A HISTORY OF SGME‘ AISPECTS OF THE
THEORY OF LOGIC. 1850 - PRESENT

Thmés fa? flue Dagree of Ph. D.
MECHIGAN STATE UNWERSITY
James Woocison Van Ewa

1966

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LIBRARY

Michigan State

University

 

#—

This is to certify that the

thesis entitled

A History of Some Aspects of the

Theory of Logic, 1850 - Present

presented by

James Woodson Van Evra

has been accepted towards fulfillment
of the requirements for

Ph.D. Philosophy

degree in

H. S. Woe—AM

Major professor

 

Date—Afinfi 1; fgéL

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ABSTRACT

A HISTORY OF SOME ASPECTS
OF THE THEORY OF LOGIC,
1850 - PRESENT

by James Woodson Van Evra

This work seeks to accomplish two objectives. First,
it is a description and explication of the theories of logic
held by those who have contributed most to logic during the
last century. As such, it deals with those philosophic
positions which, while not a part of logic, nonetheless
provide a theoretical basis for it. The two aSpects of the
theory of logic which receive the most attention are the
ontology of logic and the epistemology of logic.

The individuals who are discussed are George Boole,
Gottlob Frege, Charles S. Peirce, Bertrand Russell, Alfred
Whitehead, Ludwig Wittgenstein, C. I. Lewis, Alonzo Church,
and Willard V. O. Quine. In each case, a brief description
of the individual's contribution to logic is given, after
which a discussion of his theory of logic is undertaken.
Finally, some concluding remarks about the individual's
theory are given.

The second major purpose of the work is to indicate
what further direction work in the theory of logic might
take. In a concluding chapter, the suggestion is made that
the philosophy of logic should be recognized as a distinct
area within the field of philOSOphy (in much the same way as

the philosOphy of mathematics'is so recognized). Furthermore,

James Woodson Van Evra

the suggestion is made that all theories of logic reCOgnize
(in some way) form as an integral constituent in the subject-
matter of logic. Hence it is claimed that questions
concerning the nature of such form might serve as the cohesive
force binding all other concerns in the theory of logic

together.

A HISTORY OF SOME ASPECTS
OF THE THEORY OF LOGIC,

1850 - PRESENT

By

James Woodson Van Evra

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Philosophy

1966

Copyright by

JAMES WOODSON VAN EVRA

1967

ACKNOWLEDGEMENTS

I should like to thank all of those who have helped
me with the idea and execution of this work. In particular,
I would like to thank Professor Robert Burke Barrett, who
was initially responsible for my interest in this topic.

I am very grateful also for the assistance given to me by
Professor Gerald J. Massey; his unselfish use of time in
going over the work in detail, and his vast knowledge of
logic have rescued it from a great deal of stylistic and
substantial error.

It is impossible for me to eXpress the extent of my
indebtedness to Professor Henry 5. Leonard. The concerns
which I display in this work are largely a product of my

long acquaintance with his book, The Principles of Right

*

 

Reason. Furthermore, his many helpful suggestions have
greatly reduced the number of rough edges in the work.

I should like to thank also my wife Judy and
daughter Stephanie; their confidence and compaionship
sustained me through a long winter during which this thesis

was written.

11

TABLE OF CONTENTS

CHAPTER

I. INTRODUCTION . . . . . . . . . . . . .
II. GEORGE BOOLE . . . . . . . . . . . . .
Introduction . . . . . . . . . . . .
Boole's logic . . . . . . . . . . .
Boole's theory of logic . . . . . .
Conclusion . . . . . . . . . . . . .
III. GOTTLOB FREGE . . . . . . . . . . . .
Introduction . . . . . . . . . . . .
Frege's intellectual enviornment . .
Some basic aspects of Frege's logic
Frege's theory of logic . . . . . .
Conclusion . . . . . . . . . . . . .
IV. CHARLES SANDERS PEIRCE . . . . . . . .
Introduction . . . . . . . . . . . .
Peirce's logic . . . . . . . . . . .

Peirce's remarks on alternative
theories of logic . . . . . . . .
Peirce's theory of logic . . . . . .
Conclusion . . . . . . . . . . . . .
V. PRINCIPIA MATHEMATICA, I . . . . . . .

IntFOdUCtion o o o o o o o o o o o o

The background of the Principia Mathematica

 

iii

PAGE

15
37
38
38
ho
52
61
8h
89
89
91

101
111
127
129
129

132

CHAPTER

Some aSpects of the logic of the Principia

 

The theory of logic of the Principia

 

Mathematica» 0 o o o o o o o o o o o o 0
VI. PRINCIPIA MATHEMATICA, II . . . . . . . . .
IntrOdUCtion o o o o o o o o o o o o o o 0

Changes in the logic of Principia

 

Mathematica . . . . . . . . . . . . . .
Influences on the theory of logic of the

second edition of Principia Mathematica;

introduction . . . . . . . . . . . . . .
Influences on the second edition of

Principia Mathematica; the philosophy of

 

Ludwig Wittgenstein . . . . . . . . . .
Russell's theory of logic in the second

edition of Principia Mathematica . . . .
Conclusion to the discussion of

Principia Mathematica . .'. . . . . . .

