this predicator cannot possibly be exemplified. Never-

theless, this expression is not meaningless. It is a well-formed predicator;

it expresses the property Human And Non-Human." We shall say of

both the predicator and the property that they are L-empty (logically

empty). [There is only one L-empty property, although there are many

empty properties. If *P' and'Q' are any two L-empty predicators, then

'P s Q', that is, < (x)(Px s Q*)' fo-ib), is L-true; therefore, T' and <Q'

are L-equivalent (3-sb); hence they express the same property (4-11).]

The use of the term 'relation' in this book is analogous to that of the

term 'property' just explained. A relation is meant neither as a mental

entity nor as an expression but rather as something that is expressed by

certain designators, namely, predicators of degree two or more, and that

may hold objectively for two or more things.

The term 'concept* will be used here as a common designation for

properties, relations, and similar entities (including individual concepts, to

be explained in 9, and functions, but not propositions). For this term it

is especially important to stress the fact that it is not to be understood in

a mental sense, that is, as referring to a process of imagining, thinking,

conceiving, or the like, but rather to something objective that is found in

nature and that is expressed in language by a designator of nonsentential

form. (This does not, of course, preclude the possibility that a concept

for example, a property objectively possessed by a given thing may be

subjectively perceived, compared, thought about, etc.)

11 Compare Bennett and Baylis, [Logic], sec. 3.4: "The existence of self -inconsistent

concepts."

22 I. THE METHOD OF EXTENSION AND INTENSION

The preceding remarks are meant merely as an informal terminological

clarification. They should by no means be regarded as an attempt toward

a solution of the old controversial problem of the universals. The tradi-

tional discussions concerning this problem are, in my view, a rather

heterogeneous mixture of different components, among them logical state-

ments, psychological statements, and pseudo-statements, that is, expres-

sions which are erroneously regarded as statements but do not have cogni-

tive content, although they may have noncognitive for instance, emo-

tive meaning components. My remarks on the interpretation intended for

the term ' property' are admittedly rather vague, chiefly because of a lack

of a clear and generally accepted terminology about matters of this kind.

Nevertheless, I hope they will give sufficiently clear indications for all

practical purposes and, above all, may help to avoid certain typical mis-

understandings.

I wish to emphasize the fact that the discussions in this book about

properties, and similarly about relations, concepts in general, proposi-

tions, etc., do not involve a hypostatization. As I understand it, a hypos-

tatization or substantialization or reification consists in mistaking as

things entities which are not things. Examples of hypostatizations of

properties (or ideas, universals, or the like) in this sense are such formula-

tions as 'the ideas have an independent subsistence', 'they reside in a

super-heavenly place', 'they were in the mind of God before they became

manifested in things', and the like, provided that these formulations are

meant literally and not merely as poetical metaphors. (We leave aside

here the historical question of whether these hypostatizations are to be

attributed to Plato himself or rather to his interpreters.) These formula-

tions, if taken literally, are pseudo-statements, devoid of cognitive con-

tent, and therefore neither true nor false. Whatever is said in this book

about properties may be wrong, but it has at least cognitive content. This

follows from the fact that our statements belong to, or can be translated

into, the general language of science. We use the term 'property' in that

sense in which it is used by scientists in statements of the following form:

' These two bodies have the same chemical properties, but there are certain

physical properties in which they differ' ; 'Let us express the property . . . ,

which is exemplified by the one but not by the other of these two bodies,

by'P".

The term 'entity* is frequently used in this book. I am aware of the

metaphysical connotations associated with it, but I hope that the reader

will be able to leave them aside and to take the word in the simple sense in

which it is meant here, as a common designation for properties, proposi-

5. EXTENSIONS AND INTENSIONS 23

tions, and other intensions, on the one hand, and for classes, individuals,

and other extensions, on the other. It seems to me that there is no other

suitable term in English with this very wide range.

5. Extensions and Intensions

In analogy to the case of predicators, we shall say of two designators of any

kind that they have the same extension if they are equivalent, and that they

have the same intension if they are L-equivalent. In later sections we shall dis-

cuss the problem of finding suitable entities which might be taken as extensions

and intensions in accordance with these identity conditions. If two predicators,

say T' and *Q J , are equivalent or L-equivalent in a system S, we say also that

the properties P and Q are equivalent or L-equivalent, respectively ; and anal-

ogously with designators of other kinds and their intensions.

In the preceding section we introduced the terms 'extension' and 'inten-

sion' with respect to predicators only, in agreement with traditional us-

age. Now we shall extend the use of these terms, applying them to other

types of designators in an analogous way.

In the case of predicators, we have taken equivalence as the condition

for identity of extension, and L-equi valence for identity of intension (4-12

and 4-13). Earlier ( 3), we saw how the semantical concepts of equiva-

lence and L-equivalence can be applied to the various types of designa-

tors. Thus it seems natural to take the same conditions as defining identity

of extension or intension with respect to designators in general. This leads

to the following definitions; 4-12 and 4-13 are now regarded simply as

special cases hereof.

6-1. Definition. Two designators have the same extension (in S x ) = D*

they are equivalent (in Sj).

6-2. Definition. Two designators have the same intension (in S x ) =DI

they are L-equivalent (in S x ).

Note that the terms 'extension' and 'intension' have not been defined

hereby, but only the phrases 'have the same extension' and 'have the same

intension'. In order to speak about extensions and intensions themselves,

we have to look for entities, or at least for phrases apparently referring

to entities, which can be assigned to designators in accordance with these

definitions. In the case of predicators, we found classes and properties as

such entities. We shall see later how suitable entities can be chosen for

sentences and individual expressions.

The introduction into the metalanguage M of expressions for additional

kinds of entities is always a precarious step that must be taken with cau-

tion and with careful consideration of the consequences. We shall discuss

the problem involved in the introduction of extensions and intensions for

24 I. THE METHOD OF EXTENSION AND INTENSION

designators later ( 33 ff.). Here it may be noted that the phrases 'have

the same extension' and 'have the same intension', although apparently

referring to certain entities as extensions and intensions, are, in fact, en-

tirely free of the problematic nature of the terms 'extension' and 'inten-

sion'; for those phrases are based by the above definitions on the terms

'equivalent' and 'L-equivalent', and these go back (by 3-5) to the terms

'true' and 'L-true', which can be defined for the system S x in an exact

way, as explained earlier.

It is often convenient to apply the term 'equivalent*, and perhaps also

the term 'L-equivalent', not only to designators but likewise to the inten-

sions of these designators; thus not only to predicators (e.g., 'the predica-

tors 'H' and 'F*B' are equivalent in Si') but also to properties and rela-

tions (e.g., 'the property Human and the property Featherless Biped are

equivalent') ; and analogously not only to sentences but also to proposi-

tions. This transferred use cannot lead to any actual ambiguity or con-

fusion, for two reasons: (i) The context always makes clear whether the

term ' equivalent' is meant in the original or in the transferred sense; the

former is the case whenever the term is applied to expressions in a lan-

guage system, the latter whenever it is applied to intensions, hence to extra-

linguistic entities. (2) In the original use the term is accompanied by a

reference to a language system (e.g., 'equivalent in S,'; however, this holds

only for the complete formulation; in practice we often omit the reference

if the context makes clear which language system is meant) ; the trans-

ferred use is not so accompanied (e.g., it makes no sense to say 'these two

properties are equivalent in S/). Two designators may be equivalent in

one language and not in another, because they may have other meanings

in the second language; thus the equivalence of designators is dependent

upon the language, as all semantical concepts are. On the other hand, the

equivalence of two properties is not dependent upon language; it is a non-

semantical and, moreover, a nonlinguistic concept (e.g., it is a biological,

not a linguistic, fact that the property Human and the property Feather-

less Biped are equivalent). The term 'equivalent' in the transferred use

still belongs to the metalanguage M; not, however, to the semantical

part of M but to what we might call the object part, that is, that part of

M which contains the translations of the sentences and other expressions

of the object languages. 12 The application of the terms 'equivalent' and

12 Previously, I called terms of this kind, which are transferred from semantics to extra-

linguistic entities, absolute terms ([I], 17), in order to indicate that in their new use the

terms are no longer relative to a language. However, I now prefer to avoid the word 'absolute'

because some readers were puzzled by it and suspected behind it some sort of metaphysical

absolutism.

6. EXTENSIONS AND INTENSIONS OF SENTENCES 25

'L-equivalent' to intensions of designators, il these designators are equiva-

lent or L-equivalent in the original semantical sense, leads, in combination

with the identity conditions expressed in 5-1 and 5-2, to the following

results:

6-3. If two designators are equivalent (in Si), then we say that their ex-

tensions are identical and that their intensions are equivalent.

6-4. If two designators are L-equivalent (in Sj), then we say that their in-

tensions are L-equivalent (or identical).

Because of 5-3, ' ss 1 may be regarded as a sign both for the identity of

extensions and for the equivalence of intensions; in particular, if it stands

between predicators of degree one (as in 'H = F*B'), it is a sign of identity

of classes and a sign of equivalence of properties.

Examples. We found earlier that the predicators 'IT and '#' are

equivalent but not L-equivalent (3-8), and that 'H' and 'RA' are L-

equivalent (3-11). If we apply here the above two definitions, we obtain

the following formulations with transferred terms, in addition to the

earlier formulations in terms of identity (4-7, 4-8, and 4-9):

5-6. The property Human is equivalent to the property Featherless

Biped.

5-6. The property Human is not L-equivalent to the property Featherless

Biped.

6-7. The property Human is L-equivalent to the property Rational

Animal.

Of these three formulations, only the first is actually useful in M; the

other two serve only as preparation for analogous formulations in another

metalanguage M' ( 34). [It may be remarked incidentally that the terms

'equivalent' and 'L-equivalent' in their transferred, nonsemantical use,

which are here applied to intensions, could also be applied to extensions.

However, equivalence of extensions would be the same as identity of

extensions and hence would not be useful. And to speak of L-equivalence

of extensions would even be dangerous because it would lead to the same

consequences that we shall later find for sentences like 42-6A.]

6. Extensions and Intensions of Sentences

We take as the extension of a sentence its truth-value, and as its intension

the proposition expressed by it. This is in accord with the identity conditions

for extensions and for intensions stated in the preceding section. Propositions

are here regarded as objective, nonmental, extra-linguistic entities. It is shown

that this conception is applicable also in the case of false sentences.

26 I. THE METHOD OF EXTENSION AND INTENSION

Now let us see whether we can find entities which may be taken as

extensions and intensions of sentences in accordance with our definitions

for the identity of extensions (5-1) and of intensions (5-2).

According to 5-1, we must take as extensions of sentences something

that equivalent sentences have in common. The most natural choice seems

the truth- values:

6-1. The extension of a sentence is its truth- value.

At first glance, it may perhaps seem strange to call a truth-value an

extension, and perhaps there may be a feeling even against saying that

equivalent sentences have the same extension. The term ' extension' seems

natural enough in the case of predicators; we easily visualize the domain of

individuals as an area and the class of individuals to which a certain predi-

cator applies (e.g., the class Biped for the predicator 'B') as a subarea

which extends over a smaller or larger part of the whole. But one might

say that in the case of a truth-value there is nothing extended. However,

a closer inspection may remove the impression of strangeness. It has be-

come customary to use the term ' extensional' for truth-functional connec-

tions, i.e., for connections such that the truth- value of the full sentence is

a function of the truth-values of the components. And there is, indeed,

a strong analogy between truth-values of sentences and extensions of

predicators. This can be seen as follows: A predicator of degree n is char-

acterized by the fact that we must attach to it n argument expressions in

order to form a sentence. Therefore, a sentence might be regarded as a

predicator of degree zero. Let 2l and 8l/ be any predicators of degree

n (n ^ i); then (according to 4-12, 3-5, and 3-1 a) 2I and Sly have the

same extension if and only if (x I )(x 2 ) . . (# n )[3li#r# 2 . . # n = 2l/#i# 2 . . #J

is true. If we stipulate that this, which applies originally only to n ^ i,

is to be applied analogously to sentences as predicators of degree zero,

we find that two sentences, @, and @y, have the same extension if and

only if @ == <gy is true, hence if and only if @ and , are equivalent.

Thus we are led back to 5-1 as applied to sentences; and then it seems

natural to take the truth- values as extensions. [For the time being we may

leave aside the question of what kind of entities these truth-values are,

which are here proposed as extensions. This problem will be discussed

later (in 23).]

Now we have to decide what entities to take as intensions of sentences.

It is often said that a (declarative) sentence expresses a proposition. We

accept this use of the word i proposition' ; that is to say, we do not use this

word for sentences or for sentences together with their meaning but for

6. EXTENSIONS AND INTENSIONS OF SENTENCES 27

those entities which themselves are extra-linguistic but which, if they find

expression in a language, are expressed by (declarative) sentences/ 3 Those

authors who use the term ' proposition' in this sense are often not quite

clear as to the condition under which two sentences express the same

proposition. We decide to take L-equivalence as this condition. 14 Thus,

for example, we say that the sentences '~(Pa*QbV and '~Pa V~Qb'

express the same proposition. This seems sufficiently in agreement with

the usage of many logicians. Since we took L-equivalence as the condition

of identity for intensions (5-2), we may regard propositions as intensions:

6-2. The intension of a sentence is the proposition expressed by it.

Examples:

6-3. The extension of the sentence 'Hs' (in S z ) is the truth- value that

Scott is human, 15 which happens to be the truth.

6-4. The intension of the sentence *Hs 7 is the proposition that Scott is

human. 15

Some remarks may help to clarify the sense in which we intend to use

the term 'proposition*. Like the term ' property ' ( 4), it is used neither

for a linguistic expression nor for a subjective, mental occurrence, but

rather for something objective that may or may not be exemplified in

nature. [We might say that propositions are, like properties, of a concep-

tual nature. But it may be better to avoid this formulation, because it

might lead to a subjectivistic misinterpretation, if the fact is overlooked

that we use the term ' concept' in an objective sense (see 4).] We apply

the term ' proposition' to any entities of a certain logical type, namely,

those that may be expressed by (declarative) sentences in a language.

By the property Black we mean something that a thing may or may not

have and that this table actually has. Analogously, by the proposition that

this table is black we mean something that actually is the case with this

table, something that is exemplified by the fact of the table's being as it is.

13 On the necessity of distinguishing clearly between the two meanings of the term 'propo-

sition', compare [I], pp. 235 f .

14 Compare [I], p. 92.

'5 In analogy to 'the property Human' and 'the class Human* we might write here *th

proposition Scott-Is-Human' and 'the truth- value Scott-Is-Human'. However, this would

become rather awkward for longer sentences. Therefore, we shall instead insert 'that* after

'proposition', thus coming back to ordinary usage. For the sake of analogy, we shall likewise

write 'the truth- value that . . /, although it deviates from ordinary usage; we cannot use the

more idiomatic form 'the truth-value of the proposition that . , .' because in 6-3 we wish to

speak only about the sentence and its extension, the truth-value, not about its intension, the

proposition.

2 8 I. THE METHOD OF EXTENSION AND INTENSION

(This simple explanation is possible only in the case of a true proposition;

the problem of false propositions will soon be discussed.)

The question of whether fads are propositions of a certain kind or

entities of a different nature is controversial. Ducasse 16 identifies facts with

true propositions. Bennett and Baylis 17 say that propositions are true or

false; on the other hand, "facts themselves are neither true nor false,

but just are". The question is, to a certain extent, a terminological one and

hence to be settled by convention. Since the term 'fact' in its ordinary

use is rather vague and ambiguous, there is some freedom of choice left as

to how to turn it into an exact technical term, in other words, how to

explicate it. I am inclined to think, like Ducasse, that it would not deviate

too much from customary usage if we were to explicate the term ' fact'

as referring to a certain kind of proposition (in our objective sense of

this word). What properties must a proposition have to be a fact in this

sense? First, it must, of course, be true; second, it must be contingent (or

factual); thus it must be F-true. I think that still another requirement

should be added: The proposition must be specific or complete in a certain

sense; but I am not sure what degree of completeness should be required.

An example may illustrate the problem. The proposition that this thing

(a piece of paper I have before me) is blue is a true proposition; in other

words, this thing has the property Blue. But the property Blue has a wide

range; it is not specific but includes many different shades of blue, say

Blue x , Blue 2 , etc. This thing, on the other hand, or, more exactly speaking,

a specified position c on its surface at the present moment has only one

of these shades, say Blue s . Let p be the proposition that c is blue, and q

the more specific proposition that c is blue s . It is the truth of q that makes

p true. Therefore, the nonspecific proposition p should perhaps not be

regarded as a fact. Whether q should be so regarded remains doubtful; q is

completely specific in one respect, concerning the color, but it does not

specify the other properties of the given thing. Should we require com-

plete specificity with respect to all properties of the thing or things in-

volved, and also with respect to all relations among the given things, or

perhaps even with respect to all relations between the given things and all

other things? It seems somewhat arbitrary to draw a line at any of these

points. If we do not stop at some point but go the whole way, then we

arrive at the strongest F-true proposition p Ty which is the conjunction of

16 C. J. Ducasse, "Propositions, Opinions, Sentences, and Facts", Journal of Philosophy,

XXXVII (1940), 701-11; see also his reply to some objections (ibid., XXXDC [1942], 132-36).

" [Logic], p. 49.

6. EXTENSIONS AND INTENSIONS OF SENTENCES 29

all true propositions 18 and hence L-impli^s every true proposition. If we

require of a fact this maximum degree of completeness (short of L-falsity),

then there is only one fact, the totality of the actual world, past, present,

and future. We indicate here these various possibilities for choosing an

explicatum for the concept of fact without making a decision. We shall

not take the term * fact' as a technical term but shall use it only in informal

explanations; thus, for example, we have said ( 2) that the truth- value

of a sentence which is not L-determinate is dependent upon the facts.

The greatest difficulty in the task of explicating the concept of proposi-

tion is involved in the case of a false sentence. Since this piece of paper c

is, in fact, blue, sentences like 'c is not blue' or ^c is red' are false. They

cannot be regarded as meaningless, because we understand their meaning

before we know whether they are true or false. Therefore, these sentences,

too, express propositions. On the other hand, these propositions cannot

have the same relation to facts as the proposition expressed by the true

sentence 'c is blue'. While the latter proposition is exemplified by a fact,

the former ones are not. What, then, are these false propositions? Are

there any entities of which we can say that they are expressed by those

false sentences, but for which we cannot point out any exemplifying facts?

Russell has given a thorough discussion of the problems here involved.

He likewise decides to use the term ' proposition' for what is expressed by a

sentence, in other words, for the signification of a sentence, provided that

an entity of this kind can be found. But he despairs of finding an entity

of this kind in the objective, factual realm. He argues as follows: "Since

a significant sentence may be false, it is clear that the signification of a

sentence cannot be the fact that makes it true (or false). It must, there-

fore, be something in the person who believes the sentence, not in the ob-

ject to which the sentence refers." 19 "Propositions . . . are to be defined

as psychological and physiological occurrences of certain sorts complex

images, expectations, etc. . . . Sentences signify something other than

themselves, which can be the same when the sentences differ. That this

something must be psychological (or physiological) is made evident by

the fact that propositions can be false." 20 Thus it seems that Russell

chooses a subjective, mental explicatum for the concept of proposition

only or. mainly for the reason that, in his opinion, there is no other way of

overcoming the difficulty connected with false propositions.

18 For the concepts of disjunctions or conjunctions of infinitely many propositions see [I],

pp. 92 f .

19 Russell, [Inquiry], p. 229 (chap, xiii, in sec. A). (Page numbers refer to the American

edition; it seems that the British edition has, unfortunately, a different pagination.)

ao Ibid., pp. 237 f, (chap, xiii, end of sec. A).

30 L THE METHOD OF EXTENSION AND INTENSION

I believe that it is possible to give an objective interpretation to the

term ' proposition', which is still applicable in the case of false sentences.

Any proposition must be regarded as a complex entity, consisting of com-

ponent entities, which, in their turn, may be simple or again complex.

Even if we assume that the ultimate components of a proposition must be

exemplified, the whole complex, the proposition itself, need not be. The

situation can perhaps best be made clear by its analogy with the situation

concerning properties. As we have seen earlier ( 4), a compound predica-

tor, for example, 'HT', may express an empty property, that is, one not

exemplified by any individual. The components 'IF and "F express

properties which are exemplified. The property expressed by the com-

pound predicator is constituted out of the component properties in a

logical structure indicated by the logical particles connecting the com-

ponent predicators. Thus we see that the fact that some predicators are

empty cantiot prevent the explication of properties as objective entities.

Analogously, the fact that some sentences are false does not exclude the

explication of propositions as objective entities. Propositions, like com-

plex properties, are complex entities; even if their ultimate components

are exemplified, they themselves need not be. The difference between

propositions and complex properties or other complex concepts is merely a

difference in the logical type. Therefore, the kind of connection is different.

In the case of our example *H*T', the connection was that of conjunction.

There are other logical connections which, applied to nonpropositional

components of suitable types, result in propositions. Consider as an ex-

ample the sentence *Hs* of the system S x ; it consists of the predicator 'H'

and the individual constant V combined by juxtaposition. Therefore, it

expresses a complex intension of prepositional type. Its two components

are the intension of 'H', which is the property Human, and the intension

of V, which is, as we shall see later ( 9), the individual concept Walter

Scott. The logical connection of these two intensions is that of attribution

or predication (expressed in S, simply by juxtaposition; its converse is

expressed in certain other symbolic languages by V and in English by the

copula * is'). Thus the resultant intension of the sentence is the proposition

that Scott is human. As an example of a different structure take

'(*)(B* D Fx)\ The intension of 'E' is the property Biped, that of 'F'

is the property Featherless. These two properties are the components of

the complex intension of the whole sentence. They are connected by the

universal conditional connection, expressed, according to the rules of the

system, by the way in which ' B ' and ' F' are combined in the sentence with

the help of three occurrences of a variable, two pairs of parentheses, and

6. EXTENSIONS AND INTENSIONS OF SENTENCES 31

the conditional connective * D '. This kind of connection yields, if applied

to two properties, a proposition. Thus the complex intension expressed by

the sentence is the proposition that whatever is a biped is featherless.

Each of the two component properties is exemplified by some individuals.

Some of the sentences of the form 'Bx 3 Fx' are exemplified by facts and

hence true, but some of them are not. The whole intension is not exempli-

fied; but it is, nevertheless, a proposition because it consists of exemplified

components in a prepositional structure; just as the intension of 'HUT',

though empty, is a property because it consists of two exemplified com-

ponents in the structure of a property. Thus F-false sentences, too, express

propositions. Now we may go one step further. Consider the L-false

sentence '(H~HX. It consists of the predicator 'H*~H' and the

individual constant ' s', in the same combination as in the previous ex-

ample 'Hs'. We have seen earlier (4), that the predicator 'H^^H',

although L-empty, expresses a property, namely, the L-empty property

Human And Non-Human. Therefore, the sentence mentioned expresses a

complex intension resulting from combining this property with the indi-

vidual concept Walter Scott by attribution. Thus this intension is the

proposition that Scott is human and not human. Although this intension,

like that of 'H*~H', cannot possibly be exemplified, it still is a proposi-

tion. By going one step further in the analysis of this proposition we find

as its components the property Human and the individual concept Walter

Scott; these components are both exemplified, and they are combined in a

structure of propositional type.

Generally speaking, it must perhaps be admitted that a designator can

primarily express an intension only if it is exemplified. However, once we

have some designators which have a primary intension, we can build com-

pound designators out of them which express derivative, complex inten-

sions, no matter whether these compound designators are exemplified or

not. We do not need exemplifications in order to grasp their intensions,

because the intension of a compound designator is determined, in virtue

of the semantical rules of the system, by the intensions of the component

designators and by the way in which these designators are combined.

It has been the purpose of the preceding remarks to facilitate the under-

standing of our conception of propositions. If, however, a reader should

find these explanations more puzzling than clarifying, or even unac-

ceptable, he may simply disregard them. They are not a necessary basis

for the further discussions in this book; they will hardly be referred to

again. It will be sufficient for nearly all our discussions involving proposi-

tions to assume that they are entities of any kind fulfilling the following

32 I. THE METHOD OF EXTENSION AND INTENSION

two conditions: (i) to every sentence in a semantical system 5, exactly

one entity of this kind is assigned by the rules of S; (2) the same entity

is assigned to two sentences in 5 if and only if these sentences are L-

equivalent. If someone is in doubt as to whether there are any nonmental

and extra-linguistic entities which fulfil these conditions, he may take as

propositions certain linguistic entities which do so. We shall later see that,

for example, certain classes of sentences in 5 1 may be taken (the L-equiva-

lence classes, see remark at the end of 33) or certain classes of classes of

sentences in S (the ranges, see remark near the end of 40).

7. Individual Descriptions

An (individual) description is an expression of the form *(i^)(. . x . .)'; it

means 'the one individual such that . . x . .'. If there is one and only one

individual such that . . x . . , we say that the description satisfies the unique-

ness condition. In this case the description, i.e., the entity to which the descrip-

tion refers, is that one individual. Logicians differ in their interpretations of de-

scriptions in cases where the uniqueness condition is not satisfied. The methods

of Hilbert and Bernays and of Russell are here discussed; that of Frege will be

discussed in the next section.

We use the term 'individual* not for one particular kind of entity but,

rather, relative to a language system 5, for those entities which are taken

as the elements of the universe of discourse in 5, in other words, the enti-

ties of lowest level (we call it level zero) dealt with in 5, no matter what

these entities are. For one system the individuals may be physical things,

for another space-time points, or numbers, or anything else. Consequent-

ly, we call the variables of level zero individual variables, the constants

individual constants, and all expressions of this level, whether simple

(variables and constants) or compound, individual expressions. The

most important kinds of compound individual expressions are: (i) full

expressions of functors (e.g., '3 + 4 ? , where ' +' is a functor and '3' and

'4' are individual constants); within our systems, expressions of this kind

occur only in S 3 , not in Sj and S 2 ; (2) individual descriptions. We shall use

here the term ' description' mostly in the sense of ' individual description 1 .

Descriptions of other types do not occur in our systems; a few remarks on

them will be made at the end of 8.

A description in S x has the form *(w)(. . * . .)'; it is interpreted as

'the one individual x such that . . x . .'. ( (ix)' is called an iota-operator;

the scope ' . . x . .' is a sentential matrix with 'x' as a free variable. For

example, ' (ix) (?# ~ Qx) ' means the same as ' the one individual which

is P and not Q'.

The entity for which a description stands (if there is such an entity) will

7. INDIVIDUAL DESCRIPTIONS 33

be called its description; here, in the case of individual descriptions, the

descriptum is an individual. With respect to a given description, there are

two possible cases: either (i) there is exactly one individual which fulfils

the condition expressed by the scope, or (2) this does not hold, that is,

there are none or several such individuals. In the first case we shall say of

the scope, and also of the whole description, that it satisfies the unique-

ness condition:

7-1. Definition. Let '. . x . .' be a (sentential) matrix (in S x ) with V as

the only free variable. '. . x . .' (and '(ix) (. . x . .)') satisfies the unique-

ness condition (in Sr) = DI '(3s) (#)[.. x . . = (x = z)]' is true (in Sj).

( l x 55 z' means 'x is the same individual as s'; see 3-3.)

In the case of a description satisfying the uniqueness condition, there is

general agreement among logicians with respect to its interpretation; the

one individual satisfying the scope is taken as descriptum. In the other

case, however, there is, so far, no agreement. Various methods have been

proposed. We shall outline three of them, those proposed by Hilbert and

Bernays (I), Russell (II), and Frege (III). Then we shall adopt Frege's

method for our systems. It should be noticed that the various conceptions

now to be discussed are not to be understood as different opinions, so that

at least one of them must be wrong, but rather as different proposals. The

different interpretations of descriptions are not meant as assertions about

the meaning of phrases of the form 'the so-and-so' in English, but as pro-

posals for an interpretation and, consequently, for deductive rules, con-

cerning descriptions in symbolic systems. Therefore, there is no theo-

retical issue of right or wrong between the various conceptions, but only

the practical question of the comparative convenience of different

methods.

In order to make the following discussions more concrete, let us suppose

that two (sentential) matrices are given, each with exactly one free vari-

able; we indicate them here with the help of dots and dashes: '. . x . /

and '- - y - -' (e.g., ' Axw' and 'Hy') We construct the description with the

first as scope and substitute it for *y y in the second:

7-2. '- - (w) (. . x . .) - -'. (Example: <H(i*) (A#w)'.)

Method I. Hilbert and Bejnaysf* in a system with natural num-

bers as individuals, permit the use of a description only if it satisfies

the uniqueness condition. Since the system is constructed as a calculus,

not as a semantical system, the formula of uniqueness is required to be

C-true (provable) instead of true. It seems that this method is quite con-

31 [Grundlagen I], p. 384.

34 I. THE METHOD OF EXTENSION AND INTENSION

venient for practical work with a logico-arithmetical system; one uses

a description only after he has proved the uniqueness. However, this

method has a serious disadvantage, although of a chiefly theoretical

nature: the rules of formation become indefinite, i.e., there is no general

procedure for determining whether any given expression of the form 7-2

is a sentence of the system (no matter whether true or false, provable or

not). For systems also containing factual sentences, the disadvantage

would be still greater, because here the question of whether a given ex-

pression is a sentence or not would, in general, depend upon the con-

tingency of facts.

Method II. Russell 23 takes the whole expression 7-2 in any case as a

sentence. The uniqueness condition is here taken not as a precondition for

the sentential character of the expression but rather as one of the condi-

tions for its truth in other words, as part of its content. Thus the transla-

tion of 7-2 into M is as follows:

7-3. l There is an individual y such that y is the only individual for which

. . y . . holds, and - - y - -' (for example/ there is an individual y such that

y is the only individual which is an author of Waverley, and y is human').

Hence, 7-2 is here interpreted as meaning the same as the following (with

a certain restriction, see below) :

7-4. '(3>y) [(#)(. . x . . ss (x ss y)) - - y - -]'. (in the example,

In order to incorporate this interpretation into his system, Russell lays

down a contextual definition for descriptions; 7-2 is the definiendum, 7-4

the definiens. If we prefer to take the iota-operator as primitive instead of

defining it, we can reach the same result by framing the semantical rules

in such a way that any two sentences of the forms 7-2 and 7-4 become

L-equivalent.

In comparison with Hilbert's method, Russell's has the advantage

that an expression of the form 7-2 is always a sentence. In comparison

with Frege's method, which will soon be explained, it has the disad-

vantage that the rules for descriptions are not so simple as those for other

individual expressions, especially those for individual constants. In par-

ticular, the inferences of specification, leading from ' (y) (- - y - -) ' to

'--a--', and of existential generalization, leading from '--a--' to

' (3y) (- - y - -)', are, in general, not valid if a description takes the place

** The reasons for this method are explained in detail by Russell in [Denoting]; it has been

applied by Russell and Whitehead in the construction of the system of [P.M.], see I, 66 ff.

and 173 ff.

8. FREGE'S METHOD FOR DESCRIPTIONS 35

of the individual constant 'a'; here the uniqueness sentence for the de-

scription must be taken as an additional premise. A further disadvantage

of Russell's method is the following: A sentence like '~ Q(ix) (P#)' can be

transformed in two ways. Either this whole sentence is taken as 7-2 and

transformed into the corresponding sentence of the form 7-4; or the part

( Q(ix)(Px)' is taken as 7-2, transformed into the corresponding sentence

of the form 7-4, and then prefixed again with the sign of negation. The two

resulting sentences are not L-equivalent (in distinction to Frege's meth-

od) ; hence Russell has to lay down an additional convention, which de-

termines for each case what is to be taken as the context 7-2.

8. Frege's Method for Descriptions

We adopt for our systems a method proposed by Frege for interpreting indi-

vidual descriptions in cases of nonuniqueness. This method consists in choosing

once for all an individual to be taken as descriptum for all such cases.

Method III. Frege 33 regards it as a defect in the logical structure

of natural languages that in some cases an expression of the grammatical

form ' the so-and-so' is a name 24 of one object while in other cases it is not;

in our terminology: that some descriptions have a descriptum but others

not. Therefore, he suggests that the rules of a language system should be

constructed in such a way that every description has a descriptum. This

requires certain conventions which are more or less arbitrary; but this

disadvantage seems small in comparison with the gain in simplicity for the

rules of the system. For instance, specification and existential generaliza-

tion are here valid also for descriptions (at least in extensional contexts).

Frege's requirement can be fulfilled in various ways. The choice of a

convenient procedure depends upon the particular features of the lan-

guage system, especially upon the range of values of the variables in

question. There are chiefly two methods which deserve consideration; we

call them Ilia and Illb. We shall explain them and then use Illb for our

systems.

Method Ilia. Frege 25 himself constructs a system without type dif-

ference between individuals and classes; that is to say, he counts both

classes and their elements as objects, i.e., as values of the individual vari-

ables. To any of those descriptions which do not satisfy the condition of

uniqueness he assigns as descriptum the class of those objects which fulfil

a * [Sinn], pp. 39-42.

a * For the question of English translations for Frame's terms, see below, p. 118, n. 21,

* JGrundgesetze], I, 19.

36 I. THE METHOD OF EXTENSION AND INTENSION

the scope. Thus different descriptions of this kind may have different

descripta.

Method Illb. A simpler procedure consists in selecting, once for all, a

certain entity from the range of values of the variables in question and

assigning it as descriptum to all descriptions which do not satisfy the con-

dition of uniqueness. This has been done in various ways.

(i) If the individuals of the system are numbers, the number o seems

to be the most natural choice. Frege 26 has already mentioned this possibil-

ity. It has been applied by Godel 37 for his epsilon-operator and by myself 28

for the K-operator.

(ii) For variables to whose values the null class A belongs, this class

seems to be the most convenient choice. Such a choice has been made by

Quine, 29 in whose system there is, as in Frege's, no type difference between

individuals and classes.

(iii) How can Method Illb be applied to a language system whose

individuals are physical things or events? At first glance, it seems impos-

sible to make here an even moderately natural choice of an individual as

common descriptum for all individual descriptions which do not satisfy

the condition of uniqueness. To select, say, Napoleon would be just as

arbitrary as to select this dust particle on my paper. However, a natural

solution offers itself if we construct the system in such a way that the

spatiotemporal part-whole relation is one of its concepts. 30 Every indi-

vidual in such a system, that is, every thing or event, corresponds to a

class of space-time points in a system with space-time points as indi-

viduals. Therefore, it is possible, although not customary in the ordinary

language, to count among the things also the null thing, which corresponds

to the null class of space-time points. In the language system of things it is

characterized as that thing which is part of every thing. 31 Let us take 'a '

a6 [Sinn], p. 42 n.

37 K. Godel, "Ueber formal unentscheidbare Satze der Principia Mathematica und ver-

wandter Systeme", Monatsheftef. Math. u. Physik, XXXVIII (1931), 173-98.

* [Syntax], 7. [M.LJ, p. 147.

3 This is, for instance, the case with the following systems: a system for certain biological

concepts by J. H. Woodger (The Axiomatic Method in Biology [1937]; The Technique of Theory

Construction ["International Encyclopedia of Unified Science", Vol. II, No. 5 (1939)]); a

calculus of individuals by H. S. Leonard and N. Goodman ("The Calculus of Individuals and

Its Uses", Journal of Symbolic Logic, V [1940], 45-55); and a general system of logic recently

constructed by R. M. Martin ("A Homogeneous System for Formal Logic", Journal of Sym-

bolic Logic, VIII [1943], 1-23), where the customary symbol of inclusion and the term 'in-

clusion* apparently refer to the part-whole relation among things,

* x In the system by Martin mentioned in the preceding footnote the null thing is indeed

introduced (see op. cit., p. 3, and D7, p. 9), while in the paper by Leonard and Goodman

there is an explicit "refusal to postulate a null element" (op. cit. t p. 46).

8. FREGE'S METHOD FOR DESCRIPTIONS 37

as the name for the null thing; the other things may be called non-null

things. If a system S includes a among its individuals, then a seems a

natural and convenient choice as descriptum for those descriptions which

do not satisfy the uniqueness condition. It is true that this procedure re-

quires certain deviations from the ordinary language for the forms of

sentences in S; but these deviations are smaller than we might expect at

first glance. For most of the universal and existential sentences, the trans-

lation into S is straightforward, i.e., without change in structure; in other

cases 'non-null' must be inserted. [Examples: The sentence 'There is no

thing which is identical with the king of France in 1905' is translated into

a sentence of S of the form 'There is no non-null thing . . .'. On the other

hand, no such change in form is necessary for the sentence 'All men are

mortal' and not even for 'There is no man who is identical with the king of

France in 1905', because it follows from any suitably framed definition for

'man' that every man is a non-null thing.]

In our further discussions we assume for our system S x that Frege's

Method Illb is applied and that the individual constant 'a*' is used

for the common descriptum of all descriptions which do not satisfy the

uniqueness condition. We leave it open which individual is meant by

'a*'; it may be the null thing a , if this belongs to the individuals in S f ;

it may be o, if numbers belong to the individuals (as, for instance, in S 3 ),

but it may as well be any other individual. Consequently, a sentence con-

taining a description is now interpreted in a way different from Russell's.

The translation of 7-2 into M is now as follows (instead of 7-3):

8-1. 'Either there is an individual y such that y is the only individual for

which . . y . . holds, and - - y - -; or there is no such individual, and

- - a* - -'. [In the previous example: 'Either there is an individual y such

that y is the only author of Waverley, and y is human; or there is no such

individual y (that is to say, there is either no author or several authors

of Waverley), and a* is human'.]

Hence, the sentence 7-2 containing the description is L-equivalent in S x

to the following (instead of to 7-4) :

8-2. '(3y) [(*)(. . * . . (* y)) - - y - -] V [~(3y)(*)(. . * . . -

(a SB y)) .-- a* - -] '. (In the example: ' (3y) [(x) (Axw m (x m y))

Hy] V [~ (3y)(*) (A*w (x y)) Ha*] '.)

Here again, as in the case of Russell's method, we may set up either a

contextual definition for 7-2 with 8-2 as definiens, or semantical rules for

the iota-operator as a primitive sign such that 7-2 becomes L-equivalent

to 8-2.

I. THE METHOD OF EXTENSION AND INTENSION

The accompanying table gives a survey of the various methods just

explained for dealing with descriptions in the case of nonuniqueness. The

case of uniqueness is not represented because its treatment is the same

with all authors.

INTERPRETATION OF DESCRIPTIONS IN THE CASE OF

NONUNIQUENESS

F&EGE

SYSTEM

HlLBEIT-

BERNAYS

RUSSELL

QUINE

OP

THINGS

SYSTEM

Si

(a)

(b)

Method I

Method II

Method

Method

Method

Method

Method

Ilia

Illb

Illb

Illb

Illb

Description

No descriptum;

*(..*..)

A

Null

a*

is mean-

the sentence is

thing

ingless

meaningful

a

but false

Some brief remarks may be made on descriptions with variables of other

than individual type, especially predicator variables, functor variables,

and sentential variables. (This is a digression from the study of our sys-

tems Si, etc., which contain only individual variables.) Here it is easy to

make a natural choice of a value of the variable as a descriptum for those

descriptions which do not satisfy the condition of uniqueness. If an indi-

vidual has been chosen as a* (it may be a or o or anything else), then we

might call one entity in every type the null entity of that type, in the

following way: In the type of individuals it would be a*; in any predica-

tor type, the null class or null relation of that type, e.g., for level one and

degree one the null class A ; in the type of propositions, the L-false proposi-

tion; in any type of functions, that function which has as value for all

arguments the null entity of the type in question. Then we may take as

descriptum in the case of nonuniqueness the null entity of the type of the

description variable.

For the sake of simplicity, the following explanations are restricted to

extensional systems. Let'/ and'g' be predicator variables of level one and

degree one. Let '- - (i/)(. . / . .) - -' indicate, in analogy to 7-2, a sentence

containing a description of the type of '/', hence a description for a class

or property. This sentence is L-equivalent to the following, in analogy

to 8-2:

-A- -p.

9. INTENSIONS OF INDIVIDUAL EXPRESSIONS 39

The uniqueness condition here occurring says that there is a property g,

such that for those / and only those, which are equivalent to &;./..;

in other words, there is exactly one class g such that . . g . . . Hence

here the uniqueness applies to extensions, not to intensions. This is in

analogy to 7-1 and 7-3; for, as we shall see later, the extensions of indi-

vidual expressions are individuals.

However, if the system contains lambda-operators for the formation of

predicators, then descriptions with predicator variables are not necessary,

they can be replaced by lambda-expressions. In this case we can transform

not only a sentence containing the description as in the earlier case but

the description itself into an L-equivalent expression. The description

'(*/)( /)' is L-equivalent to the lambda-predicator ' (\x) [(3.g) ((/)

[../.. = (/*)!*)]'.

In a similar way, for every description of a function (containing an

iota-operator with a functor variable) there is an L-equivalent functor

formed with a lambda-operator. And for every description containing an

iota-operator with a sentential variable there is an L-equivalent sentence

without an iota-operator; however, in an extensional system these de-

scriptions with sentential variables are rather useless anyway.

In view of these results, it seems convenient in the primitive notation

of a system (at least in an extensional one) to use the iota-operator, if at

all, for individual descriptions only, and then to use the lambda-operator

for the formation of predicators and functors. 32

9. Extensions and Intensions of Individual Expressions

It is found to be in accord with our earlier conventions, to take as the exten-

sion of an individual expression the individual to which it refers. The intension

of an individual expression is a concept of a new kind; it is called an individual

concept.

Let us consider some examples of F-equivalence and L-equivalence of

individual expressions. We assume the following as a historical fact:

9-1. Assumption. There is one and only one individual which is an author

of Waverley, and this individual is the same as Walter Scott.

Then the descriptum of * (ix) (Axw) ' is that individual which is author of

Waverley and not a* and * (t#) (Axw) 3= s' is, according to the rule 3-3,

true, but not L-true; hence it is F-true. This leads to the following result,

according to the definitions 3-5 :

** Several forms of systems with predicators and functors built with lambda-operators

have been constructed by Church, see especially The Calculi of Lambda-Conversion ("Ann. of

Math. Studies", No. 6 [1941!).

40 I. THE METHOD OF EXTENSION AND INTENSION

9-2. ( (ix)(Axw) y is equivalent to V, but not L-equivalent, hence F-

equivalent.

On the other hand, let us compare the two descriptions ' (ix) (Hx A#w) '

and ' (ix) (RAx Axw) '. Let us see what we can find out about them if we

make use of the rules of S x , especially 1-2, but not of any historical or other

factual knowledge. If there is exactly one individual which is both human

or, which means the same, a rational animal and an author of Waver-

ley, then the descriptum of each of the two descriptions is this individual;

otherwise the descriptum of each is a*. Thus, in either case, the de-

scriptum of the first description is the same individual as that of the

second. Hence, according to rule 3-3, the sentence *(i#)(H# Axw) =

(i#)(RA# A#w)' is true; it is, moreover, L-true because we have shown

its truth by using merely the semantical rules. Therefore, the two descrip-

tions are L-equivalent.

We found earlier that individual expressions are equivalent if and only

if they are expressions for the same individual (3-12). Hence, according

to the definition of identity of extensions (5-1), individual expressions

have the same extension if and only if they are expressions for the same

individual. Therefore, it seems natural to regard as extensions of indi-

vidual expressions the individuals themselves:

9-3. The extension of an individual expression is the individual to which

it refers (hence the descriptum, if it is a description).

Since we adopted Frege's method, every description has exactly one

descriptum. Hence, on the basis of the convention just made, there is no

ambiguity with respect to the extension of an individual expression. For

instance, the extension of ' s' is the individual Walter Scott, and the same

holds for each of the three descriptions discussed above as examples. If

there were none or several authors of Waverley, then the extension of

' (ix) (A#w)' would be the individual a*.

Now let us look for entities which we might regard as intensions of

individual expressions. According to our definition for the identity of in-

tensions (5-2), the intension must be something that L-equivalent indi-

vidual expressions (for example, the two descriptions above containing

*H' and 'RA') have in common. We have earlier found entities which

seemed suitable as intensions of designators of other types; for sentences,

propositions; for predicators, properties or relations; for functors, func-

tions. Thus, in these cases, the intensions are those entities which are

sometimes regarded as the meanings of the expressions in question; and,

in the case of predicators and functors, the intensions are concepts of cer-

9. INTENSIONS OF INDIVIDUAL EXPRESSIONS 41

tain types. Now it seems to me a natural procedure, in the case of indi-

vidual expressions, likewise to speak of concepts, but of concepts of a par-

ticular type, namely, the individual type. Although it is not altogether

customary to speak here of concepts in this sense, still it does not seem to

deviate too much from ordinary usage. I propose to use the term 'indi-

vidual concept* for this type of concept. Thus we say:

9-4. The intension of an individual expression is the individual concept

expressed by it.

Examples:

9-5. The intension of V is the individual concept Walter Scott.

9-6. The intension of * (ix) (A#w) ' is the individual concept The Author

Of Waverley.

(Here, and further on, in translating descriptions into M, we omit for

brevity the phrase ' or a*, if there is not exactly one such individual'.) In-

stead of saying in the customary but ambiguous terminology that the two

L-equivalent descriptions discussed above have the same meaning, we say

now that they have the same intension and that their common intension

is the individual concept The Human Author Of Waverley, which is the

same as the individual concept The Rational Animal Author Of Waverley.

On the other hand, the following are three different individual concepts:

the one just mentioned, the individual concept Walter Scott, and the indi-

vidual concept The Author Of Waverley. Here again the intensions of given

expressions, and the identity or nonidentity of these intensions, can be

determined on the basis of the semantical rules alone.

We have* seen earlier how a sentence containing a predicator can be

translated into M, that is, English, in different ways. Thus, for the sen-

tence 'Hs', we had, in addition to the simple translation ' Scott is human',

two more explicit translations, one of which used the term property' and

the other the term 'class' (see 4-2 and 4-3). In these two explicit transla-

tions, V was still simply translated by 'Scott'. Now, however, we have

seen that, corresponding to the distinction between classes and properties,

we have in the case of individual expressions the distinction between indi-

viduals and individual concepts. Hence, we may use in M instead of

' Scott' the more explicit phrases ' the individual Scott' and ' the individual

concept Scott'. Since the distinction is perhaps clearer for a description

than for an individual constant, let us take, instead of ( Hs', the sentence

'H(i#)(A#w)'. In addition to the simple translation 'the author of

Waverley is human', we have here four more explicit translations in which

both to 'The Author Of Waverley' and to 'Human' a characterizing word

42 I. THE METHOD OF EXTENSION AND INTENSION

is added. Two of these translations are pure, two mixed. Of the two pure

translations, the first contains two references to extensions, and the second

two references to intensions; these translations are as follows:

'The individual The Author Of Waverley belongs to the class Human'.

'The individual concept The Author Of Waverley is subsumable under

the property Human'.

Since it is not customary to speak about individual concepts, there is no

word in customary usage for the relation between an individual concept

and a property corresponding to the element-relation between an indi-

vidual and a class; we have used here for this relation the word 'subsum-

able' (in the sense of 'truly subsumable'), but we shall not use it further

on. Of the two mixed translations, which contain a reference to an exten-

sion and a reference to an intension, we shall give at least one, because it is

not too far from customary usage:

'The individual The Author Of Waverley has the property Human'.

Thus we find here a multiplicity of possible translations into M, some of

them rather cumbersome and strange-looking. This multiplicity seems in-

evitable as long as we wish to distinguish explicitly between classes and

properties and between individuals and individual expressions. The prob-

lem of whether and by which means this apparent multiplicity of entities

and the corresponding multiplicity of formulations can be reduced will be

discussed later ( 33 f.).

10. Variables

We found earlier that the extension of a predicator T' is a class, and its in-

tension is a property. Therefore, a variable of the same type (e.g., */') refers

both to classes and to properties; we say that classes are its value extensions, and

properties its value intensions. Analogously, for a variable of the type of sen-

tences (e.g., 'p'), the value extensions are truth- values, and the value inten-

sions are propositions. Finally, the value extensions of an individual variable

(e.g., V) are individuals, and its value intensions are individual concepts.

Quine has repeatedly pointed out the important fact that, if we wish to

find out what kind of entities somebody recognizes, we have to look more at

the variables he uses than at the constants and closed expressions. "The

ontology to which one's use of language commits him comprises simply the

objects that he treats as falling . . . within the range of values of his vari-

ables." 33 I am essentially in agreement with this view, as I shall presently

explain. But, first, I wish to indicate a doubt concerning Quine 3 's formula-

tion; I am not quite clear whether the point raised is not perhaps of a

33 [Notes], p. 118; see also his [Designation].

10. VARIABLES 43

merely terminological nature. I should prefer not to use the word' ontology'

for the recognition of entities by the admission of variables. This use seems

to me to be at least misleading; it might be understood as implying that

the decision to use certain kinds of variables must be based on ontological,

metaphysical convictions. In my view, however, the choice of a certain

language structure and, in particular, the decision to use certain types of

variables is a practical decision like the choice of an instrument; it de-

pends chiefly upon the purposes for which the instrument here the lan-

guage is intended to be used and upon the properties of the instrument.

I admit that the choice of a language suitable for the purposes of physics

and mathematics involves problems quite different from those involved in

the choice of a suitable motor for a freight airplane; but, in a sense, both

are engineering problems, and I fail to see why metaphysics should enter

into the first any more than into the second. Furthermore, I, like many

other empiricists, regard the alleged questions and answers occurring in

the traditional realism-nominalism controversy, concerning the onto-

logical reality of universals or any other kind of entities, as pseudo-ques-

tions and pseudo-statements devoid of cognitive meaning. I agree, of

course, with Quine that the problem of "Nominalism" as he interprets

it 34 is a meaningful problem; it is the question of whether all natural sci-

ence can be expressed in a " nominalistic" language, that is, one contain-

ing only individual variables whose values are concrete objects, not

classes, properties, and the like. However, I am doubtful whether it is

advisable to transfer to this new problem in logic or semantics the label

' nominalism' which stems from an old metaphysical problem.

The sense in which I agree with Quine's thesis that " to be is to be the

value of a variable" will become clear by the following example: Suppose

somebody constructs a language not only as a subject matter of theoretical

investigations but for the purpose of communication. Suppose, further,

that he decides to use in this language variables 'm\ V, etc., for which

all (natural) numerical expressions (e.g., 'o', '3', '2 + 3', etc.) and only

those are substitutable. We see from this decision that he recognizes

natural numbers in this sense: he is willing to speak not only about par-

ticular numbers (e.g., ' 7 is a prime number') but also and this is the de-

cisive point about numbers in general. He will, for example, make state-

ments like: 'for every- m and n, m + n = n + m j and 'there is an m

between 7 and 13 which is prime*. The latter sentence speaks of the exist-

ence of a prime number. However, the concept of existence here has

nothing to do with the ontological concept of existence or reality. The sen-

** [Designation], p. 708.

44 I. THE METHOD OF EXTENSION AND INTENSION

tence mentioned means just the same as 'it is not the case that for every

m between 7 and 13, m is not prime'. By the same token, we see, furthei-

more, that the user of the language is willing to recognize the concept

Number. Generally speaking, if a language (of ordinary structure) con-

tains certain variables, then we can define in it a designator for the range

of values of those variables. In the present case, the definition is: " 'Num-

ber 1 for'(Xw)(w = m) y " or, if the language in question does not con-

tain abstraction operators, " ' Number (w)' for 'm = m' ". [In the de-

finiens, any matrix '. . m . .' may be used which is L-universal, that is,

such that ' (m) (. . m . .) ' is L-true.] It is important to emphasize the

point just made that, once you admit certain variables, you are bound

to admit the corresponding universal concept. It seems to me that some

philosophers (not Quine) overlook this fact; they do not hesitate to admit

into the language of science variables of the customary kinds, like senten-

tial variables ('/>', V, etc.), numerical variables, perhaps also predicator

variables at least of level one, and other kinds; at the same time, however,

they feel strong misgivings against words like 'proposition 7 , 'number',

'property' (or 'class'), 'function', etc., because they suspect in these

words the danger of an absolutist metaphysics. In my view, however, the

accusation of an absolutist metaphysics or of illegitimate hypostatizations

with respect to a certain kind of entities, say propositions, cannot be made

against an author, merely on the basis of the fact that he uses variables of

the type in question (e.g., '/>', etc.) and the corresponding universal word

('proposition') ; it must be based, instead, on an analysis of the statements

or pseudo-statements which he makes with the help of those signs.

Quine's thesis and my remarks in connection with it concern the lan-

guage which somebody not only analyzes but uses, hence, with respect to

semantical discussions, the metalanguage. Now let us look at the role of

variables in an object language S. If S is given, then a metalanguage M

intended for the semantical analysis of S must be rich enough in relation

to S. In particular, M must contain variables whose ranges of values cover

those of all variables in 5 (and, as Tarski has shown, even go beyond this

in order to make possible the definition of ' true in 5'). Let us further pre-

suppose here, as in the previous discussions, that M enables us to speak in

general terms about the extensions and intensions of predicators, sen-

tences, and individual expressions of S.