VII. C. I. LEWIS . . . .. .‘. . . . . . . . . . .

Introduction . . . . . . . . . . . . . . .
Lewis' logic, introduction . . . . . . . .
The general development of Lewis' logic .
The logic of classes, and the logic of

terms . . . . . . . . . . . . . . . . .
Lewis' logic; the truth—functional logic

Of‘ prOPOSitiOnS o o o o o o o o o o o 0

iv

PAGE

1&7

163

190

190

191

198

201

223

235
239
239
2&0

2&1

2&&

251

CHAPTER

Difficulties with the concept of

material implication . . . . . . . . .
The logic of strict implication . . . . .
Lewis' theory of logic . . . . . . . . .
Conclusion . . . . . . . . . . . . . . .

VIII. CONCLUSION: CHURCH, QUINE, AND THE FUTURE

Introduction . . . . . . . . . . . . . . .
Church's logic; Quine's logic . . . . . .
The theories of logic of Quine and Church

COHCIUSion o o o o o o o o o o o o o o o

BIBLIOGRAPHY O O O O O O O O O O O O O O O O O O 0

PAGE

257
260
272
290
292
292
293
298

310

317

... Logique is occupied aboute all
matters, and doeth playnly and nakedly
setfurthe with apt wordes the summe

of things ...

from
Rule 2i Reason

Thomas Wilson 1551

vi

CHAPTER I

INTRODUCTION

This work is about logicians thinking about logic.

It is not a work in logic, but rather is concerned with those
phiIOSOphical issues which constitute the foundation of the
subject. We shall deal with such questions as those con-
cerning the nature of the subject-matter of logic, and those
about the epistemological basis of logic.

The logicians with whom we are to deal are those who,
by general acclaim, have contributed most to the develOpment
of mathematical logic in the last hundred years. The discus-
sion is based on the premise that if you want to know what
logic is, a good place to begin the search is with the
pronouncements of those who have contributed most to the
subject.

Our discussion will not present a complete picture of
the development of logic in the last century; there are many
individuals who have contributed significantly to the subject
whose work we shall pass over. We shall instead concentrate
our attention on those few who not only made major advances
in the subject, but who also expressed themselves with regard
to the philosophical basis of logic.

While the develOpment of logic since Boole displays
continuity and homogeneity, our discussion will make it clear

that the development of the theory of logic has been in like

measure heterogeneous, and displays great discontinuity. The
wide range of theories provide a sharp contrast to the
orderly development of 10gic.

It is hOped that this study may provide the basis for
further work in the theory of logic. There has been a
relative neglect of the subject during the period in which
logic has advanced most. Part of our task will be to Show
the adviability of recognizing the theory of logic as a
discrete field of study in philosophy.

The format of the work is as follows: each chapter
has its own introduction. Following this, there is a brief
description of the contributions to logic made by the
individual in question. Next follows the main portion of
the work, which in each case constitutes a discussion of the
logician's theory of logic. Each chapter ends with a brief
conclusion, which constitutes an assessment of the
contribution made by the individual. The last chapter
offers a general conclusion, in which suggestions are made
as to the possible direction which further work in the

subject might take.

CHAPTER II
GEORGE BOOLE
1. INTRODUCTION.

George Boole (1815-186&) is best known for his
development of a class-logic, i.e., his algebra of logic,
and for his work in probability theory. He is generally
regarded as being the first in a direct succession of
individuals who succeeded in transforming logic, by freeing
it of the strictures imposed by what is now called traditional
or syllogistic logic, and leading it on the road to its
present highly developed state.

Assessments of Boole's accomplishments range from
those which credit him with completely establishing the new
logic, to those in which praise is more modest. E. T. Bell,
whose commentary falls into the former class, maintains that
“others before Boole, notably Leibniz and De Morgan, had
dreamed of adding logic itself to the domain of algebra;

Boole did it."1 That such descriptions are overly generous
will become apparent as we proceed. Others, like C. I. Lewis,
whose praise of Boole is more guarded, are generally in a
better position to point out his real and lasting contri-

butions to the field. Lewis maintains that while Boole did

 

1Bell, E. T., Men 23 Mathematics (New York: Simon and
Schuster, 1937), p. &35.

make significant contributions to lOgic, that even so, he was
more at home in mathematics, and was oblivious to many
central problems in logical theory.2

Many of Boole's contributions to logic have become
obscured in the intervening past hundred years; indeed,
there has been a conscious effort to preserve only the most
finished, readily comprehensible of his works —— the so—called
Boolean algebra of classes. But Boole rarely finished
anything to his own satisfaction, for he was constantly
constructing various interpretations of his calculus, and
striving for a clearer statement of the principles with which
he was concerned. All that remained constant through the
short period of his productive life was his acceptance of
the skeleton of the calculus itself - the symbols, operators,
and various operations borrowed from mathematics. Only the
most readily defensible interpretation of these symbols has
been preserved. A great deal, however, is to be gained by
looking beyond just the class interpretation (or any single
interpretation, for that matter), to a consideration of the
full scope of what amounts to Boole's theory of logic. Some
of it strikes into the heartland of epistemology, and into
various other fields. But regardless of the area, taking
full account of the theory may lead to a better understanding

of the foundations of contemporary logic.