Let S (in distinction to Si) contain not only individual variables but also

those of other types. Let us begin with variables '/', '#', etc., of the type

of predicators of level one and degree one. With respect to a predicator,

say 'H' in S,, we have distinguished between its extension, the class

10. VARIABLES 45

Human, and its intension, the property Human. A sentence '. . H . /

containing 'H' can be translated into M in different ways; we may use

either the word ' human' alone or the phrase 'the class Human' or 'the

property Human' (see, as an example, the translations of 'Hs' in 4); we

have seen that this involves merely a difference in formulation. Now in

S, we can deduce from ' . . H . .' the existential sentence ' (3f) (. . / . .)'.

For the translation of this sentence into M we have again three forms,

corresponding to the three forms mentioned for the transla tion of

'. .H. .':

(i) 'There is an/ such that ../,.',

(ii) 'There is a class/ such that ../'>

(iii) 'There is a property / such that ../..'.

As 'H' is an expression both for the class Human and for the property

Human, '/' is thus a variable both for classes and for properties. Since we

regarded the class Human as the extension of 'H', we shall now regard it

as one of the value extensions of '/'; and, analogously, we take the

property Human as one of the value intensions of '/'. Let us call the

closed expressions substitutable for a certain variable of any kind the

value expressions of that variable. Then the following holds generally,

for variables of any kind.

10-1. The extension of a value expression of a variable is one of the value

extensions of that variable.

10-2. The intension of a value expression of a variable is one of the value

intensions of that variable.

For variables of the type of sentences, say '/>', '#', etc., the situation is

analogous. Their value extensions are truth- values; their value intensions,

propositions. Let ' . . Hs . .' be a sentence containing'Hs' as a proper sub-

sentence. We may translate ' . . Hs . .' into M in various ways. One pos-

sible translation contains simply the phrase '(that) Scott is human'. Of

the two more explicit translations, one contains the phrase 'the truth-

value that Scott is human', and the other 'the proposition that Scott is

human', in accord with our earlier results concerning the extension and the

intension of 'Hs' (6-3 and 6-4). Now in 5, we may infer from' . . Hs . / the

existential sentence '(3^)(. . p . .)'. Corresponding to the three transla-

tions of ' . . Hs . .', we have three translations of this existential sentence:

(i) 'There is a p such that ../>..',

(ii) 'There is a truth- value p such that . . p . .',

(iii) 'There is a proposition p such that . . p . .'.

46 I. THE METHOD OF EXTENSION AND INTENSION

The treatment of individual variables is not essentially different from

that of the other kinds of variables. But, owing to the unfamiliarity of

individual concepts, our conception here may seem less natural at first

glance. We considered earlier the sentence 'H (i#) (A#w) ' containing a de-

scription. In addition to the simple translation ' the author of Waverley is

human', we had several more explicit translations containing the phrases

'the individual' and ' the individual concept' (at the end of the preceding

section). From the sentence with the description (or from the simpler

sentence 'Hs') we may deduce '(3#) (H#)'. Corresponding to the earlier

translations of the former sentence, we have the following translations of

this existential sentence:

(i) 'There is an x such that x is human'.

(ii) 'There is an individual x such that x belongs to the class Human',

(iii) ' There is an individual concept x such that x is subsumable under

the property Human',

(iv) 'There is an individual x such that x has the property Human'.

Thus the value extensions of individual variables are individuals, their

value intensions are individual concepts. The multiplicity of the formula-

tions and the strangeness of some of them are the same here as in the

preceding section. Our later attempt at a simplification will apply to the

present situation, too.

11. Extensional and Intensional Contexts

An expression occurring within a sentence is said to be interchangeable with

another expression if the truth-value of the sentence remains unchanged when

the first expression is replaced by the second. If, moreover, the intension of the

sentence remains unchanged, the two expressions are said to be L-inter change-

able. We say that a sentence is extensional with respect to an expression oc-

curring in it or that the expression occurs in the sentence within an extensional

context, if the expression is interchangeable at this place with every other ex-

pression equivalent to it. We say that the sentence is intensional with respect to

the expression, or that the expression occurs within an intensional context, if

the context is not extensional and the expression is L-interchangeable at this

place with every other expression L-equivalent to it. (The definitions actually

given in this section are wider than here indicated; they refer not only to

sentences but to designators of any type.) It is found, in accordance with custom-

ary conceptions, that all sentences of the system Si, which contains only the

ordinary connectives and quantifiers but no modal signs, are extensional and

that a sentence in S 3 of the form *N(. . .)', where 'N' is a sign for logical neces-

sity, is intensional.

Suppose that we replace an expression (designator or not) which occurs

within a designator by another expression. It may happen that the ex ten-

11. EXTENSIONAL AND INTENSIONAL CONTEXTS 47

sion of the designator is not thereby changed; in this case we call the two

expressions interchangeable within the designator. If, moreover, the in-

tension of the designator remains unchanged, we say that the two expres-

sions are L-interchangeable within the designator. The subsequent

definitions for these concepts in technical terms (n-ia) refer not to ex-

tension and intension but, instead, to equivalence and L-equivalence. Two

further concepts are defined (n-ib), which apply to the case in which the

conditions mentioned are fulfilled for all sentences. The system S to which

these and the later definitions (11-2) refer may be one of our systems S x ,

S^, S 3 , or a similar system with the same types of designators; it is sup-

posed that S contains descriptive predicates, and hence factual sen-

tences, 35 and also individual descriptions with those predicates. [Thus S

may be PM', but not PM, in 26; it may be ML', but not ML, in 25.]

S may, in distinction to our systems, also contain variables for the non-

individual types of designators.

11-1. Definitions

a. An occurrence of the expression 21, within the expression 21 is (i)

interchangeable, (2) L-interchangeable with 2l/ (in S) =DI 21, is

a designator and is (i) equivalent, (2) L-equivalent to the expression

2l/ constructed out of 31; by replacing the occurrence of 21, in

question by 2l/.

b. 21 y is (i) interchangeable, (2) L-inter changeable with SI/ in

the system S = Df any occurrence of 2ly within any sentence of S is

(i) interchangeable, (2) L-interchangeable with 2I/.

Consider a particular occurrence of a designator 21 y within a designator

2l. The situation may be such that the extension of 21. depends merely

upon the extension of 21,, that is to say, it remains unchanged if 21, is

replaced by any other expression with the same extension. In this case we

shall say that 21,; is extensionol with respect to that occurrence of 2ly

(n-2a). We must here refer to a particular occurrence; for, if 21* contains

35 The fact that a restriction of this kind is necessary was pointed out to me by Alonzo

Church. If S is a system of modal logic which, like Lewis' system of strict implication, con-

tains no descriptive predicates and hence no factual sentences, then any two equivalent sen-

tences are L-equivalent and hence are L-interchangeable even within a modal sentence of the

form *N(. . .)'. Thus the latter sentence would fulfil the condition of extensionality as stated

below in ii- 2b; in fact, however, a modal sentence is, of course, to be regarded as intensional

in the customary sense. To state definitions of 'extensional* and 'intensional' which are ap-

plicable also to systems containing only L-determinate sentences or no closed sentences at all,

it would be necessary to refer not only to closed designators occurring as parts but also to the

values of the designator variables and to the corresponding values of prepositional functions

expressed by matrices (for example, to the values of 'p j and the corresponding values of

'Np'). In order to avoid this complication in our present discussion, we restrict the systems

S as indicated in the text.

48 I. THE METHOD OF EXTENSION AND INTENSION

several occurrences of 81,, it may happen that one occurrence fulfils the

above condition, while another does not. If the condition is fulfilled, we

shall also say sometimes that Sly occurs within 31, at the place in question

in an extensional context.

11-2. Definitions

a. The expression SI* is extensional with respect to a certain occur-

rence of Sly within 8l (in the system S) = Df 81, and Sly are designators;

the occurrence in question of Sly within Sl is interchangeable with

any expression equivalent to Sly (in 5).

b. The expression Sl is extensional (in S) =Df Sl is a designator (in

S) ; Sli is extensional with respect to any occurrence of a designator

within SI* (in 5).

c. The semantical system S is* extensional =Df every sentence in S

is extensional.

If the condition in n-2a or b or c is not fulfilled, we shall use the term

1 nonextensionaV . The term 'intensionaV (11-3) will be used not, as is some-

times done, as synonymous with'nonextensionaP, but in a narrower sense,

namely, in those cases in which the condition of extensionality is not ful-

filled but the analogous condition with respect to intension is fulfilled.

The latter condition means that the intension of the whole remains un-

changed if the subexpression is replaced by one with the same intension;

the technical definition (11-3) does not refer to intension but uses, in-

stead, the concepts of L-equivalence and L-interchangeability.

11-3. Definitions

a. The expression Sl is intensional with respect to a certain occur-

rence of 21,- within 21, (in S) =Df Sl< and Sly are designators; Sli is not

extensional with respect to the occurrence in question of SI/ within

3l; this occurrence of Sly within SI, is L-interchangeable with any

expression L-equivalent to Sly (in S).

b. The expression SI, is intensional (in S) =DI 21, is a designator; 31;

is, with respect to any occurrence of a designator within SI,-, either

extensional or intensional, and is intensional with respect to at least

one occurrence of a designator.

c. The semantical system S is intensional =DI every sentence in S

is either extensional or intensional, and at least one is intensional.

We shall sometimes call a sentential connective or a predicator con-

stant extensional, if every full sentence of it is extensional with respect to

the argument expressions; and we shall use the term 'intensionaP Anal-

ogously.

11. EXTENSIONAL AND INTENSIONAL CONTEXTS 49

Note that the terms 'extension' and 'intension' occur only in the in-

formal explanations and not in the definitions n-i, 11-2, and 11-3 them-

selves. Thus these definitions do not presuppose any problematic entities.

They use, instead, the terms 'equivalent' and 'L-equivalent', which, as

mentioned earlier ( 5), are unproblematic and can be defined in an exact

way.

The terms ' interchangeable', ' L-interchangeable', ' extensional', and ' in-

tensionaP have been defined here in a general way so that the whole ex-

pression 2li may be a designator of any of the types occurring in our sys-

tems. These terms find their most important application, however, in

those cases in which 21 1 is a sentence; and in our further discussions we

shall use them chiefly for cases of this kind.

The concepts just defined will become clearer with some examples. The

whole expression 3l t is a sentence in all these examples. The subexpression,

21 y, is, in the first three examples, a sentence; in the later examples a

designator of another type.

Example I. A sentence '. . V - -' is extensional with respect to either

of its components. And, generally, as is well known, any full sentence of

the ordinary connectives, '~ ','V',' V ^ '> an d ' =', is extensional with

respect to its (immediate) component or components. These connectives

and the connections for which they stand are, indeed, often called exten-

sional; 36 following Russell, the connections are usually called truth-

functions.

Example II. Anticipating later explanations (chap, v), let us use here

the system S 3 , which contains the signs of B! and, in addition, 'N' as a

modal sign for logical necessity in such a way that, if ' . . .'is any L-true

sentence, 'N(. . .)' is true and, moreover, L-true; and if. . .'is any sen-

tence not L-true, then 'N(. . .)' is false and moreover L-false (see 39-3).

Let 'C' be an abbreviation for an F-true sentence (e.g., for 'Hs'); then

' C' is true but not L-true. As is well known (see the example following

t-2) 'C V~ C' is L-true. Hence:

11-4. 'C' and 'C V~C' are equivalent but not L-equivalent.

According to the given explanations for 'N', we have:

11-5. 'N(C V~C)' is true and, moreover, L-true.

On the other hand, since 'C' is not L-true, 'N(C)' is false. Therefore,

'N(C V~C)' and 'N(C)' are not equivalent. It follows, according to the

definition n-ia, that the occurrence of 'C' within 'N(C)' is not inter-

* 6 The concept of extensionality of connections and connectives and the corresponding

concept of L-extensionality are discussed in more detail hi [II], 13.

50 I. THE METHOD OF EXTENSION AND INTENSION

changeable with 'C V~C. This, together with 11-4 and the definition

n-2a, leads to the following result:

11-6. 'N(C)' is nonextensional with respect to 'C'.

This result is well known; generally, full sentences of modal signs are

nonextensional with respect to their components; in customary terms,

modalities are not truth-functions. 37 The same consideration shows that

the occurrence of 'C V~C within *N(C V~C)' is not interchangeable

with'C'. Thus we obtain (again with 11-4):

11-7. 'N(CV~C)' is nonextensional with respect to the subsentence

<CV~ C'.

Further, let 'D' be any sentence L-equivalent to 'C V~C'. Then 'D'

is likewise L-true; and hence 'N(D)', too. We found that 'N(C V C)'

is L-true (11-5). Since any two L-true sentences hold in the same state-

descriptions (2-2), they are L-equivalent to each other (2-6). Thus

'N(C V~C)' and 'N(D)' are L-equivalent. Therefore, according to the

definition n-ia, the occurrence of 'C V~C within 'N(C V C)' is L-

interchangeable with any sentence which is L-equivalent to 'C V~C'.

This, together with 11-7 and the definition n~3a, yields:

11-8. *N(C V~C)' is intentional with respect to the subsentence

<CV~C.

Example III. The sentence 'Hs' is true in S x ; it remains true if 'IV is

replaced by any equivalent predicator, for instance, by T*B J ; and like-

wise if * s' is replaced by.any equivalent individual expression, for instance,

by the description *(i#)(A#w) 7 (9-2). Therefore:

37 The results ri-6 and 11-7 refute Church's opinion that (on a certain assumption, see

below) "Carnap's definition of 'extensional' fails in that under it every language (every

semantical system) is extensional, even those which contain names of propositions and

modal operators" ([Review C.], p. 304). The definition of 'extensional' here referred to is

[I], Dio-2O and Dio-2i, p. 43; it is essentially the same as n-i and 11-2 in the present

section; however, the restriction to systems with factual sentences was omitted. Church is

right in criticizing this omission (see n. 35). However, if the definition is applied to systems

also containing factual sentences, like the example systems in my earlier book [I] and in the

present book, then the definition seems to me to be adequate; at any rate, the examples here

mentioned show that it is certainly not the case that under this definition (either in the

earlier or in the present formulation) all sentences and all semantical systems fulfill the defining

condition for extensionality. Church qualifies his statement by the following condition: "if

the designatum of a sentence is always a truth-value." [Here the term 'designatum', as

Church's preceding explanations show, is meant in the sense in which I shall use the term

'nominatum' in this book (24); this sense is different from that in which I have used the

term 'designatum' in [I], see below, 37]. However, this qualification does not change the

situation. Any assumption as to what are the designata (nominata) of sentences is irrelevant

to the question of whether the examples stated in it-6, 11-7, and 13-4 are extensional or not

on the basis of my definition, because in this definition the concept of the designatum (nomi-

natum) of a sentence is not used.

12. THE PRINCIPLES OF INTERCHANGEABILITY 51

11-9. 'Hs' is extensional with respect to both 'IF and V.

Example IV. It can easily be shown that every sentence in S, con-

structed out of predicator constants (like those mentioned in rule 1-2),

individual constants (like those mentioned in i-i), connectives of the kind

mentioned in Example I, universal and existential quantifiers, and iota-

and lambda-operators is extensional with respect to any designators con-

tained in it and hence is extensional (n-2b). Si is intended to contain

only sentences constructed in this way. Therefore, according to definition

n-2c:

11-10. Si is an extensional system.

12. The Principles of InterchangeabiUty

Some theorems are stated concerning Lnterchangeability and L-interchange-

ability in extensional and intensional contexts.

The following theorems, which we call principles of interchangeability,

follow from our previous definitions of interchangeability and L-inter-

changeability (n-i), extensionality (11-2), and intensionality (11-3). The

system S, to which the theorems of this section refer, is supposed to be

either one of our systems S,, S 2 , S 3 , or a similar system as specified earlier

(see the explanation preceding n-i).

12-1. First Principle of Interchangeability. Let . . 2Iy . . be a sen-

tence (in the system 5) which is extensional with respect to a certain

occurrence of the designator 2ly, and ..21*.. the corresponding sentence

with an occurrence of 21* instead of that of 2ly; analogously for '. . u . .'

and '. . v . / in c.

a. If 21 y and 21* are equivalent (in 5), then the occurrence in question

of Sly within . . 2ly . . is interchangeable with 21* (in 5).

b. (2ly s 21*) D (. . 2ly . . s . . 21* . .) is true (in 5).

c. Suppose that 5 contains variables for which 2ly and 21* are substitut-

able, say V and V; then'(u)(v)[(u = v) D (. . u . . ss . . v . .)]' is

true (in S).

Statement i2-ia follows immediately from the definition n-2a; and b

and c follow from a by the general definition of equivalence (3-sa). The

forms b and c have the advantage that here the principle is represented

by a sentence in the object language S itself. The form c requires suitable

variables. In the system Si, for instance, form c is applicable only with

individual variables and hence states only the interchangeability of indi-

vidual expressions, while forms a and b apply also to predicators and

sentences in S x .

52 I. THE METHOD OF EXTENSION AND INTENSION

12-2. Second Principle of Interchangeabitity. Let . . Sly . . be a sen-

tence (in 5) which is either extensional or intensional with respect to

a certain occurrence of the designator Sly, and . . 3U . . the corresponding

sentence with 81*.

a. If Hj and 81* are L-equivalent (in 5), then the occurrence in question

of 8[y within . . Sly . . is L-interchangeable and hence interchangeable

with 21* (in 5).

Formulations b and c of the second principle analogous to i2-ib and c are

possible only with the help of a modal sign, hence only with respect to a

nonextensional language system. They will be given later (39-7 b and c).

The following theorems follow from the two principles just stated, with

the help of the definitions of extensional and intensional systems (n-2c

and n-3c):

12-3. Let 5 be an extensional system (for instance, S,, see Example IV in

").

a. Equivalent expressions are interchangeable in S.

b. L-equivalent expressions are L-interchangeable in S.

Examples, a. Equivalence and therefore interchangeability in Si hold

for the following pairs of expressions: (i) 'H' and *FB' (see 3-8); (ii)

' Hs* and ' (F*B) (s) ' ; (iii) ' s' and ' (ix) (Axw) ' (see 9-2) . b. L-equivalence

and therefore L-interchangeability in Si hold for the following pairs of

expressions: (i) 'H' and ( RA' (see 3-11); (ii) <Hs' and <RAs'; (iii) '(ix)

(Hx.Axw)' md'(ix)(RAx*Axw)' (see 9).

12-4. Let S be an intensional system (for instance, S 2 with the modal sign

*N J , see Example II in n and 39).

a. Equivalent expressions are interchangeable in 5, except where they

occur in an intensional context (for example, in the system S 2 :

except in a context of the form 'N(. . .)').

b. L-equivalent expressions are L-interchangeable in 5.

Examples for S a . a. Let'C' be F-true, as in Example II, n. Then 'C'

and 'CV~C are equivalent (see 11-4). The sentence '(CV~C)

N(C V~CV is true (see 11-5). Within this sentence the first occurrence

of ' C V ~ C is interchangeable with i C', while the second is not. b. For

the pairs of L-equivalent expressions in S x mentioned above, L-equiva-

lence in S, and therefore L-interchangeability in S 3 likewise hold.

11 SENTENCES ABOUt BELIEFS S3

13. Sentences about Beliefs

We study sentences of the form 'John believes that . . .'. If here the sub-

sentence '. . .' is replaced by another sentence L-equivalent to it, then it may

be that the whole sentence changes its truth-value. Therefore, the whole belief-

sentence is neither extensional nor intensional with respect to the subsentence

'. . .'. Consequently, an interpretation of belief-sentences as referring either to

sentences or to propositions is not quite satisfactory. For a more adequate

interpretation we need a relation between sentences which is still stronger than

L-equi valence. Such a relation will be defined in the next section.

We found that ' ... V 'is extensional with respect to the subsen-

tence indicated by dots, and that* N(. . .)' is intensional. Can there be a con-

text which is neither extensional nor intensional? This would be the case if

(but not only if) the replacement of a subsentence by an L-equivalent one

changed the truth-value and hence also the intension of the whole sen-

tence. In our systems this cannot occur; every sentence in S x (and like-

wise in S 3 , to be explained later) is extensional, and every sentence in S 2

is either extensional or intensional. However, it is the case for a very im-

portant kind of sentence with psychological terms, like 'I believe that it

will rain'. Although sentences of this kind seem to be quite clear and un-

problematic at first glance and are, indeed, used and understood in every-

day life without any difficulty, they have proved very puzzling to logicians

who have tried to analyze them. Let us see whether we can throw some

light upon them with the help of our semantical concepts.

In order to formulate examples, we take here, as our object language 5,

not a symbolic system but a part of the English language. We assume that

5 is similar in structure to Sj except for containing the predicator ( . , be-

lieves that - -' and some mathematical terms. We do not specify here the

rules of S; we assume that the semantical rules of 5 are such that the

predicator mentioned has its ordinary meaning; and, further, that our

semantical concepts, especially 'true', 'L-true', ' equivalent', and *L-

equivalent', are defined for S in accord with our earlier conventions. Now

we consider the following two belief -sentences ; 'D' and'D" are here writ-

ten as abbreviations for two sentences in S to be explained presently:

(i) 'John believes that D'.

(ii) 'John believes that D".

Suppose we examine John with the help of a comprehensive list of sen-

tences which are L-true in 5; among them, for instance, are translations

into English of theorems in the system of [P.M.] and of even more com-

plicated mathematical theorems which can be proved in that system and

therefore are L-true on the basis of the accepted interpretation. We ask

54 I. THE METHOD OF EXTENSION AND INTENSION

John, for every sentence or for its negation, whether he believes what it

says or not. Since we know him to be truthful, we take his affirmative or

negative answer as evidence for his belief or nonbelief . Among the simple

L-true sentences, there will certainly be some for which John professes

belief. We take as 'D' any one of them, say ' Scott is either human or not

human'. Thus the sentence (i) is true. On the other hand, since John is a

creature with limited abilities, we shall find some L-true sentences in S

for which John cannot profess belief. This does not necessarily mean that

he commits the error of believing their negations; it may be that he, can-

not give an answer either way. We take as D' some sentence of this kind;

that is to say, D' is L-true but (ii) is false. Thus the two belief-sentences

(i) and (ii) have different truth- values; they are neither equivalent nor

Lrequivalent. Therefore, the definitions of interchangeability and L-inter-

changeability (n-ia) lead to the following two results:

13-1. The occurrence of 'D' within (i) is not interchangeable with 'D'\

13-2. The occurrence of 'D' within (i) is not L-interchangeable with 'D".

'D' and 'D" are both L-true; therefore:

13-3. 'D' and 'D" are equivalent and L-equivalent.

Examining the first belief-sentence (i) with respect to its subsentence

'D', we see from 13-1 and 13-3 that the condition of extensionality (n-2a)

is not fulfilled; and we see from 13-2 and 13-3 that the condition of inten-

sionality (n-3a) is not fulfilled either:

13-4. The belief-sentence (i) is neither extensional nor intensional with re-

spect to its subsentence 'D'.

Although 'D' and 'D" have the same intension, namely, the L-true or

necessary proposition, and hence the same extension, namely, the truth-

value truth, their interchange transforms the first belief-sentence (i) into

the second (ii), which does not have the same extension, let alone the same

intension, as the first.

The same result as 13-4 holds also if any other sentence is taken instead

of ' D', in particular, any factual sentence.

Let us now try to answer the much-discussed question as to how a

sentence reporting a belief is to be analyzed and, in particular, whether

such a sentence is about a proposition or a sentence or something else. It

seems to me that we may say, in a certain sense, that (i) is about the

sentence 'D', but also, in a certain other sense, that (i) is about the propo-

sition that D. In interpreting (i) with respect to the sentence ' D', it would,

of course, not do to transform it into 'John is disposed to an affirmative

13. SENTENCES ABOUT BELIEFS 55

response to the sentence 'D' ', because this might be false, although (i)

was assumed to be true; it might, for instance, be that John does not un-

derstand English but expresses his belief in another language. Therefore,

we may try the following more cautious formulation:

(iii) 'John is disposed to an affirmative response to some sentence in

some language, which is L-equivalent to 'D' '.

Analogously, in interpreting (i) with respect to the proposition that D, the

formulation 'John is disposed to an affirmative response to any sentence

expressing the proposition that D' would be wrong because it implies that

John understands all languages. Even if the statement is restricted to

sentences of the language or languages which John understands, it would

still be wrong, because 'D 7 ', for example, or any translation of it, likewise

expresses the proposition that D, but John does not give an affirmative

response to it. Thus we see that here again we have to use a more cautious

formulation similar to (iii) :

(iv) 'John is disposed to an affirmative response to some sentence in

some language which expresses the proposition that D'.

However, it seems to me that even the formulations (iii) and (iv), which

are L-equivalent, should not be regarded as anything more than a first

approximation to a correct interpretation of the belief-sentence (i). It is

true that each of them follows from (i), at least if we take 'belief here in

the sense of 'expressible belief', leaving aside the problem of belief in a

wider sense, interesting though it may be. However, (i) does not follow

from either of them. This is easily seen if we replace ' D ' by ' D". Then (iii)

remains true because of 13-3; on the other hand, (i) becomes (ii), which is

false. It is clear that we must interpret (i) as saying as much as (iii) but

something more; and this additional content seems difficult to formulate.

If (i) is correctly interpreted in accord with its customary meaning, then

it follows from (i) that there is a sentence to which John would respond

affirmatively and which is not only L-equivalent to 'D', as (iii) says, but

has a still stronger relation to 'D' in other words, a sentence which has

something more in common with 'D' than the intension. The two sen-

tences must, so to speak, be understood in the same way; they must not

only be L-equivalent in the whole but consist of L-equivalent parts, and

both must be built up out of these parts in the same way. If this is the

case, we shall say that the two sentences have the same intensional struc-

ture. This concept will be explicated in the next section and applied in the

analysis of belief sentences in 15.

S 6 I. THE METHOD OF EXTENSION AND INTENSION

14. Intensional Structure

If two sentences are built in the same way out of designators (or designator

matrices) such that any two corresponding designators are L-equivalent, then

we say that the two sentences are intentionally isomorphic or that they have the

same intensional structure. The concept of L-equivalence can also be used in a

wider sense for designators in different language systems; and the concept of

intensional isomorphism can then be similarly extended.

We shall discuss here what we call the analysis of the intensional struc-

tures of designators, especially sentences. This is meant as a semantical

analysis, made on the basis of the semantical rules and aimed at showing,

say for a given sentence, in which way it is built up out of designators

and what are the intensions of these designators. If two sentences are built

in the same way out of corresponding designators with the same inten-

sions, then we shall say that they have the same intensional structure.

We might perhaps also use for this relation the term ' synonymous', be-

cause it is used in a similar sense by other authors (e.g., Langford, Quine,

and Lewis), as we shall see in the next section. We shall now try to expli-

cate this concept.

Let us consider, as an example, the expressions ' 2 + 5' and 'II sum V 7

in a language 6* containing numerical expressions and arithmetical func-

tors. Let us suppose that we see from the semantical rules of S that both

' + ' and * sum' are functors for the function Sum and hence are L-equiva-

lent; and, further, that the numerical signs occurring have their ordinary

meanings and hence ' 2' and ' II' are L-equivalent to one another, and like-

wise ' 5' and ' V. Then we shall say that the two expressions are intension-

ally isomorphic or that they have the same intensional structure, because

they not only are L-equivalent as a whole, both being L-equivalent to

'7', but consist of three parts in such a way that corresponding parts are

L-equivalent to one another and hence have the same intension. Now it

seems advisable to apply the concept of intensional isomorphism in a

somewhat wider sense so that it also holds between expressions like

'2 + 5' and < sum(II,V)', because the use in the second expression of a

functor preceding the two argument signs instead of one standing between

them or of parentheses and a comma may be regarded as an inessential

syntactical device. Analogously, if ' > ' and ' Gr' are L-equivalent, and like-

wise '3' and 'III', then we regard '5 > 3' as intensionally isomorphic to

'Gr(V,III)'. Here again we regard the two predicators ' >' and 'Gr' as

corresponding to each other, irrespective of their places in the sentences;

further, we correlate the first argument expression of ' > ' with the first of

'Gr', and the second with the second. Further/ 2 + 5 > 3' is isomorphic

14. INTENSIONAL STRUCTURE 57

to < Gr[sum(II,V),III] ) , because the corresponding expressions '2+5'

and < sum(II,V) > are not only L-equivalent but isomorphic. On the other

hand, ' 7 > 3' and 'Gr[sum(II,V),III]' are not isomorphic; it is true that

here again the two predicators ' > ' and ' Gr' are L-equivalent and that cor-

responding argument expressions of them are likewise L-equivalent, but

the corresponding expressions ' f and ' sum(II,V) are not isomorphic. We

require for isomorphism of two expressions that the analysis of both down

to the smallest subdesignators lead to analogous results.

We have said earlier ( i) that it seems convenient to take as designa-

tors in a system S at least all those expressions in S, but not necessarily

only those, for which there are corresponding variables in the metalan-

guage M. For the present purpose, the comparison of intensional structures,

it seems advisable to go as far as possible and take as designators all those

expressions which serve as sentences, predicators, functors, or individual

expressions of any type, irrespective of the question of whether or not M

contains corresponding variables. Thus, for example, we certainly want

to regard as isomorphic *p V q* and ' Apq', where 'A' is the sign of disjunc-

tion (or alternation) as used by the Polish logicians in their parenthesis-

free notation, even if M , as is usual, does not contain variables of the type

of connectives. We shall then regard ' V* and 'A' as L-equivalent connec-

tives because any two full sentences of them with the same argument ex-

pressions are L-equivalent.

Frequently, we want to compare the intensional structures of two ex-

pressions which belong to different language systems. This is easily pos-

sible if the concept of L-equivalence is defined for the expressions of both

languages in such a way that the following requirement is fulfilled, in

analogy to our earlier conventions: an expression in S is L-equivalent to an

expression in S' if and only if the semantical rules of S and S' together,

without the use of any knowledge about (extra-linguistic) facts, suffice

to show that the two expressions have the same extension. Thus, L-

equivalence holds, for example, between V in S and V in S" if we see

from the rules of designation for these two individual constants that both

stand for the same individual ; likewise between ' P' and ' P", if we see from

the rules alone that these predicators apply to the same individuals; be-

tween two functors ' + ' and ' sum', if we see from the rules alone that they

assign to the same arguments the same values in other words, if their

full expressions with L-equivalent argument expressions (e.g., '2 + 5'

and ' sum(II,V)0 are L-equivalent; for two sentences, if we see from the

rules alone that they have the same truth- value (e.g., 'Rom ist gross' in

58 I. THE METHOD OF EXTENSION AND INTENSION

German, and 'Rome is large* in English). Thus, even if the sentences

( 2 + 5 > 3' and'Gr [sum (II, V), III]' belong to two different systems, we

find that they are intensionally isomorphic by establishing the L-equiva-

lence of corresponding signs.

If variables occur, the analysis becomes somewhat more complicated,

but the concept of isomorphism can still be defined. We shall not give here

exact definitions but merely indicate, with the help of some simple ex-

amples, the method to be applied in the definitions of L-equivalence and

isomorphism of matrices. Let V be a variable in 5 which can occur in a

universal quantifier l (%)' and also in an abstraction operator '(X#)'> and

'u' be a variable in S' which can occur in a universal quantifier '!!' and

also in an abstraction operator 'fi\ If V and V have the same range of

values (or, more exactly, of value intensions, 10), for example, if both

are natural number variables (have natural number concepts as value in-

tensions), we shall say that '%' and 'u' are L-equivalent, and also that

* (x)' and 'Ilu' are L-equivalent, and that ' (X#)' and '$' are L-equivalent.

If two matrices (sentential or other) of degree n are given, one in S and

the other in S', we say that they are L-equivalent with respect to a certain

correlation between the variables, if corresponding abstraction expressions

are L-equivalent predicators. Thus, for example/ x > y y in S and' Gr(w,fl)'

in S' are L-equivalent matrices (with respect to the correlation of V

with'w' and 'y' with V) because *(X#;y) [x > y}' and ' tiv[Gr(UjV)]' are

L-equivalent predicators. Intensional isomorphism of (sentential or other)

matrices can then be defined in analogy to that of closed designators, so

that it holds if the two matrices are built up in the same way out of cor-

responding expressions which are either L-equivalent designators or L-

equivalent matrices. Thus, for example, the matrices 'x + 5 > y' and

'Gr[sum (#,),?>]' are not only L-equivalent but also intensionally isomor-

phic; and so are the (L-false) sentences ' (x) (y} [x + 5 > y]' and 'IMIz;

[Gr[sum(w,V) ,*]]'.

These considerations suggest the following definition, which is recur-

sive with respect to the construction of compound designator matrices

out of simpler ones. It is formulated in general terms with respect to

designator matrices; these include closed designators and variables as

special cases. The definition presupposes an extended use of the term 'L-

equivalent' with respect to variables, matrices, and operators, which has

been indicated in the previous examples but not formally defined. The

present definition makes no claim to exactness; an exact definition would

have to refer to one or two semantical systems whose rules are stated

completely.

15. APPLICATIONS OF INTENSIONAL STRUCTURE 59

14-1. Definition of intensional isomorphism*

a. Let two designator matrices be given, either in the same or in two

different semantical systems, such that neither of them contains an-

other designator matrix as proper part. They are intensionally iso-

morphic = of they are L-equivalent.

b. Let two compound designator matrices be given, each of them con-

sisting of one main submatrix (of the type of a predicator, functor,

or connective) and n argument expressions (and possibly auxiliary

signs like parentheses, commas, etc.). The two matrices are inten-

sionally isomorphic =Df (i) the two main submatrices are intension-

ally isomorphic, and (2) for any m from i to n, the mth argument

expression within the first matrix is intensionally isomorphic to the

mth in the second matrix (' the mth 9 refers to the order in which

the argument expressions occur in the matrix).

c. Let two compound designator matrices be given, each of them con-

sisting of an operator (universal or existential quantifier, abstrac-

tion operator, or description operator) and its scope, which is a

designator matrix. The two matrices are intensionally isomor-

phic = Df (i) the two scopes are intensionally isomorphic with re-

spect to a certain correlation of the variables occurring in them, (2)

the two operators are L-equivalent and contain correlated variables.

In accord with our previous discussion of the explicandum, rule b in

this definition takes into consideration the order in which argument ex-

pressions occur but disregards the place of the main subdesignator. For

the intensional structure, in contrast to the merely syntactical structure,

only the order of application is essential, not the order and manner of

spelling.

15. Applications of the Concept of Intensional Structure

The concept of intensional structure is compared with the concepts of

synonymity discussed by Quine and Lewis. The concept is then used for giving

an interpretation of belief sentences that seems more adequate than the inter-

pretations discussed earlier ( 13). Further, the same concept helps in solving

the so-called paradox of analysis.

It has often been noticed by logicians that for the explication of certain

customary concepts a stronger meaning relation than identity of intension

seems to be required. But usually this stronger relation is not defined. It

seems that in many of these cases the relation of intensional isomorphism

could be used. For example, if we ask for an exact translation of a given

statement, say the exact translation of a scientific hypothesis or of the

60 I. THE METHOD OF EXTENSION AND INTENSION

testimony of a witness in court from French into English, we should usual-

ly require much more than agreement in the intensions of the sentences,

that is, L-equivalence of the sentences. Even if we restrict our attention to

designative (cognitive) meaning leaving aside other meaning compo-

nents like the emotive and the motivative, although they are often very

important even for the translation of theoretical texts L-equivalence of

sentences is not sufficient; it will be required that at least some of the com-

ponent designators be L-equivalent, in other words, that the intensional

structures be alike or at least similar.

Quine explains, without giving a definition, a concept of synonymity

which is different from and presumably stronger than L-equivalence. He

says: "The notion of synonymity figures implicitly also whenever we use

the method of indirect quotations. In indirect quotation we do not insist

on a literal repetition of the words of the person quoted, but we insist on

a synonymous sentence; we require reproduction of the meaning. Such

synonymity differs even from logical equivalence; and exactly what it is

remains unspecified." 38 We might perhaps think of an explicatum of this

concept of synonymity similar to our concept of intensional isomorphism.

Quine himself seems to expect that the explication will be found not in

semantics but in what we would call pragmatics, because he says that the

concept of synonymity " calls for a definition or a criterion in psycho-

logical and linguistic terms."

C. I. Lewis 39 gives a definition for the concept of synonymity which

shows a striking similarity to our concept of intensional isomorphism, al-

though the two concepts have been developed independently. Since it is

interesting to see the points of agreement and of difference, I will quote his

explanations at length. "Not every pair of expressions having the same

intension would be called synonymous; and there is good reason for this

fact. Two expressions are commonly said to be synonymous (or in the case

of propositions, equipollent) if they have the same intension, and that in-

tension is neither zero nor universal. But to say that two expressions with

the same intension have the same meaning, without qualification, would

have the anomalous consequence that any two analytic propositions would

then be equipollent, and any two self-contradictory propositions would be

equipollent." In order to overcome this difficulty, Lewis introduces a new

concept : u Two expressions are equivalent in analytic meaning, (i) if at least

one is elementary [i.e., not complex] and they have the same intension, or

(2) if, both being complex, they can be so analyzed into constituents that

** [Notes], p. 120.

* [Meaning], pp. 245 f . Other concepts used by Lewis will be discussed in the next section.

IS. APPLICATIONS OF INTENSIONAL STRUCTURE 61

(a) for every constituent distinguished in either, there is a corresponding

constituent in the other which has the same intension, (b) no constituent

distinguished in either has zero intension or universal intension, and (c)

the order of corresponding constituents is the same in both, or can be

made the same without alteration of the intension of either whole expres-

sion." As examples, Lewis states that " round excision" and "circular

hole" are equivalent in analytic meaning, while "equilateral triangle" and

"equiangular triangle" are not, although they have the same intension.

He continues: "We shall be in conformity with good usage if we say that

two expressions are synonymous or equipollent, (i) if they have the same

intension and that intension is neither zero nor universal, or (2) if, their

intension being either zero or universal, they are equivalent in analytic

meaning."

Thus Lewis' concept of synonymity is very similar to our concept of

intensional isomorphism except for one point: He applies this stronger

relation only to the two extreme cases of intension, for example, in the

field of sentences, only to L-determinate and not to factual sentences.

This discrimination seems to me somewhat arbitrary and inadvisable. Let

us consider the following examples (in a language which, in distinction to

S x , also contains expressions for finite cardinal numbers and for relations

and properties of them) :

(i) 'two is an even prime number';

(ii) 'two is between one and three';

(iii) 'the number of books on this table is an even prime number';

(iv) 'the number of books on this table is between one and three'.

The sentences (i) and (ii) have the same intension but are not equivalent

in analytic meaning (intensionally isomorphic). The same holds for (iii)

and (iv). Now, according to Lewis' definition, (i) and (ii) are not synony-

mous because they are L-true, analytic; while (iii) and (iv) are synony-

mous because they are factual, synthetic. It seems to me that it would be

more natural to regard (iii) and (iv) also as nonsynonymous, since the dif-

ference between them is essentially the same as that between (i) and (ii).

The logical operation which leads from (i) to (ii) is the same as that which

leads from (iii) to (iv) ; it is the transformation of ' n is an even prime num-

ber' into 'n is (a cardinal number) between one and three'.

Now let us go back to the problem of the analysis of belief-sentences,

and let us see how the concept of intensional structure can be utilized

there. It seems that the sentence 'John believes that D' in S can be in-

terpreted by the following semantical sentence:

62 I. THE METHOD OF EXTENSION AND INTENSION

16-1. ' There is a sentence @ in a semantical system 5' such that (a) <3*

is intensionally isomorphic to 'D' and (b) John is disposed to an affirma-

tive response to <.'

This interpretation may not yet be final, but it represents a better ap-

proximation than the interpretations discussed earlier (in 13). As an

example, suppose that John understands only German and that he re-

sponds affirmatively to the German sentence ' Die Anzahl der Einwohner

von Chicago ist grosser als 3,000,000' but neither to the sentence 'Die

Anzahl der Einwohner von Chicago ist grosser als 2 6 X 3 X 5 6> nor to

any intensionally isomorphic sentence, because he is not quick enough to

realize that the second sentence is L-equivalent to the first. Then our

interpretation of belief-sentences, as formulated in 15-1, allows us to assert

the sentence ' John believes that the number of inhabitants of Chicago is

greater than three million' and to deny the sentence 'John believes that

the number of inhabitants of Chicago is greater than 2 6 X 3 X 5 6 '. We

can do so without contradiction because the two German sentences, and

likewise their English translations just used, have different intensional

structures. [By the way, this example shows another disadvantage of

Lewis' definition of equivalence in analytic meaning. According to part

(i) of his definition, the two German sentences are equivalent in analytic

meaning if we take '3,000,000' as one sign.] On the other hand, the in-

terpretation of belief-sentences in terms of propositions as objects of be-

liefs (like (iv) in 13) would not be adequate in this case, since the two

German sentences and the two English sentences all express the same

proposition.

An analogous interpretation holds for other sentences containing psy-

chological terms about knowledge, doubt, hope, fear, astonishment, etc.,

with ' that'-clauses, hence generally about what Russell calls prepositional

attitudes and Ducasse epistemic attitudes. The problem of the logical

analysis of sentences of this kind has been much discussed, 40 but a satis-

factory solution has not been found so far. The analysis here proposed is

not yet a complete solution, but it may perhaps be regarded as a first step.

What remains to be done is, first, a refinement of the analysis in terms of

linguistic reactions here given and, further, an analysis in terms of dispo-

sitions to nonlinguistic behavior.

* Russell, [Inquiry], gives a detailed discussion of the problem in a wider sense, including

beliefs not expressed in language; he investigates the problem under both an epistemologicaJ

and a logical aspect (in our terminology, both a pragmatical and a semantical aspect), not

always distinguishing the two clearly. For C. J. Ducasse's conception see his paper "Proposi-

tions, Opinions, Sentences, and Facts," Journal of Philosophy, XXXVII (1940), 701-11,

15. APPLICATIONS OF INTENSIONAL STRUCTURE 63

The concept of intensional structure may also help in clarifying a

puzzling situation that has been called "the paradox of analysis". It was

recently stated by G. E. Moore, 41 and then discussed by C. H. Langford, 42

Max Black, 43 and Morton White. 44 Langford 45 states the paradox as fol-

lows: "If the verbal expression representing the analysandum has the

same meaning as the verbal expression representing the analysans, the

analysis states a bare identity and is trivial; but if the two verbal expres-

sions do not have the same meaning, the analysis is incorrect." Consider

the following two sentences:

'The concept Brother is identical with the concept Male Sibling.'

'The concept Brother is identical with the concept Brother/

The first is a sentence conveying fruitful information, although of a

logical, not a factual, nature; it states the result of an analysis of the

analysandum, the concept Brother. The second sentence, on the other

hand, is quite trivial. Now Moore had been puzzled by the following fact:

If the first sentence is true, then the second seems to make the same

statement as the first (presumably because, if two concepts are identical,

then a reference to the one means the same as a reference to the other,

and 'hence the one expression can be replaced by the other); "but it is

obvious that these two statements are not the same", he says. Black tries

to show that the two sentences do not express the same proposition; he

supports this assertion by pointing to the fact that the first sentence, or

rather a paraphrasing he gives for it ('the concept Brother is the conjunct

of the concept Male and the concept Sibling') refers to a certain non-

identical relation (the triadic relation Conjunct), while the second is a

mere identity. White replies that this is not a sufficient reason for the as-

sertion. None of the four authors states his criterion for the identity of

"meaning", "statement", or " proposition" ; this seems the chief cause for

the inconclusiveness of the whole discussion. If we take, as in the terminol-

ogy used in this book, L-equivalence as the condition for the identity of

propositions, then White is certainly right; since the two sentences are

L-true and hence L-equivalent to each other, they express the same

proposition in our sense. On the other hand, Black feels correctly, like

Moore and Langford, that there is an important difference in meaning

between the two sentences, because of a difference in meaning between

4* The Philosophy of G. E. Moore, ed. P. Schilpp (1942), pp. 660-67.

* "The Notion of Analysis in Moore's Philosophy", ibid.> pp. 321-42.

Mind, LIII (1944), 263-67 and LIV (1945), 372 f.

44 Mind. LIV (1945), 71 f. and 357-61. 4S Op. Ut. y p. 323.

6 4 I. THE METHOD OF EXTENSION AND INTENSION

the two expressions for the analysandum (' the concept Brother') and the

analysans ('the concept Male Sibling')- The paradox can be solved if we

can state exactly what this difference in meaning is and how it is com-

patible with the identity of meaning in another sense. The solution is

quite simple in terms of our concepts: The difference between the two

expressions, and, consequently, between the two sentences is a difference

in intensional structure, which exists in spite of the identity of intension.

Langford saw the point at which the difference lies; he says 46 that the

analysans is more articulate than the analysandum, it is a grammatical

function of more than one idea; the two expressions are not synonymous

but "cognitively equivalent in some appropriate sense". It seems to me

that this cognitive equivalence is explicated by our concept of L-equiva-

lence and that the synonymity, which does not hold for these expressions,

is explicated by intensional isomorphism.

16. Lewis' Method of Meaning Analysis

Lewis uses, in addition to the concepts of extension and intension which are

similar to ours, the concept of comprehension which presupposes the admission

of nonactual, possible things. It seems inadvisable to use this conception be-

cause it requires a new, more complicated language form. The distinction which

Lewis wants to make can better be made with respect to intensions than with re-

spect to things.

I wish to discuss briefly some concepts which have recently been pro-

posed by C. I. Lewis 47 as tools for a semantical meaning analysis. There

is a striking similarity between these concepts and our concepts of exten-

sion and intension. This similarity is due to the common aim to make some

traditional concepts, especially extension and intension, denotation and

connotation, more general in their application and, at the same time,

more clear and precise.

Lewis explains his chief semantical concepts in the following way: " All

terms have meaning in the sense or mode of denotation or extension;

and all have meaning in the mode of connotation or intension. The denota-

tion of a term is the class of all actual or existent things to which that

term correctly applies. . . . The comprehension of a term is the classifica-

* 6 0p. cit.,p. 326.

47 In [Meaning], This paper is part of a "Symposium on Meaning and Truth", published

in four parts in Philosophy and Phenomenological Research, Vols. IV (1943-44) and V (1944-

45). This symposium also contains a number of other interesting contributions to the de-

velopment and clarification of semantical concepts. I have elsewhere referred to Tarski's

paper [Truth]; I am in close agreement with his conception of the nature of semantics, but

he does not discuss the central problems of this book. Concerning these problems, I wish

especially to call attention to the papers by C. J. Ducasse (IV, 317-40; V, 320-32) and Charles

A. Baylis (V, 80-93).

16. LEWIS' METHOD OF MEANING ANALYSIS 65

tion of all consistently thinkable things to which the term would correctly

apply. . . . For example, the comprehension of "square" includes all

imaginable as well as all actual squares, but does not include round

squares. . . . The connotation or intension of a term is delimited by any

correct definition of it."

It seems that Lewis' concepts of extension and intension correspond

closely to our concepts. This is clearly the case for predicators, but per-

haps also for sentences and individual expressions. There remains the

problem of the necessity and usefulness of Lewis' third concept, that of

comprehension. It seems that Lewis follows Meinong 48 in dividing (i) all

things (in the widest sense) into impossible or inconceivable things (e.g.,

round squares) and possible things; and (2) the possible things into actual

things (e.g., Plato) and nonactual possible things (e.g., Apollo, unicorns).

[Lewis clearly makes the second division. Whether he also makes the first

and hence countenances, like Meinong, impossible things is not quite so

clear but seems indicated by the formulation that the comprehension

"does not include round squares". According to the ordinary conception,

in distinction to Meinong's, there are no round squares at all, not even in

some particular kind of objects; hence it would be redundant to say that

the comprehension "does not include round squares".] Meinong's concep-

tion has been critically discussed by Russell 49 and then rejected. Russell's

chief reason for the rejection is that the impossible objects violate the

principle of contradiction; for example, a round square is both round and

nonround, because square. Russell is certainly right in the following re-

spect: Within the logical framework of our ordinary language, we cannot

consistently apply the conception of impossible things or even that of

possible nonactual things. And, as far as I am aware, neither Meinong nor

Lewis nor any other philosopher has constructed or even outlined a

language of a new structure which would accommodate those entities.

That such a language must be different from the ordinary one is shown by

the following example: In the ordinary language we say: 'There are no

white ravens and no round squares'. In the new language we would have to

say, instead: ' There are white ravens; however, they are not actual, but

only possible. And there are round squares; however, they are neither

actual nor possible, but impossible.' I have no doubt that a resourceful

logician could easily construct a consistent language system of this kind,

if we wanted it; he would have to lay down rules for the quantifiers deviat-

ing from the ordinary rules in a way suggested by the exairfples. The

< 8 A. von Meinong, Untersuehungen zur Gegenstandstheorie und Psychologic (1904).

[Denoting], pp. 482 f .

66 I. THE METHOD OF EXTENSION AND INTENSION

decisive question is not that of the technical possibility of such a language

but rather that of its usefulness. Only if it can be shown to have great ad-

vantages in comparison to the ordinary language structure would it be

worth considering in spite of its fundamental deviation and increased

complexity.

I do not see sufficient reasons for this change. The distinctions which

Meinong and Lewis have in mind are important, but they can be taken

care of in a different way. Instead of dividing objects into (i) actual, (ii)

nonactual but possible, and (iii) impossible, we make analogous distinc-

tions, first, between three corresponding kinds of expressions and then be-

tween three corresponding kinds of intensions. Let us show this, first, for

predicators. Instead of speaking about three kinds of objects like this:

(i) '(some) horses are actual objects',

(ii) 'unicorns are nonactual but possible objects',

(iii) 'round squares are impossible objects',

we speak, rather, about three kinds of predicators:

(i) 'the predicator 'horse' is not empty',

(ii) ' the predicator ' unicorn' is F-empty, i.e., empty but not L-empty ',

(iii) 'the predicator ' round square' is L-empty '.

Then we apply the same terms to the corresponding intensions (this is a

transference of terms from a semantical to a nonsemantical use, analogous

to the transference of the terms 'equivalent' and 'L-equivalent', 5) :

(i) 'the property Horse is not empty',

(ii) 'the property Unicorn is F-empty, i.e., empty but not L-empty',

(iii) 'the property Round Square is L-empty'.

An analogous distinction can be made for individual expressions, for

instance, descriptions. (We apply here, not the special interpretation of

descriptions which we adopted in 8 because of its technical advantages,

but the customary interpretation, according to which a description has a

descriptum only if the uniqueness condition is fulfilled.) Then, instead of

using the following formulations referring to objects:

(i) 'Alexander's horse (i.e., the one horse which Alexander had at such

and such a time) is an actual object',

(ii) 'Alexander's unicorn is a nonactual but possible object',

(iii) 'Alexander's round square is an impossible object',

we use, rather, the following ones concerning individual expressions

(Lewis' singular terms) :

16, LEWIS' METHOD OF MEANING ANALYSIS 67

(i) 'the description ' Alexander's horse' is not empty',

(ii) 'the description 'Alexander's unicorn' is F-empty' (in Lewis'

terminology, it has zero denotation, but not zero comprehension) ;

(iii) 'the description 'Alexander's round square' is L-empty' (it has

zero comprehension).

And then we make analogous statements concerning the corresponding

individual concepts (in Lewis' terminology, connotations of singular

terms) :

(i) 'the individual concept Alexander's Horse is not empty',

(ii) 'the individual concept Alexander's Unicorn is F-empty',

(iii) 'the individual concept Alexander's Round Square is L-empty'.

Thus our method does not neglect the distinctions pointed out by Mei-

nong and Lewis. However, it applies the distinction to intensions, while

these philosophers apply it to objects and thereby violate the rule of ordi-

nary language which takes the addition of 'actual' to a general noun as

redundant. For example, the ordinary language takes phrases like 'actual

horses', 'real horses', 'existing horses', etc. (where 'actual', etc., does not

mean 'occurring at the present time' but 'occurring at some time, past,

present, or future'), as meaning the same as 'horses', differing from this

only in emphasis; and, likewise, ' actual unicorns' is taken as meaning the

same as 'unicorns', and hence it is said: 'there are no unicorns (at any

space- time point)'.

If we thus reject such distinctions between kinds of objects, then Lewis'

concept of comprehension can no longer be defined. Do we hereby sacrifice

a useful tool of semantical meaning analysis? I do not think so. Lewis

emphasizes rightly the difference between comprehension and extension.

But there seems not to be much difference between the purposes of the

concepts of comprehension and intension. If we accept Lewis' language

form, then these concepts are both legitimate and, of course, not identical.

But whatever is said in terms of comprehension can immediately be trans-

lated into terms of intension, because comprehension and intension deter-

mine each other logically. If you tell me the comprehension of a Chinese

word, then I know immediately what is its intension, and vice versa;

therefore there is no advantage in having both concepts. On the other

hand, if you tell me the intension of a Chinese word, I do not know its

extension (unless it is L-de terminate) ; and if you tell me only its exten-

sion, I cannot infer from this its intension. Therefore, it is useful to have

both concepts, that of intension and that of extension.