 

2Lewis, Clarence Irving, Survey 2f Symbolic Logic
(Berkeley: University of California Press, 1918), p. 51.

 

Before dealing directly with Boole's theory of logic,
let us first consider portions of his logic, and their
standard interpretations. This treatment will be by no
means complete. What follows will be a consciously hasty
review of some of the main features of Boole's logic,
offered mainly in order to clear up prevalent misoonseptions

and to set the stage for the subsequent discussion of theory.

2. BOOLE'S LOGIC.

Boole's algebra of logic is best described as a
system closely resembling a binary mathematical algebra,
but subject to non-numerical interpretations. The basic
elements of the algebra are the so-called ”elective” symbols
'x,‘ 'y,' 'z,’ etc., and symbols for operations on the
elective symbols. There is some difference of opinion on
the question» as to how many constituents of the system are

3

really basic but these two sorts of constituents are
included in all accounts.

Elective symbols are usually regarded as class
symbols, each elective standing for some class or other.

But as was mentioned above, this interpretation is only

partly correct. Boole's own conception of the place and

 

3Lewis, for example, claims that there are three
fundamental ideas: electives, operators, and the idea that
those rules of Operation hold for an algebra of the numbers
0 and 1. (Lewis, Survey gf Symbolic Logic, p. 52.) But this
last notion is not fundamental in the same sense as the other
two, for it is clearly not used as a constituent in the

 

function of elective symbols changed a good deal in the
course of his work, mostly in subtle but significant ways
which are concerned with the contents of the classes
involved, and hence will be discussed later, but also in
wahys which more directly affect the calculus. In the
Mathematical Analysis 2f Lo 1c,“ he maintains thatiif we

let 'X,' 'Y,' etc. represent individual members of classes,
electives, such as 'x,' can be used to ”select” all of the
X's from any group. The difficulty to which this gives rise
is simply that it becomes difficult to tell whether Boole
intended 'x,' 'y,’ etc., to stand for an operation, viz. that
of selection, or for the result of that operation, i.e., a
class name. As Lewis remarks, "Thus x, y, 2, etc., repre-
senting ambiguously operations of election or classes, are the
variables of this algebra. Boole speaks of them as 'elective

5

symbols' to distinguish them from coefficients.” Boole
himself is of little help in straightening this matter out,

for he changes his position on electives frequently. At one

 

formation of the system. What is more, the symbols '0' and
'l' were interpreted by Boole as being elective symbols them-
selves, and hence fall into the first class. At one point,
Boole himself incorporates identity, as a separate category,
into the basic structure of the algebra. (Boole, A3 Investi-
gation 23,322 Laws 22 Thou ht, p. 27.) His reason for doing
this will be discussed later in this section.

“Boole, George, The Mathematical Analysis 2f Logic
(London: George Bell, 18&7).

5Lewis, op. cit., p. 52.

point he uses capital letters to stand not for individual
class members, but for propositions (i.e., statements), while
at the same time he uses the small letters as mere class
symbols.6 Later, he remarks, ”... x, which we shall call an
elective symbol, represents the mental Operation of selecting
from the groupEf X's, Y'x, eta all the X's which it
contains."7 But by the time tfie L: 3‘2; Thought was written,
Boole had abandoned the use of capital letters as standing
for particular class members.

Another interesting shift in Boole's use of elective
symbols concerns the way he conceived of their relation to
language in general. Although we shall see much more of this
in the section on pure theory, in his early works, Boole
talks as though elective symbols range over things, i.e, have
things as values, while no mention is made of there being
substituends which replace the variables and name the class
members. In ngg‘gf‘Thou ht, Boole reverts to the traditional
practice of considering variables (or “literal symbols“ as
he speaks of them there) as place-markers for terms.

Taking these facts into consideration, it becomes
clear that Boole continually struggled with the interpretation
of such notions as that of an elective symbol. It should be

apparent also that the interpretation of the fundamental

 

680019, 020 Cite, Fe 510

7Boole, George, “The Calculus of Logic,“ Studies _1_q
Logic and Probability (London: Watts and Company, 1952).

constituents of Boole's logic is more difficult than we are
ordinarily led to believe. But more than that, the intro-
duction of these complexities will be of considerable use
in the discussion of Boole's theory of logic.

Under the heading of electives, Boole included two
cases of special interest, the symbol 'v' and the symbols
'1' and '0'.