We also arrive at the same result, the rejection of nonactual, possible

68 I. THE METHOD OF EXTENSION AND INTENSION

objects and of comprehension by an approach from another angle, that of

modal logic. We shall find later ( 42 f .) that the logical modalities must

be applied to intensions, not to extensions. Thus we may speak of an im-

possible (or L-false) proposition but not of an impossible truth- value; of an

impossible (or L-empty) property but not of an impossible (or L-empty)

class. Analogously, we may speak of an impossible (or L-empty) indi-

vidual concept but not of an impossible individual (object, thing), because

individuals (objects, things) are extensions, not intensions; in other

words, individuals are involved in questions of application, not in ques-

tions of meaning in the strict sense. (We take here, of course, the ordinary

conception of extensions, not that to be discussed in 23, according to

which extensions are construed as a special kind of intension.)

To sum up, I do not think that the concepts of possible and impossible

objects and of comprehension can be accused of violating logic or of lead-

ing necessarily to contradictions. However, it seems doubtful whether

these concepts are sufficiently useful to compensate for their disadvantage

the necessity of using an uncustomary and more complex language

structure.

CHAPTER II

L-DETERMINACY

We have seen ( 2) that a sentence is L-determinate if its truth-value, which

is its extension, is determined by the semantical rules. In this chapter we apply

the concept of L-determinacy also to other designators. The definitions are con-

structed so that an analogous result holds: A designator is L-determinate if the

semantical rules, independently of facts, suffice for determining its extension

(17). For the application of this concept we presuppose that the individuals

are positions in an ordered domain. An individual expression is L-determinate

if the semantical rules suffice for determining the location of the position to which

it refers ( 18, 19). A predicator is L-determinate if the semantical rules suf-

fice for determining for every position whether the predicator applies to it or

not (20). The distinction between logical and descriptive (nonlogical) signs

is discussed, and its connection with the distinction between L-determinate and

L-inde terminate designators is examined ( 21). The intension of an L-determi-

nate designator is also called L-determinate (22). There is a one-one correla-

tion between extensions and L-determinate intensions; therefore, it would be

possible, though not customary, to define extensions as L-determinate inten-

sions ( 23).

17. L-Determinate Designators

In general, factual knowledge is needed for establishing the truth-value of a

given sentence. However, if the sentence is L-determinate ( 2), the semantical

rules suffice for establishing its truth-value or, in other words, its extension.

The concept of L-determinacy will now be extended to designators of other

kinds. We stipulate that the definitions of this concept for the other kinds be

such that a designator is L-determinate if and only if the semantical rules suffice

for determining its extension. Definitions fulfilling this requirement will be con-

structed in later sections of this chapter.

We found earlier that the intension of the sentence 'Hs' in the system S x

is the proposition that Scott is human and that its extension is the truth-

value truth. Now let us consider the question of what knowledge we need

in this and other cases in order to determine the intension and the exten-

sion of a given sentence. It is clear that, for the determination of the in-

tension, only the semantical rules of the system S x are required. For every

sentence in S x these rules give an interpretation and thereby tell us what

proposition is the intension of the sentence. Thus the result mentioned

concerning the intension of f Hs' is established on the basis of those rules

which give an interpretation for ' Hs' ; these are the rules of designation for

'IT and for V (see i-i and 1-2) and the rule of truth for atomic sentences

(1-3). On the other hand, for the determination of the extension, the truth-

value, of *Hs J , knowledge of the semantical rules alone is obviously not

6 9

7 o II. L-DETERMINACY

sufficient. We need, in addition, factual knowledge. This factual knowl-

edge is based oij observations of the thing Walter Scott; these observa-

tions lead to the result that this thing has the properties characteristic of

human beings and, hence, that the sentence 'Hs' is true.

However, we have seen that there is a particular kind of sentence for the

determination of whose truth-values the semantical rules without any

factual knowledge provide a sufficient basis. These are the L-determinate

sentences, that is, the L-true and the L-false sentences (see the explana-

tion preceding 2-7). Thus, for these sentences the semantical rules suf-

fice to determine not only their intensions but also their extensions. Now

we shall extend the meaning of the term ' L-determinate' so as to make it

applicable to designators in general, in analogy to its application to sen-

tences. For this purpose it seems natural to lay down the following con-

vention for any semantical system S:

17-1. A designator is L-determinate in S if and only if its extension can be

determined on the basis of the semantical rules of S alone, without any

reference to facts.

This convention is not itself a definition of 'L-determinate'. It is meant

merely as an informal characterization of the explicandum; in other words,

a requirement which the definition should fulfil. A definition of L-deter-

minacy for sentences has already been given (2-3d). The problems of con-

structing definitions of L-determinacy for other kinds of designators will

be discussed in the subsequent sections. But even if it is regarded as

merely a requirement, the present formulation in 17-1 is found upon

examination to be insufficient. The phrase "the extension is determined

by certain rules" can be understood in two quite different senses. We have

to find out which sense is appropriate here.

The difficulty here involved can perhaps best be made clear in the case

of a predicator. The intension of the predicator 'EP can obviously be de-

termined with the help of the semantical rules alone; we see from the rule

of designation for 'H' (1-2) that its intension is the property Human. But

does the same not hold for the extension, too? Do we not also see from the

same rule that the extension of 'H' is the class Human? Should we then

say, according to our convention, that *H', and likewise every other

predicator, is L-determinate? This would obviously not be in accordance

with the intended meaning of this term.

In order to overcome this difficulty, we have to make a certain distinc-

tion which can easily be explained for sentences and then transferred to

designators of other kinds. Suppose we ask the question: "What is the

17. L-DETERMINATE DESIGNATORS 71

extension, that is, the truth- value, of the -sentence 'Hs 7 ?" Consider the

following sentences under 17-2 and 17-3, which belong to the metalan-

guage M. Let us examine whether they may be regarded as satisfactory

answers to our question.

17-2. a. 'The extension of 'Hs 7 is the truth-value truth. 7

b. ' 'Hs' is true.'

c. * Scott is human.'

d. 'The extension of 'Hs' is the same as that of 'H s HV

e. ' 'Hs' is equivalent to 'H s HV

17-3. a. 'The extension of 'Hs' is the truth-value that Scott is human/

b. ' 'Hs' is true if and only if Scott is human.'

Each of these seven sentences is true (see 6-3). And in some sense each

of them may be said to give an answer to our question. However, there is

an important difference between the sentences under 17-2 and those under

17-3. Suppose we understand the sentences of the system S t but have no

factual knowledge concerning the things referred to in these sentences;

then we do not know whether 'Hs' is true or not, in other words, whether

Scott is human or not. Suppose, further, that the purpose of our question

was to find this lacking knowledge. Then i7-2a is a completely satisfac-

tory answer because it supplies the information we want; and so is 2b,

which is merely a simpler formulation for 2a; and likewise 2C, which gives

the same information without the use of semantical terms. (For the result

that 2b and 2c mean the same, see the explanation preceding 1-7.) On the

other hand, the answer 3a, although correct, does not satisfy our purpose;

we shall reply with a modified formulation of our first question: "Yes; but

what is the truth-value that Scott is human?" Similarly, we shall reply to

3b: "Yes; but is Scott human or not?" We may formulate this difference

by saying that 2a, 2b, and 2c actually give the truth- value of 'Hs', while 3a

and 3b do not give it but merely describe it, in the sense of supplying a de-

scription for it (in Russell's sense of 'description'). We can do this by

introducing the phrase 'gives the truth- value' in the following way, which

is not meant as an exact definition. Let @/ be a true sentence in M (it

may also be a definition or a rule or a set of true sentences, definitions, or

rules). We shall say that the truth- value of a sentence <S in a system S is

given by @ ; if either the sentence ' is true (in 5)' or its negation follows

from @y (in M) without the use of any factual knowledge not supplied by

@y. [The phrase '. . . follows from (in M)' may be understood as

meaning the same as ' ... is L-implied by (in M) 7 , if we assume that

L-terms with respect to M have been defined in the metametalanguage

72 II. L-DETERMINACY

MM. For the sake of simplicity, we use the German letters with subscripts

not only in M for expressions in S but also in MM for expressions in M

and expressions in S.]

Let us now apply this criterion to the sentences under 17-2 and 17-3.

First, 2b fulfils the criterion in a trivial way; hence it gives the extension

of 'Hs'. Furthermore, each of the sentences 2a and 2c, and even 2d and

2e, gives, together with the semantical rules of S x , the truth- value of 'Hs',

because 2b follows from each of these sentences, together with the rules.

That 2b follows from 2a is obvious. Further, 2C follows from 2b, together

with the result 3b, which is based on the semantical rules for 'IT, V, and

atomic sentences (1-2, i-i, 1-3). Sentence 2d is derived from 2a and there-

by from 2b, together with the result that 'H = IF is L-true, which, in

turn, is based on the semantical rules. The same holds for 2e, which is

merely another formulation for 2d, according to the definition 5-1. On

the other hand, either 3a or 3b, together with the semantical rules, does

not give the extension of *Hs' but merely describes it, because for the

derivation of 2b we need here the factual knowledge that Scott is human.

Consider, now, in contrast to 'Hs', an L-determinate sentence, for ex-

ample, the L-true sentence 's = s' or the L-false sentence '~(s s= s)'.

Here no factual sentence @y is required in addition to the semantical rules

to give the truth- values of these two sentences. The following two sen-

tences in M follow from the semantical rules of Sj alone : l ' s s s' is true

(in S,)', ' '~(s = s)' is not true (in S,)'.

In analogy to these results for sentences, we now replace the earlier

convention 17-1 by the following:

17-4. A designator is L-determinate in S if and only if the semantical

rules of S alone, without addition of factual knowledge, give its extension.

This again does not yet constitute a definition of 'L-determinate' but

only a requirement which the definition should fulfil. For sentences, the

previous definition of L-determinacy (2-3d) is in accord with this conven-

tion on the basis of our explanation of ". . . gives the extension, i.e., the

truth-value, of the sentence ". Our task will now be to find adequate

definitions of L-determinacy for the other kinds of designators. For each

of these kinds we shall have to consider the conditions under which their

extensions are actually given, not merely described; as in the case of

sentences, "to give the extension" will only be informally explained, not

exactly defined. And then the definition of L-determinacy will be con-

structed in such a way that the requirement 17-4 is fulfilled. If a designa-

tor is not L-determinate, we call it L-inde terminate. This term has

18. L-DETERMINACY OF INDIVIDUAL EXPRESSIONS 73

been defined for sentences (2-7); however; in the case of sentences, we

usually use the synonymous term 'factual'. According to the convention

17-4, a designator is L-indeterminate if its extension can be given only by

a factual statement (in M).

18. The Problem of L-Determinacy of Individual Expressions

The conditions under which an individual expression may be regarded as

L-de terminate are examined. An attempt to base the definition of L-determi-

nacy on a distinction between (genuine) proper names and descriptions is aban-

doned as inadequate. The analysis is then applied to a coordinate language S 3 .

Its individuals are positions in a discrete, linear order. V, V, { o'", etc., are

the so-called standard individual expressions for these positions in their basic

order. Every one of these expressions indicates by its form the location within

the basic order to which it refers; hence it exhibits its own extension and may

be regarded as L-determinate. The same does not, in general, hold for a descrip-

tion (e.g., 'the one position which is blue and cold'), except when the description

is L-equivalent to a standard expression (e.g., 'the one position which is be-

tween o' and o"").

We begin with individual expressions because, as we shall see later, the

solution of the problem of L-determinacy for predicators presupposes the

solution for individual expressions.

In analogy to the earlier question, " What is the truth- value of 'Hs'?"

we now consider the question, " Which individual is the extension of

'(i#)(A#w)'?" and possible answers to it. In analogy to the earlier case,

let us imagine that we do not know whether there is exactly one author of

Waverley and, if so, who he is; and that the purpose of our question is to

find out from somebody who does know. Obviously, the answer 'the exten-

sion of the description mentioned is the author of Waverley' would not

satisfy us even though it is true; it is entirely trivial. [Note that, according

to an earlier convention, the phrase 'the author of Waverley' is to be

understood as short for * the one individual who is author of Waverley, or

a* if there are no or several such individuals'.] The answer 'the extension

sought is the author of Ivanhoe' is true and not trivial; but, nevertheless,

it would not satisfy us because it does not supply the specific information

we are looking for; we might say here again that this answer merely de-

scribes the extension but does not give it. The extension is actually and

directly given by the answer * the extension is Walter Scott'. It is indirectly

given by answers like these: 'the extension of { (ix) (A#w)' is the same as

the extension of V or * '(ix) (A#w) s s' is true'; from these we obtain

the direct answer with the help of the semantical rule (i-i), which tells

us that the extension of V is Walter Scott.

On the basis of these considerations we might perhaps be inclined to

? 4 II. L-DETERMINACY

propose the following solution : Let us say that the extension of an indi-

vidual expression is given, and not merely described, by @y if @y uses a

proper name in M (e.g., ' Walter Scott') or refers to a proper name in S

(e.g., V) in distinction to a description. However, it is easily seen that

this does not yet constitute a satisfactory solution. Suppose that 'x is a

dagger and Brutus used x for killing Caesar' can be translated into sym-

bols of Sij this symbolic expression may be indicated by ' . . x . .'. Suppose

somebody gives to our question, ' What is the extension of the description

1 (ix) (. . x . .) '?' the answer ' The extension is the same as the extension of

* b' ', where ' b' is an individual constant in Si such that ' (ix) (. . x . .) = b'

is true. Then the answer is true. According to our tentative solution, we

should say that this answer gives the extension of * (ix) (. . x . .) ', irrespec-

tive of the way in which the semantical rule for ' b' is formulated. But sup-

pose, now, that this rule says that 'b' is the symbolic translation of f the

dagger with which Brutus killed Caesar'. Then the above answer says, in

other words, that the extension of ' (ix) (. . x . .) ' is the dagger with which

Brutus killed Caesar; thus the answer merely describes the extension. The

reason for this lies in the fact that the interpretation of the constant ' b' is

given in M with the help of a description. We might perhaps say that 'b'

is therefore only an apparent proper name, not a genuine one. And we

might try to correct the solution proposed by requiring that genuine

proper names be used, not those which are defined or interpreted by de-

scriptions. This attempt, however, would lead us into serious difficulties.

A moment's reflection shows that most things have no proper names.

Some logicians for example, Russell 1 and Quine 2 do not accept indi-

vidual constants as primitive signs but only as abbreviations of compound

expressions. Thus the distinction between genuine and apparent proper

names of individuals is rather problematic. Even if there are genuine

proper names for some individuals, how should the extension of a descrip-

tion be given whose descriptum has no proper name? It is clear that the

attempted solution is inadequate in its present form.

However, I believe that another distinction will serve the purpose for

which the distinction between proper names and descriptions was in-

tended. To simplify the analysis let us take not a system like Si, whose

individual constants are names of things, but language systems of the

following kind. The individuals are positions in an ordered domain.

Among the individual expressions there are some of a special kind, called

1 Russell's language contains names for qualities but not for particulars, i.e., individuals in

our sense (see [Inquiry], p. 117).

3 Quine regards all individual constants as abbreviations for descriptions (see [M.L.], pp.

149 ff.).

18. L-DETERMINACY OF INDIVIDUAL EXPRESSIONS 75

expressions of standard form, which fulfil the following condition: (i) if

two expressions of standard form are given, then we can see from their

forms the positional relation between the two positions. For systems of a

simple structure (for example, the system S 3 discussed in this section, in

contrast to the language of physics discussed in the next section) the fol-

lowing additional condition is fulfilled: (2) for every position, there is

exactly one expression of standard form. Languages of this kind may be

called coordinate languages, in distinction to name languages like S x . 3

Let us take as an example a language system S 3 in which the basic order

of the positions has the simple structure of a progression, a discrete linear

order with an initial position but no end. Let V be taken as individual

constant for the initial position; if an individual expression of any form,

standard or not, is given as an expression for some position, an expression

for the next following position is formed from it by the adjunction of a

prime ' ". As individual expressions of standard form we take V, together

with those expressions consisting of V followed by one or several primes.

Thus 'o', V, V", 'o"", are the standard expressions for the first four

positions.

Let S 3 contain predicator signs for qualitative properties to be at-

tributed to the positions, say 'B J for the property Blue, 'C' for Cold, 'S'

for Soft. Furthermore, S 3 contains, like S r , the customary connectives,

individual variables with quantifiers, and individual descriptions. As

common descriptum for all descriptions which do not satisfy the unique-

ness condition, we take, of course, the initial position; hence V takes the

place of 'a*' (see 8). Thus, for example, the description ' dx) (Ex C*)'

means the same as ' the one position which is both blue and cold (or the

position o if no or several positions are both blue and cold)'. [As previous-

ly, we shall usually omit the phrase here included in parentheses.]

For the purpose of the subsequent examples, we presuppose this factual

assumption :

18-1. The second position (o') is the only one which is both blue and cold

and also the only one which is both blue and soft.

According to this assumption, the following holds:

18-2. '(ix) (Bx Cx) ss o" is true (and, moreover, F-true).

18-3. '(ix)(Bx Sx) ss o" is true (and, moreover, F-true).

Suppose we ask the question, 'What is the extension of the description

' (ix) (Bx Cx) '?' because we do not know the facts (18-1) and wish to find

3 [Syntax], 3. The system S 3 described in the text is similar to Language I dealt with in

[Syntax], Part I.

76 II. L-DETERMINACY

out which position is the descriptum. Let us consider the following an-

swers:

18-4. a. 'The extension of the description mentioned is the one position

which is both blue and cold/

b. "The extension of the description mentioned is the same as that

of '(i*)(B* &&)'.'

The answer i8-4a, although true, would certainly appear as unsatisfac-

tory; we should protest: ' Yes, but which position is this?'. Sentence 4b is

likewise a true answer to our question, by virtue of the fact 18-1. It is not

so trivial an answer as 4a, but it still does not supply the information we

want. It does not tell us directly which position is the descriptum but

merely refers to this position by a qualitative characterization. After re-

ceiving the answer 4b, just as in the case of 4a, we still need factual obser-

vations concerning the qualities of the positions in order to discover which

position is the descriptum of the original description.

In contradistinction to those answers, each of the following two formu-

lations tells us actually what we want to know:

18-5. a. 'The extension of the description is the second position 7 .

b. 'The extension of the description is the same as that of 'o'V

The same holds for 18-2. Each of these three answers supplies the informa-

tion directly. But there are other formulations which give the same infor-

mation in an indirect manner. In order to construct an example, let

'. . x . .' indicate a not too simple matrix in S 3 without nonlogical con-

stants, which is fulfilled only by the position o'. [We may regard the indi-

vidual expressions in S 3 as expressions of natural numbers ('o' for Zero,

'o" for One, etc.). Then we can introduce arithmetical symbols, for ex-

ample, ' > ' for the relation Greater and ' X J for the function Product, re-

spectively. 4 Let '..#..' indicate the matrix l (x > o) (x X x ss x)\

which is satisfied only by the number One, hence by o'.] Then the follow-

ing holds:

18-6. *(i#)(. . x . .) as o" is true (and, moreover, L-true).

(The sentence mentioned is L-true because it holds in all state-descrip-

tions, which differ only in the distribution of the qualitative properties.

The truth of the sentence can be shown by using only the semantical rules;

these include the rules determining the basic structure and the explicit

and recursive definitions involved.) Hence we obtain:

< These and other arithmetical symbols can be introduced in a system like S 3 with the help

of recursive definitions in the customary way (see, for instance, [Syntax], 20).

18. L-DETERMINACY OF INDIVIDUAL EXPRESSIONS 77

18-7. The extension of ' (ix) (..#..)' is the same as that of V.

Hence also the following holds, because of 18-56:

18-8. The extension of the original description ' (ix) (B# Cx) 9 is the same

as that of '('#)(. >x. .)'

May we regard this statement 18-8 as a complete answer to our question?

It must be admitted that it characterizes the extension of the original de-

scription only in an indirect way; this it has in common with 18-46. In

another respect, however, which is of a fundamental nature, 18-8 is dif-

ferent from that former answer and like those formulations which we re-

gard as complete answers, that is, i8-5a and b and 18-2. If we receive 18-8

as an answer, then, in order to derive from it the complete and direct

answer i8-5a or b, we need not make observations concerning the qualities

of the positions, as in the case of the answer i8-4b; all we have to do is to

carry out a certain logico-arithmetical procedure, namely, that which

leads to the result 18-6. Thus there is this fundamental difference: 18-6

states an L- truth, while 18-3 states an F- truth. The following two results

follow from the ones just mentioned (18-9 from 18-6, 18-10 from 18-3),

according to the definitions of L- and F-equivalence (3-$b and c) :

18-9. '('#)( .)' and V are L-equivalent.

18-10. '(ix) (Bx &*)' and V are F-equivalent.

It is because of the L-equivalence stated in 18-9 that we also say that

1 8-8 actually gives the extension, although indirectly. Thus it becomes

clear that the difference between an answer giving the extension and one

merely describing it does not simply consist in the difference between the

use of a standard expression and that of a description. If a standard ex-

pression is used, the extension is certainly given; but it may also be given

by a description, provided this description is L-equivalent to a standard

expression, as* (ix) (. . x . .)' is, according to 18-9. If, on the other hand, a

description is not L-equivalent to any standard expression, then by using

it we do not give, but merely describe, the extension in question. Note that

every individual expression is an expression of exactly one position and

hence is equivalent to exactly one standard expression. Therefore, if an

expression is F-equivalent to some standard expression, as, for example,

'(i#)(B# Sx) J is according to 18-10, then it cannot be L-equivalent to

any standard expression.

The results here found will help us in constructing, in the next section, a

definition for the L-determinacy of individual expressions.

7 8 II. L-DETERMINACY

19. Definition of L-Determinacy of Individual Expressions

For a simple coordinate language like S 3 ( 18), we define as L-determinate

those individual expressions which are L-equivalent to standard expressions.

The problem of the definition of L-determinacy for more complex coordinate

languages, like the language of physics Sp, is briefly discussed. Finally, it is

shown how the concept of L-determinacy can be applied also to name languages

if the metalanguage is a coordinate language.

In the preceding section, we analyzed the individual expressions in the

system S 3 , which was chosen as an example of a coordinate language of

simple structure. Analogous considerations hold for other systems in

which there are individual expressions of standard form which fulfil both

conditions (i) and (2), mentioned earlier. For the following definition of

L-determinacy it is presupposed that S is a system for which a standard

form has been determined which fulfils those conditions. This definition

is suggested by the results of our discussion in the preceding section.

19-1. Definition. An individual expression in the system S is L-deter-

minate = Df it is L-equivalent to an individual expression of standard form

in S. (This obviously includes the standard expressions themselves.)

That this definition satisfies our previous requirement, 17-4, is seen as

follows: If a given individual expression is L-equivalent to a standard ex-

pression, then those semantical rules on which this L-equivalence (in

other words, the L- truth of the corresponding = -sentence) is based suf-

fice to give its extension, namely, the position corresponding to the stand-

ard expression. On the other hand, if a given individual expression is not

L-equivalent to a standard expression, then it is, as we have seen, F-

equivalent to a standard expression. Therefore, in this case the semantical

rules do not suffice to give its extension; this can be given only by a factual

statement.

It should be noticed that there is, in general, no effective decision pro-

cedure for the concept of L-determinacy just defined. Still less is there a

general effective procedure for the evaluation of any given L-determinate

individual expression, that is, for its transformation into an L-equivalent

standard expression. Going back to the example of the system S 3 with

arithmetical symbols (see the explanations preceding 18-6), '(ix)(x ss

o" X o")' can be transformed into V"' simply by calculation, that is to

say, by repeated application of the recursive definitions. On the other

hand, the transformation of ' (ix) (. . x . .) ', i.e., ' (ix) [(x > o) (x X x =

#)]', into 'o" requires the proof of a universal arithmetical theorem,

which states that every number except i lacks the describing property;

and it is clear that there cannot be a fixed effective procedure for finding

19. L-DETERMINACY OF INDIVIDUAL EXPRESSIONS 79

proofs of this kind. In cases like the two examples in S 3 just given, the

L-determinacy is easily established by the fact that both descriptions do

not contain any nonlogical constants. If, however, nonlogical constants

occur, then we have, in general, no effective procedure for deciding about

L-determinacy.

The basic order of the positions in a coordinate language S may be quite

different from the simple order in S 3 ; but the procedure leading to a defini-

tion of L-determinacy will still be essentially the same. We first choose

among the individual expressions of the system those which we wish to re-

gard as of standard form. The choice is fundamentally a matter of conven-

tion, provided that, of the requirements stated earlier, at least the first is

fulfilled. The simplicity of the forms and the possibility of recognizing the

positional relations in a simple way will usually influence the choice. If the

primitive constants of the language system are divided into logical and

descriptive (i.e., nonlogical) constants (see 21), then only expressions in

which all constants are logical will be taken as standard form.

As an example of a system with a different basic order, let us briefly

consider a coordinate language of physics Sp, leaving aside the technical

details. Here the individuals are space-time points within a coordinate

system chosen by convention. First, a standard form for expressions of

real numbers in S P must be chosen. Here this is a much more complicated

task than in the case of natural numbers (as in S 3 ). The standard expres-

sions must enable us to find the location of positions and the distance be-

tween two positions with any desired degree of precision. This means that

for the representation of real numbers as systematic (e.g., decimal or dual)

fractions, we must have an effective procedure for computing any required

number of digits. 5 Since a space-time point is determined by three space

coordinates and one time coordinate, a standard individual expression in

SP will consist of four standard real-number expressions.

A continuous coordinate language like SP is, in certain respects, funda-

s This requirement can be stated in exact terms as follows. For every real number there is

a unique representation in the decimal system if we exclude decimals which, from a certain

place on, contain only the figure '9'. The integral part is a natural number; the fractional part

corresponds to a function i(n) whose value gives the nth digit after the decimal point. (For

example, for TT 3.1415 > *(*) "" *> ffa) - 4, ffo) = ii *(4) * S> etc -) # tnen > a real -

number expression consists of an expression of its integral part (say, in the ordinary decimal

notation) and an expression for the function f corresponding to its fractional part, then this

real-number expression is computable if the expression for f is computable in the sense of

A. M. Turing ("On Computable Numbers", Proc. London Math. Soc., Vol. XLII [1937]).

Turing has shown that this concept of the computability of a function coincides with Church's

lambda-definability and with the concept of general recursiveness due to Herbrand and Godel

and developed by Kleene (see Turing, "Computability and X-Definability", Journal of Sym-

bolic Logic, Vol. II [1937]).

8o II. INDETERMINACY

mentally different from a discrete coordinate language like S 3 . The first

important difference consists in the fact that no language (with expres-

sions of finite length) can contain expressions for all real numbers. 6 There-

fore, Sp cannot contain individual expressions for all individuals, that is,

space-time points let alone individual expressions of standard form.

Thus here the second of the two conditions for standard expressions can-

not be fulfilled; only the first is required. Another difference is the follow-

ing: There is no general effective method which would enable us to decide

for any two standard individual expressions whether or not they are

equivalent, that is, refer to the same position in other words, whether

or not their (four-dimensional) distance is o. However, if two standard ex-

pressions are given, we can determine their distance in the form of a com-

putable function. Hence, for any positive rational number 6, no matter

how small it may be chosen, we can establish either that the distance is ^6

and hence that the positions are distinct, or that the distance is ^8, that

is, the positions are either identical or certainly not farther apart than 5.

We cannot here go any further into the technical details of the problem

of L-determinacy for the individual expressions in Sp. The problems

which ought to be investigated are the following. It is clear that not all

individual expressions in SP can be equivalent to standard expressions.

The question should be examined as to whether the standard form can be

chosen in such a manner that at least all those individual expressions

which do not contain nonlogical constants are equivalent (and hence

L-equivalent) to standard expressions. If so, L-determinacy can be de-

fined for SP as in 19-1. Otherwise, a more complicated definition will per-

haps be necessary; but it will, in any case, be such that L-equivalence to

a standard expression is a sufficient, though perhaps not a necessary, con-

dition for L-determinacy.

So far we have applied the concept of L-de terminate individual expres-

sions only to coordinate languages. Now let us consider name languages,

as, for example, Sj. In a language of this kind we have no individual ex-

pressions which exhibit their positional relations directly by their form.

We may have individual expressions in the form of descriptions using

qualitative describing properties; furthermore, there may be individual

constants which are either primitive or perhaps introduced by definitions

as abbreviations of descriptions. However, even a primitive individual

constant in a name language S may, under certain conditions, be L-

determinate if the metalanguage M is a coordinate language. For every

primitive individual constant in S there is a rule of designation in M

* See [Syntax], 6od.

20. L-DETERMINACY OF PREDICATORS 81

which tells us which individual is meant by the expression. This rule refers

to the individual by an individual expression in M. Now if M is a coordi-

nate language and the individual expression used in the rule is L-determi-

nate in M in the sense earlier explained for coordinate languages, then we

may likewise regard the individual constant in 5 as L-determinate. This

extended use of the term 'L-determinate' seems natural, since it satisfies

our earlier requirement 17-4: The semantical rules give the extension of

the constant, that is, the location of the position to which the constant

refers. This may be illustrated by the following example. Suppose the

expressions 'o', V, V, etc., occur, not in the object language 5, which

is supposed to be a name language with individual constants V, 'b\ etc.,

but in M, and that they refer, as explained earlier for S 3 , to the positions

in a discrete linear order. Suppose, further, that the following two rules

are among the semantical rules of S formulated in M ; they are rules of

designation for the primitive constants V and V:

19-2. a. V designates the position o".

b. V designates either the one position which is both blue and cold,

or the position o if no or several positions are blue and cold.

We would in this case construct the definition of L-determinacy in such a

way that V will be called L-determinate but V not. (We omit here the

actual construction.) These results will then be in agreement with the re-

quirement 17-4. We see from rule ig-2a that the extension of V is the

third position. On the other hand, the semantical rules do not give the

extension of V but merely describe it (in rule ig-2b) ; it can be given only

by the addition of a factual statement to the rules. Thus the first part of

the factual statement 18-1, together with the rule ig-2b, tells us that the

extension of V is the second position (c/).

20. L-Determinacy of Predicators

A predicator (in a coordinate language like S 3 ) is said to be L-determinate

if every full sentence of it with individual expressions of standard form is

L-determinate. This holds if the intension of the predicator is a positional or

mathematical, rather than a qualitative, property. The analogous definition

for functors is briefly indicated.

The concept of the extension of a predicator, especially if we consider

predicators of degree one, seems entirely clear and unproblematic. For

example, the extension of the predicator 'IT in the system Si is the class

Human because its intension is the property Human. We began the ex-

planation of the method of extension and intension with the customary

and apparently clear and simple distinction between classes and proper-

8 2 II. L-DETERMINACY

ties ( 4). We took this distinction as a model and framed the distinctions

between the extension and the intension of sentences and of individual ex-

pressions in analogy to it ( 6 and 9). A closer inspection shows, however,

that a serious difficulty is involved even in the concept of the extension of

a predicator. We could leave this difficulty aside in our earlier discus-

sions, but for our present purpose we have to face it and try to overcome

it. In order to find an adequate definition for L-determinacy of predicators

we have to make clear the means by which a class can be given. We shall

see presently that this problem cannot be solved without first solving the

problem of the way in which the extension of an individual expression can

be given. This was our reason for first discussing individual expressions in

the two preceding sections.

Suppose we ask somebody for information about the extension of the

membership of Club C; that means that we want to learn who is a member

of C and who is not. The answer 'the extension is the class of the members

of C' is, although true, entirely trivial and hence would not satisfy us. Nor

would an answer like 'the class of those boys in this town who either are

between fifteen and sixteen years old or have red hair'. Although this

answer is not trivial, it still does not give the extension but merely de-

scribes it with the help of another complex property which happens to

have the same extension. What we want is not an indirect characteriza-

tion of the membership by an intension but a membership list. Would

every kind of membership list satisfy us? We see easily that some kinds

would not. Thus the problem arises: What kind of membership list does

actually give the extension? Suppose that we are given a statement which

lists all the members of the club but does so by formulations like these:

'the eldest son of Mr. Jones', 'the boy friend of Mary', etc. We should

again reject this statement, although it enumerates all members, because

it does so by descriptions. Thus we see that a certain class is not merely

described but actually given by a statement if this statement (i) refers

to each of the members of the class and (2) does so by the use of individual

expressions, which, in turn, do not merely describe but give the indi-

viduals in other words, by the use of L-determinate individual expres-

sions. This shows that the concept of L-determinacy of predicators pre-

supposes the concept of L-determinacy of individual expressions.

We presuppose for the following discussions that 5 is a coordinate lan-

guage of a simple structure similar to S 3 , as explained in the beginning of

19; that a standard form of individual expressions has been defined for

S; and that L-determinacy of individual expressions in S is defined by

our previous definition (19-1).

20. L-DETERMINACY OF PREDICATORS 83

The condition formulated above for a statement giving a class is suf-

ficient but not necessary. The statement need not give an enumeration of

all members of the class; if this were necessary, then only finite classes

could be given. It is sufficient $nd also necessary that the statement

logically imply the truth of all those true singular sentences in S which

say of an individual that it is or that it is not a member of the class, where

the individual expressions occurring are L-de terminate.

It would even be sufficient to require this merely for all the individual

expressions of standard form in S; it is easily seen that it also holds, then,

for all L-determinate individual expressions because they are L-equivalent

to standard expressions, according to the definition 19-1.

In order to give examples let us go back to the coordinate language S 3

with V, V, V", etc., as standard expressions. Suppose that the state-

ment 'the positions o and o'" and no others are blue' is true. Then it

gives the extension of the predicator 'R\ because from this statement, to-

gether with the semantical rules, we can infer that 'B(o)' and 'BCo'") 1

are true, while all other full sentences of 'B' with a standard expression

are false. Let us introduce into S 3 the customary notation '{..,..,..}'

for a finite class indicated by an enumeration of its members; the defini-

tion can be written with the help of a lambda-operator as follows:

20-1. Abbreviation. '{x l9 x 2 , . . . x n ] ' for '(\y)[(y - * x ) V (y s * a ) V . . . V

(y - *J]'.

Then the extension of 'B' in the above example can be given also by this

statement: 'the extension of *B J is the same as that of '{o, o'"}'.

These considerations suggest the following definition for L-determi-

nacy of predicators in a system S (of the kind indicated above). It presup-

poses the definition of L-determinacy for sentences (2-2d).

20-2. Definition. A predicator in S is L-determinate =Df every full sen-

tence of it with individual expressions of standard form is L-determinate.

We see easily that this definition fulfils our earlier requirement 17-4;

the concept defined applies if and only if the semantical rules alone, with-

out any factual knowledge, suffice to give the extension of the predicator

in the sense explained above, because a sentence is L-determinate if and

only if the semantical rules suffice to determine its truth-value (conven-

tion 2-1).

We see that any predicator in S 3 of the form *(X#) (..#..)', where any

molecular combination of ' ss '-matrices with V and standard expressions

stands in the place of '. . x . .', is L-determinate. Therefore, '{o, o'"}' is

L-determinate, and likewise any other predicator of the form '{. . .}'

8 4 II. L-DETERMINACY

where all individual expressions occurring are of standard form. Let us

define in the customary way the signs 'A' and 'V of the null class and

the universal class, respectively, or, more exactly, of the L-empty prop-

erty and the L-universal property, respectively:

20-3. Abbreviations.

a. <A'for'(X#)[~(x s x)]'.

b. <V'for'(X*)[x^x]'.

We see immediately that the two predicators here defined are L-determi-

nate, because all full sentences of 'A' are L-false and all of 'V' are L-true.

But there are other, more complicated predicators which likewise are

L-determinate, among them all lambda-expressions with any purely

arithmetic conditions. Take, as an example, the predicator '(X#)[Prime

(#)]', where * Prime' is defined so that it holds for all prime numbers (that

means, for all positions with a prime coordinate). 7 This example shows

that even a predicator whose extension is infinite and therefore cannot be

given by an enumeration may be L-determinate. This is the case if the in-

tension is of a mathematical, rather than of an empirical, nature; in other

words, if the intension is a positional, rather than a qualitative, property.

That, for instance, the position o'", corresponding to the number Three,

belongs to the extension of 'Prime' is found by a purely logico-mathe-

matical procedure, that is, a procedure based upon the semantical rules

and not involving the qualitative properties of that or any other position.

On the other hand, for establishing that the position o'" belongs to the

extension of *B', we need not only the semantical rules but, in addition,

an observation yielding the result that this position has the color Blue.

Here, again, for the concept of L-determinate predicators there is no

effective method of decision, since there is none for the concept of L-de-

terminate sentences on which it is based. For example, let x be called a

Fermat exponent if x > 2 and if there are positive integers u, v, and w

such that u + if = w x . A predicator for this property, say 'Fer', can

easily be defined in S 3 /Fer' is an L-determinate predicator because every

full sentence of it with a standard individual expression is an L-determi-

nate sentence. For most of these sentences it is at present unknown

whether they are true or false, and there is no decision method for deter-

mining their truth-value. Nevertheless, they are L-determinate, because

their truth-values are independent of colors or any other qualitative

properties of the corresponding positions. For the number Three and some

others it is known that they are not Fermat exponents. This has been

? Arithmetical concepts of this kind can be defined in a language similar to S 3 with the help

of recursive definitions (see, for example, [Syntax], 20).

21. LOGICAL AND DESCRIPTIVE SIGNS 85

shown by a mathematical proof; thus the result is independent of the

qualitative properties of the positions. Therefore, the sentence '~Fer(o'")'

holds in every state-description and hence is L-true in S 3 .

It may be remarked incidentally that a definition of L-determinacy for

functors and compound functor expressions can be given which is quite

analogous to that for predicators (20-2). Here it would likewise be required

that every full sentence in which the argument expressions and the value

expression are of standard form be L-de terminate. Thus, all signs or ex-

pressions for arithmetical functions are L-determinate. For example, the

functor ' + ' in S 3 is L-determinate because every full sentence with stand-

ard expressions is L-determinate; for instance, V + o" s o"" is L-true.

On the other hand, a functor for a physical magnitude, for example, tem-

perature (say, in the language of physics, Sp) is not L-determinate, because

a sentence saying that the temperature at a certain space-time point has a

certain value is not L-determinate.

21. Logical and Descriptive Signs

We make use in this section of the customary distinction between logical and

descriptive (nonlogical) signs. For the system S 3 (restricted to primitive signs)

the classification is simple: the primitive predicates are descriptive, all other

signs are logical. If a designator in S 3 contains only logical signs, then it is L-de-

terminate. A designator in S 3 is L-determinate if and only if it is L-equivalent

to a designator containing only logical signs. This could be taken as an alterna-

tive way of defining L-determinacy.

In this section we make the customary distinction between logical and

descriptive, i.e., nonlogical signs. 8 With its help we shall then make a cor-

responding distinction for expressions, which is especially important for

designators. Then we shall investigate the relation between this distinc-

tion and the distinction between L-determinate and L-indeterminate

designators. The concepts of logical and descriptive signs will seldom be

used in the rest of the book.

We shall define the concepts mentioned for two example systems, one

a coordinate language and the other a name language. As coordinate

language we take the system S 3 discussed in the preceding sections; it con-

tains 'o', 'o", etc., as individual expressions of standard form. As name

language we take a system S r ' which is like our system S t with this excep-

tion: We suppose that the individual constants in it, say 'a*', 'a', 'b', etc.,

are interpreted by the semantical rules of S^ as referring, not to things,

as in B! (see rule i-i), but to positions in an ordered domain (as, for ex-

8 For more detailed explanations see [I], 13.

86 II. L-DETERMINACY

ample, in rule ip-aa). Therefore, these constants are L-determinate, as

explained earlier (at the end of 19). Both systems are here supposed to

contain only primitive signs, not defined signs. The predicates in both sys-

tems are supposed to be interpreted by the semantical rules as designating

qualitative properties or relations like Blue, Cold, Colder, and the like

(as explained for S 3 in 18).

The distinction between logical and descriptive signs of the systems S 3

and S^ is made in the following way by enumeration of particular signs and

kinds of signs.

21-1. The following signs are regarded as logical:

a. The individual variables.

b. The connectives; the operator signs '3', Y, ' X'; the parentheses.

c. In Si, the individual constants; in S 3 , *o' and '".

21-2. The predicates are regarded as descriptive signs.

The corresponding distinction for expressions is now defined in 21-3; to

be descriptive is taken, so to speak, as a dominant property; to be logical

as a recessive property.

21-3. Definitions.

a. An expression is logical = Df it contains only logical signs.

b. An expression is descriptive =Df it contains at least one descrip-

tive sign.

Thus the standard expressions 'o', *o", etc., in S 3 are regarded as logical.

This seems justified because they refer here not to things but to positions

in a basic, presupposed order. We may even interpret them as referring

to pure numbers. In a word translation of 'B(o'")' the expression ( o" n

corresponds in this interpretation to the italicized part in ' the position

correlated to the number Three is blue', while the predicate ' B ' corresponds

to the whole nonitalicized part of this sentence. 9 This interpretation is just

as adequate as the ordinary interpretation by ' the position correlated to the

number Three is blue'. We might even say that these are merely two differ-

ent formulations for the same interpretation, since the translation of the

whole sentence is the same in both cases, and hence the truth-condition

of the sentence remains likewise the same.

In addition to the individual expressions of standard form in S 3 (e.g.,

V") and in S x ' (here we take the individual constants as standard form),

both systems contain individual descriptions.

The following results concern the system S 3 . They hold likewise for S^,

This interpretation has, furthermore, the advantage that a sentence which says that the

universe of individuals is infinite is not factual but L-true. Thus the difficulty usually connected

with the so-called Axiom of Infinity is here avoided (see (Syntax], p. 141).

21. LOGICAL AND DESCRIPTIVE SIGNS 87

provided that the basic order of its universe of individuals is either the

same as in S 3 or has a similar simple structure and provided that the rules

of designation formulated in M for the individual constants in Sj use only

individual expressions of standard form; this standard form in M may,

for example, be the same as in S 3 .

21-4. Every sentence in S 3 which contains only logical signs is either L-

true or L-false; and there is an effective decision method for determining

which of the two is the case.

21-5. Every (closed) description in S 3 is L-determinate; and there is an

effective procedure for transforming it into an individual expression of

standard form.

21-6. Every closed lambda-expression in S 3 is L-determinate; and there

is a decision method for any full sentence of the lambda-expression with

any individual expression of standard form.

The proofs of these theorems and the decision methods mentioned can-

not be given here, but they are rather simple. 10 They are based on the fol-

lowing circumstances: (i) since no predicates occur, the ultimate compo-

nents are = -matrices; (2) an = -sentence with two standard expressions is

L-true if the two standard expressions are alike, and otherwise it is

L-false.

The three results can be combined into one as follows:

21-7. Every designator in S 3 which contains only logical signs is L-

de terminate.

There are, however, also L-determinate designators which contain de-

scriptive signs. For example, 'P(o) V ~P(o)' is L-true; ' (\x) (Px V ~Px) '

is L-universal, and hence L-equivalent to 'V (2o-3b); and f (ix)

(Px V ~Px)' is L-equivalent to V; thus these three designators are all

L-determinate.

It follows from 21-7 that any designator L-equivalent to one containing

only logical signs is likewise L-determinate. Now it can be shown that the

converse of this holds too. (i) If a sentence is L-determinate, then it is

either L-true or L-false; therefore, it is L-equivalent either to *o s o'

or to the negation of this sentence, (ii) If a description is L-determinate,

it is L-equivalent to a standard expression, according to the definition

19-1. (iii) It can be shown that, if a closed lambda-expression in S 3 is

L-determinate, either its extension or the complement of its extension is

finite; therefore, the lambda-expression is L-equivalent to one of the form

'(\x)(. . x . .)', whose scope is constructed with the help of connectives

10 For further details see [Modalities], n and 12, especially Ti2-af.

88 II. L-DETERMINACY

out of = -matrices with V and standard expressions. Thus the following

holds:

21-8. A designator in S 3 is L-determinate if and only if it is L-equivalent

to one containing only logical signs.

For S 3 and similar systems, L-determinacy for designators could be

generally defined by the sufficient and necessary condition stated in 21-8.

This alternative method presupposes only the concepts of logical signs

(21-1) and of L-equivalence of designators (s-sb), hence of L-truth of

sentences (2-2); it would replace the three separate definitions of L-

determinacy for sentences, individual expressions, and predicators earlier

given (2-3d, 19-1, 20-2).

Now we can easily see that if two designators in S 3 which contain only

logical signs are equivalent, then they are L-equivalent. Since they are

equivalent, the = -sentence containing them as components is true (3~5a)

and therefore L-true, according to 21-4; hence they are L-equivalent

(3-5b). From this result the following more general theorem can be de-

rived with the help of 21-8 and the transitivity of equivalence and L-

equivalence:

21-9. If two L-determinate designators in S 3 are equivalent, then they are

L-equivalent.

22. L-Determinate Intensions

If a designator is L-determinate, then all designators L-equivalent to it are

likewise L-determinate. We shall say of the common intension of these designa-

tors that it is an L-determinate intension. For any extension, there are, in gen-

eral, many corresponding intensions; but there is among them exactly one

L-determinate intension.

The results which will be stated here can be proved in an exact way for

the system S 3 . But it can be shown in an informal way that they hold like-

wise for any system 5, provided the concepts of L-truth and L-determi-

nacy are defined for 5 in such a manner that our requirements for these

two concepts (2-1 and 17-4, respectively) are fulfilled. In the following

discussion it is presupposed that these requirements are fulfilled.

22-1. If two L-determinate designators in 5 are equivalent, then they are

L-equivalent.

Applied to S 3 , this is the same as 21-9, which was proved with the help

of the distinction between logical and descriptive signs. The general theo-

rem for a system 5 can be seen to hold in the following way, which does

not presuppose such a distinction. Since the two designators are equiva-

22. L-DETERMINATE INTENSIONS 89

lent, they have the same extension (5-1). Since they are L-determinate,

the semantical rules suffice for establishing that both have this same

extension (17-4) and hence that they are equivalent (5-1) and hence that

their ^-sentence is true (3~5a); therefore, this ^-sentence is L-true (2-1);

hence the two designators are L-equivalent (3~sb).

22-2. If a designator in S is L-equivalent to an L-determinate designator,

then it is itself L-determinate.

For S 3 , this follows from 21-8 because of the transitivity of L-equiva-

lence. That it holds generally for S is seen as follows: If the condition in

22-2 is fulfilled, the semantical rules suffice for establishing the extension

of the second designator and the identity of extension for the two desig-

nators, and thereby the extension of the first designator.

Suppose an L-determinate designator in 5 is given. It possesses a certain

intension. Any other designator having this same intension is L-equiva-

lent to the first and hence likewise L-determinate, according to 22-2. Let

us call an intension of this kind an L-determinate intension. Thus,

roughly speaking, an L-determinate intension is such that it conveys to

us its extension. For every extension, there are, in general, many cor-

responding intensions; but among them there is exactly one L-determinate

intension, which may, in a way, be regarded as the representative of this

extension (not, of course, in the sense in which a designator may be said

to represent, or refer to, its extension). This one-one correlation between

extensions and L-determinate intensions will become clearer with some

examples.

For sentences, there are only two extensions, the two truth-values,

Truth and Falsity. There are many L-determinate sentences whose ex-

tension is the truth-value Truth, namely, all the L-true sentences, e.g.,

'Pa V ~Pa' (in S r ). Since they are L-equivalent to each other, they have

the same intension, namely, the L-true or necessary proposition. Thus

this proposition is the one L-determinate intension corresponding to the

extension Truth. Analogously, the L-false or impossible proposition is the

L-determinate intension which corresponds to the extension Falsity. For

predicators, there are infinitely many extensions, namely, classes of indi-

viduals. If, as in S x and S 3 , the number of individuals is denumerably infi-

nite, the number of classes of individuals is nondenumerable; since the

number of (finite) expressions in any language system 5 is, at most, de-

numerable, not all classes of individuals can be extensions of predicators

in S. For an extension referred to by a predicator in 5 there is not neces-

sarily always a corresponding L-determinate intension expressed by a

90 II. L-DETERMINACY

predicator in S, because not every predicator has an equivalent L-determi-

ftate predicator. Whether a certain L-determinate intension is or is not

expressed by a predicator in S depends on the means of expression in S.

The L-determinate intension corresponding to the null class of individuals

is the L-empty property; in Si and S 3 this intention is expressed, for ex-

ample, by '(\x)[~(x = #)]'. The L-determinate intension corresponding

to the universal class is the L-universal property, expressed by *(X#)

[x ss #]>. The L-determinate intension corresponding to the class whose

only members are the positions o, o", and o'" is the property of being one

of these three positions, which is expressed in S 3 by '(X#)[(# s= o) V

($ = o") V (x 35 o'")]'. On the other hand, suppose that the primitive

signs of S 3 , mentioned earlier, are the only signs in S 3 and that S^ is con-

structed from S 3 by the addition of some recursively defined functors and

predicators, among them the predicator ' Prime' for the property Prime

Number. Suppose, further, that all prime number positions, and only

these, happen to be blue. Then the extension of *B' is the class of prime

number positions, and the corresponding L-determinate intension is the

property of being a prime number position. This intension is expressed in

83 by the L-determinate predicator ' Prime' ; but in S 3 it is not expressed

by any predicator.

The extensions of individual expressions are the individuals, which in

S 3 are the positions. For example, the extension of the description ' (12)

(Ex O)' in our earlier example is the second position (i.e., the position

next to the initial position, i8-5a). Therefore, the corresponding L-deter-

minate intension is the individual concept The Second Position, which is

expressed in S 3 by the L-determinate individual expression *o ;> . Generally

speaking, for every individual in S 3 there is one L-determinate intension,

namely, the individual concept of that position; this intension is expressed

in S 3 by at least one L-determinate individual expression, for instance, by

the standard expression ('o', *o", etc.).

23. Reduction of Extensions to Intensions

The one-one correlation between extensions and L-determinate intensions

suggests the identification of extensions with the corresponding L-determinate

intensions. According to this method, which is discussed in this section but will

not be used in the remainder of the book, a class is construed as a positional

property. This leads to explicit definitions of classes, in distinction to the con-

textual definitions used by Whitehead and Russell.

The method of extension and intension introduced in the first chapter

assigns to every designator an extension and an intension. Thus our

23. REDUCTION OF EXTENSIONS TO INTENSIONS 91

semantical analysis of the designators seems to assume two kinds of

entities extension and intensions. It has been mentioned earlier that

this assumption is not actually made, that, in fact, we merely use two

forms of speech which can ultimately be reduced to one. There are several

possibilities for this reduction; they fall chiefly into three kinds: (i) the

extensions are reduced to intensions; (ii) the intensions are reduced to

extensions; (iii) both extensions and intensions are reduced to entities,

which are, so to speak, neutral. We shall later explain several methods

of the first kind. The chief requirement that such a method must fulfil is

obviously this: two different but equivalent intensions must determine the

same extension. The methods of this kind to be explained later ( 33,

methods (2) and (3)) give, not an explicit definition, but only a contextual

one. That is to say, a phrase like ' the class Blue' is not itself translated

into a phrase in terms of properties; instead, a rule is given for transform-

ing any sentence containing the phrase ' the class Blue' into a sentence

referring only to properties.

Now the introduction of the concept of L-determinate intension (in the

preceding section) makes it possible to define extensions in terms of in-

tensions. This method requires that the universe of individuals in'question

exhibit a basic order so that the concept of L-determinacy may be ap-

plied. It is not required that the object language be a coordinate language;

the basic order need not be exhibited by the individual expressions of the

object language; it is sufficient that it be expressible in the metalanguage.

We suppose for the following definitions, as we did in the preceding sec-

tion, that the concepts of L-truth and L-determinacy are defined for the

system S in such a manner that our requirements for these two concepts

(2-1 and 17-4) are fulfilled. The advantage of the method to be applied

here is that it supplies explicit definitions. It is based on the following

three results, which we found earlier: (i) to every intension there cor-

responds exactly one L-determinate intension; (ii) the L-determinate in-

tensions corresponding to any two intensions which are equivalent and

hence have the same extension are identical; (iii) therefore, there is a one-

one correlation between extensions and L-determinate intensions.

The method to be proposed consists simply in identifying extensions

with the corresponding L-determinate intensions.

23-1. Definition. The extension of a designator in S = DI the one L-determi-

nate intension which is equivalent to the intension of the designator.

The concept of the equivalence of intensions used in this definition was

introduced (definition 5-3) with the help of the concept of the equivalence

9 2 II. L-DETERMINACY

of designators; the latter concept was defined (3-$a) by the truth of an

SB -sentence and hence does not presuppose the concept of extension.

Our principal requirement for extensions was that they be identical for

equivalent designators (5-1). This requirement is fulfilled by the present

definition 23-1 (see (ii) above) .

Although we have usually spoken of intensions only as intensions of

designators, occasionally reference was made to intensions independent

of the question of whether or not they were expressed by designators in the

system under discussion. Therefore, it may be useful to have the following

definition for the extension of (or, corresponding to, determined by) an

intension; here no reference is made to designators.

23-2. Definition. The extension of a given intension = DI the one L-determi-

nate intension which is equivalent to the given intension.

Let us apply these definitions to the examples in the system S 3 given in

the preceding section. Let us begin with predicator s, because in this case

the concept of extension, that is, of class, is more familiar than in the

other cases. Classes are now identified with L-determinate properties, that

is, positional properties. Let us assume, for example, that the positions

o, o", o'", and no others, are blue. On the basis of this assumption, the

extension of the predicator 'B' in S 3 is, according to the definition 23-1,

the intension of '(*x)[(x - o) V (x = o") V (x = o 7 ")]', that is, the

property of a position of being either o or o" or o'". And we say likewise,

according to the definition 23-2, that the extension of the property Blue

is the positional property just mentioned. However, it should be noted

that these two results are factual statements based on the factual assump-

tion mentioned. Our definitions do by no means say that the phrases ' the

extension of 'B J ' and 'the extension of the property Blue', to which we

may add the third synonymous phrase, 'the class Blue', mean the same

as ' the property of being either o or o" or o"". The latter phrase is merely

equivalent to each of the three former phrases. What the definition 23-1

actually says is that the phrase 'the extension of 'B J ' means the same as

'the L-determinate intension which is equivalent to the intension of

'B' ' in other words, 'the positional property which is equivalent to the

(qualitative) property Blue'. It is a matter of fact, not of logic, that the

positional property which is equivalent to the property Blue is the

property of being either o or o" or o'".

Let us assume, further, that no position is both blue and cold. Then

the extension of 'B C' in S 3 is the null class; this is now identified

with the L-empty property, which is expressed in S 3 by the predicator

23. REDUCTION OF EXTENSIONS TO INTENSIONS 93

1 C\x)[~(x =s #)]'. Suppose that all prime number positions, and no others,

are blue. Then the extension of ' B' is the class of prime number positions.

This class is now identified with the property Prime Number Position.

It may perhaps at first seem somewhat strange to regard classes not as

distinct entities corresponding somehow to properties but as properties of

a special kind. But a consideration of the examples given will remove or

mitigate the feeling of strangeness. For example, it might not seem very

unnatural to regard the intension of '(\x)[(x = o) V (x = o") V (x ss

o'")]' as a class when we consider the fact that this intension, in contrast

to L-indeterminate intensions, provides by itself an answer to the question

as to the individuals to which it applies and those to which it does not.

Now we are going to apply our definitions to sentences. If we approach

the matter naively, without careful analysis as to the nature of the

entities, we might perhaps be inclined to say that we know, at least

roughly, what we mean by the extension of a predicator (of degree one),

that is, a class. However, if it is said that the extension of a sentence is a

truth- value, it is not at all clear what entities should be regarded as truth-

values. In our earlier discussion (in 6), we left aside the difficulty here

involved; but now let us examine it and try to solve it. We consider here

languages which speak about extra-linguistic individuals, either physical

things with physical properties, as in S x , or positions, as in S 3 , with physi-

cal properties (e.g., 'the second position is cold'). Both the intensions and

the extensions of predicators are clearly extra-linguistic entities; both

properties of individuals and classes of individuals (no matter whether

regarded in the customary way or, according to the method here proposed,

as properties of a special kind) have to do with the individuals, not with

expressions in the language. The same holds for extensions and intensions

of individual expressions; both individuals and individual concepts, what-

ever their specific nature may be, are certainly extra-linguistic entities.

Therefore, it seems natural to expect, by analogy, that intensions and

extensions of designators of all kinds are extra-linguistic entities. This

holds also for the intensions of sentences, the propositions. But what about

their extensions? What kind of entities are the truth-values which we

take as the extensions of sentences? We might perhaps be inclined to

answer that the truth-values are truth and falsity and that these two

terms are to be understood in their semantical sense. However, truth in

the semantical sense is a certain property of sentences, hence a linguistic

entity. [This does not imply that truth is a merely linguistic matter; truth

is dependent upon extra-linguistic facts; therefore, its definition must refer

94 II. L-DETERMINACY

to extra-linguistic entities. However, we are here not concerned with the

question of the entities to which the definition refers, but rather with the

question of the kind (logical type) of entity to which the concept of truth

belongs. And here the answer is: It is a property of sentences.] Therefore,

truth and falsity fall outside the domain to which all other intensions and

extensions belong. Now there is nothing in the situation that compels us

to take (semantical) truth and falsity as the extensions of sentences. All

that is required is that the extension of all true sentences be the same

entity and that the extension of all false sentences be the same entity

but something different from the first. There are obviously many different

possibilities of choosing in a not too arbitrary manner two extra-linguistic

entities such that the one is connected in a simple way with all true sen-

tences and the other with all false sentences. What type of nonlinguistic

entities should we choose? It seems most natural to choose either two

properties of propositions or two propositions. Let us consider some possi-

bilities of these two kinds. The most natural properties of propositions to

be considered would obviously be truth and falsity of propositions. [In

distinction to truth or falsity of sentences, these two concepts are not

semantical but independent of language. 11 Their relation to the semantical

concepts of truth and falsity is the same as the relation of the equivalence

of intensions to the equivalence of designators; see the definition 5-3 and

the explanations preceding it, including the footnote. They are singulary,

truth-functional connections. 12 ] It would be simpler to take two proposi-

tions. We might, for example, take, on the one hand, the proposition p T

expressed by the class of all true sentences in 5, and, on the other hand, the

negation of fa. [In systems like Si and S 3 , where we have state-descrip-

tions ( 2), the proposition p T is expressed in a. simpler way by the one

true state-description.] This device might perhaps appeal to those philos-

ophers who regard truth as involving in some sense the whole universe. 13

While this method takes two factual (contingent) propositions as exten-

sions, our own method (23-1) takes the two L-determinate propositions.

Here the extension of any true sentence is the L-true (necessary) proposi-

u In the terminology of [I], they are absolute concepts; for their definitions, see [I], Diy-i

and Diy-2.

M True* in this sense is a connective with the characteristic TF and hence is redundant (e.g.,

'(the proposition) that Scott is human is true* and 'Scott is human' are L-equivalent sentences

in M); 'False* has the characteristic FT and hence is a sign of negation (compare [II], 10).

Lewis ([Meaning], p. 242) maintains a similar conception. The denotation or extension of

a proposition "is not that limited state of affairs which the proposition refers to, but the kind

of total state of affairs we call a world. ... All true propositions have the same extension, name-

ly, this actual world; and all false propositions have the same extension, namely, zero-exten-

23. REDUCTION OF EXTENSIONS TO INTENSIONS 95

tion; and the extension of any false sentence is the L-false (impossible)

proposition. Here, likewise, we probably feel, at first, some reluctance to

regard propositions as truth-values or extensions. However, the connec-

tion between the two Indeterminate propositions and what we usually

regard as the truth- values is so close and natural that it is perhaps not too

artificial to take these propositions as extensions of sentences.

Now let us apply the new method to individual expressions. Let us again

assume that only the second position o' in S 3 is both blue and cold. We

said earlier that, on the basis of this assumption, the extension of ' (ix)

(Ex Cx) J is the second position. We say now, instead, that the exten-

sion of this description is the individual concept The Second Position. In a

sense this may be regarded as merely a change in formulation. We may

even use the same formulation as before, by saying: "The extension of the

description is o'". The change appears only when we add to 'o' ? a specify-

ing noun. But this addition serves merely for greater clarity. The new

method does not lead to the result * the extension is not the individual

(or position) o". The situation is, rather, this: the new method in its pri-

mary formulation does not use the terms 'individual ', ' class', ' truth- value'

at all; thus 'o" and 'the individual concept o" are synonymous. In a

secondary formulation those terms might be reintroduced under the new

method, in analogy to the introduction of 'extension' by 23-1 and 23-2.

But then again a combination of any of these three terms with an L-deter-

minate designator is synonymous with the designator alone. Thus, for

example, on this method the phrases 'the individual (or position) o",

'o", and 'the individual concept o" all mean the same. Likewise, if 'A' is

used in M, the phrases 'the class A' (or 'the null class'), 'A', and 'the

property A' all mean the same.

I will not decide here the question of whether the method of taking

L-determinate intensions as extensions is or is not natural. It may suffice

to have shown that this method meets the formal requirements of a solu-

tion to the problem of extensions. For the further discussions in this book,

this method will not be presupposed; most of the discussions will be inde-

pendent of any particular specification of the nature of the entities chosen

as extensions, beyond the general requirement that equivalent designators

have the same extension (<M).

CHAPTER III

THE METHOD OF THE NAME-RELATION

The method of the name-relation is an alternative method of semantical

analysis, more customary than the method of extension and intension. It con-

sists in regarding expressions as names of (concrete or abstract) entities in ac-

cordance with the following principles ( 24): (i) every name has exactly one

nominatum (i.e., entity named by it); (2) any sentence speaks about the

nominata of the names occurring in it; (3) if a name occurring in a true sentence

is replaced by another name with the same nominatum, the sentence remains

true. An examination of the method shows that its basic concept involves an

essential ambiguity (25) and that it leads to an unnecessary duplication of

expressions in the object language ( 26, 27). The most serious disadvantage

of the method consists in the fact that the third of the principles mentioned,

although it seems quite plausible, leads in certain cases to a contradiction if

applied without restriction; we call this contradiction the antinomy of the

name-relation (31). It is not difficult to eliminate the contradiction; various

ways have been proposed by logicians, but all of them have certain drawbacks.

The method of Frege is discussed in detail ( 28-30). Its main feature is the

distinction between the nominatum and the sense of an expression. In many

cases these are the same as what we call the extension and the intension, re-

spectively. However, in contradistinction to these latter concepts, the nomina-

tum and the sense of an expression vary with the context in which the expres-

sion occurs. It is found that Frege's method, if applied consistently, leads to an

infinity of new entities and new expressions as names for them and thus results

in a very complicated structure of the object-language. This holds still more for

the variant of Frege's method proposed by Church. Russell and Quine avoid

the antinomy by not regarding as names certain expressions (although these

expressions are, in our method, L-equivalent to other expressions, which they

do regard as names) ; thus they require an unnecessary restriction of the field

of application of semantical meaning analysis (32). The fact that all forms of

the method of the name-relation lead to complications or restrictions makes it

appear doubtful whether this method is a suitable method of semantical

analysis.

24. The Name-Relation

The customary method of meaning analysis regards an expression as a

name for a (concrete or abstract) entity, which we call its nominatum. The

method, as customarily used, is based on three principles, usually implicit : the

principles of univocality, of subject matter, and of interchangeability.

In chapter i the concepts of equivalence and L-equivalence were intro-

duced and discussed, together with the derivative concepts of the exten-

sion and the intension of an expression. These concepts have been pro-

posed as tools for a semantical analysis of meaning. With our method of

9 6

24. THE NAME-RELATION 97

extension and intension we shall now contrast that method of analysis

which seems to be accepted by many, probably by most, logicians; it is

characterized by using as basic concept the name-relation. In the present

chapter the assumptions underlying this method of the name-relation will

be made explicit, and the consequences of its use investigated. It will be

shown that the method leads to certain difficulties, one of which will be

called the antinomy of the name-relation. Some of these difficulties have

been recognized by several logicians, and various ways have been pro-

posed to avoid them, thus leading to different forms of the method of the

name-relation. An examination of these forms will show that each of them

has serious disadvantages, e.g., an intrinsic ambiguity in the terms used,

an unnecessary multiplication of the entities leading to a complicated

language structure, or unnecessary restrictions in the construction of lan-

guages. It will be seen that the method of extension and intension is free

of the shortcomings which the customary method of the name-relation

shows, at least in its known forms.

The name-relation is customarily conceived as holding between an ex-

pression in a language and a concrete or abstract entity (object), of which

that expression is a name. Thus this relation is, in our terminology, a

semantical relation. Various phrases are used to express this relation, e.g.,

'x is a name for y\ 'x denotes 1 /, 'x designates /, f x is a designation for

y\ 'x signifies y\ etc. In this book I shall sometimes also use, besides 'x

is a name of y\ l x names y j \ this shortened form will not lead to any

ambiguity, since its customary meaning ('a person names an entity') will

hardly occur here. It is often convenient to have a short term for the con-

verse relation; I shall often say, instead of 'the entity named by (the ex-

pression) x\ 'the nominatum of #'; I shall use this term also in formu-

lating the conceptions of other authors who do not use it.

Logicians seem to differ widely with respect to the question of the

kinds of expressions which may be regarded as names. Nearly all will

1 The phrase l x denotes y 1 is often used in a quite different sense, namely, in the case where x

is a predicator for a certain property (e.g., the word 'human') and y is an entity having that

property (e.g., the man Walter Scott). This semantical relation is of a rather special kind,

since it is applicable not to designators in general but only to predicators and, moreover, only

to predicators of degree one, unless one is willing to regard a sequence of entities as the entity

denoted. As a term for this relation, perhaps ( x applies to y' and the corresponding noun 'ap-

plication' might also be considered. In any case, the word 'denotes' is at present used by many

logicians in the sense of the name-relation (see Church, [Dictionary], p. 76). Russell ([Denot-

ing]) has used the word in this sense both for the formulation of his own conception (he uses,

for instance, the term 'denoting phrases' for descriptions and similar expressions) and as a

translation for Frege's term 'bezeichnet' (see below, 28, n. 21). Church likewise uses this

word for the formulation of his conception, which is based on Frege's. Following Russell and

Church, I used the word 'denotes' for the name-relation in the first version of this book. How*

ever, in view of the ambiguity just described, I now prefer to avoid it.

9 8 IIL THE METHOD OF THE NAME-RELATION

include words like ' Napoleon' or ' Chicago' ; perhaps a majority also words

like 'green' (or 'greenness'), 'house', and 'seven'; many also (declarative)

sentences. Let us disregard at present these differences in the domain of

application of the relation and look, rather, at the way in which it is ap-

plied. It seems to me that many logicians use the name-relation for

semantical discussions, that is, for speaking about expressions and their

meanings, in such a way that the following three principles are fulfilled.

If an author fulfils these conditions, then we shall say that he uses the

method of the name-relation, irrespective of the terms he may use

for the relation. Sometimes an author may state the principles ex-

plicitly; more often we shall have to infer from the use he makes of the

relation that he regards these principles as valid.

The Principles of the Name-Relation

24-1. The principle of uniwcality. Every expression used as a name (in a

certain context) is a name of exactly one entity; we call it the nominatum

of the expression.

24-2. The principle of subject matter. A sentence is about (deals with, in-

cludes in its subject matter) the nominata of the names occurring in it.

24-3. The principle of interchangeability (or substitutivity).

This principle occurs in either of two forms:

a. If two expressions name the same entity, then a true sentence re-

mains true when the one is replaced in it by the other; in our termi-

nology (u-ib): the two expressions are interchangeable (every-

where).

b. If an identity sentence '. . . = ' (or '. . . is identical with

or '. . . is the same as ') is true, then the two argument expres-

sions ' . . .' and ' ' are interchangeable (everywhere).

The principle of univocality is, of course, applied only to a well-

constructed language without ambiguities; its fulfilment may, indeed, be

regarded as defining univocality in the sense of nonambiguity. (A lan-

guage of this kind may, for instance, be an artificially constructed system

or a modified English, where the ordinary ambiguities are eliminated,

either by assigning to an ambiguous word only one of its usual meanings

or by replacing it with several terms for the several meanings, e.g., 'prob-

ability/, 'probability/.) The principle of subject matter is rather vague

but sufficiently clear for our purposes. It is sometimes used for making

the third principle plausible. And, indeed, if somebody accepts the first

two principles, he will hardly reject the third. For, if 2l/ and 2U have the

same nominatum and if the sentence . . Sly . . says something true

24. THE NAME-RELATION 99

about this nominatum, then the sentence . . 21* . . , saying the same

about the same nominatum, must also be true. The form b oi the third

principle seems at first glance not to involve the name-relation at all.

But it does so implicitly in the concept of identity sign or identity sen-

tence. The following definitions of these concepts, it seems to me, are

tacitly presupposed in 24-3^

24-4. Definitions.

a. A predicator 2l z is an identity expression (for a certain type) =Df

for any closed expressions (names) 21 ,- and 21* of the type in ques-

tion, the full sentence of 2lj with 21,- and 2U as argument expres-

sions (i.e., 2lj(2ly, 2U) or (2l/) 2^(21*)) is true if and only if $/and

8U name the same entity.

b. is an identity sentence =Df @ is a full sentence of an identity

expression.

On the basis of these definitions, form b of the principle of interchange-

ability follows immediately from form a. Thus, granted the adequacy of

these definitions, form b is just as plausible as form a. I think that

Church 2 expresses the generally accepted conception when he says that

the interchangeability of synonymous expressions, i.e., those which

name the same entity, follows from "what seem to be the inevitable

semantical and syntactical rules for ' = ' ".

We find an example of the method of the name-relation in Frege's pro-

cedure. His distinction between nominatum and sense will later be dis-

cussed in detail ( 28-30). He formulates the principle of interchange-

ability in the first form (24-3a) in this way: 3

24-5. "The truth-value of a sentence remains unchanged if we replace

an expression in it by one which names the same [entity]."

Another example of this method is Quine's analysis in [Notes); he uses

the terms ( designates* and 'designatum' in the sense of our 'names' and

' nominatum'. The principle of interchangeability in the second form

(24-3b) is called by him the principle of substitutivity and is formulated

in this way:

24-6. "Given a true statement of identity, one of its two terms may be

substituted for the other in any true statement and the result will be

true." 4

This principle is not meant by Quine as a conventional rule for an identity

sign in an artificial system but rather as an explicit formulation of a

3 /Review C.], p. 300. * [Sinn], p. 36. (Notes], p. 113.

ioo III. THE METHOD OF THE NAME-RELATION

cedure which is customarily applied in the ordinary word language on the

basis of the customary interpretation of the words. Quine distinguishes

between the designation of an expression and its meaning; this distinction

is, as Church 5 has seen, in some respects very similar to Frege's.

The differences between the method of the name-relation and the

method of extension and intension will later be discussed in detail. Here I

wish to make only a few remarks in connection with the three principles.

The concept of the extension of an expression is, as we shall see later, in

some respects similar to the concept of its nominatum. Therefore, let us

see to what extent analogues of the three principles hold for the concept

of extension. The analogue of the principle of univocality holds; every

designator has exactly one extension. The analogue of the principle of sub-

ject matter holds, too, but with restrictions. In general, a sentence con-

taining a designator 2l/ may be interpreted as speaking about the exten-

sion of 2ly. However, it may be interpreted alternatively as speaking about

the intension of 8,-; and, as we shall see later, the latter interpretation is

sometimes more appropriate. The decisive difference emerges with respect

to the principle of interchangeability. For extensions, instead of the ana-

logue of 24-3a, only the restricted principle 12-1 holds. It says that, if two

expressions have the same extension, in other words, if they are equiva-

lent, then they are interchangeable in extensional contexts. The principle

243b speaks about identity. However, on the basis of the method of ex-

tension and intension, we cannot simply speak of identity but must dis-

tinguish between identity of extension and identity of intension, in

other words, between equivalence and L-equi valence. Therefore, instead

of the one principle 24-3b for identity, we have in our method two

principles, one for equivalence and the other for L-equivalence; these are

1 2-1 and 12-2.

25. An Ambiguity in the Method of the Name-Relation

A predicator in a word language (e.g., 'gross' in German) or in a symbolic

language (e.g., an abstraction expression in Quine's system) may be regarded as

the name of a class but also as the name of a property. This shows an intrinsic

ambiguity in the name-relation. Its consequences will be discussed later.

I shall now examine in more detail some features of the method of the

name-relation, and especially try to show that the basic concept of this

method is not so simple, clear, and unambiguous as it is usually supposed

to be.

It seems generally to be assumed that, if we understand an expression,

s [Review Q.], p. 47-

25. AN AMBIGUITY IN THE NAME-RELATION 101

we know at least to what kind of entities its nominatum belongs, and

also in some cases which entity is the nominatum, although in other cases

factual knowledge is required for this. For instance, if we understand

German, then we know that the word 'Rom' is a name of the thing Rome,

and that 'drei' is a name of the number Three. In the case of 'der Autor

von Waverley' we know at least that it names, if anything, a (physical)

thing; and if we have sufficient historical knowledge, we know that it is a

name of the man Walter Scott. Analogously, in the case of 'die Anzahl der

Planeten', we know at least that it names a number and, with the help of

astronomical knowledge, we know that it names the number Nine. Gen-

erally speaking, given a full understanding of the language in question

and, in particular, of some name in it and, in addition, all the factual

knowledge relevant to the case in question, we should expect that there

could be no doubt or controversy as to the nominatum of the name. How-

ever, it will now be shown that this is, in general, not the case.

Let G be a part of the German language, restricted to declarative sen-

tences, with all dubious expressions and ambiguities eliminated (see ex-

planation of 24-1) and, in particular, with the word 'gross' confined to its

literal meaning concerning spatial extension. We imagine two logicians,

L x and L 2 , interested in the semantical analysis of G. Before they begin

the theoretical analysis, they make certain in a practical way that they

have the same interpretation or understanding of the language G; for in-

stance, each agrees with any translation the other makes of a sentence of

G into English. Then they begin their semantical analysis of G, according

to the method of the name-relation based on the three principles (24-1,

2, 3). They examine the sentence in G: 'Rom ist gross'. They have no

doubt and no disagreement as to its meaning; this is shown by the fact

that both agree that its translation into English is: 'Rome is large'. Now

they apply to the expressions in the given sentence the analysis in terms

of the name-relation. Both agree that ' Rom' in G is a name of the thing

Rome. But now suppose that with respect to the word 'gross' (or the

phrase 'ist gross') the following controversy arises: L x says: "The sen-

tence ' Rom ist gross' means that Rome belongs to the class Large. Hence

it is about the thing Rome and the class Large. Therefore, according to

the principle of subject matter, 'gross' is a name of the class Large; and

hence, according to the principle of univocality, it cannot be a name of

any other entity". Against this, L 2 says: "The given sentence means that

Rome possesses the property Large. Hence it is about the thing Rome and

the property Large. Therefore, according to the principle of subject

matter, 'gross' is a name of the property Large; and hence, according to

102 III. THE METHOD OF THE NAME-RELATION

the principle of univocality, its nominatum cannot be any other entity ;

in particular, it cannot be the class Large."

We might perhaps try to reconcile the two logicians by pointing out

that it does not really matter whether they say ' the sentence means that

Rome belongs to the class Large' or 'the sentence means that Rome has

the property Large', since these two assertions are both true and differ

merely in their formulation. But, even if the two logicians were willing

to agree with us on this point, the controversy concerning the nominatum

of 'gross' would not be solved. Here, in distinction to the question concern-

ing the whole sentence, they cannot simply agree that they are both

right, that it does not matter whether they say that the nominatum is the

class Large or that it is the property Large; for they agree in affirming

the principles of the name-relation; therefore they must agree, according

to the principle of univocality, that 'gross' (in G) can have only one

nominatum. And, further, they agree that the class Large is not the same

as the property Large; they agree generally in recognizing the distinction

between a property and the corresponding class, as expressed, for in-

stance, by 4-7 and 4-8.

Perhaps somebody will suggest to the two logicians that their insoluble

controversy is due merely to the choice of an unsuitable object language;

that a natural language like G, even after the elimination of obvious

ambiguities, is not precise enough for univocal semantical analysis; and

that, therefore, they should restrict their analysis to a well-constructed

symbolic system with exact rules. I doubt whether the controversy is

caused merely by the imperfections of G; but let us see what will result

when the two logicians follow the suggestion. Let ML be the system con-

structed by Quine in [M.L.], and ML 7 the system constructed out of ML

by the addition, first, of the defined signs which Quine introduces in his

book but does not count as parts of his system and, second, of a few non-

logical atomic matrices. The two logicians agree on the following interpre-

tation of the system ML': the primitive notation of ML is interpreted in

accordance with Quine's explanations; on this basis the interpretations of

the defined signs in ML 7 are determined by their definitions; for the inter-

pretation of the nonlogical atomic matrices, the following rule (similar to

1-2) is laid down:

2&-1. Rules of designation (for ML').

a. 'Hx' is the translation of 'x is a human thing'.

b. -'Fx'x is a featherless thing'.

c. 'Btf'-^ 'x is a biped thing'.

25. AN AMBIGUITY IN THE NAME-RELATION 103

'Thing' is here meant in the sense of ' physical thing'. ML 7 is interpreted

in such a way that things are taken as individuals in Quine's sense. 6 Ac-

cording to the rules 25-1, the three atomic matrices mentioned are fulfilled

only by entities which are things, and hence both individuals and elements

in Quine's sense. 7

The two logicians agree not to take the signs 'HP, 'F', and 'B', intro-

duced by 25-1, as names, because it is obvious that otherwise they would

immediately get into the same controversy concerning the nominata as

they did with respect to the word ' gross 5 in G (compare the translations

4-2 and 4-3 of 'Hs y ). They agree to take as names only those expressions

which Quine calls closed terms, and among them especially the closed

abstraction expressions, i.e., expressions of the form '(. . x . .)' without

free variables.

Now the two logicians examine the following sentence in ML':

( (H.x) C x(B%y, which we call @ x . There is no doubt and no di^agree-

ment between them as to its meaning. They agree that, according to the

rules of ML', 8 @ x is L-equivalent to ' (x) (H# 3 Ex) ' and hence may be

translated into 'for every x, if x is human then x is a biped' (see 4-4; we

assume here that 'human' means as much as 'human thing', and 'biped'

as 'biped thing'). However, as soon as they raise the question as to what

is the nominatum of the abstraction expression ( (HLx)', as it occurs in

@r, a controversy starts which is perfectly analogous to the earlier one

with respect to ' gross' in G, in spite of the fact that we have here the exact

system ML'. L t says: "We agree about the meaning of *, namely,

that it is translatable as just stated; but it is likewise translatable into

' the class Human is a subclass of the class Biped' (4-6). Hence @ x is about

the class Human and the class Biped. Therefore, according to the principle

of subject matter, '(H#)' is a name of the class Human; hence, accord-

ing to the principle of univocality, it cannot be a name of any other

entity". L 2 replies: "Since @ x is translatable as previously stated, it is

likewise translatable into: 'the property Human implies (materially) the

property Biped' (4-5). Hence @ x is about the property Human and the

property Biped. Therefore, according to the principle of subject matter,

'(H#)' is a name of the property Human; and hence, according to the

principle of univocality, it cannot be a name of any other entity; in par-

ticular, it cannot be a name of the class Human." Since both logicians

agree that the class Human is not the same as the property Human, they

<[M.L.],p. 135- ' /&*., p. 131.

8 In particular, the definitions Dai and DQ in [M.L.], pp. 185 and 133, apply here; note also

the above remark on 25-1 concerning things.

io 4 HI. THE METHOD OF THE NAME-RELATION

must regard their statements concerning the nominatum of '^(H#)' as

incompatible on the basis of the principle of univocality. In support of his

statement, L x may point to the fact that Quine, the author of the system

ML, says himself that the terms are names of classes, 9 that ' C ' is a sign

of class inclusion, 10 and that the whole language deals with classes. L 2 may

reply that he admits that the mode of speech used by Quine and by L x

can be applied consistently; his point is that the same holds for the other

mode of speech, which he uses. However, what makes the controversy

insoluble is this: The divergence between Lj and L 2 , which is at the start

nothing but a difference in the mode of speech, namely, between the trans-

lations of @ x in terms of classes and in terms of properties, leads, on the

basis of the principles of the name-relation, to two statements which are

incompatible, namely, those concerning the nominatum of '&(Hxy.

Now L x discovers a new way which, he thinks, must lead to an un-

ambiguous solution of the puzzling problem. Since the difference between

classes and properties has its root in the difference of the identity condi-

tions, an identity sentence Sl t = SI/ in ML' should be analyzed where

91 and SI/ are abstraction expressions; by determining the truth-condi-

tion of this sentence, we should be able to see, he thinks, whether the two

expressions Sl and SI/ are names of classes or of properties. Therefore, he

proposes to examine the following sentence in ML' : ' A (Hx) = (Fx Ex) ',

which we call @ 2 . There is again complete agreement between the two

logicians as to the meaning of this sentence. They agree that, according to

the rules of ML', 11 the sentence @ 2 is L-equivalent to ' (x) (H# = Fx Ex) '

and hence, on the basis of the biological fact 3-6, @ 2 is true. Further, both

agree that the two classes in question are, in fact, identical (see 4-7),

while the two properties are not (see 4-8). Now Lj argues as follows: "The

identity sentence @ 2 can only refer to the two classes; for, if it referred

to the two properties, it would be false because they are nonidentical". L 2

replies: " You, like the author of the system, take ' = ' as a sign of identity

of classes. I admit that this is in accordance with the rules of the system

ML 7 . But then, l = ' may just as well be called a sign of equivalence of

properties (like t ss' in S x ; see remark on 5-3). And since the two proper-

ties in question, though not identical, are indeed equivalent (see 5-5),

@ 2 is also true on the basis of this analysis, which interprets the two ab-

straction expressions as names of properties."

Lj will perhaps ask whether the character of ' = ' in the system ML' as

[M.LJ,p. 119.

">/taf.,p. 185.

" See, in particular, the definitions Dio and DQ in [M.L.], pp. 136 and 133.

25. AN AMBIGUITY IN THE NAME-RELATION 105

a genuine sign of identity and not merely a sign of equivalence, like * s='

in S x , is not assured by the fact that ML' contains a principle of inter-

changeability (called principle of substitutivity of identity"). To this, L 2

will give a negative answer. Interchangeability on the basis of i = ' holds

likewise in S x (see i2-3a); thus, in this respect also, ' = ' in ML' is like

' ==' (between predicators) in Si. It is true that general interchangeability

on the basis of ' ss' does not hold in some systems, for example, in S 2 ; but

it holds in all extensional systems (i2-3a). Thus the effect of the principle

of interchangeability in ML 7 (and ML) is simply to make ML' (and ML)

an extensional language like S x ; the principle prevents the introduction

into ML' of intensional predicators or connectives, for instance, of a sign

of logical necessity (like *N' in S a , see n, Example II). But it does not

prevent in any way the interpretation of abstraction expressions in ML'

(or ML) as names of properties.

Now let us draw the conclusion from our examination of the contro-

versy between the two logicians. Note that this controversy is not an

instance of the well-known multiplicity of interpretations, that is, of the

fact that for a given logical system (calculus) there are, in general, several

interpretations, all of them in accordance with the rules of the system.

Lj and L 2 apply the same interpretation to their object language G, and

then likewise to the language system ML'. Even when L x says that the

sign ' = ' in ML' is a sign of identity of classes while L 2 says that it is a

sign of equivalence of properties, this does not show a difference in inter-

pretation but merely a difference in the choice of semantical terms used

for describing one and the same interpretation; for equivalence of proper-

ties is just the same as identity of classes (or, speaking more exactly, ' the

properties expressed by two predicators are equivalent' and 'the corre-

sponding classes are identical' are L-equivalent sentences in M). That L,

and L 2 apply the same interpretation to ML' (as well as to G) means that

to any given sentence in ML' they attribute the same meaning or, in

other words, the same truth-condition. The decisive point is rather this:

In spite of their agreement in the interpretation, it is possible for L x

and L 2 to maintain different conceptions as to what are the nominata of

the names occurring conceptions which are incompatible with each

other, though each is consistent in itself. This shows, it seems to me, that

the method of the name-relation involves an intrinsic ambiguity, inas-

much as the fundamental term of this method, namely, 'is a name of,

"Quine, [M.L.], 29, *2oi; for the corresponding principle with respect to the word

language, see above, 24-6.

io6 III. THE METHOD OF THE NAME-RELATION

is ambiguous, although it is generally believed to be quite clear and un-

ambiguous. This is not to say that, in general, a logician uses these

terms ambiguously, but only that several logicians may use them in

different ways. For instance, Lj uses the method consistently and un-

ambiguously, and so does L 2 . The trouble is that, if one logician thinks

that the results which he has found on the basis of his conception must be

accepted by everybody else, he is mistaken, because it may be that the

results do not hold for another conception of the name-relation.

We have discussed the ambiguity only with respect to predicators,

where either classes or properties may be taken as nominata. Analogously,

for a designator of another kind, either its extension or its intension may

be taken as its nominatum. Thus there are, in fact, many more than two

ways for using the method of the name-relation. And the multiplicity of

ways is, further, considerably increased by the fact that some logicians

take some predicators as names of classes and other predicators of the

same type as names of properties (see 26) ; and that some logicians even

take the same expression as a name of an extension in one context and in

another as a name of an intension (for example, Frege, see below, 28,

29). For the present, it will suffice to point out the great multiplicity of

different ways of using a method of the name-relation, in other words, the

many different senses in which the term 'name' or similar terms are used.

Some of these ways will be discussed later in order to show the complica-

tions which they involve.

26. The Unnecessary Duplication of Names

Many systems have different names for properties and for the corresponding

classes. This is discussed with respect to examples from the system of Principia

Mathematica. Analyzing these names by the method of extension and intension,

we find that a name for the property Human and a different name for the class

Human have not only the same extension but also the same intension. There-

fore, the duplication of names to which the method of the name-relation leads

is superfluous.

Another consequence of the customary way of using the method of

the name-relation will now be discussed. The principle of subject matter

(24-2) says that if a sentence contains a name of an entity, then it says

something about this entity. And the method is usually conceived in such

a way that, conversely, if a sentence is intended to be about a certain

entity, then it must contain a name of this entity. Then it follows, in

virtue of the principle of univocality (24-1), that, in order to speak about

two different entities, we have to use two different expressions as their

names.

26. THE UNNECESSARY DUPLICATION OF NAMES 107

On the basis of the method of extension and intension, on the other

hand, the situation is quite different. A designator is here regarded as

having a close semantical relation not to one but to two entities, namely,

its extension and its intension, in such a way that a sentence containing

the designator may be construed as being about both the one and the

other entity. Thus here, if a sentence is intended to speak about an entity

which is an extension, an expression is needed whose extension is that

entity; and if we wish to speak about an entity which is an intension, an

expression is needed whose intension is that entity. Therefore, in order to

speak first about a certain intension and then about the corresponding ex-

tension, this method requires only one expression, while the method of the

name-relation would require two and hence lead to an unnecessary dupli-

cation in symbolism.

This duplication can best be made clear in the case of predicators. The

method of extension and intension needs only one predicator to speak

both about a certain property and about the corresponding class. The

method of the name-relation in its customary form, however, needs for

this purpose two different expressions, a property name and a class name.

As an example, let us take the symbolic system PM constructed by White-

head and Russell in [P.M.]; PM includes not only the primitive signs but

also the (logical) signs introduced by the definitions as given by the au-

thors. Let PM' consist of PM and, in addition, a few nonlogical predica-

tors or atomic matrices. Let PM' be interpreted in the following way: The

primitive logical signs are interpreted in accordance with the explanations

of the authors of [P.M.]; the interpretations of the defined signs are then

determined by their definitions; the nonlogical signs are interpreted by

25-1 as a rule of designation for PM'.

The system PM' uses different expressions as names for properties

(construed as prepositional functions) and as names for classes. Take, as

examples, the following four statements concerning two pairs of expres-

sions in PM':

26-1. 'H^' is a name of the property Human.

26-2. '&(Hx)' is a name of the class Human.

26-3. 'Ftf B' is a name of the property Featherless Biped.

26-4. *(Fx Ex) 9 is a name of the class Featherless Biped.

[For the present discussion we may leave aside the fact that Russell does

not assume that there are classes as separate entities, in addition to

properties; he introduces class expressions by contextual definitions on

the basis of property expressions. The problem of this and the converse

108 III. THE METHOD OF THE NAME-RELATION

reduction will be discussed later (33). For our present problem it is suf-

ficient that an author speaks in his metalanguage both of properties

(qualities, propositional functions of degree one) and of classes (distin-

guished in the customary way) ; that he uses in his object language two

different kinds of expressions; and that he declares that those of the first

kind are meant as expressions of properties and those of the second kind

as expressions of classes.]

The four statements given express results of a semantical analysis of

certain expressions in PM', according to the method of the name-relation.

If, instead, we analyze PM 7 by the method of extension and intension, we

arrive at the following results, which contain counterparts of the earlier

results, supplemented by new ones. Instead of 26-1, we have here:

26-6. The intension of 'EW is the property Human.

To this statement, however, another statement is added, which follows

from it:

26-6. The extension of 'H#' is the class Human.

Instead of 26-2, we have here:

26-7. The extension of 'f (Ho:)' is the class Human.

To this we add:

26-8. The intension of ':(H#)' is the property Human.

While 26-6 follows directly from 26-5, the same is not true for 26-8 and

26-7 ; every intension uniquely determines an extension, but the converse

does not hold. Statement 26-8 is based, rather, on the rule 25-ia and

the circumstance that, according to the rules of PM', the sentence

'(y)\yc(Hx) = Hy]' is L-true in PM'. The results corresponding to

26-3 and 26-4 are, of course, analogous.

Thus the outcome, from the point of view of our method, is that the

two expressions t Hff and '^(H#)' in PM' have the same extension and

also the same intension. Therefore, it is unnecessary to have both forms

in the system. The two expressions are, in a certain sense, L-equivalent

predicators. It is true that one of them cannot simply be replaced by the

other; this is the effect of certain restricting rules concerning the two

kinds of predicators. First, there is the following unessential difference,

which is merely an accidental syntactical feature of the systems PM

and PM 7 . The rules require that an argument expression for a predicator

of the first kind (e.g., 'H' or 'Hf ') succeeds it (resulting in 'Hs'), while one

for a predicator of the second kind precedes it with a copula ' e' interposed

(e.g., 's#(H#)'). Another difference is more important. It concerns

26. THE UNNECESSARY DUPLICATION OF NAMES 109

identity sentences built with ' = '. Consider the following two sentences as

examples:

26-9. l A(Hx) = A(Fx Ex) 9 .

26-10. 'fit = F.E'.

According to the explanation given in [P.M.], the sentence 26-9 says that

the two classes in question are identical; hence this sentence is true (see

4-7). On the other hand, the sentence 26-10 says that the two properties

in question are identical; hence this sentence is false (see 4-8). Thus, 26-9

is in notation and meaning just like a sentence in ML' previously dis-

cussed (@ 3 in 25). Likewise, its L-equivalence to '(x)(Hx ^Fx Ex)'

holds for PM'. Therefore, the contention of L 2 that ' = ' in 26-9 is like ' as'

in Si (or S 2 ) and, hence, is simply a sign of equivalence applies here as

well. On the other hand, ' = ' in 26-10 is a sign of identity or L-equivalence

of properties; it is therefore, in distinction to ' =' in 26-9, a nonexten-

sional sign. (This is recognized by Whitehead and Russell.) 13 Hence it

cannot correspond to any sign in the extensional language S,; but it cor-

responds exactly to the modal sign ' m ' in S 2 , which will be introduced

later (see 39-6; accordingly, the false sentence 26-10 is L-equivalent to

42-2bA without the sign of negation). Thus the method of extension and

intension by no means overlooks the difference between 26-9 and 26-10.

On the basis of this method, in distinction to the method of the name-

relation, the first components in the two sentences (i.e., the predicators

'H' and *#(H#) J ) are equalized in certain respects, and so are the second

components. Nevertheless, the difference is preserved because the occur-

rences of ' = ' in 26-9 and in 26-10 are here construed as having different

meanings. The first is interpreted as a sign of equivalence or, in other

words, of identity of extensions; the second as a sign of L-equivalence

or, in other words, of identity of intensions.

We see that the situation with respect to the two methods under dis-

cussion is this: At the beginning, there is merely a difference of pro-

cedure in describing the semantical features of given language systems.

The customary method does it in terms of nominata; our method does it,

instead, in terms of extensions and intensions. At first glance, one might

think that both methods were neutral with respect to the structure of the

language systems, in the sense that either method is as applicable to any

system as the other. If so, the choice of the one or the other method of

semantical analysis would not have any effect upon the choice of a struc-

ture for a system to be constructed. However, this is not so. According to

'3 [P.M.], I, 8 4 .

no m. THE METHOD OF THE NAME-RELATION

the first method, the two expressions 'EW and ^(Hx)' are said to have

different nominata; and this circumstance is then naturally regarded as

justification for the decision to incorporate both expressions into the sys-

tem, as is done in the system PM 7 . According to the second method, on the

other hand, the two expressions are said to have the same extension and

the same intension. This leads to the view that the inclusion of both

would be an unnecessary duplication, and hence to the decision to con-

struct the system in such a way that it contains, instead of those two ex-

pressions, only one, as in the systems S t and S 2 (and in many systems con-

structed by other logicians 14 ). Corresponding to the two expressions in

PM 7 , S x and S 2 have the one predicator c (\x)(Hxy (of course, either of

the two notations in PM 7 could be taken, instead, just as well). That we

could do in previous examples (e.g., 3-8) without lambda-expressions was

merely due to the simplicity of the examples. In general, an identity sen-

tence for classes in PM 7 (like 26-9) will be translated into Si and S 2 in the

form s (\x)(. . .) 35 (X#)( --- )', and the corresponding identity sentence

for properties (like 26-10) will be translated into S a in the form

*(X#)(. . .) s (X#)( --- )', with the same two lambda-expressions as the

first sentence.

Our conclusion that the duplication of predicators in PM and PM' is

unnecessary holds likewise for systems which use two different kinds of

operators for class abstraction (e.g., '(..%. .)') and for functional ab-

straction, 15 that is, formation of abstraction expressions for properties,

here construed as propositional functions (e.g., '(\x)(. . x ..)') Here

again, if the same matrix ( . . x . .' occurs as scope in both expressions, they

have the same extension and the same intension; however, they have dif-

ferent conditions of identity. Thus they are analogous to ^(Hte)' and

( K' y respectively, in PM 7 .

Since the choice of a semantical method and the choice of a form of

language are interconnected, we may also reason in the inverse direction:

our preference for a language structure may influence our preference for

one of the two semantical methods. If a language system with only one

kind of predicator is, in fact, not only as effective (for the purposes of both

mathematics and empirical science) as a system with two kinds like PM 7

** That it is unnecessary to have special class expressions in addition either to simple

predicator signs and their combinations or to property expressions has already been seen by

several logicians. Concerning the historical development of this insight and concerning the

possibility of a form of language without special class expressions, see [Syntax], 38 and 37.

The discussion in the present book confirms this conception by basing it on a more general con-

ception, namely, that of the method of extension and intension for designators in general.

x * See, for instance, Church, [Dictionary], p. 3.

27. NAMES OF CLASSES in

but also simpler and hence more convenient, then I think the method of

the name-relation must be regarded as at least misleading, if not in-

adequate.

27. Names of Classes

A name for a class must be introduced by a rule which refers to exactly one

property; otherwise, the meaning of the new sign and of the sentences in which

it occurs is not uniquely determined. This shows that a semantical rule for

a sign determines primarily its intension; only secondarily, with the help of

relevant facts, its extension. The customary use of different kinds of variables

for properties and for classes is shown to be as unnecessary as that of different

names. The duplication of names and variables on the first level leads to a still

greater multiplication of names and variables on higher levels. The concepts of

mathematics can be defined without the use of special class expressions and

class variables. This is shown by definitions of '2* and of 'cardinal number*.

We have seen in the preceding section that those expressions in the sys-

tem PM' which are regarded as names of certain classes by the authors of

the system do not only have these classes as their extensions but, at the

same time, have certain properties as intensions (see 26-8). Here the ques-

tion might be raised as to whether it could not happen in some system that

a predicator has only an extension, not an intension; in other words, that

it refers to a class without referring to any of those properties which have

that class as an extension. I think that this is not possible in a semantical

system, that is, in a system whose interpretation is completely given. To

begin with, it is not possible to refer to a class without referring to at least

one of the corresponding properties. This holds, even if the class is speci-

fied by an enumeration of its members, e.g., by a phrase like 'the class of

the individuals a, b, and c', or in the symbolic language S x : ' (\x)[(x = a)

V (x = b) V (x 2= c)]\ This predicator does not lack an intension; it

is the property of being (identical with) either a or b or c. The feeling

which we might have, that this is not a property in the same sense as

properties like Blue or Human, is right; it is (if 'a', 'b', and V are inter-

preted as L-determinate constants for positions in an ordered domain,

19) a positional, not a qualitative, property; in our earlier terminology

(22), it is an L-determinate property; but, in any case, it is an intension.

One might perhaps think a class name without an intension could be

introduced into a system by stipulating that it is to be a name for the class

which such and such equivalent properties have in common; this reference

to several properties would have the effect that none of them would be

the intension of the name. Consider, for instance, the following as a

semantical rule for the class name 'K J in S x :

ii2 III. THE METHOD OF THE NAME-RELATION

27-1. * K' is to be a name at once for the class Human and for the class

Featherless Biped, which is the same class.

This rule does not involve an inconsistency, since the classes mentioned

are indeed identical (see 4-7). However, it is not sufficient as a semantical

rule for 'K'; the interpretation of 'K' or, in ordinary words, its meaning,

is not completely given by 27-1 but merely confined to certain possibilities.

It is true that this rule, together with rules for the other signs in Sj and

knowledge of the relevant facts, is sufficient to determine the truth-value

of any sentence in S x in which 'K' occurs. For instance, ' Ks' is found to be

true in S x on the basis of the historical facts which make the two sentences

'Hs' and *Fs Bs' true. The decisive point is that, although the truth-

values, the extensions, of the sentences containing 'K' are determined,

their intensions are, in general, not. For instance, it remains undetermined

what proposition is expressed by 'Ks'; is it the same as that expressed by

'Hs', or by 'Fs Bs', or by their disjunction, or their conjunction? These

are four different propositions. To express it in other terms, the given

K-rule (27-1), together with the rules for other signs, does not suffice for

the application of the L-concepts to the sentences containing 'K'. For in-

stance, it is not determined whether ' Ks = Hs' is L-true or F-true. There-

fore, strictly speaking, on the basis of the K-rule and the other rules we

cannot understand sentences like 'Ks' or 'Ks = Hs', although we can

establish their truth-values. The reason for the objection here raised

against the K-rule is not the fact that it introduces ' K' as a name for a

class, but rather the fact that it does not do this by reference to exactly

one property. In contradistinction to 27-1, the following would be a com-

plete semantical rule for 'K':

27-2. 'K' is to be a name for the class Human.

For this would say the same as: ' 'K' is to be a name for the class which

is the extension of the property Human' ; and this, in turn, may be under-

stood as saying: ' ' K' is to be a sign whose intension is the property Hu-

man; therefore, its extension is the class Human.' The first part of this

last sentence would suffice as a rule; the second part (' therefore . . .') is

a semantical statement following from the rule. This shows that the

semantical rule for a sign has to state primarily its intension; the extension is

secondary, in the sense that it can be found if the intension and the rele-

vant facts are given. On the other hand, if merely the extension were

given, together with all relevant facts, the intension would not be unique-

ly determined.

27. NAMES OF CLASSES 113

We have seen in the preceding section how the method of the name-

relation leads to the use of two kinds of predicators within the same type

(for example, level one and degree one). On the basis of this method, es-

pecially of the principle of subject matter, this duplication of predicators

is regarded as necessary if we wish to speak both about classes and about

properties. An analogous situation arises with respect to variables. For

speaking about particular entities, names are used; and thus the method

leads to class names and property names. On the other hand, for speaking

about entities of some kind in a general way, variables are used; thus here

the method of the name-relation leads to the introduction of two kinds of

predicator variables for the same type; the values of variables of the first

kind are classes, the values of those of the second are properties. Thus,

for example, the system PM uses V, '/?', etc., as class variables and

'<', '\l/', etc., as variables for properties (prepositional functions). From

the point of view of the method of extension and intension, this duplica-

tion is analogous to that of closed predicators and just as superfluous. In

the system PMV^(H#)' is a value expression for V. We have seen that,

on the basis of our method, '^(Ha?)' has not only an extension, namely, the

class Human (see 26-7), but also an intension, the property Human (see

26-8). Therefore, not only does the class Human belong to the value ex-

tensions of 'a' according to 10-1, but it is also the case that the property

Human belongs to the value intensions of 'a' according to 10-2. But

exactly the same holds for '$' because of 26-6 and 26-5, since 'H^' is a

value expression for * <t> J . Thus both kinds of variables have the same value

extensions, namely, classes of individuals, and the same value intensions,

namely, properties of individuals. Therefore, the duplication of variables

is as unnecessary as that of closed predicators. It is sufficient to use one

kind of variable for the predicator type in question; their value extensions

are classes, their value intensions are properties (see 10). Therefore, they

serve for speaking in a general way both about classes and about proper-

ties. [Thus, for instance, with respect to the examples in 10 preced-

ing 10-1, sentences of both the forms (ii) and (iii) are translated into

a symbolic language with the help of the same variable l f in the form

'(a/) (/..);]

The situation with respect to variables of other kinds is theoretically

the same but practically different; while many logicians use different vari-

ables for classes and for properties, it seems that hardly anybody pro-

poses to use different variables for propositions and for truth-values, or

ii 4 III. THE METHOD OF THE NAME-RELATION

different variables for individuals and for individual concepts. Thus our

method does not deviate here from the customary procedure.

If the reasoning on the basis of the method of the name-relation, which

leads to the use of two kinds of predicators within the simplest type, is

carried to higher levels, then it results in an immense multiplication of

predicators of the same type. From our point of view this multiplication

is as unnecessary as the duplication with which it starts. For the sake of

simplicity, let us restrict the discussion to predicators of degree one, that

is to say, let us speak only of classes and properties, leaving relations

aside. If on the first level a distinction is made between names of classes

and names of properties, then, on the second level, four kinds of predica-

tors must be distinguished, namely:

names of classes of classes

names of properties of classes

names of classes of properties

names of properties of properties

To form examples in the system PM, let us start with the following

matrix, which contains the class variable 'a' as the only free variable:

<(3s)(3y)[~ (x = y) . (z)(zea. s : z = x. V .z = y)]\

As shorthand for this in the subsequent examples, let us simply write

' . . a . .'. This matrix says that the class a has exactly two members, or,

as we may say for short, that a is a pair-class. Let ' . . <t> . .' be taken as

shorthand for that matrix in PM which is analogous to the one mentioned

but which contains the property variable ^' instead of 'a' (that is to say,

'zea' is replaced by ( <t>z'). Hence, ' . . < . / says that there are exactly

two individuals which have the property 0, or, as we may say, that < is

a pair-property. Now let us examine the following four expressions in

PM:

(i) '(. . a . .)>,

(ii) '. . a . .',

(iii) '$(. . . .)',

(hr) '..*..',

where the dots indicate the matrices just described. Expression (i) is a

name of the class Pair-Class and hence belongs to the first of the four kinds

of predicators on the second level mentioned above; (ii) is a name of the

property Pair-Class and hence belongs to the second kind; (iii) is a name

27, NAMES OF CLASSES 115

of the class Pair-Property and hence belongs to the third kind; (iv) is a

name of the property Pair-Property and hence belongs to the fourth kind.

The nominatum of (i), that is, the class of all classes which have exactly

two members, is in PM taken as the cardinal number Two, and therefore

1 2* is introduced as abbreviation for (i). The expressions (ii), (iii), and (iv)

do not, it seems, actually occur in the book [P.M.], but they are formed

according to the rules of the system PM. The four expressions belong to

the same type; they are predicators of level two and degree one. If we were

to construct, on the basis of our method of extension and intension, a

system with a predicator variable '/, then it would contain, instead of the

four expressions of PM, only one, namely, '(A/) ( / )'

The multiplication of kinds of predicators on the basis of the method

of the name-relation increases with higher levels. On the level n, there are

2* different kinds of predicators within the same type. They are supposed

to be required as names of 2 n kinds of entities. On the basis of our method,

there is only one kind of predicator in each type; and the 2 n corresponding

predicators in the other method are here replaced by one.

On the basis of our method, all the mathematical concepts can be

defined in a way that is analogous to that in [P.M.] except that no special

class expressions and class variables are used. Let us suppose that 5 is a

system which contains not only individual variables but also variables for

which predicators of various levels can be substituted, say '/' and 'g j as

variables of level one and ' m? and 'n' as variables of level two. Then, for

example, the cardinal number Two can be defined in 5 as a property of

properties as follows:

27-3. V for < (X/)[(a^)(3y)[-(a; m y} . (z)(fz m (z ** x) V (* m y))]]'.

It is true that a certain requirement of extensionality must be fulfilled

by any explicatum for the concept of cardinal number in order to be ade-

quate. However, it is not necessary to require that the cardinal numbers

be extensions; it is sufficient to require that any statement attributing a

cardinal number to a given property (or class) be extensional. This re-

quirement is also fulfilled by our method, because the cardinal numbers

are here defined as properties of properties which are extensional. That, for

example, 2 as defined by 27-3 is an extensional property of properties is

not explicitly stated in the definition, but it is seen from the fact that the

following sentence is provable with the help of the definition 27-3 :

ii6 III. THE METHOD OF THE NAME-RELATION

The general concept of cardinal number can likewise be defined in the

system 5 without the use of special class expressions. While Russell

explicates cardinal numbers as classes of classes, Frege takes them as

classes of properties. Since we wish to take them as properties of proper-

ties, we may follow Frege's procedure half the way. We say, like Frege, 16

that the property / is equinumerous to the property g (in symbols:

*Equ(/, g)') if there is a one-to-one correlation between those individuals

which have the property /and those which have the property g. Then we

define the cardinal number of the property / as the property (of second

level) Equinumerous To /:

27-4. <Nc'/' for '(\g)[Equ(gJ)]\

[Frege takes as definiens not 'the property Equinumerous To/, but 'the

extension of the property Equinumerous To/', which means the same as

'the class Equinumerous To f. Now it is interesting to see that Frege

adds to this definition a footnote (op. cit., p. 80) which says: "I believe

that instead of ' extension of the property' we might say simply ' property '.