The symbol 'v' is used by Boole to signify the class
whose only necessary characteristic is that it have members.8
Regarding this symbol, Boole at one point remarks, "the term
Some X‘s will be expressed by vx, in which v is an elective
symbol appropriate to a class V, some members of which are
X5, but which is in other reSpects arbitrary."9 This
symbol is of interest in that it differs from the other
electives in various significant reSpects. Perhaps the most
interesting respect concerns the way Boole employs it in what
in traditional logic was called inference by sub-alternation.
Kneale, for instance, remarks that Boole maintains that from
x(l-y)=0 (which means all x is y), we can deduce vx(1-y)=0
only if x and y are not null, or as Boole himself put it,

only on the condition that v(l-x)=0, and v(l-y)=0 are both

8Cf. Boole, op. cit., The Mathematical Analysis 22_
L0 10, p. 210

9Boole, OE. cit., "The Calculus of Logic," p. 128.

true.10 In this manner Boole comes to grips with the

problem of existential import, which plagues the traditional
square of opposition. It is interesting that Boole intro-
duced the notion of a class whose only necessary character-
istic was that it have members, in order to accomplish the
task of clearing up questions of import. The symbol 'v' with
this interpretation is an ancestor of the existential
quantifier, but because it is an elective or class symbol,

it remains a distant ancestor.

The symbols '1' and '0' form another interesting
ppecies of elective. 0f the symbol '1,‘ Boole remarks, ”let
us understand it as comprehending every conceivable class of
objects whether actually existing or not."11 '0' is employed
as the complement of 'l,' or nothing. This is the most
straightforward statanent of the significance of these two
symbols which Boole gives anywhere, although throughout his

work it is clear that he intended to use it for the most

part as an elective.

 

10The point may be reformulated as follows: 'x(l-y)=0'
means that the class formed by the intersection of the class
of all x's and the class of non-y's is empty. Further,
'vx(l-y)=0' means that the intersection of the non-empty
class of x's (symbolized as 'vx', which is alternatively
translatable as 'some x') and the class of non-y's is empty.
Boole maintains that we cannot infer the second preposition
from the first unless both classes have members, i.e., are
not null. Cf. Kneale, William and Kneale, Martha, The DeveloE-
ment 2E Logic (Oxford: Oxford University Press, 196277

11Boole, op. cit., The Mathematical Analysis 2;; Lo ic,
p. 15.

10

Concerning Boole's use of the symbold 'l' and '0',
Kneale maintains that ”in practice, Boole takes his sign 1
to signify what De Morgan called the finiverse 2f discourse,
that is to say, not the totalitytcf all conceivable objects
of any kind whatsoever, but rather the whole of some
definite category of things which are under discussion."12
This is clearly, however, not the case. Apparently Boole
went totally unaware of the difficulties inherent in the
notion of a class with everything as its members, for he
used '1' as if there were no such trouble at all. Because
he was oblivious of any such difficulty, his interpretation
of the symbol embodied no effort to avoid the difficulty.

Perhaps the most interesting feature of this pair of
symbols concerns the subject-matter interpretations which
they received at various times in Boole's logical works. A
discussion of this topic~ will appear in the next section.

Much of the rest of Boole's logic is composed of signs
of operations, such as '+', '-', 'x', etc. which are used to
combine elective symbols into formulae. The formulae of the
system are invariably equational, for Boole translates
ordinary predicative instances of 'to be' by means of the
identity sign, maintaining that all discourse can be so

reduced.

 

12Knea10, OE. Cite, pa “080

ll

xxy, or in its more usual form xy, serves as the sign
of what is now known as the logical product of x and y. x+y
serves as the logical sum, with one minor variation from
present-daywusage: Boole maintains that the classes x and y
must be discrete for 'x+y' to be significant. That is, he
uses strong disjunction. He cites the following reason for
this restriction: “in strictness, the words 'and,' aor,'
interposed between the terms descriptive of two or more
Classes or objects, ”imply that those classes are quite
distinct, so that no member of one it found in another. In
this and in all other respects the words 'and,' 'or,' are
analoguous with the + in algebra, and their laws identical."13
This feature of Boole's algebra, which most commentators
view as a defect in his system, was eliminated from later
algebras of logic, largely through the intercession of
W. S. Jevons, whose analysis of disjunction closely approxi-
mates present logical usage.

The sign of subtraction is used by Boole to indicate
exclusion. His example of its use is as follows: I"All
states except those which are monarchial' will be expressed

by x-xy,'15 where x stands for 'all states,' and y for

 

13Boole, George, An Investi ation of the Laws of
nought (London: Walton, 1855), p. 32.

1I‘Jevons, W. Stanley, The Princi les of Science (London:
Macmillan and Company, 1879),— pp. 33 ff.

15Boo1e, op. cit., p. at.

12

'monarchial states.' Curiously enough, Bochenski maintains
that Boole's use of subtraction is a ”procedure which admits
of no logical interpretation or of only a complicated and

”16 This, however, is simply not

scarcely interesting one.
the case. Not only is it immediately apparent that subtrac-
tion is allowable in that it serves as a symbol for exclusion,
but more than that, it is impossible to tell what similarity
Bochenski sees as holding between Boole's use of the symbol
for subtraction, and his use of, say, division, or of numbers
greater than 1, which are Operations that are genuinely
uninterpretable in the system.

It is difficult to determine accurately just how Boole
thought that the sign of identity differed from the signs of
the Operations just discussed - but it is clear that he did.
The signs mentioned above, he calls "signs of those mental
operations whereby we collect parts into a whole."17
Speaking of identity, Boole says that it is a stgn "by which
relation is expressed, and by which we form propositions.”
Certainly one difference which must have occurred to Boole

was that identity is the only indispensable element in all

of the propositions of the algebra.