But two objections would be raised: ... . I am of the opinion that both

these objections could be removed; but that might lead here too far."

Thus Frege considers here the simpler procedure which we now adopt. He

seems to regard it as feasible but does not pursue it any further. In his

later work 17 he again defines cardinal number in the way stated above,

without even mentioning an alternative possibility. His chief reason for

regarding cardinal numbers as classes of properties rather than as proper-

ties of properties seems to be his view 18 that cardinal numbers are inde-

pendent entities, in combination with his general conception that classes

are independent entities, while properties are not. However, I find his

reasoning on this question not quite clear and far from convincing.]

Finally, we define, like Frege, 19 'n is a cardinal number' (in symbols:

'NC(w)') by ' there is a property/ such that n is the cardinal number of/ :

27-5. 'NC for'(*)[(3/)( = Nc'/)]'.

Suppose that the properties /and g are equinumerous. Frege shows on

the basis of his definitions that in this case the cardinal number of / is

equal to that of g. The latter statement is interpreted by him as saying

that the class Equinumerous To /is the same as the class Equinumerous

To g. Thus he explicates equality of numbers as identity. Here our defini-

16 [Grundlagen], pp. 73-79, 83-85. l8 [Grundlagen], pp. 67-72.

*' [Grundgesetze], I, 57. t Ibid., p. 85.

27. NAMES OF CLASSES 117

tion 27-4 may seem to involve a difficulty, because, even if / and g are

equinumerous, the property Equinumerous To /need not be the same as

the property Equinumerous To g. However, although these two proper-

ties, which in our method are regarded as cardinal numbers, are not

identical, they are equivalent (in the sense of 5-3; see the example 5-5).

Thus the difficulty disappears if we explicate equality of numbers as

equivalence rather than as identity and hence symbolize it by ' s'. Thus,

for example, the sentence

( the number of planets = 9'

would be translated into the system S as follows, if we take ' P' as predica-

tor for the property Planet:

27-6, 'Nc'P s 9'.

(The definition of '9' is, of course, analogous to that of '2' in 27-3.)

We have said that we explicate cardinal numbers as properties of

second level, in contrast to Frege and Russell, who take them as classes of

second level. But this formulation is a concession to the customary view

based on the name-relation, according to which a predicator is a name

either of a class or of a property and cannot refer to both of them at once.

According to the method of extension and intension, it would be more ade-

quate to say that we introduce cardinal number expressions as predica-

tors of second level and that these predicators have as intensions proper-

ties of second level and as extensions classes of second level. Thus, for

example, '2' is a predicator of second level; its intension is the property

(of second level) Two, which we might call the number intension Two

or the number concept Two; and its extension is the class (of second level)

Two, which we might call the number extension Two. Since the sentence

'Nc'P s 9' is true but not L-true, the predicators 'Nc'P' and '9' are

equivalent but not L-equivalent. Therefore, the number extension The

Number Of Planets is the same as the number extension Nine, while the

number intension The Number Of Planets is not the same as, but equiva-

lent to, the number intension Nine. Thus we see that in our method, too,

as in those of Frege and Russell, equality of numbers can be regarded as

identity of certain entities, not of number intensions but of number

extensions.

In this way the whole system of mathematics constructed on the basis

of logic by Frege and Russell can be reconstructed in a simpler form with-

out the use of class expressions distinct from property expressions and of

class variables distinct from property variables.

n8 III. THE METHOD OF THE NAME-RELATION

28. Frege's Distinction between Nominatum and Sense

Frege distinguishes for any name between its nominatum, i.e., the object

named, and its sense, i.e., the way in which the object is given by it. We see

from Frege's discussion that his concept of nominatum fulfils the principles of

the name-relation stated earlier (24); thus his method of semantical analysis

is a particular form of what we call the method of the name-relation. According

to Frege, the nominatum of an isolated sentence is its truth-value, and its sense

is the proposition expressed by it. However, if the sentence stands in an oblique

(i.e., nonextensional) context, then its nominatum is that same proposition.

Frege 20 has made a very interesting distinction between the nomina-

tum of an expression and its sense. 21 This distinction will now be ex-

plained and then, in the next section, compared with our distinction be-

tween extension and intension. It will be seen that in some respects there

is a close similarity between the two kinds of distinctions; and it was,

indeed, Frege's pair of concepts that first suggested to me the concepts of

extension and intension as applied to designators in general. On the other

hand, we shall find differences between the two conceptions, based chiefly

upon the fact that Frege's conception is a particular form of what I have

previously called the method of the name-relation.

The purpose of Frege's paper, described here in modern terminology, is

to carry out a semantical analysis of certain kinds of expressions in the

ordinary word language and to propose, examine, and apply semantical

concepts as instruments for this analysis. His discussions seem to me of

great importance for the method of logical analysis; but, like his other

works, this paper has not found the attention it deserves. Except for

Russell, [Denoting], who has discussed Frege's analysis in detail but re-

jected most of it, Frege's paper seems to have been neglected for about

half a century, until Alonzo Church 22 began, several years ago, to point

20 [Sinn].

21 1 list here the English terms which I shall use as translations of Frege 1 s terms, following,

in most cases, Russell, [Denoting], and Church (see n. 22). 'Ausdriicken' is translated into 'to

express 1 ('to connote* might perhaps also be taken into consideration, in analogy to 'to denote 1 ,

although it often has in ordinary usage a quite different sense which concerns not the designa-

tive meaning component but other ones, especially the associative and emotive); 'Sinn*

'sense* (so Church; Russell uses 'meaning'; 'connotatum' or 'connotation' might also be con-

sidered); 'bezeichnen' 'to be a name of or 'to name* (Russell and Church: 'to denote'; see

the remark on the ambiguity of this term in n. i, 24); 'Bedeutung' 'nominatum' (Russell

and Church: 'denotation'); 'Begriff' 'property' (Frege uses 'Begriff' for attributes of degree

one only; for attributes in general he uses the phrase 'Begriff oder Beziehung'); 'Gedanke'

'proposition' (see Church's justification for this translation, [Review Q.], p. 47) ; 'gewdhnlich

(Rede, Bedeutung, Sinn)' 'ordinary'; 'ungerade (Rede, Bedeutung, Sinn) ''oblique' ;

'Gegenstand' 'object'; 'Wertverlauf'- 'value distribution'; 'Behauptungssatz' ' (declara-

tive) sentence'.

aa In reviews in the Journal of Symbolic Logic, V (1940), 162, 163; VII (1942), 101; see also

an abstract of a paper of his, ibid., VII, 47; further, more in detail, in [Dictionary] article,

"Descriptions", [Review C], and [Review Q.].

28. FREGE'S NOMINATUM AND SENSE 119

out repeatedly the importance of Frege's conception, defending its basic

idea while beginning to develop further the details of its application,

Frege's distinction between nominatum and sense is made in the fol-

lowing way: Certain expressions are names of objects (this term is to be

understood in a wide sense, including abstract, as well as concrete, ob-

jects) and are said to name ('bezeichnen') the objects. From the nomina-

tum of an expression, that is, the object named by it, we must distinguish

its sense; this is the way in which the nominatum is given by the expres-

sion. This is illustrated by the following example:

28-1. The two expressions ( the morning star' and 'the evening star' have

the same nominatum.

This holds because both are names of the same thing, a certain planet; in

other words, the following is a true statement of an astronomical fact:

28-2. The morning star is the same as the evening star.

On the other hand, the following holds:

28-3. The expressions ' the morning star' and ' the evening star' do not

have the same sense.

The reason for this is that the two expressions refer to their common

nominatum, that planet, in different ways. If we understand the language,

then we can grasp the sense of the expressions; for instance, we are then

aware that the sense of 'the morning star' is the same as that of the

phrase 'the body which sometimes appears in the morning before sunrise

in the eastern sky as a brightly shining point'. The nominatum is not,

however, given by the sense but only, as Frege puts it, illuminated from

one side ("einseitig beleuchtet"). To find the result 28-1, more is required

than merely to understand the sense of the expressions (namely, observa-

tion of facts).

After having explained the distinction in a general way, Frege proceeds

to apply it to sentences. In a (declarative) sentence we express a proposi-

tion ('Gedanke'). Is the proposition expressed by a sentence its sense or

its nominatum? By a long and careful analysis, Frege arrives at the fol-

lowing two results:

28-4. The (ordinary) sense of a sentence is the proposition expressed by it.

28-5. The (ordinary) nominatum of a sentence is its truth value.

These are the results for ordinary cases; they hold, in particular, for any

isolated sentence, that is, one which is not a part of a larger sentence; the

exceptions will be discussed later. For our purposes the most important

120 III. THE METHOD OF THE NAME-RELATION

question to be raised here concerns the method by which Frege arrives at

these two results (and at the exceptions to them). They are clearly not

meant simply as conventions, as, so to speak, part of the definitions of the

terms ' sense' and 'nominatum'. If this had been Frege's intention, he

probably would have chosen a simple general rule not complicated by

exceptions. It becomes clear from his discussion that the situation is other-

wise. Frege assumes that he knows quite clearly what he means by 'sense'

and 'nominatum', that is, that he knows the way in which he intends to

use these terms. On the basis of this knowledge, he investigates how these

terms apply to various kinds of expressions. Thereby he discovers objec-

tive results, and these he reports as he finds them, whether they are

simple or complicated. For the reader, however, it is not so clear as for

Frege himself what is to be understood by his two terms. The preliminary

explanations which he gives are certainly not sufficient to lead to the re-

sults, or even to make them plausible. The nominatum of an expression,

for instance, is explained as that of which the expression is a name. This

explanation, however, by no means succeeds in making the result 28-5

plausible. I think any unprepared reader would be inclined to regard a

sentence as a name of a proposition rather than as a name of a truth-

value if, indeed, he is at all willing to regard a sentence as a name of any-

thing. Another explanation for ' nominatum ' which Frege gives is that a

sentence is about the nominata of the expressions occurring in it (we have

previously called this the principle of subject matter, 24-2). But this

explanation, it seems to me, does not make 28-5 any more plausible. Take

as an example the false sentence 'Hw' (see rules i-i and 1-2) as part of

*~Hw'. (According to Frege, this is an ordinary case, that is to say, 28-4

and 28-5 also hold for 'Hw' in this context.) The question here is whether

the nominatum of 'Hw' as part of '~Hw' is (i) falsity or (ii) the (false)

proposition that the book Waverley is a human being. According to the

principle of subject matter, the sentence ' ^Hw' is in case (i) about falsity

(presumably saying that falsity does not hold), and in case (ii) about the

proposition mentioned (presumably saying that it does not hold). I be-

lieve that the first alternative, which is Frege's result 28-5, would appear

to any unprepared reader far less natural than the second.

The foregoing considerations are by no means intended as refutations

of or objections to Frege 's results. They are merely meant to show that

Frege's preliminary explanations of his terms are not sufficient as a basis

for his results. In order to understand the specific sense in which Frege

means his terms, we have to look not so much at his preliminary explana-

28. FREGE'S NOMINATUM AND SENSE 121

tions as at the reasoning by which he reaches his results. When we do this,

we find that Frege makes use of certain assumptions as if they were self-

evident or at least familiar and plausible, without formulating them

explicitly as the basic principles of his method. These assumptions can be

formulated as principles of interchangeability in the following way:

Frege' s Principles of Interchangeability

Let . . Sly . . be a complex name containing an occurrence of the

name Sly, and . . 21* . . the corresponding expression with the name 21*

instead of Sly.

28-6. First principle. If Sly and 21* have the same nominatum, then

. . Sly . . and . . 21* . . have the same nominatum. In other words, the

nominatum of the whole expression is a function of the nominata of the

names occurring in it.

28-7. Second principle. If Sly and 31* have the same sense, then . . Sly . .

and . . 21* . . have the same sense. In other words, the sense of the whole

expression is a function of the senses of the names occurring in it.

Now let us see how Frege reaches his results 28-4 and 28-5 with the help

of the first principle. His problem is: What is the nominatum and what is

the sense of an (isolated) sentence? He says: "If we replace a word in a

sentence by another word with the same nominatum but a different

sense, then this change cannot have any influence upon the nominatum

of the whole sentence." 23 Here, the first principle seems to be tacitly pre-

supposed. Let us take two sentences which are alike except for the oc-

currence of the phrases 'the morning star' in the one and 'the evening

star' in the other. According to our earlier statements (28-1 and 28-3),

this is a case in question. Hence, according to Frege's reasoning just

quoted, the two sentences have the same nominatum. What, then, could

be regarded as this common nominatum? The propositions expressed by

the two sentences may, obviously, be different. Hence they cannot be the

nominata; therefore, Frege reasons, they must be the senses of the sen-

tences. (Here another assumption seems to be tacitly made, namely, that

the proposition expressed by a sentence, because it has clearly a close

(semantical) relation to the sentence, must be either its nominatum or its

sense.) On the other hand, the two sentences have the same truth-value

(at least in ordinary cases). Therefore, the truth- value may be regarded

a * [Sinn], p. 32.

122 III. THE METHOD OF THE NAME-RELATION

as the common nomination. Thus the results 28-4 and 28-5 are reached

(for ordinary cases).

The most important application of Frege's two principles is to cases in

which the whole expression . . 2l/ . . is an isolated sentence (while 21,- may

be either a sentence or a name of another form). For these cases the prin-

ciples take the following special forms, if the results 28-4 and 28-5 are

applied to the whole sentences:

Frege's Principles of Interchangeability within Sentences

Let . . 2l/ . . be an isolated sentence containing an occurrence of the

name 21,-, and . . 2t* . . the corresponding sentence with the name 21* in-

stead of 21,-.

28-8. First principle. If &/ and 21* have the same nominatum, then

. . S, . . and . . 21* . . have the same truth-value. In our terminology

(n-i) : Names which have the same nominatum are interchangeable with

one another.

28-9. Second principle. If 2l/ and 21* have the same sense, then . . 2ly . .

and . . 21* . . express the same proposition. In our terminology: Names

which have the same sense are L-interchangeable with one another.

Our references in what follows are to these specialized forms of Frege's

two principles.

What Frege means by ' nominatum 7 and 'sense' is shown more clearly

by these principles than by his preliminary explanations. Frege's first

principle 28-8 is the same as 24~3a, the principle of interchangeability for

the name-relation. Since Frege's discussion shows that the principles 24-1

and 24-2 also hold for his concept of nominatum, his method is a particu-

lar form of what we have called the method of the name-relation. As we

have seen earlier, 24-3a is quite plausible; hence Frege's first principle is

plausible. Whether this is also true for his second principle is hard to

say. But I think it does not seem implausible if we regard it as revealing

the fact that Frege understands the term i sense' in such a way that the

sense of a compound expression and, in particular, of a sentence is some-

thing which is determined by the senses of the names occurring in it.

Frege's principles lead him, on the one hand, to the results 28-4 and

28-5 for ordinary cases for example, for isolated sentences as we have

seen. On the other hand, these same principles compel him to regard

certain cases as exceptions to these results and thereby to make his whole

scheme rather complicated. These exceptions are the cases in which a

name occurs in an oblique context (which is about the same as a non-

28. FREGE'S NOMINATUM AND SENSE 123

extensional context in our terminology, n-za). Take, for example, the

occurrence of the (false) sentence

(i) ' the planetary orbits are circles'

within the oblique context

(ii) * Copernicus asserts that the planetary orbits are circles'.

The problems involved here would, of course, be the same if, instead of

'asserts', a term like 'believes' were to occur; hence this example is similar

to the belief-sentences discussed earlier ( 13). According to Frege's re-

sults (28-5 and 28-4), the ordinary nominatum of (i), that is, that nomina-

tum which this sentence has when occurring either isolated or in an ordi-

nary, nonoblique context, is its truth- value, which happens to be falsity;

and the ordinary sense of (i) is the proposition that the planetary orbits

are circles. Now Frege says that the sentence (i) within the oblique con-

text (ii) has not its ordinary nominatum but a different one, which he

calls its oblique nominatum, and not its ordinary sense but a different one,

which he calls its oblique sense. Concerning the oblique nominatum, Frege

makes the following two statements; the second is a special case following

from the first:

28-10. The oblique nominatum of a name is the same as its ordinary

sense.

28-11. The oblique nominatum of a sentence is not its truth- value but the

proposition which is its ordinary sense.

Thus, for the above example the following result holds:

28-12. The oblique nominatum of the sentence (i), that is, the entity

named by (i) in an oblique context like (ii), is the proposition that the

planetary orbits are circles.

For this result, Frege gives two reasons at different places in his paper, (i)

"In the oblique mode of speech, one speaks about the sense, for example,

of the utterance of another person. Hence it is clear that ... in this

mode of speech a word does not have its ordinary nominatum, but names

that which ordinarily is its sense." 24 1 understand Frege's reasoning here

in the following way, if applied to the above example. He seems to presup-

pose tacitly the principle of subject matter (24-2). According to it, the

whole sentence (ii) speaks about the nominatum of the subsentence (i).

Now it is clear that (ii) does not speak about the sentence (i), because

Copernicus may have used other words than (i) and even another lan-

a * Ibid., p. 28.

124 HI. THE METHOD OF THE NAME-RELATION

guage. Nor does (ii) speak about the truth- value of Copernicus' statement

but rather about its sense, because (ii) says that Copernicus asserted a

certain sense, a certain proposition, namely, that proposition which is the

ordinary sense of (i). Therefore, this proposition must be the nomina-

tum of (i) in (ii). (2) That the nominatum of a sentence in an oblique

context is not the truth- value but the proposition, is, Frege says, "also to

be seen from [the circumstance] that it is irrelevant for the truth of the

whole sentence whether that proposition is true or false." 25 This is pre-

sumably meant in the following way: According to Frege's first principle,

the nominatum, that is, the truth-value, of the whole sentence (ii) is a

function of the nominatum of the subsentence (i). Now if the latter

nominatum were the truth- value, then the truth- value of (ii) would depend

upon that of (i). This, however, is not the case; in order to establish that

(ii) is true we need not know whether (i) is true or false. Hence the nomi-

natum of (i) in (ii) cannot be its truth- value; therefore, it must be the

proposition. (For this last step, again, a certain assumption seems tacitly

presupposed.)

In one respect, Frege's concept of proposition ('Gedanke') is not quite

clear; he does not state an identity condition for propositions. In the fore-

going discussion I have assumed that he takes the same identity condition

that we take, namely, L-equivalence (see 6 and [I], p. 92). However, in

this case, Frege's analysis of sentences with terms like 'asserts'/ believes',

etc., is not quite correct; because a sentence of this kind may change its

truth- value and hence, a fortiori, its sense if the subsentence is replaced by

an L-equivalent one (see, for example, the discussion of belief-sentences

in 13, especially 13-4). His analysis would be correct if he had in mind a

condition stronger than L-equivalence, something similar to the concept

of intensional structure explained above ( 14). In this case our second

formulation of 28-9, which was meant as a translation of Frege's second

principle into our terminology, must be omitted.

29. Nominatum and Sense: Extension and Intension

Frege's pair of concepts (nominatum and sense) is compared with our pair

(extension and intension). The two pairs coincide in ordinary (extensional)

contexts, but not in oblique (nonextensional) contexts. This does not constitute

an incompatibility, a theoretical difference of opinion, but merely a practical

difference of methods. Frege's pair of concepts is intended as an explicatum for

a certain traditional distinction, and our pair as an explicatum for another

distinction.

. 37.

29. NOMINATUM AND SENSE 125

We shall now compare Frege's distinction between the nomination and

the sense of an expression with our distinction between the extension and

the intension of an expression.

Our pair of concepts is, like Frege's, intended to serve for the purposes

of semantical meaning analysis. Our two concepts may be regarded, like

Frege's, as representing two components of meaning (in a wide sense).

The concepts of sense and of intension refer to meaning in a strict sense,

as that which is grasped when we understand an expression without know-

ing the facts; the concepts of nominatum and of extension refer to the ap-

plication of the expression, depending upon facts.

A decisive difference between our method and Frege's consists in the

fact that our concepts, in distinction to Frege's, are independent of the

context. An expression in a well-constructed language system always has

the same extension and the same intension; but in some contexts it has

its ordinary nominatum and its ordinary sense, in other contexts its

oblique nominatum and its oblique sense.

Let us, first, compare the extension of an expression with its ordinary

nominatum; it seems that these concepts coincide. With respect to predi-

cators, Frege does not seem to have explained how his concepts are to be

applied; however, I think that Church 26 is in accord with Frege's inten-

tions when he regards a class as the (ordinary) nominatum of a predicator

(of degree one) for instance, a common noun and a property as its

(ordinary) sense. As an example, Church states that the nominatum of

' unicorn' is the null class, and its sense is the property of unicorn-hood.

And here the extension is likewise the class in question. With respect to a

sentence, its truth-value is both the ordinary nominatum and the exten-

sion. And in the case of an individual expression the ordinary nominatum

and the extension is the individual in question. Thus we have this result:

29-1. For any expression, its ordinary nominatum (in Frege's method) is

the same as its extension (in our method).

It is more difficult to see clearly what constitutes the ordinary sense in

Frege's method. As mentioned before, this is due to the lack of precise

explanation and especially of a statement as to the condition of identity

of sense; we shall assume here again that Frege would agree to take

L-equivalence as this condition. Then, for a sentence, its ordinary sense is

the proposition expressed by it, hence it is the same as its intension. For

a predicator (of degree one) its ordinary sense is the property in question,

and its intension is the same. Frege does not use any special term for the

* 6 [Review C.J, p. 301.

126 III. THE METHOD OF THE NAME-RELATION

sense of an individual expression. 37 But he says that the sense of a sen-

tence is not changed if an individual expression occurring in an ordinary

context is replaced by another one with the same sense. Therefore, it

seems reasonable to assume that what he means by the sense of an indi-

vidual expression is about the same as what we mean by an individual con-

cept. Hence, on the basis of our understanding of Frege's explanations,

the following seems to hold:

29-2. For any expression, its ordinary sense (in Frege's method) is the

same as its intension (in our method).

Thus, for ordinary occurrences of expressions, our two concepts coincide

with those of Frege. The differences arise only with respect to expressions

in an oblique context. Here our concepts lead to the same entities as for

the ordinary occurrences of the same expressions, while Frege's concepts

lead to different entities. As we have seen earlier, this complication is

not introduced by Frege arbitrarily but is an inevitable consequence of his

general principles, especially the first.

It seems that Frege, in introducing the distinction between nominatum

and sense, had the intention of making more precise a certain distinction

which had been made in various forms in traditional logic. Thus his task

was one of explication (in the sense explained in the beginning of 2).

The explicata proposed by him are the concepts of nominatum and sense.

Now the question is: What were his explicanda, that is, for which pair of

traditional concepts did Frege propose his explicata? Church 28 refers in

this connection, first, to the distinction between ' ex tension' and 'compre-

hension' in the Port-Royal Logic, and, second, to the distinction between

'denotation' and 'connotation' made by John Stuart Mill. It seems to me

that we find in the historical development two pairs of correlated concepts,

appearing in various forms. These pairs are closely related to each other

and may sometimes even merge. Nevertheless, I think that it is, in gen-

eral, possible to distinguish them, (i) In traditional logic we often find

two correlated concepts: on the one hand, what was called the 'extension'

or 'denotation' (in the sense of J. S. Mill) of a term or a concept; on the

other hand, what was called its 'in tension'/ comprehension', 'meaning', or

'connotation'. 29 It seems to me that Frege intended an explication of this

*i Church uses the term 'description', which is, however, more customary for an individual

expression constructed with an iota-operator than for its sense.

28 [Review C], p. 301.

39 For a detailed discussion and comparison of the conceptions of Mill and other authors

see Ralph M. Eaton, General Logic (1931), chap. vi.

29. NOMINATUM AND SENSE 127

pair of concepts by his distinction between the value-distribution of a

prepositional function and the prepositional function itself; in the case of

degree one, this distinction is the familiar one between a class and a

property. Our distinction between extension and intension is likewise

meant as an explication of the same pair of concepts, as far as predicators

are concerned, and simultaneously as an enlargement of the domain of

application of the customary concepts to other kinds of designators. (2)

The second pair of concepts starts with the name-relation. In everyday

language, it is said, for instance, that ' Walter Scott' is a name of the man

Walter Scott. Logicians extend the application of this relation. They also

regard individual descriptions as names, e.g., 'the author of Waverley' as

a name of the same man Walter Scott, a usage not admitted by everyday

language. Going further, they even construe expressions of another than

the individual type as names; they regard them as names of abstract enti-

ties, e.g., of classes or properties, relations, functions, propositions, etc.

(Other terms used as synonyms of 'is a name of were mentioned at the

beginning of 24.) With respect to any expression regarded as a name, a

distinction is made here between that entity whose name the expression is

and the meaning or sense of the expression. It seems that the second con-

cept in this pair is very similar to the second in the first pair; for both of

them the term 'meaning' is sometimes used.

Now it seems to me that the explicandum which Frege intended to

explicate by his distinction between nominatum and sense was the second

pair of concepts rather than the first. And I interpret also some of Quine's

discussions in [Notes] as an endeavor toward a clarification and explica-

tion of the concepts of the second pair. Since Church's discussions in re-

cent publications, especially [Review C.] and [Review Q.J, are intended to

defend and develop Frege's distinction, I regard them, too, as belonging

more to the second historical line than to the first. However, the two his-

torical lines, the two pairs of concepts taken as explicanda, are closely

related to each other. I have emphasized the difference between them only

in order to make clearer the difference between the problem which Frege

intended to solve and my problem or, more exactly, the difference between

the explicandum which Frege took as the basis of his distinction between

nominatum and sense (if I understand him correctly) and the explican-

dum for which my distinction between extension and intension is in-

tended.

Thus it becomes clear and I wish to emphasize this point that the

difference between Frege's method and that here proposed is not a dif-

ference of opinion. In other words, it is not the case that there is one

128 III. THE METHOD OF THE NAME-RELATION

question to which different and incompatible answers have been given.

There are two questions, and, more precisely, these are not even theo-

retical questions but merely practical aims. While the general aim is the

same, namely, the construction of a pair of concepts suitable as instru-

ments for semantical analysis, the specific aims are different. Frege tries

to achieve the general aim by an explication of one pair of concepts, I by

the explication of another pair. Frege's principles are not assertions which

are open to refutation or doubt. They are to be regarded rather as part of

the characterization of his two concepts and hence hold analytically for

these concepts. If someone were to say as I do not that he disagrees

with Frege's principles, he would merely be saying in effect that he under-

stands the two terms 'nominatum' and 'sense' in a way different from

Frege in other words, that he uses different concepts and there would

be thus no genuine disagreement. The results found by Frege, including

the complication in the case of oblique contexts, are consequences of his

principles and hence share their analytic validity (assuming that Frege

made no mistake in reasoning from the principles to the results). There-

fore, I am in complete agreement with Frege's results in this sense: they

are valid for his concepts. The same holds for Church's results on the same

(or a somewhat modified) basis.

The two concepts used in our method coincide, as we have seen earlier,

in certain cases with Frege's concepts, while in other cases they do not.

This is not a contradiction between two theories, since our concepts are

admittedly different from Frege's. The situation is, rather, similar to the

following: Suppose someone divides all animals into aquatic, aerial, and

terrestrial animals; someone else divides them into fishes, birds, and the

rest. The two classifications coincide to some extent because fishes are

aquatic animals and birds are aerial; but they do not coincide entirely.

The one man puts whales into his first class, while the other does not.

This fact, however, does not constitute a difference of opinion, a theoreti-

cal contradiction, because the two concepts in question are admittedly

different. Since the two classifications and the assertions made on their

bases are not incompatible, it would be theoretically possible to use both

simultaneously. However, if the simultaneous use of both seems unneces-

sarily complicated, there is a kind of practical incompatibility or competi-

tion. In this case the decisive question is this: which of the two triples of

concepts is more fruitful for the purpose for which both are proposed,

namely, a classification of animals?

The situation with regard to Frege's pair of concepts and that proposed

Jiere seems to me to be analogous. I have the feeling, without being quite

30. THE DISADVANTAGES OF FREGE'S METHOD 129

certain, that it would not be very fruitful to use simultaneously both pairs

of concepts for semantical analysis. If so, then there is, in spite of the

theoretical compatibility, a practical competition or conflict. This con-

flict might, for instance, appear over the following point, which has been

mentioned earlier: A logician, thinking in terms of Frege's concepts,

might be inclined, though not compelled, to construct a logical system in

such a way that it contains different expressions for classes and for

properties, while a logician, thinking, instead, in terms of extension and

intension, would probably be less inclined to do so.

30. The Disadvantages of Frege's Method

Frege's special form of the method of the name-relation involves additional

complications. Starting with any ordinary name, it leads to an infinite number

of entities and an infinite number of expressions as names for them, while the

method of extension and intension needs only one expression and speaks only of

two entities. Furthermore, according to Frege's method, the same name, when

occurring in different contexts, may have an infinite number of different

nominata; and sometimes even the same occurrence of a name may simul-

taneously have several nominata.

The disadvantages of Frege's pair of concepts in comparison with the

pair here proposed all belong to the concept of nominatum. Frege's con-

cept of sense is very similar to that of intension; we might even say that,

when we consider simply these two concepts, it is difficult to see any

reason that there should be a difference between them. The difference is

brought about by Frege's differentiation between the ordinary and the

oblique sense of a name. It is not easy to say what his reasons were for re-

garding them as different. Perhaps he was led to make this distinction be-

cause of his original distinction between the ordinary and the oblique

nominatum. It does not appear, at least not to me, that it would be un-

natural or implausible to ascribe its ordinary sense to a name in an oblique

context. However, Frege could not do this because he had already used

this ordinary sense as nominatum in the oblique context. And since he

assumes that nominatum and sense must always be different, he had thus

to introduce a third entity as the oblique sense. Incidentally, it seems that

Frege nowhere explains in more customary terms what this third entity is.

Since Frege's method is a special form of what we have called the meth-

od of the name-relation, it also possesses the disadvantages which we

have previously found in this method. We found ( 25) that the concept

of nominatum involves a certain ambiguity, which is also transferred to

other semantical concepts, for instance, those of identity sentence and

identity sign.

130

III. THE METHOD OF THE NAME-RELATION

Further, we saw ( 26, 27) that the method of the name-relation may

lead to a complicated duplication or multiplicity of names within the

same type. If Frege's form of the method is adopted, the situation becomes

even more complicated. We shall illustrate this by two examples. (See the

diagram, where an arrow with 'N* indicates the name-relation and an

arrow with 'S' the sense-relation.) Example (/) : Let us start with a name

Names:

Example (I):

Example (2):

Entities:

Example (I):

Example (2):

truth-value Truth

class Human

proposition that .

property Human

n,, say the sentence 'Hs'. According to Frege's method, there is an entity,

e,, named by this name; this is the truth- value of 'Hs'. And there is an-

other entity, e a , which is the sense of 'Hs'; this is the proposition that

Scott is human. This proposition e 2 may also have a name; if we wish to

speak about it, we need a name for it. This name is different from n, be-

cause the latter is the name of e x and hence, in a well-constructed lan-

guage, should not be used simultaneously as a name of another entity. Let

the new name be n 2 . Like any name, n 2 has a sense. This sense of n 2 must

be different from the nominatum of n 2 ; it is a new entity, e 3 , not oc-

curring in customary analyses. In order to speak about e 3 , we need a new

name, n 3 . The sense of n 3 is a new entity e 4 ; and so forth ad infinitum.

Example (2) : The situation is analogous if the first name n x is of another

type, for instance, a predicator, say 'IT. The entity d named by n x is here

the class Human; the sense e 2 is the property Human. The name n 2 is

introduced as a name for the property Human; and the new entity e 3 is

the sense of this name. The name n 3 is a name of this sense e 3 ; e 4 is the

sense of this name n 3 , and so on. Generally speaking, if we start with any

name of a customary form, we have, first, two entities familiar to us: its

ordinary nominatum and its ordinary sense; they are the same as its

extension and its intension, respectively. Then Frege's method leads, fur-

ther, to an infinite number of entities of new and unfamiliar kinds; and,

if we wish to be able to speak about all of them, the language must con-

tain an infinite number of names for these entities. To provide for this

infinite sequence of names seems, thus, a natural decision on the basis of

Frege's method. And Church does, indeed, take this decision in his de-

30. THE DISADVANTAGES OF FREGE'S METHOD 131

velopment of Frege's method by declaring it desirable "that the object

language should contain for every name in it a name of the associated

sense." 30 On the basis of the method of extension and intension, on the

other hand, we need in the object language, instead of an infinite sequence

of expressions, only one expression (for instance, in the first example 'Hs',

in the second 'H') ; and we speak in the metalanguage only of two entities

in connection with the one expression, namely, its extension and its in-

tension (and even these are, as we shall see later, merely alternative ways

of saying the same thing).

The fact that, according to Frege's method, the same name may have

different nominata in different contexts has already been mentioned as a

disadvantage. But the multiplication of entities goes far beyond Frege's

initial distinction between the ordinary and the oblique nominatum of a

name. Actually, these two nominata constitute only the beginning of an

infinite sequence of nominata for the same name. If we apply Frege's

method to sentences with multiple obliqueness, then we have to distin-

guish the ordinary nominatum of the name, its first oblique nominatum,

its second oblique nominatum, and so forth. In order to construct an

example, let us suppose that the system S contains not only, like S 2 (see

n, Example II), modal signs, say 'N^' for 'it is necessary that p y and

' Qp' for 'it is possible that />', but also psychological terms, say ' J/>' for

' John believes that p\ Now let us consider a series of sentences in 5, each

occurring within the next in a simple oblique context:

(i) 'Hs' (' Scott is human');

(ii) ' 0(Hs)' ('it is possible that Scott is human');

(iii) ' J(<} (Hs)) ' C John believes that it is possible that Scott is human') ;

(iv) t ~N(](() (Hs)))' ('it is not necessary that John believes that it is

possible that Scott is human') ; etc.

Let us see what the nominatum of the original sentence 'Hs' is in these

various contexts. According to our previous explanation of Frege's

method, the nominatum of 'Hs' in isolation is its truth-value, hence the

entity d in the above diagram; and the nominatum of its occurrence

within (ii) is the proposition that Scott is human, hence the entity e a in

the diagram. It can further be shown, by an analysis which we shall not

describe here in detail, that the nominatum of 'Hs' within (iii) is e 3 , its

nominatum within (iv) is e 4 , and so on. Thus the same expression 'Hs'

has an infinite number of different entities as nominata when it occurs in

different contexts.

30 [Review Q.], p. 47.

i 3 2 III. THE METHOD OF THE NAME-RELATION

This fact that different occurrences of a name may have different

nominata is certainly a disadvantage. It is the reason that Church pro-

poses a certain modification of Frege's method whereby this multiplicity

of nominata is avoided (see 32, Method III).

Worse than the multiplicity of nominata for different occurrences of a

name is the fact that within certain contexts, according to Frege's own

analysis, one occurrence of a sentence has simultaneously two different

nominata. Frege takes as an example a sentence 'Bebel wahnt, dass . . .',

that is (writing 'A' as an abbreviation for a long subsentence), 'Bebel has

the illusion that A', or 'Bebel believes erroneously that A'. Frege inter-

prets this sentence, no doubt correctly, as 'Bebel believes that A; and not

A'. Now here we have two occurrences of 'A', the first in an oblique con-

text, the second in an ordinary one, with therefore different nominata.

Thus Frege comes to the conclusion that, in the original sentence 'Bebel

believes erroneously that A', the subsentence ' A' "strictly speaking, must

be taken twice with different nominata of which the one is a proposition,

the other a truth- value". 31 The situation is analogous in a case like ' John

knows that A', because this implies 'John believes that A; and A'.

This double nominatum of a name, not, as in the earlier cases, for dif-

ferent occurrences but for the same occurrence, seems a startling result of

Frege's method. The sentences in question seem perfectly clear. At first

glance it will not seem plausible that the subsentence ' A' should simul-

taneously name two distinct entities. It can easily be seen that the feature

here discussed has nothing to do with the ordinary ambiguities so fre-

quently met with in natural word languages, but is likewise to be found in

an exact, symbolic system of modal logic. A modal sign 'CT' for con-

tingent truth of propositions (which is a nonsemantical concept, see 23)

can be introduced in S 2 on the basis of 'N' (see n, Example II) in this

way:

30-1. Abbreviation. ( CT(py for 'p .

On this basis, the sentence 'CT(Hs)' is L-equivalent to 'Hs ~ N(Hs)';

in words: ' Scott is human, but it is not necessary that Scott is human';

or, briefly: ' Scott happens to be human'. According to Frege's analysis,

the sentence 'Hs' within 'CT(Hs)' has at once two different nominata, as

have the signs 'H' and V; and the same holds for the words ' Scott' and

'human' in the sentence 'Scott happens to be human'. This seems a rather

unsatisfactory result.

If, instead of Frege's method, the method of extension and intension is

[Sinn], p. 48.

31. THE ANTINOMY OF THE NAME-RELATION 133

used, then the situation becomes much simpler. Every expression has al-

ways the same extension and the same intension, independent of the

context. The problems connected with modal contexts will be discussed

later (chap. v).

31. The Antinomy of the Name-Relation

The third principle of the name-relation (24-3) permits replacing a name with

another name of the same entity. Although this principle seems quite plausible,

it is not always valid. This has been pointed out by Frege, Russell, and Quine,

The contradiction which sometimes arises if such a replacement is made hi a

nonextensional context is called here the antinomy of the name-relation.

The principles which characterize the method of the name-relation

(24-1, 2, and 3) seem quite plausible; and this holds for either form of the

principle of interchangeability , the one using the concept of name-relation

(24-3a) and the other using the concept of identity (24-3 b). Therefore, in

a naive approach without a closer investigation, we might be tempted

to regard these principles as generally valid without any restrictions. How-

ever, if we do so and, in particular, if we apply the principle of inter-

changeability in either form to nonextensional contexts, we arrive at a

contradiction. I propose to call this contradiction the antinomy of the

name-relation. [My choice of this term is, of course, motivated by the fact

that, from my point of view, the method of the name-relation is respon-

sible for the antinomy. Others, who regard this method as harmless and

unobjectionable and who feel that the source of the difficulty lies, rather,

in the use of modal contexts or, more generally, intensional contexts or,

still more generally, oblique (i.e., nonextensional) contexts, will perhaps

prefer to call it the antinomy of modality or of intensionality or of

obliquity.]

The antinomy of the name-relation can be constructed, as we shall see,

in either of two forms; the first uses the first form of the principle of inter-

changeability (24-3a), the second uses its second form (24-3^. The second

form of the antinomy may perhaps also be called antinomy of identity or

antinomy of identical nominata or antinomy of synonymity (provided the

term ' synonymous' is understood, not in the sense of 'intensionally iso-

morphic' (14-1), but as 'having the same nomination').

Frege was the first to point out the circumstance that the principle of

interchangeability (see 24-5) if applied to the ordinary nominata of names

does not hold for oblique contexts. Although Frege's formulation was not

presented in terms of a contradiction, his result constitutes the basis of

what I propose to call the antinomy of the name-relation.

i 3 4 HI. THE METHOD OF THE NAME-RELATION

It seems that the antinomic, paradoxical character of the situation was

first seen by Russell. 33 He explains the antinomy in its second form with

respect to an interchange of individual expressions as the first of the three

" puzzles' ' which he says every theory of denoting (name-relation) must

solve. He states the second form of the principle of interchangeability

(24-3^ in the following words : " If a is identical with b, whatever is true of

the one is true of the other, and either may be substituted for the other

in any proposition without altering the truth or falsehood of that proposi-

tion." 33 He takes as an example the sentence * George IV wished to know

whether Scott was the author of Waverley'. If in this sentence, on the basis

of the true identity sentence ' the author of Waverley is identical with

Scott' (9-1), the description 'the author of Waverley' is replaced by

' Scott', the resulting sentence is presumably false.

Quine 34 likewise points out the second form of the antinomy with

respect to individual expressions. His first examples are psychological

sentences with the phrases' is unaware that' and 'believes that'; 35 they are

similar to Frege's example, 'Copernicus asserts that. . .' (see above,

28), and Russell's example just mentioned. Further examples given by

Quine are modal sentences. 36 The first is: 'Necessarily, if there is life on

the evening star, then there is life on the evening star'. If here, on the

basis of the identity sentence, 'The morning star is the same as the evening

star' (28-2), which is found to be true by astronomical observations, one

occurrence of 'the evening star' is replaced by 'the morning star', a false

sentence results. (If, instead of the truth of the identity sentence 28-2, the

semantical statement 28-1 is used, we have the first form of the antinomy.)

In another example of a modal sentence, Quine uses numerical expres-

sions:

'9 is necessarily greater than 7'.

If here, on the basis of the true identity sentence 'The number of

planets = 9', '9' is replaced by 'the number of planets', the following

false sentence results:

'The number of planets is necessarily greater than 7'.

I shall now give an example of the antinomy in both forms with respect

to predicators. We found earlier an ambiguity in the concept of the nomi-

natum of a predicator (for example, the German word 'gross' may be re-

3* [Denoting], p. 485.

"Ibid. "Ibid.

34 [Notes], p. 115. *lbid., p. 121.

31. THE ANTINOMY OF THE NAME-RELATION 135

garded as a name of the class Large or of the property Large, see 25).

In order to show that the antinomy of the name-relation is independent

of this ambiguity, the example will be formulated with phrases of the form

'the class . . .' and only classes taken as nominata of these phrases. The

following sentence is true ('necessary' is here, as in earlier examples, used

in the sense of 'logically necessary') :

'It is necessary that the class Featherless Biped is a subclass of the

class Biped'.

Now we replace in this sentence ' the class Featherless Biped' by ' the class

Human'; this replacement may be based either, according to 24-3^ on the

circumstance that the identity sentence ' the class Featherless Biped is the

same as the class Human' is true (4-7) or, according to 24-3a, on the cir-

cumstance that the phrases ' the class Featherless Biped' and ' the class

Human' have the same nominatum. The result of the replacement is the

sentence

'It is necessary that the class Human is a subclass of the class

Biped'.

Since, however, the fact that human beings have two legs is a contingent

biological fact and not logically necessary, the following is true :

'It is not necessary that the class Human is a subclass of the class

Biped'.

The contradiction between these two results constitutes an instance of the

antinomy of the name-relation.

Those logical situations which are called logical antinomies (in the

modern, not the Kantian sense) or logical paradoxes are characterized by

the fact that there are two methods of reasoning, which, although both

plausible and in accordance with customary ways of thinking, lead to

contradictory conclusions. Any solution of an antinomy, that is, the elimi-

nation of the contradiction, consists, therefore, in making suitable changes

in the reasoning procedure; at least one of its assumptions or rules must, in

spite of its plausibility, be abolished or restricted in such a way that it

is no longer possible to reach the two incompatible conclusions. Some-

times a certain form of inference is abolished or restricted. Sometimes a

more radical step is taken by abandoning certain forms of sentences which

were previously regarded as meaningful and harmless. Thus, for instance,

Russell's solution of the antinomy known by his name consisted in the

rejection of sentences of the form ' aca'. Sometimes several different wftys

for solving a given antinomy are found. It is a matter of theoretical

i 3 6 III. THE METHOD OF THE NAME-RELATION

investigation to discover the consequences to which each of the solutions

leads and, especially, what sacrifices of customary and plausible ways of

expression or deduction each of them entails. But which of the solutions

we choose for the construction of a language system is ultimately a matter

of practical decision, influenced, of course, by the results of the theoretical

investigation.

32. Solutions of the Antinomy

Six procedures for the solution of the antinomy of the name-relation are dis

cussed. The first five still apply the method of the name-relation. Frege and

Church develop particular forms of this method by introducing certain dis-

tinctions, which, however, lead to a more complicated language. Russell re-

stricts to a considerable degree the application of the method of the name-

relation and thereby of the semantical analysis of the meaning of expressions.

Quine does the same to a smaller degree. The antinomy would also be elimi-

nated by restricting the language to extensional sentences; but it is not known

at present whether the whole of logic and science is expressible in a language

of this kind. Finally, the method of extension and intension avoids the antinomy

by avoiding the concept of nominatum. The concept of extension, though simi-

lar to that of nominatum, eliminates the contradiction without unnatural

restrictions and complications.

We shall now explain some of the solutions for the antinomy of the

name-relation which have been proposed or considered by logicians; we

call them Methods I-VI. First, we discuss five solutions which preserve

the method of the name-relation, at least to some extent. They may be

regarded as particular forms of this method. We shall find that each of

them has serious disadvantages. Then we shall consider the possibility of

solving the antinomy by giving up the method of the name-relation.

Method I, Frege. It seems that Frege was aware of the fact that the

principle of interchangeability (in the form 24-3%) would lead to a con-

tradiction if the ordinary nominata of names were ascribed also to their

oblique occurrences and that the contradiction does not arise if different

nominata are ascribed to these occurrences. In this sense we may say that

Frege offers a solution for the antinomy of the name-relation. It is true

that Frege does not speak explicitly of the necessity of avoiding a contra-

diction; he gives other reasons for his distinction between the ordinary

nominatum and the oblique nominatum of a name. His reasoning gives

the impression that this distinction appeared to him natural in itself,

without regard to any possible contradiction. However, I think that to

many readers it will scarcely appear very natural and that they, like my-

self, will see the strongest argument in favor of Frege's method rather in

the fact that it is a way of solving the antinomy.

32. SOLUTIONS OF THE ANTINOMY 137

The disadvantages of Frege's method have been explained earlier ( 30).

We have seen that the unnecessary multiplicity of entities and names

which is generally a consequence of the method of the name-relation is

here even much greater. Furthermore, occurrences of the same name may

have different nominata indeed, an infinite number of them; and in

certain contexts even the same occurrence of a name may have simul-

taneously several nominata.

Method II, Quine. Quine 37 uses the term * designation' for the name-

relation. He says of an occurrence of an expression in a nonextensional

context (as, for instance, ' the evening star' in the first and ' 9' in the second

of the two examples of his, quoted in the preceding section) that it is' 'not

purely designative" and that it does not refer simply to the object

designated (the nominatum). He thinks that nonextensional contexts are

fundamentally different from extensional contexts and more similar to

contexts in quotation marks; and, in particular, that the customary

logical rules of specification and existential generalization are not valid for

nonextensional contexts (this will be discussed later, 44). Thus his solu-

tion agrees with Frege's in not ascribing the ordinary nominatum to an

occurrence of a name in a nonextensional context. But where Frege

ascribes a different nominatum, Quine ascribes no nominatum at all.

Consequently, the principle of interchangeability (see his formulation

24-6) is declared by Quine not to be applicable to these occurrences, and

thus the antinomy is eliminated.

The advantage of Quine's method in comparison with Frege's consists

in avoiding the immense multiplication of entities and corresponding

names to which the latter method leads. But Quine's method pays a high

price for this simplification by restricting the name-relation ('designa-

tion') to extensional contexts and grouping all nonextensional contexts

together with contexts in quotation marks and, further, by imposing nar-

row restrictions upon the use of variables in modal sentences. Those

logicians in particular who are interested in constructing or in semantical-

ly analyzing systems of modal logic will hardly be inclined to adopt this

method.

Method III, Church. Church 38 regards Frege's method as prefer-

able to Quine's in two respects: first, because it provides that a name al-

ways has a nominatum 39 even in nonextensional contexts and, second,

because Frege's conception of the sense of names as something outside

the language (e.g., propositions or properties) seems more natural than

Quine's way of construing the sense (meaning) of a name as its L-equiva-

37 [Notes]. " [Review Q.]. Ibid., p. 46.

138 III. THE METHOD OF THE NAME-RELATION

lence class (see end of 33). However, Church does not simply adopt

Frege's method in its original form; he proposes important modifications

in it. He agrees with Frege's conclusion that the nominatum of an oblique

(nonextensional) occurrence of a name must be different from its ordi-

nary nominatum and must be the same as its ordinary sense. But Church

seems to accept this only as a result of an analysis of nonextensional

sentences as they occur in natural word languages and in systems of modal

logic of the customary form. In a well-constructed language, however,

this multiplicity of nominata for the same name should be avoided.

Therefore, Church proposes, for semantical discussions in the natural

word languages, "to adopt some notational device to distinguish the

oblique use of a name from its ordinary use"; 40 this would be analogous

to the customary use of quotation marks. Mere distinguishing marks are

not sufficient, however, in a symbolic language system; here we should go

one step further, as we do when we use not quotation marks but special

symbols as names of signs. "In a formalized logical system, a name would

be represented by a distinct symbol in its ordinary and in its oblique

use".

I agree that, if the method of the name-relation is used, then the

changes in the notation proposed by Church are indeed an improvement.

On the other hand, it seems that these changes would cause an additional

complication in a system of modal logic. For example, there would be

an infinite number of types corresponding to the one type of sentences

in the method of extension and intension.

Although Church's method avoids the multiplicity of nominata for the

same name, it shares the other complications of the original form of

Frege's method explained in 30. This fact, however, is not an argument

against Church's method in comparison to the other forms of the method

of the name-relation. On the contrary, I think that Church's form of the

method may well be regarded as that which carries out the basic ideas

of the method of the name-relation in the most consistent and thorough

way, eliminating features not tolerable in a well-constructed system and

not restricting unduly the domain of application of the fundamental con-

cepts of the method. Therefore, the great complications to which it leads

are to be regarded, rather, as an argument against the method of the

name-relation in general provided that there is some other convenient

method which avoids them.

Method IV, Russell. Russell 41 constructs the antinomy of the name-

relation with respect to individual expressions; in his example (see the

* Ibid., p. 46. <* [Denoting].

31 SOLUTIONS OF THE ANTINOMY ttf

preceding section) the description 'the author of Waverley' is replaced

by the proper name ' Scott'. According to Russell's conception, a descrip-

tion has no meaning in itself, but a sentence containing a description has

a meaning, 42 and this meaning can be expressed without using the descrip-

tion. The contextual definition of a description (see above, 7, Method

II) is a rule for transforming a sentence containing a description into a

sentence with the same meaning which no longer contains the description.

Although in the case of an individual description which fulfils the unique-

ness condition we may regard the one individual (the descriptum) as the

nominatum of the description, nevertheless, a sentence containing this

description is not about this individual. (Thus the principle of subject

matter, 24-2, is rejected with respect to descriptions.) What the sentence

actually means is shown only in its expanded form. Proper names (e.g.,

' Walter Scott 7 ) are regarded as abbreviations of descriptions. Thus, in the

primitive notation, neither proper names nor descriptions occur. There-

fore, the principle of interchangeability for individual expressions is not

applicable, and that form of the antinomy which arises from an inter-

change of individual expressions is eliminated. The situation is quite

analogous for abstraction expressions of classes (for example, '(H#)'; see

the explanations above, at the beginning of 26). These expressions are

likewise introduced by contextual definitions and not regarded as having

any meaning in themselves. The meaning of a sentence containing a class

expression is shown by its expansion in primitive notation, where no

class expression occurs. Thus, also with respect to class expressions the

principle of interchangeability is inapplicable, and the antinomy does not

appear.

If Russell regards sentences as names at all, then presumably he re-

gards them as names of propositions; in any case, he does not regard them

as names of truth-values. Thus the final result with respect to Russell's

application of the name-relation may be summed up in the following way :

Although individual expressions and class expressions may, in a certain

sense, be regarded as naming individuals or classes, they do not occur in

the primitive notation but are incomplete symbols without independent

meaning. As nominata in the strict sense, neither individuals nor classes

nor truth- values occur, hence none of those entities which we call exten-

sions. The antinomy of the name-relation arises from an interchange of

two expressions with the same nominatum. In all the chief kinds of in-

stances of the antinomy including all instances mentioned in this book

and all instances given by the authors mentioned the common nomina-

* Ibid., p. 480.

i 4 o III. THE METHOD OF THE NAME-RELATION

turn is an extension. Therefore, RusselPs method, by excluding extensions

from the realm of nominata in the strict sense, eliminates at least the most

important instances of the antinomy.

A few remarks may be made on Russell's objections to Frege's method.

The chief objection 43 concerns the case of a description which does not

fulfil the uniqueness condition. Frege says that in this case the description

has a sense but no nominatum. Russell regards it as unsatisfactory that

expressions of the same syntactical form should in one case have a

nominatum and in another case not. Since, according to Frege, a sentence

is about the nominata of the expressions occurring in it (24-2), in the case

in which the uniqueness condition is not fulfilled the sentence is about no

entity at all; hence, Russell says, 44 one would suppose that the sentence

"ought to be nonsense; but it is not nonsense, since it is plainly false".

This reasoning seems to me convincing; moreover, I suppose that Frege

himself would agree with it because he regards the feature mentioned as a

defect of natural languages. 45 This is the reason for his demand that in a

well-constructed language every description should have a nominatum by

virtue of a suitable convention. 46 RusselPs objection here is that this pro-

cedure is artificial and does not give an exact analysis of the actual use

of descriptions. However, Frege's convention had a different purpose. He

first gave an analysis of the natural language and then proposed the con-

vention as a step not in the exact reconstruction of the natural language

but rather in the construction of a new language system intended to be

technically superior to the natural language.