 

16Bochenski, I. M., A History 2f Formal Lo ic, trans.
ed. Thomag,Ivo (South Bend: Notre Dame University Press,

17Boole, Op. cit., p. 32.

13

It is also worth noting how Boole justifies what
amounts to the pure extensionality of his system, through
the interpretation of the identity sign. He remarks that
every verb can be resolved into 'is' or 'are' for the
purposes at hand, and that such an analysis does justice to

18

what is actually expressed in any proposition. Thus he

allows a shift from ”Caesar conquered the Gauls' to "Caesar
is he who conquered the Gauls,” that is, in more recent terms,19
he permits the shift from an attributive term to its first
denotative correlate, the extfimsional contents being the
same. Thus it can be seen how strictly Boole's algebra
adheres to an extensional interpretation. Lewis regards
this restrictive interpretation as an advantage, in that it
allowed Boole to avoid the muddles in which Leibniz found
himself due to the introduction of intensional notions into
the attempt to apply mathematical symbolism to logic.20
Besides the constituents of Boole's algebra of logic
mentioned thus far, there is one other distinct sort, which
is recognized as a basic feature neither by Boole, nor by

any of his successors or commentators. The constituents in

question are the processes of development and elimination,

 

18Ibid., p. 3&.

 

19Such as the convention concerning these terms employed
in Henry S. Leonard's Principles 23 Right Reason (New York;
Holt, Rinehart and Winston, 1957).

20
Lewis, op. cit., p. 12.

l&

which in Boole's calculus serve as devices for transfor-
mation, but are not recognized as such. These Operate in
his system in exactly the same way they operate in standard
algebra; elimination allowing for the systematic reduction
of components from each side of an equation, develOpment
allowing for the exhaustive assignment of values to the

21 Inference in Boole's system

components of equations.
depends directly on the availability of these two Operations.
Furthermore, Boole never discusses their nature with respect
to what could in any sense be regarded as a theoretical
description. That is, he uses the Operations as standard
mathematical procedures while refraining from speculating on
any logico-philosophical significance which they might have.
These, then, are the major constituents in Boole's
logic. Our description is by no means complete, and is not

intended to be. We have merely discussed those features

which are either of crucial import to the following

 

21Development can be described in general as follows:
"any function f(x) is expressable as a sum ax+b!, wherein the
coefficients a and b are respectively f(l) and f(O); likewise
any function f(x,y) is expressable as a sum:

(1) axy+b§y+cxy+d§y

wherein the coefficients are respectively f(l,l), f(0,l),
f(l,0), and f(0,0); and correspondingly for higher functions
....” Quine, Willard Van Orman, ”Whitehead and the Rise of
Modern Logic,” Thg_Philgggphy 2; Alfred Ngzth‘flnitghggd,

ed. P. A. Schilpp (New York: Tudor Publishing Co., l9&l),
Pp. 130-131.

15

discussion of theory, or are such that the subsequent discus-

sion could not be understood without a knowledge of them.

3. BOOLE'S THEORY OF LOGIC.

Boole was first a mathematician, second a logician,
and third, (a distant third) a philosopher. He only rarely
took up questions concerning topics such as the interpre-
tation and subject-matter of the calculus. His main goal
was apparently the completion of the calculus while paying
only minimal attention to theory. Nonetheless, through
the accumulation of bits and pieces of pronouncements made
by Boole on various topics, a coherent view of his theory of
logic can be pieced together. What follows is a discussion
of Boole's theory of logic as it has been gathered from
various places throughout the entirety of his work.22

Boole's theory of logic is composed of three over-
lapping parts; the first is Boole's concern with what he took
to be the basic subject-matteriof logic - the laws of thought.
As we shall see, he considered these laws to be the basic
subject-matter of logic, in that logic is both governed by
them, while at the same time describing, or mirroring them.

The second part of the theory deals with what might be called

 

22Attention has been paid to chronological sequence
only when necessary, in that more attention has been paid to
reconstructing his theory than to describing his intellectual
development.

16

the secondary subject-matter of logic. According to Boole,
not only is logic about the laws of thought, but derivatively
it also deals with what Boole calls elements, which are the
sort of things pOpulating the classes described by the
elective symbols. The third phase deals with the scape of
logic -- Boole's pronouncements On the place of the logic of
class, and other logical enterprises in the general field of
logic.

Boole believed logic to be the science of the laws of
thought.23 While this fact is widely known, few realize what
is actually involved in this position. It is, in fact, much
more complex than one might suspect.

_To get to the heart of Boole's conception of the
nature and function of the laws of thought, and how they
serve as the subject-matter of logic, let us consider his
views on the relation of logic to language, of language to
the mind, of language and the mind to extra-mental entities,
and finally, his ideas on the nature of the mind.

When Boole said that logic is the science of the laws
Of thought, he did not mean that logic is itself basically a
mental science; i.e., it does not deal directly with these
laws. His logic was not a system for manipulating laws of

thought, but rather for manipulating language, which $2 turn

 

ZBCf. 30019, George, "Logic and Reasoning" (ca. 1857).