Russell's general objections 47 against Frege's distinction between nomi-

natum and sense are rather obscure. This is due chiefly to RusselPs con-

fusion between use and mention of expressions, which has already been

criticized by Church. 48

The disadvantage of RusselPs method lies in the fact that meaning is

denied to individual expressions and class expressions. That these kinds of

expressions can be introduced by contextual definitions and hence that

what is said with their help can also be said without them is certainly a

result of greatest importance but does not seem a sufficient justification

for excluding these expressions from the domain of semantical meaning

analysis. It must be admitted, I think, that descriptions and class expres-

sions do not possess a meaning of the highest degree of independence; but

that holds also for all other kinds of expressions except sentences (see re-

* Ibid., pp. 483 f. 4* Ibid., p. 41; see above, 8.

Ibid., p. 484. 47 Op. cit., pp. 485-88.

45 [Sinn], p. 40. * 8 [Review CJ, p. 302.

32. SOLUTIONS OF THE ANTINOMY 141

marks at the end of i). And it is certainly useful for the semantical

analysis of the meanings of sentences to apply that analysis also to the

meanings, however derivative, of the other expressions, in order to show

how out of them the independent meanings of the sentences are con-

stituted.

Method V, Extensional Language. The most radical method for

eliminating any antinomy arising in connection with certain forms of ex-

pression consists in excluding these forms entirely. In the case of the

antinomy of the name-relation this solution would consist in excluding

all nonextensional contexts in other words, in using a purely extensional

language (see the definition n-2c). To construct an extensional language

system for certain restricted purposes involves, of course, no difficulties

(as examples of such systems, see, e.g., Quine's language system ML and

my systems I and II in [Syntax]). But this is not sufficient for the present

purpose. In order to eliminate the antinomy by excluding all nonexten-

sional contexts, it would be necessary to show that for the purposes of

any logical or empirical field of investigation an extensional language

system can be constructed; in other words, that for any nonextensional

system there is an extensional system into which the former can be

translated. The assertion to this effect is known as the thesis of extension-

ality.* 9 The problem of whether it holds or not is still unsolved. Translat-

ability into extensional sentences has been shown for certain kinds of non-

extensional sentences. Thus, for instance, any simple modal sentence is

L-equivalent to a semantical sentence in an extensional metalanguage

using L-terms, as we shall see later ( 39) . s For example, the modal sen-

tence 'N(A) J , in words: 'it is necessary that A', is L-equivalent to the

semantical sentence ' ' A' is L-true' (according to a convention to be dis-

cussed later). The application of this method of translation to sentences

with iterated modalities (e.g., 'it is necessary that it is possible that . . .')

involves a certain difficulty; this, however, can be overcome, as I have

shown at another place. 51 The translation of nonextensional sentences

with psychological terms like 'believes', 'knows', etc., is presumably like-

wise possible, although at present it is not yet clear how it can best be

< See [Syntax], 67; [I], p. 249; Russell, [Inquiry], chap. xix.

s For this translation see [Syntax], 69; I would now define the L-concepts not as syn-

tactical but as semantical concepts (see above, 2). Note that in this translation the two

sentences, although L-equivalent, are not intensionally isomorphic ( 14). A translation in the

stronger sense, preserving intensional structure, is obviously impossible between a nonexten-

sional and an extensional sentence.

s* [Modalities].

i 4 2 III. THE METHOD OF THE NAME-RELATION

made (see the discussions in 13 and 15). The question of whether an

extensional language is sufficient for the purposes of semantics will be dis-

cussed later ( 38); an affirmative answer does not seem implausible, but

the question is not yet definitely settled.

If we could prove the thesis of extensionality and if we decided to ex-

clude all nonextensional sentence forms, then obviously the antinomy of

the name-relation would be eliminated. Furthermore, the difference be-

tween the method of the name-relation and the method of extension and

intension would disappear, since, with respect to extensional occurrences,

the nominatum of an expression is the same as its extension, and its sense

the same as its intension (29-1 and 2). Attractive though these conse-

quences may appear, it seems to me that it would be at least premature to

propose Method V as a solution of the antinomy at the present time.

Even if the thesis of extensionality were proved, this would not be suf-

ficient as a justification for Method V. We should have to show, in addi-

tion, that an extensional language for the whole of logic and science is not

only possible but also technically more efficient than nonextensional forms

of language. Though extensional sentences follow simpler rules of deduc-

tion than nonextensional ones, a nonextensional language often supplies

simpler forms of expression; consequently, even the deductive manipula-

tion of a nonextensional sentence is often simpler than that of the compli-

cated extensional sentence into which it would be translated. Thus both

forms of language have their advantages; and the problem of where the

greater over-all simplicity and efficiency is to be found is still in the bal-

ance. Much more investigation of nonextensional, and especially of modal,

language systems will have to be done before this problem can be decided.

Therefore, for the time being, Method V as a solution of the antinomy has

to be left aside.

Method VI, Extension and Intension. If, instead of the method of the

name-relation, the method of extension and intension is used for semanti-

cal analysis, then the concept of nominatum does not occur, and hence the

antinomy of the name-relation in its original form cannot arise. Since,

however, the concept of extension is in many respects similar to and part-

ly coincides with the concept of nominatum, there might arise, under cer-

tain conditions, an antinomy of the identity of extension analogous to that

of the identity of nominatum. The antinomy would arise if for the concept

of extension a principle analogous to the principle of interchangeability of

names (24-3) were laid down. The form which we have chosen for the

method of extension and intension excludes the antinomy by prescribing

32. SOLUTIONS OF THE ANTINOMY 143

for expressions with the same extension, in other words, for equivalent

expressions, a principle of interchangeability which is restricted to e#te?i-

sional contexts (12-1). Our second principle (12-2) concerns L-equivalent

expressions, hence those with the same intension; thus it is related to

Frege's second principle (28-9).

Perhaps it will occur to the reader at this point to ask why, if a restric-

tion of interchangeability to extensional contexts assures the elimination

of the antinomy, we might not simply keep Frege's two concepts and re-

strict his first principle to extensional (nonoblique) contexts. The reply is

that Frege's concept ('bezeichnen') is meant in the sense of a name-rela-

tion, that is, as a relation characterized by the principles 24-1 and 2;

therefore it would be quite implausible and unnatural, as we have seen

earlier, not to maintain the principle of interchangeability 24-3 in its unre-

stricted form. Or, to put it the other way round, if somebody uses a con-

cept for which the principle 24-3 does not hold unrestrictedly, then this

concept is not a name-relation and is not the concept meant by Frege

and many other logicians, for example, Church and Quine.

It is easy to see that the method of extension and intension avoids those

features of the other methods which we have found to be disadvantages.

In our general discussion of the method of the name-relation, we have first

explained the ambiguity in the concept of nominatum ( 25); for in-

stance, even if we understand clearly what is meant by a given predicator,

we may regard either the property or the class as its nominatum. The con-

cept of extension does not involve any analogous ambiguity; the exten-

sion of any predicator of level one and degree one is the class of those indi-

viduals to which the predicator can be truly applied. Further, we have

shown the multiplicity of expressions in the object language to which the

method of the name-relation leads ( 26) ; we have seen that, if our meth-

od is used, this multiplicity is replaced by one expression. Further, the

complications caused by the particular form of the method introduced by

Frege have been explained ( 30). Their common root is the fact that

different occurrences of the same expression may have different nominata.

Since the extension of an expression is always the same, independent of

the context, no analogous complications are caused by our method. The

disadvantage of Quine's method is the restriction of the name-relation to

extensional contexts; there is no analogous restriction of the ap-

plication of the concept of extension. While Church's method avoids

some of the disadvantages of Frege's original method, it shares most of

them; further, his modification of Frege's method, necessary though it is,

144 HI. THE METHOD OF THE NAME-RELATION

causes a new complication, which does not occur in our method. The dis-

advantage of Russell's method is its denial of meaning to individual ex-

pressions and class expressions. In our method there is no such restriction ;

to every expression of these kinds an extension and an intension are

ascribed (for class expressions in the system PM, see above, 26).

Let us sum up the result of our discussion of the method of the name-

relation in this chapter. The method appears in various forms with dif-

ferent authors. Most authors who use the concept of the name-relation do

not seem to be aware of the antinomy and do not develop the method in a

sufficiently explicit form to enable us to see whether and how they avoid

the contradiction. All procedures that have been proposed for the elimi-

nation of the antinomy have serious disadvantages; some of these pro-

cedures lead to great complications, others restrict considerably the field

of application of the semantical meaning analysis. Thus it seems doubtful

whether the method of the name-relation is a suitable method for se-

mantical analysis.

CHAPTER IV

ON METALANGUAGES FOR SEMANTICS

In the metalanguage M, which we have used so far, we have spoken about

extensions and intensions, for instance, about classes and properties. It is the

main purpose of this chapter to show that this distinction does not actually

presuppose two kinds of entities but is merely a distinction between two ways

of speaking. First, we discuss possible methods for denning extensions in terms

of intensions or vice versa, without adopting any of them ( 33). Then we con-

struct a new metalanguage M' ( 34-36). While M contains distinct expres-

sions for an extension (e.g., 'the class Human') and an intension (e.g., 'the

property Human'), M' contains only one expression (e.g., 'Human'), which is,

so to speak, neutral, like the expressions in the symbolic system S x (e.g., 'H').

Therefore, we call M' a neutral metalanguage. By this elimination of the dupli-

cation of expressions, the apparent duplication of entities disappears. It is

shown that all sentences of M can be translated into M', including the semantics

of systems like S x (37). Finally, the question is examined as to whether a com-

plete semantical description of a system, even a nonextensionai system like S 2 ,

can be formulated in a metalanguage which, in distinction to M and M', is

extensional; it seems that this is the case (38).

33. The Problem of a Reduction of the Entities

In the metalanguage M we have so far spoken as if there were two kinds of

entities in each type, extensions and intensions, for example, classes and

properties. Here the question is discussed as to whether we can get rid of this

apparent duplication of entities by defining one kind in terms of the other. Four

methods for defining extensions in terms of intensions are discussed: the concep-

tion of extensions as L-determinate intensions ( 23) ; Russell's contextual

definition of classes in terms of properties, which is shown to involve a certain

difficulty; a modified version of Russell's definition, which avoids the difficulty;

and, finally, a method which uses property expressions themselves as class

expressions but presupposes a particular structure of the language. It does not

seem possible to define intensions themselves in terms of extensions. However,

the class of all designators L-equivalent to a given designator might be taken as

a representative of its intension.

We have used as metalanguage M a part of English, modified and

supplemented in a certain way ( i). Throughout our discussions we have

used in M terms like ' class', ' property ', 'truth-value', 'proposition', 'indi-

vidual', 'individual concept', and the more general terms 'extension' and

'intension'. This manner of speaking gives the appearance of dealing

with a great variety of entities and, in particular, with two kinds of enti-

ties within each type. As stated at the beginning ( 4) , we have used the

terms mentioned only because they help to facilitate understanding, but

us

146 IV, ON METALANGUAGES FOR SEMANTICS

our theory is not based on the assumption that there are entities of all

these kinds. Now, mindful of Occam's razor, we shall try to show how the

number of apparent entities can be reduced to half. Since the apparent

duplication of entities was actually only a duplication of terminology, all

we have to do is to construct another way of speaking which avoids the

terminological split into extensions and intensions.

Let us begin with the discussion of predicators, because here the dis-

tinction between extension and intension is customary and familiar. If we

wish to have a language which is not, like S r , restricted to elementary

statements about things but contains a more comprehensive system of

logic and especially of mathematics, then we must introduce means for

speaking in general terms not only about things but also about entities of

higher levels, say classes or properties. So much is admitted even by those

logicians who are most wary in admitting abstract entities. 1 The question

is whether it is necessary to admit both kinds of entities, classes and

properties, or whether those of the one kind are definable with the help of

those of the other. For instance, is one of the two phrases (in M) ' the class

Human' and ' the property Human' definable with the help of the other?

Explicit definition is not necessary; a contextual definition would suffice

to make one of the two phrases dispensable in the primitive formulation.

Let us first look for methods which define class expressions in terms of

property expressions.

1. If the concept of L-determinate intensions ( 22) is available, we

can define ' the class/' as 'the L-determinate property which is equivalent

to the property/' ( 23).

2. If we do not wish to make use of the concept of L-determinate in-

tensions, we may consider the possibility of a contextual definition for ( the

class/ by a generalized reference to the properties which are equivalent to

the property /. Since all these properties determine the same class, the

most natural procedure seems to be to interpret a statement about the

class / as a statement about all these properties. Thus, for a system 5,

containing predicate variables'/', 'g\ etc., we could lay down the following

contextual definition for the class-expression '&(fx) J \

33-1. '. . *(*) . .' fo

This definition must be supplemented by a rule specifying what is to be

taken, in any given case in which '$(/#)' occurs, as the context ' . . $(fx) . .'

1 See, for example, Quine, [Notes], p. 125: "Anyone who cares to explore the foundations

of mathematics must, whatever his private ontological dogma, begin with a provisional toler-

ance of classes or attributes [i.e., properties]".

33. THE PROBLEM OF A REDUCTION OF THE ENTITIES 147

to which the definition is to be applied. Following Quine 3 rather than

Russell (see below), we stipulate that the definition is to be applied to the

smallest sentence or matrix in the primitive notation in which the class

expression occurs. Thus, before applying the definition, we have to trans-

form the given sentence containing a class expression by eliminating all

previously defined signs with the help of their definitions; then, with the

help of definition 33-1, we expand each smallest matrix in which the class

expression occurs.

3. Russell 3 was the first to propose a contextual definition of class ex-

pressions on the basis of property expressions. Whitehead and Russell

used this definition in their construction of the system of mathematics in

[P.M.]. 4 Though the method has been able to supply a good working

basis for this construction, there is one feature of the definition which

seems to me disadvantageous. The definition given above (33-1) is, in-

deed, nothing else than a variant of Russell's definition, changed, however,

with respect to the point in question. The definition in [P.M.], transcribed

in our notation, 5 is as follows:

33-2. '. . z(fz) . .' for'(3*)[fe -/)...*. .]'.

The definiens here contains an existential quantifier, not a universal one,

as does 33-1. Thus a statement about the class/ is here interpreted as a

statement not about all properties but about at least one property equiva-

lent to the property/ (in the terminology of [P.M.], "at least one proposi-

tional function formally equivalent to the prepositional function ft").

Russell does not explain his reasons for the form of the definition chosen,

except for saying, correctly, that the definiens ought to be extensional;

this, however, is likewise the case if a universal quantifier is used, as in

33-i-

The form of the definition with the existential quantifier seems to me not

only to be less natural but also to lead to serious difficulties, which make

[M.L.], 26.

3 Russell, "Mathematical Logic as Based on the Theory of Types", American Journal of

Mathematics, XXX (1908), 222-62; for the definition, see p. 249.

4 [P.M.], I, 71 ff., 187 ff.

s The definition in the original notation (p. 249 of the article mentioned above, and [P.M.],

I, 76, 1 88) is as follows:

*2o.oi./{zOM) . - :(3W>) : 4>\x . **x .tx:f{<t>\z\ Df

Our transcription 33-2 is changed in inessential respects only. The exclamation point is omitted

because it is necessary only on the basis of the ramified system of types, which is now generally

regarded as unnecessary, and because it is at any rate inessential for the problem under dis-

cussion. The context is indicated only by dots instead of by a second-level variable, in order to

make the definition applicable also to systems not containing such variables. The biconditional

sign is used according to our abbreviation 3-1.

i 4 8 IV. ON METALANGUAGES FOR SEMANTICS

it appear doubtful whether the definition fulfils the purpose intended. To

show this, let us consider two nonextensional properties of properties, say,

$j and * 2 , such that $ 2 is the contradictory of $,; hence $ 3 holds in all

cases, and only in those, in which * t does not hold. Since 4>j is nonexten-

sional, there are different, but equivalent, properties, say f x and f 2 , such

that * x holds for f x and not for f 2 , and hence $ 2 holds for f 2 . Then, accord-

ing to definition 33-2, both ^ and <&.* hold for the class s(fi3), although

4> x and $ 2 are contradictories and hence logically incompatible. This would

be an awkward result, although it does not constitute a formal contra-

diction, since 4> x and $> 2 are logically exclusive only with respect to proper-

ties, while their application to classes is introduced merely as a certain

mode of speech, which in the formal system itself, as distinguished from

the informal interpretation in terms of classes, is merely a device of ab-

breviation.

In order to see the situation more clearly, let us try to construct a con-

crete example. As earlier ( 26), let PM be the system constructed in

[P.M.], and PM' be the same system with some nonlogical constants

added on the basis of rule 25-1. In order to find something like ^ and

$> 2 in PM or PM', we have to look for nonextensional signs. Among the

very few such signs occurring in the system PM itself, there are the signs

of identity * = ' and nonidentity ' ^ J when standing either between

property expressions or between a property expression and a class expres-

sion. The sign ' = ' is actually used in [P.M.] in this way; 6 and the authors

are aware that it is nonextensional in these contexts. 7 We shall first use the

system PM'. We take as premises the following two sentences of this

system: 8

(i) '(x)(Fx Ex & H*)', or briefly, 'F B = H'.

(ii) 'Fz.Bz *m>.

These sentences say that the property Featherless Biped and the property

Human are equivalent but not identical. Hence they are true. Now we

shall examine the following two sentences:

(jii) '*(&) = Hz'.

(iv) l

6 See [P.M.], I, 191, the proofs of *2o.i3 and *2o.i4.

bid., p. 84.

8 For the convenience of the reader, we transcribe the notation of [PM.] into our notation

by writing the quantifier in the form '(*)' instead of in the form of a subscript and by using

parentheses instead of dots. We keep, however, the notation 'Hz* for a property expression be-

cause this is an essential feature of the notation in [P.M.] (see above, 26).

33. THE PROBLEM OF A REDUCTION OF THE ENTITIES 149

We shall expand these sentences by applying Russell's definition 33-2 in

order to eliminate the class expression ^(Hz)'. We substitute in this defi-

nition <H' for'/'; as'. . z(Rz) . .' we take (iii) and (iv) in turn. Thus (iii)

is expanded into

This sentence is provable in PM', because it follows by existential general-

ization from the instance with 'H' for 'g'. Therefore, (iii) is provable and

fyence true on the basis of the interpretation assumed ( 26). Now let us

expand (iv). Here we have to take into consideration Russell's rule of con-

text, according to which the smallest sentence or matrix in the actually

given abbreviated notation is to be taken as corresponding to the left

side in the definition 33-2. In other words, * 5^' is not to be eliminated be-

fore the elimination of the class expression, and hence the whole of (iv)

is to be taken as ' . . 2(Hz) . .'. 9 Thus we obtain as the expansion of (iv):

This sentence is derivable from the conjunction of our premises (i) and

(ii) by existential generalization with respect to ' F B2'. Hence, (iv) is

derivable from the premises and therefore likewise true. Thus the result

is that the sentences (iii) and (iv) are both true, although they look like

contradictories. They do not actually constitute a contradiction because

(iv) is not meant as the negation of (iii) ; this is shown by the fact that,

according to the rules of the system PM', (iv) is expanded not into

*~(2(Hs) = H)' but into (vi). Nevertheless, our result shows that the

notation of the system PM' is here misleading, because it suggests the

interpretation of (iv) as "(Hs) is not identical with Hz", which would be

in contradiction to (iii). It is true that Russell warns repeatedly that the

class expressions are incomplete and have no meaning in isolation. On the

other hand, the notation has been constructed with this aim in mind:

The class expressions should be such that they can be manipulated as if

they were names of entities; and Russell seems to assume that this aim

has been reached. 10 Our result makes this assumption doubtful.

In the system PM itself, without the use of nonlogical constants, we

can reach a similar result. Here we take as premise the assumption that

there are two properties which are equivalent but nonidentical. Any par-

9 It is stated ([P.M.], p. 188) that with regard to the scope of class expressions the same

conventions are adopted as for descriptions. That the sign '?*' when occurring in combination

with a description is not eliminated before the elimination of the description is seen from the

example in [P.M.], p. 173, line 2 from bottom.

10 [P.M.], p. 188, 11. 3-5 and 14-16; and the text of p. 198.

i5o IV. ON METALANGUAGES FOR SEMANTICS

ticular instance for example, the conjunction of (i) and (ii) can be

formulated only in PM', not in PM. But the existential assumption can

be formulated in PM itself as follows:

In a way similar to the above we can derive from this premise in PM the

following:

(viii) W)[(W =/*) (/*) */)]'-

This sentence is not provable in PM, but it is derivable from the premise

(vii), which is, no doubt, true on the basis of the interpretation intended

in [P.M.]; this work itself mentions the example of the properties Feather-

less Biped and Human. Although (viii) is not actually self-contradictory,

still it looks as if it were. This shows again that the way the class expres-

sions are introduced by Russell's definition is not quite in agreement with

the intended purpose.

If, instead of Russell's definition 33-2, a definition involving a universal

quantifier like 33-1 is used, then (iii) is not provable. In this case, both

(iii) and (iv) are false. This apparently, but not actually, violates the

principle of excluded middle; however, this seems less disturbing than the

previous apparent violation of the principle of contradiction. If, further-

more, the rule of the context of the class expression is changed from

Russell's form (the smallest sentence in the actually given abbreviated

notation) to Quine's form (the smallest sentence in the primitive notation),

as was done in 33-1, then (iv) is expanded into the negation of (iii). In this

case, (iii) is false and (iv) is true, and thus there are no longer any puzzles. If

in [P.M.] the definition of classes were changed according to 33-1, then

only some of the proofs in a few subsections referring to the definition

would need to be changed. It seems that later only extensional contexts

occur; therefore, the theorems and proofs throughout the bulk of the work

would remain unchanged.

4. Suppose that the language system S in question is such that every

smallest matrix, that is, one which does not contain another matrix as a

proper part, is extensional. This is the case, for instance, if modal operators

are the only nonextensional signs. [Therefore, it is the case in S 2 , where

'N' is the only nonextensional sign. Here, every nonextensional matrix

contains a (proper or improper) part of the form 'N(. . .)' and hence a

matrix ' . . / as a proper part. On the other hand, it is not the case in the

system PM' if we take it as including the sign ' = ' between property

expressions. For example, the sentence 'H = H* is of smallest size, but

it is intensional (see above).] Then every class expression in S stands, after

33. THE PROBLEM OF A REDUCTION OF THE ENTITIES 151

the elimination of all other defined signs, within a smallest matrix which

is extensional. Therefore, the class expression can here simply be replaced

by the corresponding property expression, even if the smallest matrix in

question stands within a wider nonextensional context. [For example,

*N[a *(. . x . .)]' or 'N[(. . x . .)(*)]' is L-equivalent to, and hence L-

interchangeable with, *N[(X#)(. . x . .)(a)] J , and hence also with

*N(. . a . .)'] The reason for this is as follows. Let the smallest matrix

containing a certain occurrence of '&(Hxy be represented by '<!>(: (H#))'.

This is, according to our definition 33-1, L-equivalent to ' (g) [(g = H) D

$#]'. This obviously L-implies '^H'; but the latter also L-implies the

former (12-1), since, according to our assumption with respect to 5, '$'

is extensional. Therefore, the two sentences are L-equivalent, and hence

also L-interchangeable even in intensional contexts (12-2).

This shows that, in a system S of the kind described, we may simply

take the property expressions themselves as class expressions also. This

procedure is still simpler than procedure (2) explained above, which uses

the contextual definition 33-1 for class expressions.

We have discussed four methods for the definition of classes in terms of

properties. They can be used more generally for the definition of exten-

sions of any kind in terms of intensions. These methods as here explained

apply to symbolic object languages. The same methods can, of course, be

applied in an analogous way to a word language and, in particular, to our

metalanguage M. This latter application would be more important for us,

because in our symbolic object languages we do not want to have class

expressions in addition to property expressions, for the reasons explained

earlier (26), while in M we have phrases of both forms ' the class Human'

and ' the property Human' and we should like to dispense with one of

these forms in the primitive formulations in M. Since M contains identity

sentences for properties (like PM'), it does not fulfil the condition re-

quired for S in method (4). But we could apply method (2) to M. This

would consist in laying down the following three definitions; the first cor-

responds to 33-1, the second and the third are analogous to it:

33-3. ... the class /. . . =DI for every property g equivalent to the

property/, ... the property g . . . .

33*4. . . . the truth- value p . . . =Df for every proposition q equivalent

to the proposition p, . . . the proposition q . . . .

33-5. ... the individual x . . . = DI for every individual concept y equiva-

lent to the individual concept #, . . . the individual concept y . . . .

1 5 2 IV. ON METALANGUAGES FOR SEMANTICS

A convention determining the context indicated by dots would here be

laid down similar to that for 33-1. (We may disregard here inessential

changes of this context required by the accidents of idiom; for example,

'* belongs to the class/' is changed to 'x has the property/.)

The three definitions here mentioned will not actually be adopted for

M, because we shall find another, simpler form of a metalanguage which

avoids even the apparent duplication of entities in M by entirely avoiding

the duplication of expressions. This will be explained in the next section.

Would it be better to take properties as primitive and to define classes in

terms of properties or to take classes as primitive and to define properties

in terms of classes? We have explained four methods for the first alterna-

tive. Quine 11 rejects it for the reason that a property is even more obscure

than a class. Which of the two is more obscure and which intuitively

clearer is a controversial question. I shall not discuss this question here; it

seems to be more psychological than logical. However, I think that most

logicians agree that, if the terms ' class' and ' property' are understood

in their customary sense, classes can be defined by properties, but it is

hardly possible to define properties by classes (unless these classes are, in

turn, characterized by properties); for a property determines its class

uniquely, while many properties may correspond to a given class. It is,

however, possible to define in terms of classes certain entities which stand

in a one-one correlation to properties or other intensions and therefore

may represent them for many purposes. We defined earlier the L-equiva-

lence class of a designator in S as the class of all designators in S L-equiva-

lent to it (3-1 sb). It is easily seen that there is a one-one correlation be-

tween the L-equivalence classes in S and the intensions expressible in S.

Therefore, the L-equivalence class of a designator in 5 may be taken as

its intension or at least as a representative for its intension. Procedures of

this kind have been indicated by Russell and by Quine. Russell 12 mentions

as a possibility the definition of a proposition as " the class of all sentences

having the same significance as a given sentence". Quine 13 defines the

meaning of an expression as the class of those expressions which are

synonymous with it. Russell's concept of having the same significance and

Quine's concept of synonymity correspond at least approximately to our

concept of L-equivalence; if a stronger relation than L-equivalence is

meant, for example, something like intensional isomorphism (14), the

concepts are, of course, analogous.

" [Notes], p. 126. w [Inquiry], p. 209. x * [Notes], p. 120.

34. THE NEUTRAL METALANGUAGE M' 153

34. The Neutral Metalanguage M 7

While some symbolic systems (e.g., Russell's) have different expressions for

properties and for classes, our systems (Sj and S 3 ) have only one kind of expres-

sion. Analogously, we now introduce a "neutral" metalanguage M'. While M

contains phrases like 'the property Human' and 'the class Human', M' con-

tains only the neutral expression 'Human'; and similarly with other types of

designators. In this way the duplication of expressions in M is eliminated in

M', and thus the apparent duplication of entities disappears.

If, of the two phrases ' the class Human' and ' the property Human* in

M, either the first were defined by the second or vice versa, then in the

primitive notation of M we should have only one phrase instead of two,

and hence the number of entities would be cut in half. I think that the

same aim can also be reached in another and even simpler way. We have

seen earlier ( 26) that, on the basis of the method of extension and inten-

sion, the notation in a symbolic object language can be simplified. Instead

of one expression as a name of a property (e.g., 'H' in PM') and another

expression as a name of the corresponding class (e.g., ':(H#)'), it is suf-

ficient to use one expression (e.g., ' (X#) (H#) ' or 'H' in S x ). This expression

is, so to speak, neutral in the sense that it is regarded neither as a name of

the property nor as a name of the class but rather as an expression whose

intension is the property and whose extension is the class. If we apply an

analogous procedure to the word language M, then our aim will be at-

tained. Thus we have to look for a language form M 7 in which we use, in-

stead of the two phrases 'the class Human' and 'the property Human',

only one phrase; this phrase, however, is not to be one of the two but

rather another one which is neutral in containing neither the word ' class'

nor the word 'property'. The simplest procedure is to take the word

' human' or * Human' alone (the capitalized form to be used, as previously,

at places where English grammar does not permit an ordinary adjective).

We take M' as the neutral metalanguage which results from M by these

changes, that is, by eliminating the terms 'class',' property', etc., in favor

of neutral formulations. Our tasks is now to find suitable forms for formu-

lations in M / . In this discussion we shall speak about M and M', and hence

we shall speak in a metametalanguage MM. For easier understanding, we

take MM similar to M rather than to the less familiar M 7 ; that is to say,

we shall use terms like 'class', 'property '/extension', 'intension', etc., in

speaking about M ; , although these terms cannot occur in M' itself. The

very next sentence will, in fact, be an example of this use. The term

'Human' in M 7 is neutral in the same sense in which 'H' is neutral in S x :

'Human' is regarded neither as a name of a class nor as a name of a prop-

154 IV. ON METALANGUAGES FOR SEMANTICS

erty ; it is, so to speak, at once a class expression and a property expression

in the following way :

34-1. The extension of ' Human' in M' is the class Human.

34-2. The intension of ' Human' in M' is the property Human.

Analogously, instead of the two phrases ' the individual Scott ' and ' the

individual concept Scott 7 in M, we have in M ; the one neutral term ( Scott '.

Here we have:

34-3. The extension of 'Scott' in M' is the individual Scott.

34-4. The intension of ' Scott' in M' is the individual concept Scott.

Since classes and properties have different identity conditions, a diffi-

culty arises in the translation of identity sentences into M'. Take as an

example the following sentences in M (see 4) :

34-6. The class Human is the same as the class Featherless Biped.

34-6. The property Human is not the same as the property Featherless

Biped.

34-7. The property Human is the same as the property Rational Animal.

We translated, above, two phrases in M into 'Human' by simply omitting

the words 'the class' and 'the property'. However, if we were to do the

same with 34-5 and 34-6, a contradiction would obviously result. Gen-

erally speaking, since identity is different for extensions and intensions, a

neutral formulation cannot speak about identity. Hence, identity phrases

like 'is identical with' or 'is the same as' are not admissible in M 7 . How,

then, to translate identity sentences into M'? Here the terms l equivalent 5

and 'L-equivalent' in their nonsemantical use, as defined by 5-3 and 5-4,

will help; note that in this use the terms stand for relations, not between

designators, but between intensions. The definitions show that identity of

extensions coincides with equivalence of intensions, and identity of inten-

sions coincides with L-equivalence of intensions. Here in M', the terms

'equivalent' and 'L-equivalent' can be used in connection with neutral

phrases instead of phrases for intensions without any difficulty ; therefore,

we shall speak of equivalence and L-equivalence of neutral entities. Thus

the general rules for the translation of identity sentences (in M or in a

non-neutral object language, e.g., PM') into neutral formulations in M'

are as follows:

34-8. A sentence stating identity of extensions is translated into M' as a

sentence stating equivalence of neutral entities.

34-9. A sentence stating identity of intensions is translated into M 7 as a

sentence stating L-equivalence of neutral entities.

34. THE NEUTRAL METALANGUAGE M' 155

Accordingly, we translate the identity sentences 34-5, 34-6, and 34*7 in M

into the following sentences in M':

34-10. Human is equivalent to Featherless Biped.

34-11. Human is not L-equivalent to Featherless Biped.

34-12. Human is L-equivalent to Rational Animal.

These three sentences can be obtained from 5-5, 5-6, and 5-7 in M by

simply dropping the phrase * the property'.

The sentences 34-10, 34-11, and 34-12 must be clearly distinguished

from the following sentences, which look similar but are fundamentally

different in their nature :

'Human' is equivalent to ' Featherless Biped' in M'.

* Human' is not L-equivalent to ' Featherless Biped' in M'.

i Human' is L-equivalent to ' Rational Animal' in M'.

These sentences are semantical sentences in MM concerning certain predi-

cators in M'. Therefore, the predicators are included in quotation marks,

and the sentences contain references to the language M'. They are perfect

analogues to the sentences 3-8 and 3-11, which are semantical sentences

in M (or M') concerning predicators in S^ On the other hand, the sen-

tences 34-10, 34-11, and 34-12 are not semantical sentences; they do not

speak about the predicators but use the predicators in order to speak

about nonlinguistic entities. Therefore, the predicators are not included

here in quotation marks, and there is no reference to a language system.

The sentences belong to the nonsemantical (and, moreover, to the non-

semiotical) part of M', to that part into which the sentences of the object

languages can be translated. Sentence 34-10 is not only a translation of

the sentence 34-5 in M, but also of the corresponding identity sentence

26-9 in PM' and in ML' ( 25) ; 34-10 is, furthermore, an exact translation

of the likewise neutral sentence 'H = F B' of Si. Since 34-11 and 34-12

are intensional (in the sense of n-3b), there cannot be sentences exactly

corresponding to them in the extensional language S x ( n, Example IV).

But there are such sentences in the modal language S 2 , as we shall see

later; thus 'H m RA' corresponds to 34-12.

Now let us see how neutral formulations of sentences are to be framed

in M'. The translation of simple sentences, especially atomic sentences,

into M' involves no difficulty, since it corresponds closely to the custom-

ary formulation. Thus, for instance, as a translation of 'Hs' we take

in M' the simplest of the translations in M, namely, * Scott is human'

(which is 4-1). The other translations into M earlier discussed are not

neutral (for instance, 4-2, 4-3, and those with 'individual' and 'individual

156 IV. ON METALANGUAGES FOR SEMANTICS

concept' analogous to those mentioned in 9) ; hence they are excluded

from M'. The neutral formulation, 'Scott is human', in M' replaces not

only the non-neutral sentences in M just mentioned but also the two non-

neutral phrases, ' the truth-value that Scott is human' and ' the proposition

that Scott is human' in M (see 6-3 and 6-4). In some cases the simple

formulation ' Scott is human' does not comply with ordinary English

grammar, for instance, when occurring as a grammatical subject. In these

cases we might, in analogy to ' Human', capitalize all words: 'Scott-Is-

Human'; but this would be rather awkward for longer sentences. Another

alternative is the addition of ' that' (see remark on 6-3 and 6-4) : ' that Scott

is human'. This formulation is to be used only as part of larger sentences,

especially in the translation of sentences of M containing one of the

phrases 'the truth- value that Scott is human' or 'the proposition that

Scott is human'. In some cases, this formulation agrees with ordinary

usage, in others not; but we shall admit it into M' in all cases. Thus the

(false) sentence 'N(Hs)' (in S 2 ) is the translated into 'It is necessary that

Scott is human'. Since 'Hs' is equivalent to '(F B)s', the following is

true in M 7 :

34-13. That Scott is human is equivalent to that Scott is a featherless

biped.

This formulation is admittedly somewhat awkward. The more customary

formulations in M with 'the proposition' or 'the truth- value' inserted

after 'to' are not possible here in M' because they are not neutral; and

there is no customary neutral noun. Therefore, we decide to admit the

form 34-13 in M', and likewise the analogous form 34-14 below.

Since 'Hs' is L-equivalent to 'RAs', the following is true in M':

34-14. That Scott is human is L-equivalent to that Scott is a rational

animal.

The use of 'equivalent' and 'L-equivalent' as nonsemantical terms stand-

ing between sentences, as in 34-13 and 34-14, is in analogy to the use of

these terms between predicators (as in 34-10, etc.) and individual expres-

sions, but here, between sentences, it is still more at odds with ordinary

grammar. Fortunately, there is another formulation which is customary

and grammatically correct; but it has the disadvantage that it is ap-

plicable only in connection with sentences, not with other designators.

Instead of 'equivalent' we may use here 'if and only if, and instead of

'L-equivalent' 'that ... if and only if , is necessary'. (Here the

phrase 'is necessary' is placed at the end only for the reason that English

provides no other simple means to indicate that the argument of this

35. M' IS NOT POORER THAN M 157

phrase is the whole 'if and only if sentence and not only its first compo-

nent.) In this way, the following sentences take the place of 34-13 and

34-15* Scott is human if and only if Scott is a featherless biped.

34-16. That Scott is human if and only if Scott is a rational animal, is

necessary.

35. M' Is Not Poorer than M

The question is raised as to whether the designators in M' are correctly

described as neutral or whether they are, perhaps, actually names of intensions

in disguise. If somebody wishes to regard 'Human' in M' (or 'IT in S a ) as the

name of a property, there is no essential objection. But it would be wrong to

say that a language like S a or M' contains only names of properties and no names

of classes and therefore lacks important means of expression. Actually, all

sentences of M are translatable into M'. That M' is not poorer in means of

expression than M is also shown by the possibility of reintroducing the non-

neutral formulations of M into M' with the help of contextual definitions.

Perhaps a reader who is accustomed to the usual method of the name-

relation will have some doubts as to whether the language M' or any other

language can possibly be genuinely neutral; he will say that the allegedly

neutral word 'Human' in M', and likewise the corresponding sign 'IT in

Si and S 2 , in order to be unambiguous, must mean either as much as * the

property Human* or as 'the class Human'; in other words, it must be a

name either of the property or of the class (compare Quine's comments

below, 44). I cannot quite agree with this either-or formulation. I think

we should rather say that the word 'Human', and likewise 'H', stand

both for the property as its intension and for the class as its extension.

However, it must be admitted that the neutrality is not quite sym-

metrical. As we have seen earlier ( 27), a designator stands primarily for

its intension; the intension is what is actually conveyed by the designator

from the speaker to the listener, it is what the listener understands. The

reference to the extension, on the other hand, is secondary; the extension

concerns the location of application of the designator, so that, in general,

it cannot be determined by the listener merely on the basis of his under-

standing of the designator, but only with the help of factual knowledge.

Therefore, if somebody insists on regarding a designator as a name either

of its intension or of its extension, then the first would be more adequate,

especially with respect to intensional languages like M' and S 2 . I think

there is no essential objection against an application of the name-relation

to the extent just described, for example, against regarding 'Human 1 in

M ; and 'H' in S 2 as names of the property Human. The only reason I

158 IV. ON METALANGUAGES FOR SEMANTICS

would prefer not to use the name-relation even here is the danger that this

use might mislead us to the next step, which is no longer unobjectionable.

In accordance with the customary conception of the name-relation, we

might be tempted to say : " If ' Human' (or ' H') is a name for the property

Human, where do we find a name for the class Human? We wish to speak,

not only about properties, but also about classes; therefore, we are not

satisfied with a language like M' or S 2 , which does not provide names for

classes and other extensions." This I should regard as a misconception of

the situation. M 1 is not poorer than M by not containing the phrase 'the

class Human'. Whatever is expressed in M with the help of this phrase is

translatable into M' with the help of ' Human 7 ; and whatever is expressed

in a non-neutral symbolic language like PM' with the help of the class

expression ' ^(H#)' is translatable into S 2 with the help of ' H'. The simplest

method for the translation into S 2 is based on the method (4) explained in

the preceding section. For the sake of an example, let us take, not the

system PM', but the system PM", which is like PM' except for containing

our form of the contextual definition of classes (33-1) instead of Russell's

(33" 2 )- The rules of translation from PM" into S 2 with respect to class

expressions are as follows:

36-1. a. For the translation of a smallest sentence (or matrix) which is

extensional and does not contain * = ', both a property expression

(e.g., *H:6) and a class expression (e.g., 'i(H#)0 in PM" are

translated into S 2 by the corresponding neutral expression (e.g.,

'H' or ' (\x) (Hx) '). (This rule is based on method (4), explained

in 33-)

b. An identity sentence in PM" with two class expressions (e.g.,

'i(Ha) = (Fx !}#)') is translated into an = -sentence with

the corresponding neutral expressions (e.g., 'H == F B').

c. An identity sentence with two property expressions (e.g.,

( H& = RA:f) is translated into the corresponding = -sentence

(e.g.,'HRA').

(We leave aside here identity sentences with one class expression

and one property expression; all such sentences are L-false.)

The translation from M into M' is analogous. We may assume that any

sentence of smallest size in M which is not an identity sentence is exten-

sional. Then the rules are as follows:

35-2. a. In all contexts except identity sentences, both class expressions

(e.g., 'the class Human') and property expressions (e.g., 'the

35. M' IS NOT POORER THAN M 159

property Human') are translated by the corresponding neutral

expressions (e.g., 'Human').

b. A sentence stating the identity of classes is translated into a

sentence stating the equivalence of the corresponding neutral

entities.

c. A sentence stating the identity of properties is translated into a

sentence stating the L-equivalence of the corresponding neutral

entities.

Rules 3S-2b and c are special cases of the general rules 34-8 and 34-9 for

extensions and intensions (see examples 34-10, 34-11, and 34-12).

Thus we see that the view that M' is poorer than M is a misconception.

Since the formulation "The designators in M' are names for intensions,

and there are no names for extensions in M"', may easily lead to this mis-

conception, it seems to me inadvisable. It seems more adequate and less

misleading to say either "every designator in M' has an intension and an

extension" or "the designators in M' are neutral".

In the translations by rule 35-2a, characterizing phrases like ' the class',

'the property', etc., are simply dropped. This might give the impression,

perhaps, that in the transition from M to M' certain important distinc-

tions disappear. This, however, is not the case. All the distinctions made

in M are preserved in M'; they are only formulated in a different and, in

general, in a simpler way. This is shown by the fact that all the non-

neutral ways of speaking in M with terms like 'class', 'property', etc.,

could be reintroduced into M' by contextual definitions if we wanted them

there. (In fact, of course, we do not want to destroy the neutrality of

M'.) Thus the terms 'class' and ' property' could be introduced by the fol-

lowing contextual definitions:

36-3. a. ... the class / . . . = Df for every g, if g is equivalent to /, then

. . . g . . . .

b. . . . the property/ . . . =Df for every g, if g is L-equivalent to/,

then . . . g . . . .

(Concerning the context indicated by dots, see the remarks following

33-5.) If the context indicated by dots is extensional, we may take, in-

stead of (a), the simpler definition:

a'. ... the class/ . . . =DI .../... .

If the context is either extensional or intensional, we may take, instead of

(b), the simpler form:

b'. ... the property/ . . . =D* .../* .

i6o IV. ON METALANGUAGES FOR SEMANTICS

Contextual definitions for the terms 'individual', 'individual concept',

'truth-value', and 'proposition' are analogous. It is admitted that these

definitions lead in some cases to unusual formulations. However, they do

not lead to false results. The decisive point is that they also yield the origi-

nal non-neutral formulations in M.

Identity sentences like those in M can likewise be reintroduced into M'

by a procedure the reverse of that described in 34-8 and 34-9:

36*4* a. The class /is the same as the class g = Df/is equivalent to g.

b. The property /is the same as the property g = Df/is L-equivalent

tog.

The possibility of these definitions in M' for the non-neutral formula-

tions in M shows that all distinctions in M are actually preserved in M'

in a different form. In other words, M' is not poorer in means of expression

than M.

36* Neutral Variables in M ;

Some symbolic systems have different variables for classes and for properties;

we have seen earlier ( 27) that this is unnecessary. Similarly, the phrases

'for every class' and 'for every property* in M constitute an unnecessary dupli-

cation. They are replaced in M' by 'for every/', where '/' is a neutral variable

whose value-intensions are properties and whose value-extensions are classes.

Neutral variables for other types are introduced analogously.

There are still other non-neutral expressions in M which have to be re-

placed by neutral expressions in M', namely, those phrases by which we

refer in a general way to entities of some kind, for instance, pronouns

like 'every', 'any', 'all', 'some', 'there is', 'none', in combination with

words like 'class', 'property', etc. In a symbolic language, phrases of this

kind are translated with the help of variables in quantifiers. We have seen

earlier ( 27) that in a symbolic language not only the use of different ex-

pressions for classes and for properties is an unnecessary duplication, but

so is likewise the use of different variables for classes and for properties

(as, for instance, 'a' and '<' in the system PM). Instead, we may use

neutral variables, whose value-extensions are classes and whose value-

intensions are properties. Now we shall do the same in M', in order to

make possible the neutral formulation of general sentences. We supple-

ment the word language in M' by the following letters as variables: '/',

'g', etc., for predicators of level one and degree one as value expressions;

V, '/> etc., for individual expressions; '^', '<?', etc., for sentences. Thus, a

non-neutral formulation of a universal sentence in M containing one of

the two phrases 'every class' (or 'all classes') and 'every property' (or 'all

37. SEMANTICS IN THE NEUTRAL METALANGUAGE M' 161

properties') is translated into a neutral sentence of M' with the help of

the phrase ' for every/', corresponding to a universal quantifier. Likewise,

an existential sentence in M containing one of the phrases 'some class' (or

' there is a class') and ' some property' (or ' there is a property') is translated

into M' with the help of 'for some/' (or 'there is an/'). Analogously, a

general sentence concerning propositions or truth-values is translated

into M' with the help of ' for every p j or ' for some p* (or ' there is a p y ). And

a general sentence concerning individuals or individual concepts is trans-

lated with the help of 'for every x' or 'for some x' (or 'there is an #').

Examples will be given later (see 43-4).

Universal sentences in M about extensions or intensions in general can

likewise be translated into M' if we introduce general variables for which

designators of all types are value expressions. To avoid contradictions,

suitable restrictive rules have to be laid down for the use of these general

variables; this can be done in different ways. 14

37. On the Formulation of Semantics in the Neutral Metalanguage M'

Two semantical relations between expressions and neutral entities, designa-

tion and L-designation, are introduced into M'. It is shown how semantical

rules and statements in M can then be translated into M'. The relation of

designation is extensional; it is used for the translation of statements concerning

the extension of given expressions. The relation of L-designation is nonexten-

sional; it serves for the translation of statements concerning the intension of

expressions. Thus the whole semantics of a system (e.g., S x ) can be translated

from M into M'.

In the preceding sections we have discussed only the nonsemantical

part of the metalanguage, that part into which the sentences of object

languages can be translated. We come now to the more important

semantical part of the metalanguage, that part in which we speak about

the sentences and other expressions of the object languages, applying to

them semantical terms like 'true', 'L-true', 'equivalent', 'L-equivalent',

etc. Most of the discussions in the earlier chapters of this book are formu-

lated in this semantical part of the metalanguage M. This holds, in par-

ticular, for those statements which speak about classes, properties, prop-

ositions, etc., not only in a general way but in relation to expressions of

an object language for instance, the following two ( 4) :

37-1. The extension of 'H' in S x is the class Human.

37-2. The intension of 'H' in Si is the property Human.

14 For a historical survey of different methods of avoiding the antinomies, sec Quine,

[MX.], 29.

162 IV. ON METALANGUAGES FOR SEMANTICS

The important question now is whether it is possible also to translate

these semantical statements of M into the neutral metalanguage M', that

is to say, into formulations which, instead of phrases like 'the class Hu-

man' and 'the property Human', use only neutral phrases like 'Human'.

Only if this is possible can we say that we have overcome the duplication

of entities.

We shall see that it is indeed possible to translate semantics from M

into M'. The sentence 37-1 states that the relation of extension holds be-

tween the class Human and the predicator 'H' (in S x ), and 37-2 states

that the relation of intension holds between the property Human and the

same predicator. How can we obtain neutral formulations in M' referring

to the neutral entity Human instead of to the class and the property?

It would, of course, not do simply to drop the phrases ' the class' and ' the

property' in those sentences, because then the same entity would be as-

serted to be at once the extension and the intension of the same predica-

tor, and that would not be in accordance with the intended meaning of

the terms 'extension' and 'intension'. Instead, we must make use of a

relation which holds between the neutral entity Human and the predica-

tor 'H', a relation which can be neither the relation of extension nor that

of intension, although it is similar to them. A closer investigation of the

situation shows that we need here two new relations, both holding between

'H' and Human; the first of them is related to the second as a radical

semantical concept (e.g., truth) to the corresponding L-concept (e.g.,

L-truth). Therefore, it seems natural, if we find a suitable word for the

first relation, to take the same word with the prefix 'L-' for the second.

The first relation is here meant in such a way that it is definable also in an

extensional metalanguage; but the second relation is intensional, as we

shall see. Since the first relation holds between an expression (e.g., 'H')

and an entity (e.g., Human) for which that expression stands, a word like

' means', ' signifies', ' expresses', ' designates', ' denotes', or something similar

would seem suitable. I do not wish to make a specific suggestion. Let us

tentatively use the term 'designates' for the first relation, and hence

'L-designates' for the second. Then, instead of 37-1 and 37-2 in M, we

have in M', with respect to Si, the following:

37-3. 'H' designates Human.

This may be regarded as the formulation in M' of a rule of designation for

the system S x (corresponding to the first item in the previous rule 1-2).

The first relation is intended to be extensional; that is to say, any full

sentence of it is extensional with respect to each of the two argument

37. SEMANTICS IN THE NEUTRAL METALANGUAGE M' 163

expressions. Hence, 37-3 is extensional with respect to ' Human'; that is

to say, the occurrence of l Human' in this sentence is interchangeable with

any predicator which is equivalent to * Human' in M'. Thus we obtain the

following two results, according to the equivalences stated in 34:

37-4. 'H' designates Featherless Biped.

37-5. 'H' designates Rational Animal.

By using a neutral predicator variable ( f (see 36) and 'equivalent' as

a nonsemantical term (see 5-3 and 34), we can express the result in a

general form :

37-6. For every /, if / is equivalent to Human, then ' H' designates /

(in S,).

If a suitable definition for 'designates in S t ' is laid down, which has not

been done here, then the converse of 37-6 also holds:

37-7. For every/, 'H' designates/ (in SO if and only if / is equivalent to

Human.

We have decided to use the term 'L-designates' for the second relation.

We shall not give a definition for it. We assume for the following discus-

sion that it is defined with respect to a given system, say S t , in such a way

that the following condition 37-8 is fulfilled; an analogous condition holds

for L-truth, according to our convention 2-1, and for the other L-concepts.

37-8. An expression Sl L-designates an entity u in Si if and only if it can

be shown that 2li designates u in Si by merely using the semantical rules

of Si, without any reference to facts.

(The variable 'u y here used in M' is a general, that is, not type-restricted,

variable; see the remarks at the end of 36.) Now let us apply 37-8 to

37-3, 37-4, and 37-5 in turn. Statement 37-3 can be established on the basis

of the semantical rules of Si alone in a trivial way, since it is itself one of

these rules. This yields, with respect to S x :

37-9. 'H' L-designates Human.

The same, however, does not hold for 37-4. In order to show that this

statement holds, we have used and must use not only the semantical rule

37-3 but also the result that the predicate rs 'Human' and 'Featherless

Biped' are equivalent in M'; this equivalence, like that of the correspond-

ing predicators 'H' and *F B' in S x , is not an L-equivalence (see 34)

but is based on biological fact (3-6). Hence, according to 37-8, the follow-

ing is true in M' :

37-10. 'H' does not L-designate Featherless Biped.

164 IV. ON METALANGUAGES FOR SEMANTICS'

Since ' Human 1 and ' Featherless Biped' are equivalent in M', we see from

37-9 and 37-10 that the relation of L-designation is nonextensional.

Statement 37-5 can again be established on the basis of rule 37-3 alone,

without reference to facts, because 'Human* and 'Rational Animal' are

supposed to mean the same (see remark on 1-2). Hence, according to 37-8,

the following is true:

37-11. 'H' L-designates Rational Animal.

We can formulate the result in a general form with a neutral variable ( f

and 'L-equivalent' as a nonsemantical term:

37-12* For every/, if /is L-equivalent to Human, then 'H' L-designates/.

If a suitable definition for 'L-designates' is laid down in accordance with

the convention 37-8, then also the converse of 37-12 holds:

37-13. For every/, 'H' L-designates / if and only if / is L-equivalent to

Human.

Statement37-3 maybe regarded as a translation of 37-1 into M',and like-

wise 37-9 as a translation of 37-2. It is true that the explicit reference to a

class in 37-1 is not directly mirrored by any expression in 37-3, but it is

indirectly represented by the extensionality of 37-3 with respect to 'Hu-

man', which is shown by the instance 37-4 and generally by 37-6. Thus,

37-6 may also be regarded as a translation of 37-1. Similarly, the explicit

reference to a property in 37-2 is indirectly represented by the intensional-

ity of 37-9 with respect to 'Human', which is exhibited in instances like

37-10 and 37-11 and generally in 37-12. Thus, 37-12 may also be regarded

as a translation of 37-2.

We have shown the application of the relations of designation and L-

designation to predicators. The application to designators of other types is

quite analogous. As examples with respect to individual expressions in Si,

in analogy to 37-3, 37-4, 37-6, and 37-7, the following sentences are true

inM':

37-14. 's' designates Walter Scott.

37-16. 's' designates The Author Of Waverley.

37-16. For every x, if x is equivalent to Walter Scott, then * s' designates x.

37-17. For every x, ' s' designates x if and only if x is equivalent to Walter

Scott.

Rule 37-14 is a rule of designation of the system S x , corresponding to

the first item in i-i. Sentence 37-15 is derived from 37-14 with the help

of a historical fact (9-1). Further, in analogy to 37-9, 37-10, 37-12, and

37-13, the following sentences are true in M':

37. SEMANTICS IN THE NEUTRAL METALANGUAGE M' 165

37-18. 's' L-designates Walter Scott,

37-19. 's' does not L-designate The Author Of Waverley.

37-20. For every #, if x is L-equivalent to Walter Scott, then 's' L-desig-

nates x.

37-21. For every x, 's' L-designates x if and only if x is L-equivalent to

Walter Scott.

Sentences 37-14 and 37-1 6 maybe regarded as translations of the follow-

ing sentence in M ( 9) :

'The extension of 's' is the individual Walter Scott'.

Sentences 37-18 and 37-20 may be regarded as translations of:

'The intension of 's' is the individual concept Water Scott'.

Remarks analogous to those made above on 37-3 and 37-9 hold here.

Analogously, with respect to sentences in S x , the following statements

are true in M'; we use a that-clause for the neutral formulation ( 34):

37-22. 'Hs' designates that Scott is human.

This statement, in distinction to 37-3 and 37-14, is itself not a semantical

rule but follows from these rules with the help of a suitable definition for

'designates in Si', as applied to sentences. The following is a consequence

of 37-22, because 'Scott is human' and 'Scott is a featherless biped' are

equivalent in M':

37-23. 'Hs' designates that Scott is a featherless biped.

Generally, with the neutral variable '/>' ( 36):

37-24. For every p, if p is equivalent to that Scott is human, then 'Hs'

designates p.

37-25. For every p, 'Hs' designates p if and only if p is equivalent to that

Scott is human.

(In these two statements, the nonidiomatic phrase 'is equivalent to that 1

may be replaced by 'if and only if; see the explanations to 34-13 and

34-IS-)

Furthermore, for L-designation, the following statements are true in

M':

37-26. 'Hs' L-designates that Scott is human.

37-27. 'Hs' does not L-designate that Scott is a featherless biped,

37-28. For every />, if p is L-equivalent to that Scott is human, then

'Hs' L-designates p.

i66 IV. ON METALANGUAGES FOR SEMANTICS

37-29. For every p, 'Hs' L-designates p if and only if p is L-equivalent to

that Scott is human.

(In the last two statements, the nonidiomatic phrase 'is L-equivalent to

that' can be avoided by a transformation analogous to that of 34-14 into

34-16.)

Sentences 37-22 and 37-24 may be regarded as translations of the sen-

tence 6-3 in M concerning the truth- value as extension; likewise, 37-26

and 37-28 as translations of the sentence 6-4 concerning the proposition as

intension. Remarks analogous to the earlier ones hold here.

We have previously seen that it would be possible to reintroduce the

non-neutral terms 'class', 'property', etc. into M' by contextual defini-

tions. If we were to apply these terms in the formulation of semantical

statements in M', these statements would become quite similar to. those

in M. For example, by applying the definition of 'class' (3S~3a) to 37-6,

we obtain:

37-30. 'H 5 designates the class Human.

Likewise, by applying the definition of 'property' (35-3!)) to 37-12, we

obtain:

37-31, 'IF L-designates the property Human.

Analogous results would be obtained for individual expressions and

sentences. These results show that the relation of designation in M' cor-

responds to the relation between a designator and its extension in M, and

the relation of L-designation in M' corresponds to the relation between

a designator and its intension in M. IS

1 5 My use of the terms 'designation* and 'designation' in [I] was, as I realize now, not quite

uniform, because at that time I did not yet see clearly the distinction which I make now in M

with the help of the terms 'extension' and 'intension*, and in M' with the help of the terms

'designation* and 'L-designation'. The use of 'designatum' in [I] corresponds in most cases to

the present use of 'intension' in M (or 'L-designatum' in M'). Thus, in the Table of Designata

([I], p. 1 8) and hi later examples of Rules of Designation, the following kinds of entities are

taken as designata: properties, relations, attributes, functions, concepts, and propositions. It

is only with respect to individual expressions that I used the term hi a different way, taking as

designata hi the table and in the examples not individual concepts but individuals. Since it is

not customary to speak of individual concepts under any term, I was not aware of the fact

that they, and not individuals, belong to the same category as properties, propositions, etc.

Thus, in the case of individual expressions, what I took as designata were the same as what

would be taken as nominata by the method of the name r relation. It is probably due to this fact

that Church ([Review C.]) understood my term 'designatum' in all cases in the sense of

'nominatum'; and presumably Quine ([Notes]) likewise believes himself to be in accord with

my use when he applies 'designatum' in this sense. I regret that the lack of a clear explanation

hi [I] has caused these misunderstandings. This lack was not accidental but was caused by an

obscurity of long standing in some of the fundamental semantical concepts. If I see it correctly,

this obscurity has been overcome only by the analysis made in this book. Church's statement

([Review C.], pp. 299 f .) that the designatum of a sentence is not a proposition but a truth-

value is on the basis of Frege's method of the name-relation correct for Church's use of

37. SEMANTICS IN THE NEUTRAL METALANGUAGE M' 167

The examples in this section show how semantical sentences in M, Stat-

ing the extensions or intensions of predicators, individual expressions,

and sentences in Si, can be translated into neutral formulations in M'.

The translation of semantical sentences which refer not to nonlinguistic

entities but only to expressions in the object language, for instance, sen-

tences about truth, L-truth, equivalence, and L-equivalence, does, of

course, not involve any difficulty. Thus the whole of semantics, with re-

spect to Si or any other system, can be translated from M into H'.

The reasons for our use of the two metalanguages, M and M', may be

briefly summarized. Metalanguage M was used in the first three chapters

of this book in an uncritical way, so to speak. It supplies pairs of terms

'class'-'. property ', atid the like, and the general terms ' extension' and ' in-

tension'. The use of these terms constituted what we have called the meth-

od of extension and intension. The chief reason for using these pairs of

terms is that they correspond to familiar concepts, usually regarded as

kinds of entities. In the present chapter we constructed the neutral meta-

language M', which has no such pairs of terms and thus avoids the appear-

ance of a duplication of entities. Although the terms 'extension* and * in-

tension' do not occur in M', the essential features of the method used in

M' are still the same as in M; therefore, we might still call the method

used in M' the neutral form of the method of extension and intension, or

else the (neutral) method of equivalence and L-equivalence, or the

(neutral) method of designation and L-designation. The distinctions made

in M are not neglected in M' but are represented in a different form. In-

stead of an apparent duplication of entities, we have here a distinction

between two relations among expressions, namely, equivalence and L-

equivalence, and, based upon it, a distinction between two relations be-

tween expressions and entities, namely, designation and L-designation.

We have seen that it is possible to construct in M' contextual definitions

for the non-neutral terms ' class', * property', etc., which lead to formula-

tions like those in M. This result shows, on the one hand, that the neutral

method in M' does indeed preserve all distinctions originally made in M

and hence is an effective substitute for the original form of the method. On

the other hand, the result is a justification for M, since it shows that the

'designatum' in the sense of 'nominatum'; not, however, for my use of 'designatum 1 in [I] in

the sense of 'intension*.

In [I], I occasionally used the terms 'synonymous 1 and 'L-synonymous'. The distinction

which I had in mind but did not grasp satisfactorily is now expressed more adequately by the

terms 'equivalent' and 'L-equivalent' in their application to designators in general.

i68 IV. ON METALANGUAGES FOR SEMANTICS

apparent duplication of entities in M is, in fact, only a duplication of

modes of speech.

Since the non-neutral mode of speech in M and the neutral mode of

speech in M 7 cover the same domain, the choice between them is a matter

of practical preference. The neutral formulation is much simpler and

avoids even the appearance of a duplication of entities. Therefore, this

formulation might be preferable in cases in which a metalanguage for

semantical purposes is to be constructed in a strict, systematized way, for

instance, in a symbolic language or in words whose use is regulated by

explicit rules. On the other hand, the non-neutral formulation is in most

cases more familiar, more in accordance with ordinary usage. Therefore,

this formulation may seem preferable for semantical discussions which

are not on a highly technical level, especially for purposes of introductory

explanations. That is the reason for its use in the first part of thig book.

38. On the Possibility of an Extensional Metalanguage for Semantics

The question is discussed as to whether a complete semantical description

of a system, even a nonextensional system like S 2 , can be formulated in an

extensional metalanguage, for instance, the sublanguage M e of M' contain-

ing only the extensional sentences of M 7 . It is found that most of the semantical

rules (rules of formation, of truth, and of ranges) can be formulated in M e

without any difficulty. The situation is not so simple with respect to the rules

of designation; but it seems that these rules can also be adequately formulated

in M e .

We have formulated semantical sentences in two different metalan-

guages, M and M'. Both these languages are nonextensional. The question

arises as to whether semantics can be formulated in an extensional meta-

language more exactly, whether it is possible to construct an extensional

metalanguage sufficient for the formulation of a complete semantical de-

scription even of a nonextensional object language (as, for instance, S 2 ).

A semantical description of an object language is complete if it, given as

the only information about the language, enables us to understand every

sentence of the language and hence to determine whether or not it is

L-equivalent to any given sentence of our metalanguage. The answer to

the question is not at present known. However, on the basis of some

studies I have made, an affirmative answer seems to me not improbable.

Here I shall give a few indications only.

It is easily seen that a sentence in M which says what the intension of a

certain expression is, is nonextensional. For example, the sentence 'the

intension of 'H' in S x is the property Human' (4-17) is nonextensional with

respect to 'the property Human', because if this predicator is replaced by

38. AN EXTENSIONAL METALANGUAGE FOR SEMANTICS 169

the equivalent one, 'the property Featherless Biped', then the true sen-

tence is changed into a false one. Sentences of this kind are essential for

the use of our method in M. Therefore, if we wish to find extensional

semantical sentences, it seems more promising o look at the neutral

formulations in M'. The term 'intension' does not occur in M'; nor do

those intensional sentences of M which state the identity or the non-

identity of properties or other intensions (for instance, 4-8 and 4-9).

Nevertheless, M' is not extensional; the semantical formulations which

we used in M' contain the following three nonextensional (and, moreover,

intensional) terms and no others. The first is the modal term 'necessary'

(see, for instance, 34-16). The second is the term 'L-equivalent' in its

nonsemantical use, as occurring, for instance, in 34-11, 34-12, and 34-14;

it is easily seen that each of these sentences is nonextensional with respect

to both argument expressions. This term is definable on the basis of

'necessary' (compare, for instance, 34-14 and 34-16). [Note, incidentally,

that the semantical term 'L-equivalent in the system 5' is extensional.

For example, ' @ x is L-equivalent to @ 2 in the system Si' is extensional; in

contradistinction to 34-14, it does not contain sentences as parts, but only

names of sentences.] The third nonextensional term in M' is'L-designates'

(see the remark following 37-10).

Let M e be the metalanguage which contains all the extensional sen-

tences of M' and no others; we can construct it out of M' by omitting all

sentences containing the three nonextensional terms mentioned. Our ques-

tion is: How much of the semantics, say of the extensional system S x and

the intensional system S 2 , can be formulated in M e ?

A complete system of semantical rules for S x or S 2 , which is not given in

this book, would consist of the following kinds of rules:

(i) Rules of formation, on the basis of a classification of the signs;

these rules constitute a definition of 'sentence',

(ii) Rules of designation for the primitive descriptive constants,

namely, individual constants and predicates.

(iii) Rules of truth.

(iv) Rules of ranges.

It is easy to see that the rules of kinds (i), (iii), and (iv) can be formu-

lated in an extensional metalanguage like M e . We must here consider these

rules in their exact formulation. The designations of expressions of the

object language must be formed, not with the help of quotation marks, as

we did for the sake of convenience in the previous examples of semantical

rules and statements, but as descriptions with the help of German letters.

iyo IV. ON METALANGUAGES FOR SEMANTICS

Let us add here, for this purpose, the letter 'W as designation in M e of the

modal sign ' N J in S 2 . As an example of a rule of formation for S 2 in M e , let

us take the rule for N-matrices: 'If 21, is a matrix in S 2 , then 3l($ii) is a

matrix in S 2 .' In application to the instance 'Hs', this rule says that, if 'Hs'

is a matrix, as it is, indeed, according to another rule, then 'N(Hs)' is a

matrix. Note, however, that the rule itself does not contain the expres-

sion 8, for instance, 'Hs', but only refers to this expression by using a

name'Sl/ for it (actually, a variable for which a name, say %', may be

substituted). Among the rules of truth we leave aside for the moment that

for atomic sentences because it contains the term 'designates' (or 'refers

to', see 1-3) which will be discussed later. The following is an example of

one of the other rules of truth (1-5): 'A disjunction of two sentences @

and @y (that is to say, a sentence consisting of @ included in parentheses

followed by the wedge followed by @/ included in parentheses) is true if

and only if either @ t or @/ or both are true.' It is clear that this formula-

tion is extensional. The same holds for the rules of ranges for S 2 , which

will be given in 41 . These rules define ' the sentence @ holds in the state-

description $ n '; $ n is a class of sentences. Note that the sentence @ t ,

let alone the class $ n , does not itself occur in the rule; only the names (or

variables) ' @/ and ' $ w ' occur. Thus it is clear that the relation of holding

is extensional. The rules of ranges refer, moreover, to assignments; an as-

signment is a function which assigns to a variable and a state-description

as arguments an individual constant as value. Only the extensions of

these functions are essential for the rules and the statements based upon

the rules; that is to say, if a reference to one assignment in a true state-

ment is replaced by a reference to another equivalent assignment (i.e.,

one which assigns to all pairs of arguments the same values as the first

assignment), then the resulting statement is likewise true. Note, further,

that the exact formulation of the rule concerning ' N' (4i-2g) has the form :

'A matrix 9t(8Ii) holds . . .'; thus it does not contain the modal sign 'N'

itself but only its name '31'. Thus we see that all rules of ranges for S 2 ,

including the rule concerning 'N', are extensional.

Now we go back to the rules of designation. Here is the one critical

point for our problem of the expressibility of the semantics of S 2 in M e .

In M', we distinguished two relations between designators and neutral

entities, namely, designation and L-designation. The relation of designa-

tion is extensional and hence does occur in M e ; but the relation of L-

designation does not. Thus we have to examine the question as to whether

the relation of designation suffices for describing the meanings of the ex-

pressions in the object languages. For instance, the meaning of 'IT in St

38. AN EXTENSIONAL METALANGUAGE FOR SEMANTICS 171

and S 2 is (the property) Human, not Featherless Biped; the meaning of

V is Walter Scott, not The Author Of Waverley. In M', we can easily

express this distinction with the help of the term 'L-designation' by the

statements 37-9 and 37-10, 37-18 and 37-19. But how can we do it in M e ,

where we have only the term 'designation'? The difficulty consists in the

fact that, with respect to designation, the following two statements are

both true (37-3 and 37-4):

38-1. 'H' designates Human.

38-2. 'H' designates Featherless Biped.

And the same holds for the following two statements (37-14 and 37-15) :

38-3. V designates Walter Scott.

38-4. V designates The Author Of Waverley.

In view of this fact, it might seem at first glance as though it were impos-

sible to give in M e the information about the meanings intended for 'H'

and '$'. However, I believe that this is not impossible. In M e we lay down

38-1 and 38-3 among the rules of designation for S x and S 2 . Then the state-

ment 38-2, although it is likewise true, is fundamentally different from

38-1, for it is neither a semantical rule, nor derivable from the semantical

rules alone; it was derived from rule 38-1 together with a biological fact

(3-6). If the metametalanguage MM, in which we are speaking here about

M e and the other metalanguages, contains L-terms, then we can formu-

late the difference in this way: 38-1 is L-true in M e but 38-2 is only F-true.

The relation between 38-3 and 38-4 is analogous. But even in M e itself we

can describe the situation in more explicit terms. If we wish to add to 38-1

a negative statement in M e , the following may be taken (3-8) :

38-6. 'H' and 'F B' are not L-equivalent (in S x and S a ).

This statement, together with 38-1 and some other semantical rules, cor-

responds in a certain sense to the negative statement 37-10 in M'.

The rules of designation themselves refer only to the primitive indi-

vidual constants and predicator constants. But the extensional relation of

designation can also be defined in M e in a wider sense so as to apply to all

designators, including compound individual expressions, predicators, and

sentences, also intensional sentences in S 2 . Then, for example, the fol-

lowing two statements hold in M e (37-22 and 37-23):

38-6. 'Hs' designates that Scott is human.

38-7. 'Hs' designates that Scott is a featherless biped.

The difference between these two statements is analogous to that be-

tween 38-1 and 38-2: Statement 38-6, though not itself a rule, follows

172 IV. ON METALANGUAGES FOR SEMANTICS

from the semantical rules alone, while for the derivation of 38-7 a factual

premise is needed.

The foregoing discussion shows that, even if somebody possesses no

other information concerning S! and S 2 than the semantical rules for these

systems formulated in M e , he is, nevertheless, in a position to know the

meanings that is to say, not only the extensions but also the intensions

which are intended, first, for the primitive descriptive constants and,

second, for all designators. All he has to do is to look, first, at the rules of

designation themselves and, second, at those statements about designa-

tion which follow from the semantical rules alone, leaving aside all those

statements in M e which, although true, can be arrived at only with the

help of factual knowledge. In other words, he has to consider only those

statements about designation which are L-true in M e .

It is sometimes said that a metalanguage, in which the semantics of an

object language S is to be formulated, must contain translations of all

expressions or at least of all designators in S. If this were right, M e would

not suffice as a semantics language for S 2 , because M e cannot, of course,

contain an expression L-equivalent to the intensional sign 'N' in S 2 . But

the requirement mentioned is only approximately right; strictly speaking,

it is too strong. The metalanguage must, indeed, contain for every sen-

tence in S an L-equivalent sentence; furthermore, it must be sufficiently

equipped with variables and descriptive expressions. It is, however, not

necessary that it contain an L-equivalent expression for every logical sign

in S. Although M e cannot contain a translation of 'N', it can contain a

semantical rule for 'N', for instance, the rule of ranges mentioned above.

If @ is a sentence in S 2 containing 'N', then an extensional language like

Si or M e cannot, of course, contain a translation of @ in the strong sense

of a sentence with the same intensional structure ( 14). But it can be

shown that S x , and hence M e , too, always contains a sentence L-equivalent

to ,. [For full sentences of 'N', this follows simply from the circumstance

that they are either L-true or L-false (see 39-2) ; however, since sentences

may contain several occurrences of ' N' and quantifiers in any combina-

tion, the general proof is rather complicated.] Further, S x and S 2 contain

the same variables and descriptive signs. Hence, if M e is sufficient for the

formulation of the semantics of Si, it is likewise sufficient for that of S 2 .

On the basis of these considerations, I am inclined to believe that it is

possible to give a complete semantical description even of an intensional

language system like S 2 in an extensional metalanguage like M a . However,

this problem requires further investigation.

CHAPTER V

ON THE LOGIC OF MODALITIES

In this chapter we study logical modalities like necessity, possibility, im-

possibility. We introduce 'N* as a symbol of necessity; the other modal con-

cepts, including necessary implication and necessary equivalence, can be de-

fined with its help. The modal system S 2 is constructed by adding *N' to our

previous system Sj (39); and the semantical rules for S 2 are stated (41).

An analysis of the variables occurring in modal sentences shows that they have

to be interpreted as referring to intensions ( 40) ; hence a translation in words

must be given either in terms of intensions (in the metalanguage M) or in

neutral terms (in MO (43). Quine's views on the possibility of combining

modalities and variables are discussed ( 44). Finally, the main results of the

discussions in this book are briefly summarized ( 45).

39. Logical Modalities

We form the modal system S 2 from our earlier system Sj by the addition

of the modal sign 'N' for logical necessity. We regard a proposition as necessary

if any sentence expressing it is L-true. Other modalities can be defined in terms

of necessity, for example, impossibility, possibility, contingency. With the help

of 'N', we define symbols for necessary implication and necessary equivalence;

the latter symbol may be regarded as an identity sign for intensions.

In the earlier chapters, modal sentences have sometimes been taken

as examples, especially sentences about necessity or possibility, either in

words (for instance, in 30 and 31) or in symbols (for instance, n,

Example II). We use *N' as a sign for logical necessity; 'N(A)' is the sym-

bolic notation for 'it is (logically) necessary that A'.

Quite a number of different systems of modal logic have been con-

structed, by C. I. Lewis (see Bibliography) and others. 1 These systems

differ from one another in their basic assumptions concerning modalities.

There is, for instance, the question of whether all sentences of the form

*N/> 3 NNp' are true, in words: 'if it is necessary that p, then it is neces-

sary that it is necessary that p\ Some of the systems give an affirmative

answer to this question, other systems give a negative answer or leave it

undecided. Not only do logicians disagree among themselves on this ques-

tion, but sometimes also one logician constructs systems which differ in

this point, probably because he is doubtful whether he should regard the

sentences mentioned as true or false. There are several further points of

1 For bibliographical references up to 1938, see Church's bibliography in Journal of Symbolic

Logic, Vols. I and III; the pertinent references are listed in III, 199 ("Modality") and aoa

("Strict Implication").

173

i 74 V. ON THE LOGIC OF MODALITIES

difference between the systems. All these differences are, I think, due to

the fact that the concept of logical necessity is not sufficiently clear; it can,

for instance, be conceived in such a way that the sentences mentioned

are true, but also in another way such that they, or some of them, are false.

Our task will be to find clear and exact concepts to replace the vague

concepts of the modalities as used in common language and in traditional

logic. In other words, we are looking for explicata for the modalities. It

seems to me that a simple and convenient way of explication consists in

basing the modalities on the semantical L-concepts. The concept of logical

necessity, as explicandum, seems to be commonly understood in such a

way that it applies to a proposition p if and only if the truth of p is based

on purely logical reasons and is not dependent upon the contingency of

facts; in other words, if the assumption of not-p would lead to a logical

contradiction, independent of facts. Thus we see a close similarity between

two explicanda, the logical necessity of a proposition and the logical

truth of a sentence. Now for the latter concept we possess an exact ex-

plicatum in the semantical concept of L-truth, defined on the basis of the

concepts of state-description and range (2-2). Therefore, the most natural

way seems to me to take as the explicatum for logical necessity that

property of propositions which corresponds to the L-truth of sentences.

Accordingly, we lay down the following convention for 'N' :

39-1. For any sentence '. . .', 'N(. . .)' is true if and only if '. . .' is L-true.

We shall construct the system S 2 by adding to the system S x the sign 'N'

with suitable rules such that the convention just stated is fulfilled ( 41).

This convention may be regarded as a rule of truth for the full sentences

of TNP. S 2 thus contains all the signs and the sentences of Si.

On the basis of our interpretation of 'N', as given by the convention

39-1, the old controversies can be solved. Suppose that 'L-true in S 2 ; is

defined in such a way that our earlier convention 2-1, which says that a

sentence is L-true if and only if it is true in virtue of the semantical rules

alone, independently of any extra-linguistic facts, is fulfilled. Let 'A' be

an abbreviation for an L-true sentence in S a (for example, 'Hs V ~ Hs').

Then *N(A)' is true, according to 39-1. And, moreover^ it is L-true, be-

cause its truth is established by the semantical rules which determine the

truth and thereby the L-truth of 'A', together with the semantical rule for

'N', say 39-1. Thus, generally, if 'N(. . .)' is true* then 'NN(. . .)' is true;

hence any sentence of the form 'Np D NN>' is true. This constitutes an

affirmative answer to the controversial question mentioned in the be-

ginning. It can be shown in a similar way that every sentence of the

39. LOGICAL MODALITIES

175

form '~ Np D N ~ Np* is true. This settles another one of the contro-

versial questions. 2

This analysis leads to the result that, if *N(. . .)' is true, it is L-true;

and if it is false, it is L-false; hence:

39-2. Every sentence of the form ( N(. . .)' is L-determinate.

Therefore, the convention 39-1 may be replaced by the following more

specific one:

39-3. For any sentence '. . .' in S 3 , *N(. . .)' is L-true if '. . .' is L-true;

and otherwise *N(. . .)' is L-false.

On the basis of the concept of logical necessity, the other logical

modalities can easily be defined, as is well known. For example, 'p is im-

possible' means *non-/> is necessary'; ( p is contingent* means 'p is neither

necessary nor impossible' ; l p is possible' means 'p is not impossible' (we

adopt this interpretation in agreement with the majority of contemporary

logicians, in distinction to other philosophers who use 'possible' in the

sense of our 'contingent'). Let us use the diamond, 'Q ', as a sign of possi-

bility; we define it on the basis of *N':

39-4. Abbreviation. <<>( )' for '~ N ~(. . .)'.

It would also be possible to take '<> ' as primitive, as Lewis does, and then

to define 'N(. . .)' by ( ~ <> ~(. . .)'.

There are six modalities, that is, purely modal properties of proposi-

tions (as distinguished from mixed modal properties, for instance, con-

tingent truth, see 30-1). The accompanying table shows how they can be

THE SIX MODALITIES

Modal Property of

a Proposition

With 'N'

With'O'

Semantical Prop-

erty of a Sentence

Necessary

N^

^ ^ 6

L-true

Impossible

N ~ p

~ P

L-false

Contingent

Non-necessary . . .

Possible

~ N/> ~ N ~ p

~N/

~N~p

~p* Op

~P

00

Factual

Non-L-true

Non-Irfalse

Noncontingent. .

NpVN~p

^o^^v^o^

L-determinate

expressed in terms of 'N' and in terms of 'Q'. The last column gives the

corresponding semantical concepts; a proposition has one of the modal

* The two questions and the reasons for our affirmative answers are discussed in more detail

in [Modalities], i.

176 V. ON THE LOGIC OF MODALITIES

properties if and only if any sentence expressing the proposition has the

corresponding semantical property.

Every proposition with respect to a given system 5 is either necessary

or impossible or contingent. This classification is, according to our inter-

pretation of the modalities, analogous to the classification of the sentences

of S into the three classes of L-true, L-false, and factual sentences. There

is, however, one important difference between the two classifications. The

number of L-true sentences may be infinite, and it is, indeed, infinite

for each of the systems discussed in this book. On the other hand, there is

only one necessary proposition, because all L-true sentences are L-equiva-

lent with one another and hence have the same intension. [This result

holds only for that use of the term 'proposition' which is based on L-

equivalence as the condition of identity. It is, of course, possible to phoose

a stronger requirement for identity, for instance, intensional isomorphism.

In this case the intensional structures are called 'propositions'. And their

number is infinite.] Likewise, there is only one impossible proposition, be-

cause all L-false sentences are L-equivalent. But the number of con-

tingent propositions (with respect to a system with an infinite number of

individuals) is infinite, like that of factual sentences.

It should be noted that the two sentences 'N(A)' and 'the sentence 'A'

is L-true in S a ' correspond to each other merely in the sense that, if one

of them is true, the other must also be true; in other words, they are L-

equivalent (assuming that L-terms are defined in a suitable way so as

to apply also to the metalanguage). This correspondence cannot be used

as a definition for 'N', because the second sentence belongs, not to the

object language S 2 as the first one does, but to the metalanguage M. The

second sentence is not even a translation of the first in the strict sense

which requires not only L-equivalence but intensional isomorphism

( 14). If M contains the modal term 'necessary', then 'N(A)' can be

translated into M by a sentence of the form 'it is necessary that . . .'

(where '. . .' is the translation of 'A'). If M contains no modal terms,

then there is no strict translation for 'N(A)'. But the correspondence

stated makes it possible in any case to give an interpretation for 'N(A)' in

M with the help of the concept of L-truth, for example, by laying down

the truth-rule, 39-1.

On the basis of 'N', we introduce two further modal signs for modal

relations between propositions:

39-5. Abbreviation. Let '. . .' and ' ' be sentences in S a . '. . . 3 '

for'N(. .. D --)'.

840. MODALITIES AND VARIABLES 177

39-8. Abbreviation. Let '. . .' and ' ' be any designators in S 2 (sentences

or otherwise). '. . . n - - -' for 'N(. ..==-- -)'.

Thus ' 5 ' is a sign for necessary implication between propositions (Lewis'

strict implication). The symbol 'm' is a sign for necessary equivalence.

The sign ' m ' in S 2 is the analogue to the term 'L-equivalent' in its non-

semantical use in M (5-4) or M' ( 34), where it designates a relation be-

tween intensions, not between designators. When standing between

sentences, it corresponds to Lewis' sign '=' for strict equivalence. We

have seen earlier that '==', standing between designators of any type, is a

sign for the identity of extensions (see remark on 5-3). Here in S a , ( m' is,

similarly, a sign for the identity of intensions. For example, *H s RA' is

short for 'N(H = RA)'. Hence, according to the rule 39-1, *H m RA' is

true if and only if *H s= RA' is L-true, hence if and only if *H' and 'RA'

are L-equivalent, in other words, have the same intension.

We have earlier formulated the two principles of interchangeability

(12-1 and 12-2). For the first principle we have given, in addition to the

chief formulation in semantical terms (i2-ia), alternative formulations

with the help of sentences of the object language containing ' = ' (i2-ib

and c). Now, with the help of ' m ', we can provide analogous formulations

for the second principle. The following theorems 3Q-yb and c, which may

be added to i2-2a as i2-2b and c, follow from i2-2a because Sly and 2U

are L-equivalent if and only if Sly 3U is true.

Second Principle of Interchangeability (alternative formulations) :

39-7. Under the conditions of 12-2, the following holds:

b. (i2-2b). (Sly B JU) 3 (. . Sly . . . . SI* . .) is true (in 5).

c. (i2-2c). Suppose the system 5 contains variables for which Sly

and SI* are substitu table, say V and V; then '(u)(v)[(u m v) D

(. . u . . m . . v . .)]' is true (in 5).

40. Modalities and Variables

Problems concerning the interpretation of variables in modal sentences are

discussed, in preparation for the semantical rules given in the next section. It is

found that a universal quantifier preceding *N* is to be interpreted as if it fol-

lowed the *N'. It is generally shown that variables in modal sentences are to be

understood as referring to intensions rather than to extensions. Thus an indi-

vidual variable in S a is interpreted as referring to individual concepts rather

than to individuals. We decide to take as values of these variables not only

those individual concepts which are expressible by descriptions in S but the

wider class of all individual concepts with respect to S a . A concept of this kind

is represented by any assignment of exactly one individual constant to each

state-description in S a .

178 V. ON THE LOGIC OF MODALITIES

So far we have given an interpretation for 'N' only in the case in which

the argument-expression of 'N' is a sentence. But in a system which con-

tains variables we also have to solve the problem of interpreting occur-

rences of 'N' followed by a matrix with free variables, e.g., c N(Po?)'. Let

us investigate this problem in a general way for a system S containing a

variable V of any type. How should we interpret the sentence '(w)[N

(. . u . .)]', where '. . it . .' is a matrix containing V as the only free vari-

able? Let us first consider the case in which V has only a finite number of

values, say n, and all these values are expressible in 5, say by the designa-

tors 'IV, 'UY, . . . 'UV. (As we shall see later, the interpretation of a

variable in a modal sentence has to be given in terms of value-intensions,

not value-extensions. Therefore, the statement just made is to be under-

stood as saying that there are n value-intensions for 'u' and that they are

the intensions of the designators 'IV, etc.) Now any universal sentence,

whether in an extensional or in a modal language, always means that all

values of the variable possess the property expressed by the matrix.

Therefore, if the number of values is n, the universal sentence means the

same as the conjunction of the n substitution instances of the matrix.

In our example, '(#)[N(. . u . .)]' means the same as 'N(. . U x . .)

N(. .U 2 .. )... N(..U n ..)>.

A conjunction of n components (n ^ 2) is L-true if and only if every one

of the components is L-true. Therefore, the following holds, in virtue of

the correspondence between necessity and L-truth (39-1) :

40-1. If %', . . . 'An' are any sentences, 'N(Ax A 2 . . . A n )' is L-

euivalent to 'NA, NA 2 . . . NA n '.

equivalent to 'N(A,) N(A 2 ) . . . N(A n )'.

If we apply this to the above result, we find that '(w)[N(. . u . .)]' means

the same as 'N[(. . U x ..)(.. U a ..)( U n . .)]' and hence

the same as 'N [()(. . u . .)]'. Thus the result is that '()' and 'N' may

exchange their places.

Next, let us consider the case in which the variable V has an infinite,

but denumerable, number of values, all of which are expressible in 5, say

by the designators 'UY, 'Ua', etc. Here we cannot form a conjunction of

the substitution instances, but we can still consider their class. If we

interpret a class of sentences as a joint assertion of its sentences, in accord

with the usual procedure, then we can apply semantical concepts to it in

the following way: We define the range of a class of sentences as the prod-

uct of the ranges of the sentences. This leads to the following two re-

sults:

40. MODALITIES AND .VARIABLES 179

(i) A class of sentences is true if and only if all its sentences arfc true.

(ii) A class of sentences is L-true if and only if all its sentences are

L-true.

Now the sentence '(w)[N(. . u . .)]' is true if and only if the class of the

instances *N(. . U n . .)' for n = i, 2, etc., is true; hence, according to (i),

if and only if every sentence of the form *N(. . U n . .)' is true; hence, ac-

cording to 39-1, if and only if every sentence of the form '. . U n . .' is

L-true; hence, according to (ii), if and only if the class of these sentences is

L-true; hence, if and only if '(u) (. . u . .)' is L-true; hence, according to

39-1, if and only if *N[(w) (. . u . .)]' is true. Thus the result is that, in

the case of infinitely many values also, the quantifier '(#)' and the modal

sign 'N' in the original sentence may exchange places.

It seems natural to apply the same result to the case in which not all

values of 'u' are expressible in 5, that is to say, to interpret a sentence of

the form '(w)[N(. . u . .)]' in any case, irrespective of the number and ex-

pressibility of the values of V, as meaning the same as 'N[(#) (. . u ..)]'.

In particular, we shall construct the semantical rules of the system S 2 in

such a way that any two sentences of the forms just stated are L-equiva-

lent ( 41). In S 2 'u' must, of course, be an individual variable.

Since a modal system contains not only extensional but also intensional

contexts, a designator may, in general, be replaced by another one only if

they are not merely equivalent but L-equivalent. Thus, in general, we

have to take into consideration the intensions of the designators, not

merely their extensions. Similarly, we have to consider for a given vari-

able its value-intensions in the first place. If the system contains variables

of the type of sentences, say l p\ 'q\ etc., then a quantifier with a variable

of this kind occurring in a modal sentence must be interpreted as re-

ferring to propositions, not to truth-values. For example, the sentence

'(ftp) (~N^>) ' must be understood as saying that there is a non-necessary

proposition. It would hardly make sense to interpret it as saying that

there is a non-necessary truth-value, because there are propositions

with the same truth-value such that one of them fulfils the matrix

*~N/>', while another one does not. This interpretation in terms of

propositions seems generally accepted. C. I. Lewis, as well as the other

logicians who have discussed his systems of modal logic or have con-

structed new ones, have used interpretations in terms of propositions. If

variables of the type of predicators of degree one occur in a modal system,

it is clear that they must be interpreted analogously in terms of proper-

ties, not of classes. Here, again, I think that most logicians would agree;

i8o V. ON THE LOGIC OF MODALITIES

however, modal sentences with variables of this kind have not been dis-

cussed frequently.

In my view the situation with respect to individual variables is quite

analogous, although this is usually not recognized. I think that individual

variables in modal sentences, for example, in S 2 , must be interpreted as

referring, not to individuals, but to individual concepts. The difficulties

which would otherwise arise will be explained later ( 43) . Thus a sentence

of the form '(#)(. . x . .)' in S 2 is to be interpreted as referring to all

individual concepts. Therefore, we now have to study the question as to

what is to be regarded as the totality of all individual concepts with re-

spect to S 3 .

We shall assume for the following discussions that the individual con-

stants in S 2 are L-determinate (19), that is to say that they are inter-

preted by the rules of designation as referring to positions in an ordered

domain and that any two different constants refer to different positions.

[For this purpose, it would be more natural to construct S 2 on the basis of

S 3 ( 18) rather than of S x . The reason for taking S x as the basis is merely

the possibility of using the earlier examples. But we must then suppose

that, for example, the rule of designation for V does not use the phrase

'the man who was known by the name of Walter Scott ' ', but rather:

'the man who was born at such and such a place at such and such a

time'; and even this formulation would not be entirely adequate.] Conse-

quently, we take any sentence of the form 'a = b' as L-false. However,

= -sentences with one or two descriptions (for example, *()#) (Arw) = s')

are still, in general, factual.

A description 21, in S 2 , say '(?#) (. . x . .)', characterizes ope of the indi-

vidual positions with the help of the property expressed by the matrix

'. . x . .'. If exactly one position has this property, then this position

is the descriptum; otherwise, a* is the descriptum ( 8). Thus for the

determination of the descriptum, the extension of Sl, factual investiga-

tion is required (unless the description is L-determinate). On the other

hand, the intension of ?!, the individual concept expressed by 2J t -, must be

something that can be determined by logical analysis alone. In order to

understand more clearly what kind of entity an individual concept is,

let us see what we can find out about the description 2l by logical analysis

alone. Suppose a state-description $ in S 2 is given (which is an infinite

class of sentences in S a ). Then the question of whether or not there is

exactly one individual position in $ n fulfilling the matrix '. . x . .' in

other words, whether or not there is exactly one.substitution instance of

40. MODALITIES AND VARIABLES 181

the matrix with an individual constant which holds in $ w is a purely

logical question. If the answer is in the affirmative, the descriptum of 8<

with respect to $ n is represented by that one individual constant; other-

wise it is represented by 'a*'. Thus the description %' assigns to every

state-description exactly one individual constant; any individual con-

stant may be assigned to several state-descriptions. If 8U and Sly are L-

equivalent and hence express the same individual concept, then both

assign to any state-description the same individual constant. Therefore,

we might say that an individual concept with respect to S 2 is an assign-

ment of exactly one individual to every state (which is a proposition ex-

pressed by a state-description). However, we shall actually take not these

states but the state-descriptions; and not the individuals but the indi-

vidual constants. The latter is possible because we have assumed that

these constants are L-determinate and that there is a one-one correlation

between the individuals and the individual constants. Thus we shall take

any assignment of exactly one individual constant to each state-descrip-

tion in S 2 (in other words, any function from state-descriptions to indi-

vidual constants) as representing an individual concept with respect to S 2 .

Only a small part (a denumerable class) of the individual concepts repre-

sented by assignments of this kind are expressible by descriptions in S 2 .

Now we decide to take as values of the individual variables in S 2 not only

the individual concepts expressible by descriptions in S 2 but all individual

concepts represented by assignments of the kind described; we call them

individual concepts with respect to S 2 . In the next section we shall lay down

the semantical rules for S 2 in accord with this decision; a universal quanti-

fier will be interpreted as referring to all individual concepts with respect

toS 2 .

Some remarks may, incidentally, be made concerning the interpreta-

tion of variables of other than individual type. Let S be a modal system

which also contains propositional variables *p\ etc., and variables '/', etc.,

for properties of level one, that is, properties of individuals. As values for

propositional variables we should take not only those propositions which

are expressed by sentences in 5, but all propositions with respect to 5.

They are represented by the ranges in 5, that is, the classes of state-

descriptions in 5. And as values for '/', etc., we should take not only those

properties which are expressed by predicators (including lambda-expres-

sions) in 5, but all properties with respect to 5. Since the attribution of a

property to an individual results in a proposition, we may regard a

i8 2 V. ON THE LOGIC OF MODALITIES

property as an assignment of exactly one proposition to each individual.

Therefore, we may represent the properties with respect to 5 by the as-

signments of ranges (classes of state-descriptions) in S to the individual

constants in S. Similarly, assignments of ranges in S to ordered pairs of

individual constants in S may be taken as representing the relations with

respect to S as values of relation variables in S. [In analogy to the rules of

ranges for matrices containing individual variables in S a , which will be

given in the next section, rules for variables of other types in S might be

stated as follows: (i) The matrix 'p* holds in the state-description $ n for

a certain range as value if and only if $ belongs to this range, (ii) The

matrix '/a' holds in $ for a given assignment of the kind described as

value of '/' if and only if $ n belongs to that range which is assigned to

'a'.]

41. Semantical Rules for the Modal System S 2

On the basis of our previous decisions concerning the interpretation of *N*

( 39) an( l f the individual variables in S 2 ( 40), we lay down semantical rules

for S 2 . The most important rules are the rules of ranges, which are here some-

what more complicated than for Si because individual concepts rather than indi-

viduals must here be taken as values of the variables. The L-concepts for S 2

have the same definitions as for Si. Some examples of L-true modal sentences in

S 2 are given.

The signs of the modal system S 3 comprise those of Si and, in addition,

the modal sign 'N'. In Si, compound designators and designator matrices

are formed out of atomic matrices with the help of the following means:

the ordinary (i.e., nonmodal) connectives, quantifiers, the iota-operator,

and the lambda-operator. In S 2 a rule of formation for *N' is added,

which says that, if '. . .' is any matrix, 'N(. . .)' is a matrix.

Now we have to construct the rules of ranges for S 2 . The state-descrip-

tions in S 2 are the same as in B! ( 2), because S 2 does not contain any

new descriptive constants. If we had only sentences without variables, we

could simply take the rules of ranges for S x (see the examples in 2,

omitting the rule for a universal sentence) and add the following rule:

41-1. N(@) holds in every state-description if @< holds in every state-

description; otherwise, N(@) holds in no state-description.

This rule is clearly in accord with our convention 39-3 (see 2-2 and 2-4).

However, in order to accommodate sentences with variables, we have to

use, instead, more complicated rules of ranges. They must apply not only

to sentences, like the rules of ranges for S x ( 2), but to matrices, and they

41. SEMANTICAL RULES FOR THE MODAL SYSTEM S 183

must refer to values of the individual variables occurring in the matrix.

According to our analysis in the preceding section, we take as values of

the variables all individual concepts with respect to S 3 ; every one of these

concepts is represented by an assignment of individual constants to state-

descriptions. Suppose that we have chosen as a value of the variable 'x*

occurring in the atomic matrix { Px' an assignment of this kind and that

the individual constant assigned to a given state-description is 'b'.

Then the question of whether the matrix 'Px* for the chosen value of V

holds in $ n means simply whether the sentence Tb' holds in $; and this

is, of course, the case if Tb' belongs to ff n (compare the example (i)

of the rules of ranges for Si in 2). This analysis suggests the first of the

subsequent rules of ranges (4i-2a). The other rules are analogous to the

rules of ranges for S x ( 2), together with the rule 41-1 for 'N', except that

the present rules apply to matrices and therefore have to refer to assign-

ments as values of the free variables. 3 Note that sentences are matrices

without free variables ( i) ; therefore, these rules apply also to sentences,

in which case the references to values are dropped.

41-2. Rules of ranges for the modal system S 2 . Let ?! be a matrix and

$ n be a state-description in S 2 . By a value of a variable we mean

any assignment of the kind described earlier.

a. Let 2l be of atomic form. 2l holds in $ n for given values of the

individual variables occurring in Sl, if and only if $ n contains

the atomic sentence formed from S< by substituting for every

free variable the constant assigned to $ n by the value of the

variable.

b. Let 2l be an s -matrix with individual signs (constants or vari-

ables). ?! holds in $ n for given values of the variables occurring

in Sli, if the individual constant for the left side (that is, either

J The system MFL described in [Modalities], 9, is similar to, but somewhat simpler than,

our present system S a . Sentences of the form 'a b' in MFL are regarded as L-false, like the

corresponding sentences of the form *a m b* in S 2 ; this shows that the individual constants in

MFL are, in terms of our present theory, L-de terminate like those in Sa. The state-descriptions

are the same in both systems. The differences are as follows: MFL does not contain lambda-

expressions and individual descriptions; this difference is not essential, since both kinds of

expressions in Sa can be eliminated, as we have seen. More essential is the difference in the

interpretation of individual variables. A universal sentence '(*)(. . x . .)' in MFL is regarded

as L-equivalent to the class of substitution instances of the matrix '..*..' with all individual

constants; thus, in terms of our present theory, the universal quantifier refers to all L-determi-

nate individual concepts and to no others. A universal quantifier hi Sa, on the other hand, refers

to all individual concepts (with respect to Sa). This wider range of values for the individual

variables in Sa seems more adequate; but it makes necessary the somewhat more complicated

form of the rules of ranges as given in the text, while the rules of ranges for MFL are as simple

as those for Si, together with the rule 41-1 for 'N'.

i8 4 V. ON THE LOGIC OF MODALITIES

the individual constant standing on the left side or the indi-

vidual constant assigned to $ w by the value of the variable

standing on the left side) is the same as that for the right side.

c. Let 2li be ~2l ,-. 21, holds in $ n for given values of the variables

occurring freely in 21,, if 21, does not hold in $ n for these values.

d. Let SI* be 81, V 21*. 21, holds in f for given values of the free

variables, if either 21,- or 2U or both hold in $ w for these values.

e. Let 2l be 21, 2U. 2l holds in $ n for given values of the free

variables, if both 21, and 2U hold in $ n for these values.

. Let 2lt consist of a universal quantifier followed by the matrix 2l/

as its scope. 2l< holds in $ n for given values of the variables

occurring freely in 21, (hence not including the variable occurring

in the initial quantifier), if 21, holds in $ n for every value of the

variable of the initial quantifier and the given values of the

other free variables.

g. Let 21, be N(8/). 21, holds in $ n for given values of the free vari-

ables, if 8y holds in every state-description for these values.

The following two theorems are simple consequences of these rules;

they may be used instead of the rules for the determination of the range

of a nonmodal matrix or sentence in S 2 .

41-3. Let 2l< be a matrix of any form without *N' in S 2 . 21* holds in $ n for

given values of the free variables, if and only if the sentence formed from

21, by substituting for every free variable the constant assigned to $ n by

the value of the variable holds in $ n .

41-4. If a sentence in S 2 does not contain 'N', then it holds in S 2 in the

same state-descriptions as in Sj.

In order to avoid certain complications, which cannot be explained

here, it seems advisable to admit in S 2 only descriptions which do not con-

tain 'N'. But any description may, of course, occur within the scope of an

*N'. The smallest matrix in which a description occurs (in the primitive

notation) is always a nonmodal context, because the description must be

an argument expression either of a primitive predicator constant or of

' m\ This smallest matrix is then taken as the context '- -(ix) (. . x . .) - -',

which can be transformed into 8-2. In this way every description can be

eliminated. Since L-equivalent sentences are L-interchangeable also

within modal contexts, according to the second principle of interchange-

ability (12-2), the result of the elimination is L-equivalent to the original

sentence; or, rather, we lay down a rule to the effect that any sentence con-

41. SEMANTICAL RULES FOR THE MODAL SYSTEM S 2 185

taining descriptions holds in the same state-descriptions as the sentence

resulting from the described elimination of the descriptions, and hence the

two sentences become L-equivalent.

Another point is worth noting. Although we interpret the individual

variables in S a as referring to individual concepts, not to individuals,

nevertheless a description in S 2 characterizes, not one individual concept,

but mutually equivalent individual concepts in other words, one indi-

vidual. This follows from the rule just mentioned, which permits the trans-

formation into 8-2. The first part of 8-2 says, in words: ' there is an indi-

vidual concept y such that, for every individual concept x, x has the de-

scriptional property if and only if x is equivalent (not 'L-equivalent' or

'identical'!) to y'; in other words, 'all individual concepts equivalent to y,

and only these, have the property'; or, 'the individual y is the only indi-

vidual which has the property'. This is as it should be, because the purpose

of a description, even in a modal language, is to refer to one individual

with the help of a property possessed by that individual alone. Neverthe-

less, the description has, of course, a unique intension, which is an indi-

vidual concept. This individual concept is not the only one possessing

the descriptional property, since, as we have seen, all equivalent ones do

likewise; but it is uniquely determined by the descriptional property; as

Frege puts it, it is not the individual but the way in which the description

refers to the individual.

For lambda-expressions we do not impose the restriction stated for

descriptions; they may also contain 'N'. Any lambda-operator can be

eliminated in S 3 by conversion in the same way as in S x ( i). Here, again,

a rule would be laid down saying that a sentence containing lambda-

operators holds in the same state-descriptions as the sentence resulting

from their elimination.

The L-concepts are defined for S a in the same way as for S t ( 2). The

following theorems give a few results, which hold on the basis of the rules

of ranges stated above.

41-6. Any sentence of one of the following forms is L-true in S a . (The

variables '^', l q\ . . '/', do not occur in S a but are here used merely

to describe forms of sentences in S 2 . A sentence in S 2 is said to have

one of the forms described if it is formed by substituting for y or

y any sentence in S a and for ( fx' any matrix containing V as the

only free variable.)

186 V. ON THE LOGIC OF MODALITIES

a. 'N/> 5 p'.

b. 'p 3 0^'-

c. <(/> 3 q) 3 (N/> 3 N?)'

d. 'N(/> . q) m N/> Ng'.

e.

f. 'NN/>

g. 'N

h. <oO/> = <>/>'

k. '

1. '(3*)N( 3 N (3*) (/*)'

m. <(3*)OO) - 0(3*) (/*)'

We see from these theorems that 'N' is quite similar to a universal

quantifier and '() ' to an existential quantifier. This seems plausible, since

N@ is true if @,- holds in every state-description, and 0@ is true if @

holds in at least one state-description.

42. Modalities in the Word Language

The problem of the translation of modal sentences of S 2 into the metalan-

guages M and M' is discussed. It is shown that it is advisable to use for the

translations either terms of intensions in M or neutral terms in M'. The use of

terms of extensions within modal sentences in M is not in itself incorrect, pro-

vided that certain restrictions are observed; but it involves the danger of mak-

ing wrong inferences by overlooking the restrictions.

We shall examine here the problem of the formulation of modal sen-

tences in words and, in particular, the problem of the translation of

modal sentences into our metalanguages M and M'. It is worth while to

study this problem because, it seems to me, certain difficulties which have

sometimes been found in connection with modal sentences are due chiefly

to their inadequate or misleading formulation in the word language.

Since modal sentences, for instance, in S 2 or in a richer language with

several types of variables, are not semantical, their translations are like-

wise not semantical sentences and hence belong to the nonsemantical part

of M and M' (this part of M 7 was explained in 34-36). As translation

of 'N*, we take 'it is necessary that'; hence, this is an intensional phrase.

We shall discuss three examples A, B, and C. In A, we have predica-

tors as argument expressions of '*&' or ''; in B, sentences; in C, indi-

vidual expressions. Otherwise, the three examples are perfectly analogous.

Therefore, we arrange them in three parallel columns, This facilitates the

42. MODALITIES IN THE WQRD LANGUAGE. 187

comparison of corresponding expressions in the three examples and the

recognition of their analogy.

Because of the perfect analogy, any one of the three examples would

theoretically be sufficient. However, for practical reasons it seems advis-

able to give all three. The purpose of the analysis of the examples is to

show that it is advisable to formulate modal sentences either in terms of

intensions or in neutral terms, while formulation in terms of extensions

involves certain dangers. Now this result is easily seen in the case of

predicators; presumably, most readers will agree in this case. Then the

analogy will make it easier to recognize the same situation in the case of

sentences and, finally, in the case of individual expressions. In this last

case the inhibitions against a translation in terms of intensions are strong-

est because it is not customary to speak of individual concepts. Therefore,

here the help of the two other examples seems necessary for practical,

psychological reasons, although theoretically the situation is here as clear

and simple as in the first two cases.

The example A (the conjunction of 43-iA and 42-2aA) is similar to one

given by Church; 4 our '~N(. . .)' corresponds to his C 0~C )' la the

example C, we use 'au' as abbreviation for '(ix) (Axw)\ In the translation

of this description into the word language, we omit, for the sake of brevity,

the phrase 'or a*, if there is not exactly one such individual' (as we did

earlier, 9).

The following sentences in S 2 are true but not L-true (see 3-7 and 9-2) :

42-1. A B C

T.B B H'. '(F.B)s * Hs'. 'au s'.

Therefore, according to 39-1, prefixing 'N' yields false sentences; hence

the following is true:

42-2a. A B C

<~N(F*B m H)'; <~N[(FB)s s Hs]'; <~N(au ^ s)';

or, abbreviated with ' m ' (39-6) :

42-2b. A B C

<~(F*B B H)'. <~[(F*B)s Hs]'. <~(au s)'.

Now let us examine the question of the translations of these sentences

of S 2 into M. The first sentence, 42-1 (in each of the three examples), is a

nonmodal sentence. It can be translated in two different ways, either into

42-3 in terms of intensions with the nonsemantical term 'equivalent' (see

5-3 and 5-5) or into 42-4 in terms of extensions with the identity phrase

'is the same as' (see 4-7 and 9-1) :

(Review Q.), p. 46.

iSS

V. ON THE LOGIC OF MODALITIES

42*3.

The property Feath-

erless Biped is equiv-

alent to the property

Human'.

42-4. A

The class Feather-

less Biped is the

same as the class

Human'.

B

The proposition

that Scott is a

featherless biped is

equivalent to the

proposition that

Scott is human'.

B

The truth- value

that Scott is a

featherless biped is

the same as the

truth-value that

Scott is human'.

The individual

concept The Au-

thor Of Waverley

is equivalent to the

individual concept

Walter Scott'.

C

The individual The

Author Of Waver-

ley is the same as

the individual

Walter Scott'.