Mes. is. legals 225.1. Probabilit . R. Rhee: (ed.)(London:
Watts and Company, 1952 , p. 212.

17

bears a peculiarly reflexive relation to these laws; language

rgpresents the laws of thought, and is at the same time

 

governed by them.2u Language, then, serves as a medium for
logic, a medium capable of mirroring formal properties and
relations.

Boole's recognition of the fact that language plays
a distinct and singularly important role in logic is itself
a step forward. While he was not the only logician of his
day to do this, earlier would-be logicians (with the obvious
exception of Leibniz) suffered from the inability to
recognize the vital relation between language and lOgic,
and it was not until the time of De Morgan, Mill and Boole
that logic gained a firm footing - it was provided with a
medium which could be shared by all logicians.

Language, then, came into its own as a proper study
for logicians in the mid-nineteenth century. While some,
like De Morgan, were of the opinion that logic was solely
about language,25 Boole, while able to separate language and
its functions from the mind and its functions, was neverthe—
less determined to try to reach beyond the medium of language.
This led him to think that by observing features of language,

he was in fact laying bare the fundamental operations of the

 

2&
Cf. Boole, op. cit., "The Calculus of Logic," p. 1&0.

25Thus, "De Morgan revived Hobbes' heresy: logic is
about names.” Passmore, John, A Hundred Years 2: Philosophy
(London: Gerald Duckworth and Company, 1957), p. 12&.

18

mind. As we shall see, perhaps Boole was reaching too far,
for his explanations along these mentalistic lines frequently
slip into obscurity. But obscure or not, many of these same
notions persisted for more than a half-century.

Boole expressed his attitude toward the relation
between logic and language as follows: "That which renders
logic possible is the existence in our minds of general
notions, - our ability to conceive of a class and to designate
its individual members by a common name. The theory of
logic is thus intimately connected with that of language."26
Though his ideas on the nature of language are few and
scattered, there arise a few persistent themes which are in
no manner unusual, even in the present day. Language, Boole
thought, is a system of expressions,27 in that language is
composed of propositions (i.e., statements), each proposition
expressing relations between what he called elements,28
which in turn are either facts, or things. Besides these
meager pronouncements, Boole at another place speaks briefly

of the uses of language, maintaining that it can be used

for communication, or as an organon for reasoning; but

 

26Boole, op. cit., The Mathematical Analysis 2;
Lo ic, p. &.

27

28
Cf. Boole, George, “Sketch of a Theory and Method of

Probabilities, Founded Upon the Calculus of Logic,” (ca. 1850),
§329333,33Hg2513,522 Probability, R. Rhees (ed.)(London:
Watts and Company, 1952), p. 1&5.

Bf. Boole, Op. cit., "The Calculus of Logic," p. 1&0.

19

always, the important feature of language, whatever else it
may be, is that it mirrors the mind:

”In the processes of reasoning, signs stand

in the place and fulfill the office of the

conceptions and operations of the mind; but

as those conceptions and operations represent

things, so signs represent things with their

connections and relations; and lastly, that

as signs stand in the place of the conceptions

and Operations of the mind, they are subject 29

to the laws of those conceptions and operations.”
In this way language serves as an organon; a calculus of
thought.

Boole shows almost no interest in extending his
ideas on language beyond the points already mentioned. He
goes just far enough to show how language is related overall
to reasoning, and how it can be thus employed, but never
goes far enough to display interest in the peculiarly
interesting problems concerning language to which logicians
were only later to address themselves. Lewis in fact
remarks that perhaps it was for the best that Boole gave
such problems as intension and extension, etc. a vaction
long enough 'to get the subject 6090, logig started in terms
30

of a simple and general procedure.”

 

29Boole, op. cit., Ag Investigation 2f the Laws 2f
Thou ht, p. 26.

30Lewis, op. cit., p. 51.

20

While sparse, Boole's treatment of language is also
very tidy. A recurring notion in Boéle's theory is that the
relations and operations of the mind, of language, and of
facts and things are formally the same, differing only in
level. Facts and things, and their relations are supposedly
mirrored by signs and their relations, which also mirror the
mind and its relations, which in turn mirrors, through
conceptions, things and their relations. The difference
between these levels, besides simply being different kinds
of things, has to do with the question as to which of them
might be said to be primary with respect to the others.
Formally, language, the mind, and facts and things are all
the same, i.e., their formal properties are indistinguishable.
Thus, says Boole, “Whether we regard signs as the repre-
sentatives of things and of their relations, or as repre-
sentations of the conceptions and.operations of the human
intellect, in studying signs, we are in effect studying the
manifested laws of reasoning.'31 To the same point, he also
says that ”operators are signs of those mental Operations
whereby we collect parts into a whole, or separate a whole
into its parts."32 The real difference, with regard to logic,

between language and the mind seems to be that the formal

 

31Boole, op. cit., Ag Investigation 23 the Laws 2;
Thou ht, p. 2&.

321bid., p. 32.

21

properties of the mind form the ground, i.e., the raison
d'etre for the formal properties of language. But just
because thought serves as the spring from which language
flows, the hard and fast formal correspondence between the
two is not diminished in the least. Boole recognizes this
correspondence, saying at one point, ”though in investi-
gating the laws of signs, g_posteriori, the immediate
subject of examination is language, with the rules which
govern its use, while in making the internal processes of
thought the direct object of inquiry, we appeal in a more
immediate way to our personal consciousness, - it will be
found that in both cases the results obtained are formally
equivalent."33

In sum, Boole never really attains perfect clarity in
identifying the roles played by the mind, language, and
other sorts of things with reSpect to questions concerning
the representation, etc., of forms. But it would be quite
unfair to be critical of Boole for not attaining such clarity.
Boole, after all, was primarily an innovator and an experi-
menter in that he was continually testing alternative
interpretations, and was for that reason often immersed in
difficulty. He was unwilling to remain within the bounds
of the neutrality of language in his search for the subject-

matter of logic. He was convinced that there was more than

 

Balbide , p. 25.

22

a mere linguistic significance to logic, and yet he found
himself in difficulty whenever he tried to go beneath the
surface, and unravel the various relations mentioned above.

We shall now consider another feature of the theory,
which has been already briefly touched on: Boole's con—
ception of the nature and function of the mind. Here again
his Opinions vary a good deal, and are often halting and
obscure. Nonetheless, they form the basis for any under-
standing of the laws themselves.

In a paper written in 1851, entitled ”The Claims of
Science," Boole presents a comprehensive account of his
ideas on the nature of the mind and of thought, and their
relation to logic. He begins by advocating what amounts to
traditional empiricism by maintaining that all scientific
truths are grounded in the observation of facts, although
ultimately, observation is not the only element, saying
that

“from the contemplation of facts, the mind

rises to the perception of their relations.

While in the former state it is little more

than a passive recipient of the impressions

of the external world, in the latter it

exercises an unborrowed activity. The

faculties of judgment, of abstraction, of

comparison, of reason, are an agency of _

strength and power from within, which it

brings to bear on the lifeless elements
before it, shaping them into order, and

23

extracting from them their hidden meaning and

significance.'39
In this and other similar passages, we see Boole attempting
to hold a position which reflects both the tradition of
British empiricism, and also that position, concerning the
relation of the given element in experience to the mind,
which had its origin in the works of Kant, and would appear
later in the works of C. I. Lewis. Indeed, there is a good
deal of similarity between Boole and Lewis along these lines,
although the difficulty which plagues any such comparison
is that the views held by Boole are never fully enough
developed.

In the passage just quoted, careful attention shows
Boole to be lodged in difficulty already; he wants, at one
and the same time, to hold that the functions of the mind
are the sole contributors of mind to experience, and that
it is the laws which govern such contributions we study
whenever we study logic. Notice, however, the last line of
the quotation - the mind extracts from the given in
experience "their hidden meaning and significance.“ It
very much seems that Boole wanted to hold tenaciously to the
idea that it is really the elements of experience (i.e., the

given) which make the major contribution to experience. The

 

3“Boole, George, ”The Claims of Science,” (1851),
Studies $2.Logic and Probabilit , R. Rhees (ed.)(London:
Watts and Company, 1952), p. 189.

2&

question to which this gives rise is simply, where do these
laws of thought come from? - are they, like everything else
in the mind, a product of the accomodation which occurs
between the mind and the elements of experience, or are they
somehow special? It is clear that he wished to hold that
in the mind are found the formal principles which logic
studies. It is not clear, however, how or from where they
arise. To add to the uncertainty, Boole continues, saying,
“there is ... a correspondence between the powers of the
human understanding and the outward scenes and circumstances
which press upon its regard. In this agreement alone is
science made possible to us. The naive powers of the mind,
cast abroad amid a world of mere chance and disorder, could
never have raised the conception of law."35 Hence Boole is
offering a sort of accomodation or agreement theory, maintain-
ing that the mind is not purely active, impressing form on
totally chaotic data, nor on the other hand are the data
completely formed, so that the mind has nothing to do but
accept them in some manner or other. But where, between
these extremes, the true origin of form lies, we are not to
discover.

Before becoming thoroughly entangled in the diffi-
culties just noted, Boole attempts to extricate himself

by maintaining that the laws of thought are really different

 

351bid., p. 190.

25

from all other sorts of generalizations, and hence that
there is no point in trying to decide the balance obtaining
between the mind and the given element in experience. He
now maintains that in the other forms of science, the mind
bonds experiences together by borrowing something from each.
Then, however, he says that "there are also certain other
principles which are of a more special character, yet,
equally with the fonmer[£.e., principles of the other
scienceé], have their seat in the mind. In these principles
together are involved the laws of our intellectual nature,
even as in the final generalizations of physical science
we discern the laws of the material universe."'36 The laws
of thought which logic studies seem to be the laws governing
the laws constructed out of the elements of experience.
Boole never states the situation quite this directly, but his
intent is clear; 22! we construct generalizations from
experience is determined by the true laws of thought - these
generalizations from experience may indeed be laws, and
might be called laws £222 thought.