For the modal sentences 42-2, however, the situation is different. First,

we shall give the translation into M in terms of intensions. We base the

translation 42-5 on the second of the two notations a and b given for 42-2,

utilizing the fact that 'm' is a sign for the identity of intensions (39).

(For A, see 4-8; for B, 6-4; for C, 9).

42-5. ABC

The property

Featherless Biped is

not the same as the

property Human'.

The proposition that The individual

Scott is a feather-

less biped is not the

same as the propo-

sition that Scott is

human'.

concept The Au-

thor Of Waverley

is not the same as

the individual con-

cept Walter Scott'.

This translation is adequate and unobjectionable. Not so, however, the

following translation in terms of extensions; here we base the translation

on the first notation 42-2a and regard '==' as a sign for the identity of

extensions (see remark on 5-3).

42-6. A

'It is not necessary

that the class Feath-

erless Biped is the

same as the class Hu-

man'.

B

'It is not necessary

that the truth- value

that Scott is a

featherless biped is

the same as the

truth- value that

Scott is human*.

'It is not necessary

that the individual

The Author Of

Waverley is the

same as the individ-

ual Walter Scott'.

42. MODALITIES IN THE WORD LANGUAGE

189

Formulations of this kind might perhaps be admitted as sentences in M; if

so, they would presumably be regarded as true and as correct translations

of 42-2a. However, these formulations are dangerous; if we apply custom-

ary ways of thinking to them, we obtain false results. In the ordinary word

language, we are accustomed to using the principle of interchangeability

(24-3b) implicitly. If in any of the three examples we apply this principle

to 42-6 on the basis of the true identity sentence 42-4, we obtain the

following result, 42-7. This, however, if admitted at all as a sentence, will

certainly be regarded as false.

42-7.

'It is not necessary

that the class Hu-

man is the same as

the class Human'.

B

'It is not necessary

that the truth-value

that Scott is human

is the same as the

truth-value that

Scott is human'.

'It is not necessary

that the individual

Walter Scott is the

same as the indi-

vidual Walter

Scott'.

These are instances of the antinomy of the name-relation in its second

form, similar to our previous example (31). In spite of this result, we

may admit the formulations 42-6, provided that we are willing to prohibit

the use of the principle of interchangeability in cases of nonextensional

contexts. However, since the unrestricted use of this principle is custom-

ary and plausible, there would always be the danger of forgetting the

prohibiting rule and using the principle inadvertently. Therefore, it seems

more advisable to avoid formulations like 42-6 and, in general, formula-

tions in terms of extensions within modal or other nonextensional con-

texts.

Now let us see how the given symbolic sentences of S 2 are to be trans-

lated into the neutral metalanguage M'. As explained earlier, there are no

identity phrases in M'; instead, the terms 'equivalent* and 'L-equivalent'

are applied in their nonsemantical use (see 34-8 and 34-9). As 'equivalent'

is a direct translation of the symbol s ', so is 'L-equivalent' of ' m '. (This

shows again that the nonsemantical term 'L-equivalent' is intensional;

this holds for all nonsemantical (absolute) L-terms, see [I], 17.) Thus

the translation of 42-1 into M! is as follows (see 34-10 and 34-13) :

igo

V. ON THE LOGIC OF MODALITIES

42-8. A

'Featherless Biped

is equivalent to Hu-

man .

The Author Of

Waverley is equiv-

alent to Walter

Scott'.

B

u That Scott is a

featherless biped,

is equivalent to

that Scott is hu-

man'.

b. 'Scott is a

featherless biped

if and only if

Scott is human '.

In B we add here the alternative form b because it sounds more natural

(see end of 34).

There are two ways of translating 42-2 into M'. The first is based on

42-2a and translates 'N' by 'it is necessary that'. (In B we use again the

more natural phrase 'if and only if instead of 'is equivalent to'; concern-

ing the reason for the word order, see remark at the end of 34.)

42-9a. A

'It is not necessary

that Featherless Bi-

ped is equivalent to

Human'.

B

'That Scott is a

featherless biped if

and only if Scott is

human, is not nec-

essary'.

The second alternative is based on the notation 42-2b and translates

( m' by 'L-equivalent' (see 34-11):

'It is not necessary

that The Author

Of Waverley is

equivalent to Wal-

ter Scott'.

42-9b. A

'Featherless Biped

is not L-equiva-

lent to Human'.

B

'That Scott is a

featherless biped,

is not L-equivalent

to that Scott is hu-

man'.

'The Author Of

Waverley is not

L-equivalent to

Walter Scott'.

This translation does not involve any difficulty analogous to that con-

nected with 42-6.

Thus the final result is as follows: It seems advisable to frame the

formulation of modal and other nonextensional sentences in the word

language, not in terms of extensions, but either (i) in terms of intensions

or (ii) in neutral terms. Which of the two formulations (i) and (ii) one

prefers is a matter of practical decision (see the discussion at the end of

37). The formulation in neutral terms is simpler, but the nonsemantical

43. VARIABLES IN THE WORD LANGUAGE 191

use of the terms 'equivalent' and 'L-equivalent' . is not customary.

Formulations in terms of intensions, like 42-5, are, in general, more

customary, except for the reference to individual concepts in case C.

But this reference will perhaps appear less strange if we recognize the es-

sential analogy in 42-5 between C, on the one hand, and A and B, on the

other.

43. Modalities and Variables in the Word Language

Translations of symbolic modal sentences with variables into M and M'

are examined. The result is analogous to that in the preceding section. It is

advisable to avoid terms of extensions and to use either terms of intensions in

M or the neutral terms in M'. The translation in terms of propositions and

properties is customary, but that in terms of individual concepts instead of

individuals may at first appear strange.

We have seen earlier ( 10) that, as a designator has both an extension

and an intension, a variable has both value-extensions and value-inten-

sions. Therefore, a sentence with a variable can be translated into M either

in terms of its value-extensions or in terms of its value-intensions. Fur-

thermore, it can be translated into M' in neutral terms ( 36) . In analogy

to the result in the preceding section, we shall find here that it is advis-

able to avoid the formulation in terms of value-extensions and to use

either terms of value-intensions or neutral terms.

For the same reason as in the preceding section, we use here three

analogous examples, A, B, and C. They are existential sentences with the

variables '/', 'p\ and V in a modal system S containing variables of these

types and the modal sign 'N'.

The following sentences 43-ia and b differ only in their notation. In

each of the three examples, A, B, and C, 43-ia is derived by existential

generalization from the conjunction of the sentences 42-1 and 42-2a; and

likewise 43-ib from 42-1 and 42-2b.

43-la. ABC

H) W)[(* -Hs) '(3*)[(* - s)

H)]'. ~N(p s Hs)]'. ~N(* s s )]'.

43-lb. A B C

- Hs) '(3a)[(* - s)

We shall now examine the possibilities for the translation of these sen-

tences into M. If it were a question of an extensional existential sentence

for instance, 43-ia with the second conjunctive component omitted

then translations in terms of value-intensions and of value^xtensions

XQ2

V. ON THE LOGIC OF MODALITIES

would be equally acceptable. This, however, is not the case for these

modal sentences. We shall first give a translation in terms of value-inten-

sions, in analogy to 42-3 and 42-5, taking notation 43-ib and translating

'm* by identity of intensions:

43-2. ABC

There is a property

/which is equivalent

to but not the same

as the property Hu-

man'.

There is a proposi-

tion p which is

equivalent to but

not the same as the

proposition that

Scott is human'.

There is an indi-

vidual concept x

which is equiva-

lent to but not the

same as the indi-

vidual concept

Walter Scott'.

In each of the three examples, this sentence can be derived by existential'

generalization from the conjunction of 42-3 and 42-5.

Now we shall translate 43-ia in terms of value-extensions, in analogy

to 42-4 and 42-6, translating 's j by identity of extensions:

43-3.

B

There is a class /

which is the same

but not necessarily

the same as the class

Human'.

There is a truth-

value p which is the

same but not nec-

essarily the same as

the truth-value

that Scott is hu-

There is an indi-

vidual x which is

the same but not

necessarily the

same as the indi-

vidual Walter

man'. Scott'.

In each of the three examples, this sentence can be derived by existential

generalization from the conjunction of 42-4 and 42-6. We have seen in the

preceding section that formulations of modal sentences in terms of exten-

sions, like 42-6, are dangerous because they lead to the antinomy of the

name-relation unless special restrictions are imposed and that it is there-

fore advisable to avoid these formulations. The same holds for formula-

tions like 43-3.

The translation of 43-1 into neutral formulations in M', in analogy to

42-8 and 42-pb, is as follows:

43-4* A

There is an / such

that / is equivalent

but not L-equiva-

lent to Human'.

B

There is a p such

that p is equiva-

lent but not L-

equivalent to that

Scott is human'.

There is an x such

that x is equivalent

but not L-equiva-

lent to Walter

Scott'.

44. QUINE ON MODALITIES 193

(Use of ' -equivalent' as a nonsemantical term would provide a Shorter

formulation.) In each of the three examples this sentence can be de-

rived by existential generalization from the conjunction of 42-8 and 42-9^

The formulations 43-4 are free of the dangers involved in 43-3.

Now let us compare the three examples, A, B, and C. Our proposal not

to translate variables in modal sentences in terms of extensions seems

quite natural in cases B and A. As remarked earlier ( 40), it seems that

all logicians interpret modal sentences in terms of propositions rather than

of truth- values, and most of them use terms of properties rather than of

classes. Only in case C does our interpretation deviate from the custom-

ary one. The reference to individual concepts may first appear some-

what strange; and the alternative translation in neutral terms (e.g.,

43 -40), which avoids the reference to individual concepts, uses the un-

familiar terms 'equivalent 7 and X-equivalent'. However, I believe that,

once we are aware of the perfect analogy between the three cases, we

recognize the inadequacy of the formulations in terms of individuals; and

the impression of strangeness which the formulation in terms of individual

concepts and, to a lesser degree, the neutral formulation may first give will

perhaps disappear. Modal sentences with variables are of a quite peculiar

logical nature, and it should not be surprising that an adequate and cor-

rect rendering for them in the word language is not always possible in

entirely customary and natural terms.

44. Quine on Modalities

Quine's article [Notes] explained his view that, under customary conditions,

modalities and quantification cannot be combined. A new statement by Quine

is quoted here, in which he says that my language succeeds in combining

modalities with quantification but only at the price of repudiating all exten-

sions, for instance, classes and individuals. I try to show that my modal lan-

guage does not exclude anything that is admitted by a corresponding exten-

sional language.

Quine 5 illustrates the difficulty which we have called the antinomy

of the name- relation by the following example among others (as men-

tioned above, 31). We find as an arithmetical and hence logical truth:

(i) 'g is necessarily greater than 7'.

The following is a true statement of astronomy:

(ii) 'The number of planets = 9'.

5 Quine [Notes] (18) p, 121, (15) p. 119, (23) p. 121.

194 V. ON THE LOGIC OF MODALITIES

If, in (i), V is replaced by 'the number of planets 1 in virtue of the true

identity statement (ii), we obtain the false statement:

(iii) The number of planets is necessarily greater than 7'.

Quine's method for solving the antinomy has been explained earlier

( 32, Method II). According to our method, the following sentence takes

the place of (ii) in M':

(iv) The number of planets is equivalent to 9'.

The sentences (i) and (iii) occur also in M'. But now it is not possible to

infer the false sentence (iii) from the true sentence (i) together with (iv).

According to the first principle of interchangeability (12-1), the expres-

sions 'the number of planets' and '9' are interchangeable on the basis of

(iv) in extensional contexts only, hence not in (i). Thus the difficulty

disappears, and the designators occurring in nonextensional contexts still

function, according to our conception, as normal designators.

An even more serious problem is raised by Quine's objection to modal

sentences with variables. He discusses the following expression :

(v) There is something which is necessarily greater than 7'.

He says 6 that this expression "is meaningless. For, would 9, that is, the

number of planets, be one of the numbers necessarily greater than 7?

But such an affirmation would be at once true in the form . . . [our

(i)] and false in the form . . . [our (iii)]." Quine does not regard (i) and

(iii) as meaningless. As explained earlier ( 32, Method II), he regards

occurrences of designators in nonextensional contexts, e.g., V m (0 an d

'the number of planets' in (iii), as "not purely designative" ; in other

words, these occurrences do not function as names, and hence the principle

of interchangeability is not applicable. For the same reason, according to

Quine's view, the rule of existential generalization is not applicable to

these occurrences. Therefore, there is no valid inference from (i) to (v),

and, moreover, (v) has no meaning and hence cannot be admitted as a

sentence. Thus Quine arrives at the following conclusions, which are

stated at the end of his paper: "A substantive word or phrase which desig-

nates an object may occur purely designatively in some contexts and not

purely designatively in others. This second type of context, though not

less "correct" than the first, is not subject to the law of substitutivity of

identity nor to the laws of application and existential generalization. More-

over, no pronoun (or variable of quantification) within a context of this

6 Ibid., p. 124.

44. QUINE ON MODALITIES 195

second type can refer back to an antecedent (or quantifier) prior to that

context. This circumstance imposes serious restrictions, commonly un-

heeded, upon the significant use of modal operators, as well as challeng-

ing that philosophy of mathematics which assumes as basic a theory of

attributes [i.e., properties] in a sense distinct from classes." 7

To Quine's contexts of the second kind belong all those which we call

nonextensional. He discusses, in particular, contexts within quotes and

modal contexts. With respect to contexts within quotes his conclusions

are no doubt correct. I cannot agree, however, with Quine's conclusion

concerning modal contexts. We have combined modalities and variables

both in symbolic object languages ( 40) and in word formulations in our

metalanguages (43).

Church likewise does not accept Quine's result. He says in the review

of Quine's paper that he "would question strongly the conclusion which

the author draws that no variable within an intensional context . . . can

refer back to a quantifier prior to that context .... The conclusion

should rather be that in order to do this a variable must have an inten-

sional range a range, for instance, composed of attributes [properties]

rather than classes." 8 Up to this point I am in agreement with Church.

His solution is as follows: He distinguishes, like the system PM (see 27),

between class variables, e.g., 'a', and property variables, e.g., '<'. He takes

as example a sentence which is essentially the same as a conjunction of

42-iA and 42-2aA. In distinction to Quine, he regards it as admissible to

infer from this sentence by existential generalization an existential sen-

tence; the latter, however, must not have the form < (3a)(. . a . .)' but

rather the form '(3<)(. . <j> . .)'. It seems to me that this procedure is

correct and, indeed, solves completely the difficulty pointed out by Quine.

I believe, however, that there is a simpler way to achieve this. It is simi-

lar to that of Church but avoids the use of two kinds of variables for the

same type. This use is, as explained earlier ( 27), an unnecessary duplica-

tion. It is sufficient to use variables of one kind which are neutral in the

sense that they have classes as value-extensions and properties as value-

intensions; this is done in 43-iaA. The use of different variables for exten-

sions and intensions within all types would lead in the case of Quine's

example (v) to the introduction of variables for number concepts different

from the variables for numbers. This, however, would be both unneces-

sary and unusual.

The problem of whether or not it is possible to combine modalities and

7 Ibid., p. 127. 8 [Review Q.], p. 46.

i 9 6 V. ON THE LOGIC OF MODALITIES

variables in such a way that the customary inferences of the logic of quan-

tification in particular, specification and existential generalization re-

main valid is, of course, of greatest importance. Any system of modal

logic without quantification is of interest only as a basis for a wider sys-

tem including quantification. If such a wider system were found to be im-

possible, logicians would probably abandon modal logic entirely. There-

fore, it is essential to clarify the situation created by Quine's analysis and

objections. For this reason I have asked Quine, who has read an earlier

version of the manuscript of this book, for a statement of his present view

on the problem mentioned and, in particular, his reaction to my method

for combining modalities and variables as explained in the preceding sec-

tion. With his kind permission, I am quoting here his statement in full: 9

Every language system, insofar at least as it uses quantifiers, assumes one or

another realm of entities which it talks about. The determination of this realm

is not contingent upon varying metalinguistic usage of the term 'designation'

or 'denotation', since the entities are simply the values of the variables of

quantification. This is evident from the meaning of the quantifiers '(*)', '(/)',

'(#)', '(3*)', '(a/)', '(%PY themselves: 'Every (or, Some) entity x (or /or p) is

such that'. The question what there is from the point of view of a given language

the question of the ontology of the language is the question of the range

of values of its variables.

Usually the question will turn out to be in part an a priori question regard-

ing the nature and intended interpretation of the language itself, and in part

an empirical question about the world. The general question whether for ex-

ample individuals, or classes, or properties, etc., are admitted among the values

of the variables of a given language, will be an a priori question regarding the

nature and intended interpretation of the language itself. On the other hand,

supposing individuals admitted among the values, the further question whether

the values comprise any unicorns will be empirical. It is the former type of in-

quiry ontology in a philosophical rather than empirical sense that interests

me here. Let us turn our attention to the ontology, in this sense, of your

object language.

An apparent complication confronts us in the so-called duality of M' as be-

tween intensional and extensional values of variables; for it would appear then

that we must inquire into two alternative ontologies of the object language.

This, however, I consider to be illusory; since the duality in question is a

peculiarity only of a special metalinguistic idiom and not of the object language

itself, there is nothing to prevent our examining the object language from the

old point of view and asking what the values of its variables are in the old-

fashioned non-dual sense of the term.

It is now readily seen that those values are merely intensions, rather than

extensions or both. For, we have:

(*)(* = *),

i.e., every entity is L-equivalent to itself. This is the same as saying

that entities between which L-equivalence fails are distinct entities a

9 The first two-thirds of Quine's statement as here quoted is dated October 23, 1945; the

remainder January i, 1946.

44. QUINE ON MODALITIES 197

clear indication that the values (in the ordinary non-dual sense of the term)

of the variables are properties rather than classes, propositions rather than

truth-values, individual concepts rather than individuals. (I neglect the further

possibility of distinctness among L-equivalent entities themselves, which would

compel the entities to be somehow "ultra-intensional" ; for it is evident that

you have no cause in the present connection to go so far.)

I agree that such adherence to an intensional ontology, with extrusion of

extensional entities altogether from the range of values of the variables, is in-

deed an effective way of reconciling quantification and modality. The cases of

conflict between quantification and modality depend on extensions as values of

variables. In your object language we may unhesitatingly quantify modalities

because extensions have been dropped from among the values of the variables;

even the individuals of the concrete world have disappeared, leaving only their

concepts behind them.

I find this intensional language interesting, for it illustrates what it would be

like to be able to give the modalities free rein. But this repudiation of the con-

crete and extensional is a more radical move, in general, than a mere compari-

son of 43-3 with 43-2 might suggest. The strangeness of the intensional language

becomes more evident when we try to reformulate statements such as these:

(1) The number of planets is a power of three,

(2) The wives of two of the directors are deaf.

In the familiar logic, (i) and (2) would be analyzed in part as follows:

(3) (3w) (n is a natural number , the number of planets = 3"),

(4) (3*)(3:y)(32)(3^)[# is a director . y is a director . ^ (x = y) . z is

wife of x . w is wife of y . z is deaf . w is deaf].

But the formulation (3) depends on there being numbers (extensions, presum-

ably classes of classes) as values of the bound variable; and the formulation (4)

depends on there being persons (extensions, individuals) as values of the four

bound variables. Failing such values, (3) and (4) would have to be reformu-

lated in terms of number concepts and individual concepts. The logical predi-

cate ' = ' of identity in (3) and (4) would thereupon have to give way to a logical

predicate of extensional equivalence of concepts. The logical predicate 'is a

natural number* in (3) would have to give way to a logical predicate having the

sense 'is a natural-number-concept'. The empirical predicates 'is a director',

'is wife of, and 'is deaf, in (4), would have to give way to some new predicates

whose senses are more readily imagined than put into words. These examples do

not prove your language-structure inadequate, but they give some hint of the

unusual character which a development of it adequate to general purposes

would have to assume.

The first important point to be noticed in Quine's statement is that he

agrees that the form of modal language explained in the present chapter

"is indeed an effective way of reconciling quantification and modality".

Some readers of Quine's article believed that it proved the impossibility

of a logical system combining modalities with variables. Quine's statement

now shows that this is not the case.

However, there are still some serious problems involved. Quine, while

admitting the possibility of modal systems with quantification, believes

ig8 V. ON THE LOGIC OF MODALITIES

that these systems have certain peculiar features which he regards as

disadvantages. Let us now examine these problems.

I have previously explained (at the beginning of 10) that I agree with

Quine's view that an author who uses variables of some kind thereby

indicates that he recognizes those entities which are values of the vari-

ables. (I have simultaneously expressed some doubts concerning the ad-

visability of applying the term 'ontology' to this recognition; but for our

present discussion we may leave aside this question.) It is the counterpart

of this thesis that is of importance for our problem; it says that, if some-

one uses a language which does not. contain any variables with certain

entities as values, he thereby indicates that he does not recognize these

entities or at least that he does not intend to speak about them as long as

he restricts himself to the use of this language. In a certain sense, I can

agree also with this thesis. As an example, let us compare the following

two languages SP and S P . Let Sp be the ordinary language of physics

( 19). It contains variables which have real numbers, both rational and

irrational, as values. Suppose somebody proposes another language Sp

for physics which contains variables for rational numbers, but no vari-

ables to whose values irrational numbers belong. Here I would be willing

to say, like Quine, that the user of this language S P excludes or "repudi-

ates" the irrational numbers and that these numbers "have disappeared"

from the universe of discourse. Now Quine says that the variables in the

modal language have as values only intensions, not extensions, and that

therefore, as far as this language is concerned, all extensions, for example,

classes and "the individuals of the concrete world", "have disappeared".

With this I cannot agree. At the first glance, the situation here may seem

to be similar to that in the example of the irrational numbers; but actually

it is fundamentally different.

In order to clarify the situation, we shall contrast in the following dis-

cussion our two language systems, the extensional language S x and the

modal language S 3 . We shall further consider the following two extended

languages. The language S,' is extensional like S x but contains additional

kinds of variables, say '/', 'g\ etc., for which predicators of level one (and

degree one) are substitutable, f m\ W, etc., for predicators of level two,

and 'p', ( q', etc., for sentences. The language S 2 ' is constructed from S x ' by

the addition of 'N'; hence it is a modal language like S 2 . According to

Quine's view, the values of 'f in S 2 ' are not classes but properties, because

'(/)(/" B /)' holds. In the extensional system S,', on the other hand, we

have only '(/)(/ /)' Therefore, Quine will presumably regard classes

as the values of '/ in this system, as he does for the variables of his ex-

44. QUINE ON MODALITIES 199

tensional system ML (see above, 25). Similarly, Quine says that the

values of individual variables (e.g., V) in modal systems like S 3 and S 2 '

are individual concepts; on the other hand, he presumably regards indi-

viduals (concrete things or positions) as the values of individual variables

in extensional systems like S r and S,'. Now the decisive point is the follow-

ing: As explained previously (35), there is no objection against regard-

ing designators in a modal language as names of intensions and regarding

variables as having intensions as values, provided we are not misled by

this formulation into the erroneous conception that the extensions have

disappeared from the universe of discourse of the language. As explained

earlier ( 27), it is not possible for a predicator in an interpreted language

to possess only an extension and not an intension or, in customary terms,

to refer only to a class and not to a property. Similarly, it is impossible for

a variable to be merely a class variable and not also a property variable.

On the other hand, it is, of course, possible for a variable to have as values

only properties and no relations, or only rational numbers and no irra-

tional numbers. This shows the difference between the two cases. For

example, the so-called class variables in the system PM' (e.g., V) are, as

we have seen ( 27), also property variables, that is to say, they have

properties as value intensions. The same holds now for variables like l f

in S x . Languages of Quine J s form ML' or of Russell's form PM' or of our

form 8j speak also about properties. The restriction of these extensional

languages in comparison with modal languages like S 2 consists merely in

the fact that whatever is said in any of these languages about a property

is either true for all equivalent properties or false for all equivalent

properties; in technical terms, all properties of properties expressible in

these languages (by a matrix with a free variable of the kind mentioned)

are extensional. This makes it possible to paraphrase all sentences of

these languages in terms of classes. An analogous result holds for indi-

vidual variables. These variables in an extensional language like S x and

S[ refer not only to individuals but also, and even primarily, to individual

concepts. The restriction is again merely this: Whatever is said in these

languages about individual concepts is either true for all equivalent indi-

vidual concepts or false for all of them; in technical terms, it is extensional.

Therefore, whatever is said in these languages about individual concepts

can be paraphrased in terms of individuals.

Although the sentences of an extensional language (S t or S^) can thus

be interpreted as speaking about individuals and classes, they can be

translated into the corresponding modal language (S a or S,, respectively).

This translation fulfils not only the requirement of Inequivalence but

200 V. ON THE LOGIC OF MODALITIES

also the requirement of intensional isomorphism, the strictest requirement

that any translation can fulfil ( 14). Any given sentence in S,' is trans-

lated into 83 by that sentence itself, that is, by the same sequence of signs

now taken as signs in S a . Any two corresponding designators, that is, any

designator in S,' and the same expression in S,, are L-equivalent to one

another. This follows from the following two results:

(i) The rules of designation for the descriptive signs are the same in

both systems S, and S, (for example, the rules 1-2 for primitive predica-

tors).

(ii) Any sentence in S^ has the same range in both systems S and S a '

(see 41-4 concerning S x and S 2 ). Since the range is the same, the truth-

conditions are the same; therefore, the sentence means exactly the same

in S 2 ' as in S,'.

Thus the decisive difference between the situation here and that in the

earlier example concerning the irrational numbers becomes clear. In the

transition from SP to Sp the irrational numbers actually disappear, be-

cause a sentence in Sp of the form 'there is an irrational number such

that . . .' is not translatable into Sp. On the other hand, in the transition

from an extensional to a modal language the individuals and classes do by

no means disappear. A sentence in S x (or S,') which says that there is an

individual of a certain kind is translatable into S 2 (or 83) ; and a sentence

in Sj which says that there is a class of a certain kind is translatable into

s;.

In order to illustrate this result by an example, let us take Quine's

sentence (2). Since this sentence requires only individual variables, it can

be translated into S x . Let us assume that S x contains the following predica-

tors, either as primitive signs or as defined in a suitable way: *W for the

relation Wife, 'D' for the property Director, and T' for the property

Deaf. Then (2) is translated into Si by the following sentence:

(5) '(Zx)(3y)(?Lz)('Zw)['Dx . Vy ~(x = y).Wzx. Wwy . Fz . Fw]'.

Now this same sentence is also the translation of (2) into S 2 . It would be

an error to think that it was necessary for the translation into S a either

to use new predicators or to assign a new meaning to the old predicators,

as though, for example, 'Dx* in S, said that the individual x has the prop-

erty Director while 'Dx' in S 3 said that the individual concept x has a

strange new property somehow analogous but not quite the same as the

property Director. The matrix 'Da' expresses in both languages the

property Director; it may be defined in both languages in exactly the same

44. QUINE ON MODALITIES 201

way. Suppose a speaker X t uses the language S z and X 2 uses S 2 . Then the

question of whether a given full sentence, say 'Db', is true, may be de-

cided by both speakers in the same way. Both confirm or disconfirm this

sentence on the basis of observations of the person b, using the same

empirical criteria for the property Director. Nothing in the semantical

analysis of this sentence or in the procedure of empirical confirmation or

in the expectation of possible future experiences implied by the sentence

needs to be different for the two speakers. The same holds for the existen-

tial sentence (5) and for any other sentence occurring in both languages.

Therefore, I cannot agree with the view that, while the speaker Xj

recognizes the individuals of the concrete world, they have disappeared

for X 2 , leaving only their concepts behind them.

The situation with respect to Quine's other example (i) is analogous,

except that cardinal numbers are involved and therefore a variable of

second level, say V, is used. We have seen earlier ( 27) that, for the in-

troduction of particular cardinal numbers and of the general concept of

cardinal number, it is not necessary to use special class expressions and

class variables, as Frege and Russell did; we may, instead, regard cardinal

numbers as properties of second level or, rather, introduce cardinal num-

ber expressions as predicators of second level, whose intensions are proper-

ties of second level and whose extensions are classes of second level. Equal-

ity of cardinal numbers is then expressed with the help of ' = '. Thus we

translated the sentence

(6) 'the number of planets = 9'

into the following sentence of Sj :

(7) 'Nc'P s g'.

Similarly, Quine's sentence (i) can be translated into S,' as follows, if we

assume that exponentiation has been defined by a suitable procedure

(analogous to that of Cantor or Russell, [P.M.], Vol. II, *n6):

(8) <(3w)[NCM.Nc'P5E 3 7.

(If we wish to say that n is finite, we may use the concept of inductive

cardinal number with a definition analogous to Russell's). Here, again, the

given sentence (i) can likewise be translated into the modal language S a ',

namely, by the same sentence (8), hence without the use of any strange

new concepts. The translation is by no means dependent upon the occur-

rence of class variables as distinct from property variables. *NC(n)'

means in S a , just as in S,, that n is a cardinal number; thus in S a ', just

as in S,', sentences like C NC(2)' and 'NC(Nc'P)' are L-true. That the

202 V. ON THE LOGIC OF MODALITIES

sentence (8) has in S a ' the same factual content as in S, is seen by con-

siderations similar to those concerning the previous example (5) . The same

astronomical observations confirm the sentence in the one as in the other

language; it gives rise to the same expectations of future observations in

both languages. Thus there cannot be any difference in meaning.

The preceding discussion shows that a modal language is not inadequate

in comparison with the corresponding extensional language, that is to

say that we can express in the former whatever is expressed in the latter.

(So much Quine seems to admit.) We have seen, moreover, that the ex-

pressions used in a modal language for translations from the extensional

language do not have any unusual character with respect to either their

form or their meaning. Every designator and every sentence in the exten-

sional language has exactly the same meaning in the modal language

more exactly speaking, it has both the same intension and the same exten-

sion. The world of concrete things and the conceptual world of numbers

are dealt with in the modal language just as well as in the extensional one.

In order to see correctly the functions of these languages, and generally

of any languages, it is essential to abandon the old prejudice that a predi-

cator must stand either for a class or for a property but cannot stand for

both and that an individual expression must stand either for an indi-

vidual or for an individual concept but cannot stand for both. To under-

stand how language works, we must realize that every designator has both

an intension and an extension.

45. Conclusions

The main conclusions of the discussions in this book are briefly summarized.

The difference between the two operations understanding the meaning of a

given expression and investigating whether and how it applies to the actual

state of the world suggests a distinction between two different semantical

factors, which our method tries to explicate by the concepts of the intension and

the extension of an expression.

The chief purpose of this book is to develop a method for the analysis

of meaning in language, hence a semantical method. We may distinguish

two operations with respect to a given linguistic expression, in particu-

lar, a (declarative) sentence and its parts. The first operation is the

analysis of the expression with the aim of understanding it, of grasping its

meaning. This operation is a logical or semantical one; in its technical

form it is based on the semantical rules concerning the given expression.

The second operation consists in investigations concerning the factual

situation referred to by the given expression. Its aim is the establishment

45. CONCLUSIONS 203

of factual truth. This operation is not of a purely logical, but of an em-

pirical, nature. We can distinguish two sides or factors in the given ex-

pression with regard to these two operations. The first factor is that side

of the expression which we can establish by the first operation alone, that

is, by understanding without using factual knowledge. This is what is

usually called the meaning of the expression. In our method it is expli-

cated by the technical concept of intension. The second factor is estab-

lished by both operations together. Knowing the meaning, we discover

by an investigation of facts to which locations, if any, the expression ap-

plies in the actual state of the world. This factor is explicated in our

method by the technical concept of extension. Thus, for every expression

which we can understand, there is the question of meaning and the ques-

tion of actual application; therefore, the expression has primarily an in-

tension and secondarily an extension.

The method of intension and extension stands in contrast to the

customary method of the name-relation. The basic weakness of the latter

method is its failure to realize the fundamental distinction between mean-

iiig and application. This leads to the conception that an expression must

be the name of exactly one of the two semantical factors involved. For

example, properties and classes are regarded as entities of equal standing;

this leads to the view that a language ought to contain both names of

properties and names of classes. This conception is the ultimate source

of the various difficulties which we found involved in the method of the

name-relation. They center around the well-known difficulty which we

have called the antinomy of the name-relation. We have seen how the

various methods of keeping the name-relation but avoiding the antinomy

lead either to great complications in the language structure or to serious

restrictions in the use of the language or in the application of the semanti-

cal method.

The formulations in terms of 'extension' and 'intension', 'class' and

'property', etc., seem to refer to two kinds of entities in each type. We

have seen, however, that, in fact, no such duplication of entities is presup-

posed by our method and that those formulations involve only a con-

venient duplication of modes of speech. As it was shown to be unnecessary

to use different expressions for classes and properties in a symbolic object

language, it likewise turned out to be unnecessary to use those pairs of

terms in the word language as a metalanguage. A new metalanguage was

constructed, in which instead of the pair of phrases 'the class Human' and

'the property Human' only the neutral term 'Human' is used. It was

shown that the ordinary formulations can be translated into this neutral

204 V. ON THE LOGIC OF MODALITIES

metalanguage and that the latter language preserves all previous distinc-

tions, though in different formulations.

Our semantical method also helps in the clarification of the problems of

the modalities. It suggests a certain interpretation of the logical modali-

ties which supplies a suitable basis for a system of modal logic. In par-

ticular, the distinction between intensions and extensions enables us to

overcome the difficulties involved in combining modalities with quantified

variables.

The different conceptions of other authors discussed in this book, for

instance, those of Frege, Russell, Church, and Quine, concerning semanti-

cal problems, that is, problems of meaning, extension, naming, denotation,

and the like, have sometimes been regarded as different theories so that

one of them at most could be right while all others must be false. I regard

these conceptions and my own rather as different methods, methods of

semantical analysis characterized chiefly by the concepts used. Of course,

once a method has been chosen, the question of whether or not certain re-

sults are valid on its basis is a theoretical one. But there is hardly any

question of this kind on which I disagree with one of the other authors.

Our differences are mainly practical differences concerning the choice of

a method for semantical analysis. Methods, unlike logical statements, are

never final. For any method of semantical analysis which someone pro-

poses, somebody else will find improvements, that is, changes which will

seem preferable to him and many others. This will certainly hold for the

method which I have proposed here, no less than for the others.

Let me conclude our discussions by borrowing the words with which

Russell concludes his paper. 10 It seems to me that his remarks, although

written more than forty years ago, still apply to the present situation

(except, perhaps, that instead of 'the true theory' I might prefer to say

'the best method'):

"Of the many other consequences of the view I have been advocating,

I will say nothing. I will only beg the reader not to make up his mind

against the view as he might be tempted to do, on account of its ap-

parently excessive complication until he has attempted to construct

a theory of his own on the subject of denotation. This attempt, I believe,

will convince him that, whatever the true theory may be, it cannot have

such a simplicity as one might have expected beforehand/'

10 [Denoting], p. 493.

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[The abbreviated titles in brackets are used in citations throughout this book.]

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Introduction. New York, 1939.

CARNAP, RUDOLF. [Syntax]. Logical Syntax of Language. Orig., Vienna, 1934; English

trans. London and New York, 1937.

. [I]. Introduction to Semantics. "Studies in Semantics", Vol. I. Cambridge,

Mass., 1942.

-. [II], Formalization of Logic. "Studies in Semantics", Vol. II. Cambridge,

Mass., 1943.

. [Inductive]. "On Inductive Logic", Philosophy of Science , XIJ (1945), 72-97.

-. [Remarks]. "Remarks on Induction and Truth", Philosophy and Phenom-

enological Research, VI (1946), 590-602.

.-. [Modalities]. "Modalities and Quantification", Journal of Symbolic Logic, XI

(1946), 33-64.

CHURCH, ALONZO. [Dictionary]. Articles in D. D. RUNES (ed.), The Dictionary of

Philosophy. New York, 1942.

. [Review C.]. "Carnap's Introduction to Semantics" (a review of Carnap, [I]),

Philosophical Review, LII (1943), 298-304.

-. [Review Q.]. A review of Quine, [Notes], Journal of Symbolic Logic, VIII

(1943), 45-47-

FREGE, GOTTLOB. [Sinn]. "Ueber Sinn und Bedeutung", Zeitschrift fiir Philosophic

und philosophische Kritik, C (new ser., 1892), 25-50.

. [Grundlagen]. Die Grundlagen der Arithmetik: Eine logisch-mathematische

Untersuchung Uber den Begriff der Zahl. Breslau, 1884. Reprinted, 1934.

-. [Grundgesetze]. Grundgesetze der Arithmetik, Vols. I, II. Jena, 1893; 1903.

HILBERT, DAVID, and BERNAYS, PAUL. [Grundlagen I]. Grundlagen der Mathematik,

Vol. I. Berlin, 1934.

LEWIS, C. I. A Survey of Symbolic Logic. Berkeley, 1918.

. [Meaning]. "The Modes of Meaning", Philosophy and Phenomenological Re-

search, IV (1943-44), 236-50.

LEWIS, C. I., and LANGFORD, C. H. Symbolic Logic. New York, 1932.

MORRIS, CHARLES. [Signs]. Signs, Language, and Behavior. New York, 1946.

QUINE, W. V. [Designation]. "Designation and Existence", Journal of Philosophy,

XXXVI (1939), 701-9.

. [M.L.]. Mathematical Logic. New York, 1940.

. [Notes]. "Notes on Existence and Necessity", Journal of Philosophy, XL

(1943), 113-27.

RUSSELL, BERTRAND. [Denoting]. "On Denoting", Mind, XIV (new ser., 1905), 479-

93-

. [P.M.]. See WHITEHEAD.

. [Inquiry], An Inquiry into Meaning and Truth. New York, 1940.

TARSKI, ALFRED. [Wahrheitsbegriff], "Der Wahrheitsbegriff in den formalisierten

Sprachen", Studia philosophica, I (1936), 261-405. Originally published in 1933.

. [Truth]. "The Semantic Conception of Truth and the Foundations of Seman-

tics", Philosophy and Phenomenological Research, IV (1944), 341-76.

WHITEHEAD, A. N., and RUSSELL, B. [P.M.]. Principia Mathematica. 3 vols. Cam-

bridge, England, 1910-13. 2d ed., 1925-27.

WITTGENSTEIN, LUDWIG. [Tractatus]. Tractatus logico-philosophicus. London, 1922.

30$

INDEX

[The numbers refer to pages. The most important terms, names, and references are indi-

cated by boldface type.]

A, a, S, 4

a , 3 6

a*, 37

* Absolute' use of terms, 24 n . ; see also Non-

semantical use

Abstraction expressions, 3

Actual and nonactual things, 65

Adequacy: for F-concepts, 12; for L-con-

cepts, ii f.; for L-truth, 10 f .

Ambiguity of the name-relation, looff.

Analysis,' paradox of, 63

Analytic sentences, 8; see also L-truth

Antinomies, nature of, 135

Antinomy of the name-relation, i33ff 189;

solutions for, 136 If.

Arithmetical concepts, 84 f ., 90, 1 15 ff ,

Atomic sentences, 5

au, 187

Autonymous use, 4

Axiom of infinity, 86 n .

B, 4

BAYLIS, C. A., 21 n., 28, 64 n., 205

Belief-sentences, 53ff* 61 f. 123 f., 141

BENNETT, A. A., 21 n., 28, 205

BERNAYS, P., 33, 38, 205

Biconditional, 3

BLACK, M,, 63

Blanks, 4

CANTOR, G., 201

Capitalization, 17 n., 153

Cardinal number, 116, 201; see also Number

CHURCH, A.: Bibliography by , 173 n.; on de-

noting, 97 n.; 'description,' i26n.; on

designatum, son., i66n.; explicanda of,

127; on extensionality, 47 n., son.; on

Frege, 118, 125, 126, 132, 13?*-; infinity

of names, 130 f., 138; on lambda-de-

finability, 79 n.; on lambda-operator,

39 n.; method of, of name-relation, 137 f.,

143, 204; on modal sentence, 187; publica-

tions of, 3 205; on Quine, 100, i37t 195;

results of, are valid, 128; on Russell, 140;

on synonymity, 99; on two abstraction

operators, no; on two kinds of variables,

i9S

Class names, 107, in f.

Class variables, 113, 160, 201

Classes, i6ff., contextual definition of, 114,

146, 147, 159; as extensions, i6ff.; as in-

tensions, 92; Russell's definition of, 107

Classes of sentences, 178 f.

Closed sentences, 3

Cognitive meaning, 6

Computable numbers, 79 n .

Concept, 21

Conditional, 3

Conjunction, 3

Connectives, 3

Connotation 65, 118 n., 126

Contingent propositions, 175

Contingent truth, 12, 132

Conversion, 3

Coordinate languages, 75 79

CT, 132

Degree, 5

Denotation, 126

Denoting, 97 n., 118 n.

Describing an extension, 71

Description: (i) individual, 3, 32 ff.; in S a ,

184 f .; (2) other types, 38 f.

Descriptive expressions, 3, 86

Descriptum, 33

Designation, 97, 99, 162 f., 171

Designators, 6 f .

'Designatum/ i66n.

Disjunction, 3

DUCASSE, C. J.: articles by, 64 n,; on epis-

temic attitudes, 62; on facts, 28

Duplication: of names, io6ff.; on higher

levels, ii4f.; of variables, 113

EATON, R. M., 126 n.

Empty predicators, 21

Entities, 20, 22, 42 f., 91, 146, 167 f., 196;

see also Multiplicity of entities

Enumeration, 83, in

Equality of numbers, 1 1 7

Equinumerous properties, 116

Equivalence: (i) semantical, of expressions:

of designators, 14; of individual expres-

sions, 15; of predicators, 14!.; of sen-

tences, 6; (2) nonsemantical, of entities:

of extensions, 25; of intensions, 24; of

neutral entities, 154; of properties, 24; of

propositions, 24

Equivalence, logical; see L-equi valence

Equivalence class, 16

Exemplification, 21

Existence, 43

Explicandum, explication, explicatum, 8

Extension: concerns application, 157, 203; of

designators, 23; of individual expressions,

40; of predicators, 18 f .; of sentences, 26

Extensional expressions, 47 f ,

307

208

INDEX

Extensional language: avoids antinomy,

141 f.; definition of, 48; refers to proper-

ties, 199; for semantics, 168 ff .

Extensional metalanguage, 169

Extensionality, thesis of, 141

F,4

/, 38.44 f-

F-concepts, ia

F-equi valence: of designators, 14; of indi-

vidual expressions, 40; of predicators, 15;

of sentences, 12

F-falsity, la

F-implication, la

F-truth, ia

Fact, 12, a8

Factual sentences, la

Falsity, 6

FREGE, G.: on antinomy of name-relation,

133, 136; on belief-sentences, 123 f., 132; in

bibliography, 305; on descriptions, 8, 34,

35 ff,, 38, 140; disadvantages of method of,

1 39 If., 137, 143; on double nominatum,

133; explicanda, 136 f.; infinity of names

and entities, 130 f.; on nominatum,

119 f., 125; on number, 8, n6f., 201; on

oblique context, 133; principles of inter-

changeability, 99, i3i f., 143; on 'propo-

sition,' u8n., 134; results are valid, 128;

semantical method of, 119, 204; on sense,

119, 125; terminology of, 118 n.

Functors, 7

G, 101

German letters, 4, 72

Giving an extension, 71

GOEDEL, K., 36, 79 n.

GOODMAN, N., 36 n.

H,4

HERBRAND, J., 79 n.

HlLBERT, D., 33, 34, 38, 3Q5

Hold, 9, 170

Hypostatization, 33, 44

Identity, 14

Identity condition: for classes, 18; for ex-

tensions, 33; for intensions, 33; for proper-

ties, 18; for propositions, 37, 124

Identity expression, sentence, sign, 99

Implication, logical; see L-implication

Implication, material; see Conditional

Impossibility, 175

Impossible things, 65

Inouvidual concepts, 41, 180 f.

Individual constants, 4* 32

Individual description; see Description (i)

Individual expressions, 33

Individual variables, 3, 32

Individuals, 33; as numbers, 86; as positions,

74

Inductive logic, 9n.

Intension: of designators, 33; of individual

expressions, 41; of predicators, i8f.; as

primary, 113, 157, 303; of sentences, 37

Intensional expressions, 48

Intensional isomorphism, 56, 59

Intensional structure, 56

Interchangeably, 47; principles of, 51 f.

Interpretation, 5, 9

Iota-operator, 3, 9, 33

K, 112

KANT, I,, 8, 12

KLEENE, S. C., 79 n.

L-concepts, 7 ff.

L-designation, 163

L-detenninacy: (i) semantical: of designa-

tors, 70, 73, 88; of functors, 85; of indi-

vidual expressions, 78, 80; of predicators,

83; of sentences, n; (2) nonsemantical:

of intensions, 89

L-empty predicators, 21, 84

L-equivalence: (i) semantical, of expres-

sions: of designators, 14; of individual

expressions, 40; of matrices, 58; of predi-

cators, 15; of sentences, ii; in two sys-

tems, 57; (2) nonsemantical, of entities:

of extensions, 25; of intensions, 24 f.; of

neutral entities, 154; of properties, 35; of

propositions, 24

L-equivaience class, 16, 152

L-falsity, u, 31

L-implicatipn, n

L-indeterminacy: of designators, 72; of

sentences, 12

L-interchangeability, 47

L-range; see Range

L-state; see State

L- truth, 8, 10

L-universality, 44, 84

Lambda-operator, 3, 9 39

LANGFORD, C. H., 8n., 305; on analysis,

63 f.

Language of physics, 79

LEIBNIZ, G. W., 8, 9, 10

LEONARD, H. S., 36 n.

LEWIS, C. L: in bibliography, 305; on com-

prehension, connotation, denotation, 64 f.;

definition of necessity, 175; n denotation

of true propositions, 94 f.; on equivalence

in analytic meaning, 60 f., 62; on modal

logic, 173; on nonactual things, 65 f.; on

propositions, 179; on strict equivalence,

177; on strict implication, 4711., 177; on

synonymity, 60 f .

Logical expressions, 86

Logical truth; see L- truth

M, 4, i4S

M', 153, 167

M., 169

MARTIN, R. M., 36 n.

Material implication; see Conditional

Mathematical concepts; see Arithmetical

concepts

Matrix, 5

Meaning, 6 f., 10, 60, 63, 64, 118 n., 303 f.

INDEX

209

MEINONO, A. VON, on impossible things, 65

Mental entities, 20

Metalanguage, 4, 44

Metametalanguage, 71, 153

Metaphysics, 43

Methods: of extension and intension, 2, 23,

107, 142 f., 167, 202 f.; of the name-rela-

tion, 98, 144, 203; comparison of the two,

106!., 109 f., 117, 131, 143

MFL, 183 n.

MILL, JOHN STUART, 126

Mixed modalities, 175

ML, 102

ML/, 102, 199

MM, 153

Modalities: analogous to quantifiers, 186;

correspondence to L-concepts, 174 f.;

definitions of, 175; intensional, 50; inter-

pretation of, 176, 178; Lewis on, 173;

logic of, 173, 186; multiple, 131, 141, 172;

' Quine on, 193 f?.; translation of, 141, 176,

187;' and variables, i77ff., i94ff.; and

variables in word language, 191 ff.; in

word language, 186 ff.

MOORE, G. E., on analysis, 63

MORRIS, C. W., 6 n., 205

Multiplicity of entities : in Frege and Church,

130 f ., 138; in our method, 17, 145 f., 167 f .

N, 49, 173

ft, 170

Name, 97 f., 127

Name languages, 75

Name-relation, 97

Names of expressions, 4

NC, Nc, 116

Necessary equivalence, 177

Necessary implication, 177

Necessity, 49, 174

Negation, 3

Neutral expressions, 153, 157

Neutral metalanguage, 153

Neutral variables, 160

Nominalism, 43

Nominatum, 97

Nonextensional expressions, 48

Nonsemantical (absolute) use of semantical

terms, 24, 66; see also Equivalence (2);

L-determinacy (2); L-equivalence (2);

Truth (2)

Null class, 84, 92

Null thing, 36

Number, 43 f., 76, 115, 201

Number concept, Number extension, Num-

ber intension, 117

Object language, 3 f .

Oblique context, 122, 131

Ontology, 22, 43, 196 f., 198

Operand; see Scope

Operators, 3; see also Quantifiers

p T , 28, 94

Paradox of analysis, 63

Paradoxes; see Antinomies

Part-whole relation, 36

Physical magnitude, 85

Platonic ideas, 22

PM, 107, 113, 114 f., 148 ff.

PM', 107, "3, 148 f., 199

PM", 158

Port-Royal Logic, 1 26

Positional properties and relations, 75, 80,

84,92, in

Positions, 74

Possibility, 175

Possible and impossible things, 65

Practical questions, 33, 43, 128, 136, 168, 190,

204

Predicates, 4

Predicators, 6

Principles: of interchangeability, 51 f., 177

(see also FREGE, QUINE, RUSSELL); of the

name-relation, 98; of subject matter, 98;

of uniyocality, 98

Probability, 9 n.

Progression, 75

Proper names, 74

Properties, i6ff., i8if.; nonmental, 20;

physical, 20

Property names, 107

Propositions: false, 28 ff.; as intensions, 27;

nonmental, 27; as ranges, 181

Pseudo-statements, 22, 43

Psychological sentences, 62

Pure modalities, 175

Quantifiers, 3

QUINE, W. V.: on abstraction expressions,

103; on antinomy of name-relation, 134,

i37 193; on classes, 104, 146 n.; on de-

scriptions, 36, 38; on 'designation/ 137;

on 'designatum/ 166 n.; on elimination of

antinomies, 1 6 1 n . ; explicanda, 127; on

identity, io4f.; on individual concepts

197; on individual constants, 74; on indi-

viduals, 103; on meaning, 100, 152; method

of name-relation, 137, 143; on modalities,

193 ff.; on neutrality, 157; on nominalism,

43; on nonextensional contexts, 137;

'not purely designative,' 137, 194; on

number, 197; on ontology, 42!., 146 n.,

196 f.; on properties, 152; publications of,

3, 205; on "repudiation" of extensions and

individuals, 197 ff.; rule of context, 147,

150; statement of, 196 f.; on substitutivity,

99, 105; on synonymity, 60, 152; on sys-

tem ML, 102, 141 ; on values of variables,

7, 42, 196

Quotation marks, 4, i7n., 169

RA, 4

Range, 9

Real-number expressions, 79 f .

Recognition of entities, 22, 42 f., 196 ff.

Reduction of entities, 91 ; of classes to proper-

ties, 146 ff.; of extensions to intensions,

91, 151

210

INDEX

Relations, 21, 182

"Repudiation": of extensions, 197**.; of

individuals, 197 ff.; of irrational numbers,

198 ff.

Rules, semantical, 5, 169; of context, 146 f.,

149, 150; of designation, 4, 5, 170; of

formation, 5, 169 f.; of ranges, 9, 182 ff.;

of truth, 5, 170

RUSSELL, B.: antinomy of, 135; on antino-

my of name-relation, 134, i38ff .; in bibliog-

raphy, 205; on class variables, 113, 201 ; on

definition of classes, 107, 147; denial of

meaning, 140, 144; on denoting, 97 n.; on

descriptions, 8, 34 .,38, 139; duplication of

names, 1071!.; on extensionaiity, 141 n.;

on Frege, 118, 140; on identity, 14; on im-

possible objects, 65; on incomplete sym-

bols, 139, 149; on individual constants,

74; on name-relation, 139 f.; on number,

8, 115 f., 201; principle of interchange-

ability, 134; on prepositional attitudes, 62;

on propositions, 29, 152; rule of context,

149, 150; semantical method of, 138 ff.,

204; on truth-functions, 49

S,, 3 ff.

S a , 49, 174. 182

Sj,75

Sp, 79

s, 4

@,4

Scope, 32

Semantical methods; see Methods

Semantical rules; see Rules, semantical

Semantical systems, 3, 169; see also S x , etc.

Semantics: in extensional language, i68ff.;

in neutral language, 161 ff.

Sentences, 5

Sentential function or matrix; see Matrix

Solutions of antinomy of name-relation,

136 ff.

Specific propositions, 28

Standard forms: of individual expressions,

75, 79J of real-number expressions, 79 f.

State-descriptions, 9, 182

States, 9

Strict implication.; see Lewis

Substitutivity; see Interchangeability

'Syncategprematk expressions,' 7

Synonymity, 56, 60, 133

Synthetic judgments, 12

TARSKI, A.: in bibliography, 205; on meta-

language, 44; on semantics, 64 n.; on

truth, 5 n., 6 n.

Terminology, 6 n., 20 fL

That-clauses, 27 n., 156

-tor, 6 n.

Translation, 59; from M into M', 158

Truth: (i) semantical, of sentences, 5 f., 93;

(2) nonsemantical, of propositions, 94, 132

Truth-conditions, 5

Truth-functions, 49

Truth- values: as extensions, 26; as extra-

linguistic entities, 93 f.; as propositions,

94 f-

TURING, A. M., 79 n.

Two, 115

Understanding, 20, 119, 202 f.

Uniqueness condition, 33

Universal class, 84

Universal words, 44

Universals, problem of, 22

Univocality, 98

Value extensions, 45

Value intensions, 45

Values of variables, 42

Variables, 3, 42 ff.; and modalities, 177 ff.,

191 ff., I94ff.