The special nature of the laws of thought consists in
the fact that they are underived, which is evident from what
was said above, and that they are necessary truths, i.e.,

they are 2_priori and analytic, in contradistinction to what

 

36Boole, op. cit., “The Claims of Science," p. l9&.

26

Boole later calls natural laws, which are derived from
experience and are presumably synthetic.37
After arriving at this point, Boole abruptly steps.
After arguing for the separation of laws into natural laws
and laws of thought, and after barely touching on the basis
of the difference between the two sorts of laws, he proceeds
no further in an attempt to explain the nature and function
of the mind. Instead, he adopts the following position: it
is possible for the mind to "attain a knowledge of the laws
to which itself is subject, without palpg 5212 pp understand
‘Egpi£_ground pp origin."38 What is the nature of these
laws? What sort of a thing must the mind be to be subject to
them? Boole is saying, in effect, that it doesn't matter;
we can know these various laws and their interconnections
without having to commit ourselves to any theory concerning
the nature of the mind, or to any theory concerning the
status of laws. Boole gives forceful voice to this position
early in the £313 pf Thought.39 First he claims that there
are various ways of expressing the laws of reasoning,
according to the metaphysical theory held: the idealist

would speak in one fashion, the skeptic in another, the

faculty psychologist in yet another. Then he says, “Now

 

37On this point, of. Boole, Ag Investigation p£ the
Laws 2£_Thou ht, p. &20.

381bid., p. 11. Italics mine.

39cr. ibid., p. 39 ff.

27

the principle which I would here assert, as affording us

the only ground of confidence and stability amid so much of
seeming and of real diversity, is the following, viz., that
if the laws in question are really deduced from observation,
they have a real existence as laws of the human mind,
independently of any metaphysical theory which may seem to

&0

be involved in the mode of their statement.” He goes on
to say that the laws of thought contain truths which are not
affected by any metaphysical argument, and that we can
discover and use them without any concern for metaphysical
theory.

Boole never again abandons this position of neutrality.
In the writings subsequent to the £2!p_p£ Thought, he never
again comes to grips with such questions of theory. He
speaks of the laws of reasoning with perfect ease, but never-
alludes to any of the philosophical issues which surround
them.

Since in subsequent chapters we shall meet this
neutrality with respect to theory again and again (with
some variation in form), it is advisable to restate Boole's
position in somewhat different terms than he permitted
himself: he maintained that we have direct, empirical

evidence of the existence of the laws of thought. Such

evidence is obtained by taking direct notice of the results,

 

“Orbid., p. to.

28

cast in language, of various acts of reason. We know that
these laws are laws of thought because our evidence of them
comes from language, which is essentially an organon of
reason. He is saying, in effect, ”whatever the mind is, and
whatever the laws are, the laws we are considering govern
the mind.”

By holding such a view, Boole avoided enmeshing
himself in philosOphic controversy. For him, it was enough
(after a good deal of experimentation), to locate the laws
of thought and briefly to describe their hold on language.
He saw no reason for taking sides on the question as to how
the mind functions, but he came to this position only after
travelling the course described above.

There are other sides to Boole's theory - the other
two phases mentioned at the beginning of this section.

While they do not lead around the impasse dealt with above,
they are both quite enlightening, and provide a good deal of
depth to his theory. First we shall discuss what was earlier
called the secondary, or derivative, subject-matter of

logic, which has to do with how Boole conceived of the
constituents of classes which serve as the values of elective
symbols.

As has already been pointed out, most surviving
versions of Boole's algebra are couched in terms of classes
of things. Boole, in fact, is regarded by some as being an
extreme nominalist. While it is true that he was concerned

with classes, the population of these classes presented him

29

with a difficulty which he never completely overcame; this
difficulty has to do with the question, are the constituents
of these classes things, or facts?; do the elective symbols
replace terms, or prOpositions? Even in his earliest work
in logic (2gp Mathematical Analysis BEHEEELE) Boole
attempted to hold a mid-way position, by saying that "every
proposition which language can express may be represented
by elective symbols, and the laws of the combination of
these symbols are in all cases the same; but in one class
of instances the symbols have reference to collections of
objects, in the other, to the truths of constituent propo-
sitions.'u1 He goes on to remark that he disagrees with
those like Latham, who maintain that only prOpositions, not
words, can be connected in logic.

It is at first unclear whether Boole means that
electives can have propositions or terms as substituends,
or whether terms are substituends, and propositions are,
somehow, values. This difficulty in interpretation arises
as a result of his apparent ignorance of the difficulties
surrounding the ambiguity in the interpretation of the
operators of his system. But while he does not come to
terms with this particular difficulty (and perhaps it is
best that he did not), yet this area remains one of the

most original and significant in his theory of logic. He

 

ulBoole, op. cit., The Mathematical Analysis pg
L0 1c, p. 590

30

recognized, that is, not only that '+' can be interpreted
as a class-operator, standing between terms, but also that
its interpretation in common usage is "or”, which can be
used as a statement connective. Boole was thus becoming
aware of the true breadth of logic; and while he never .
succeeded in actually working out the complexities involved
in the symbolism itself, he was at least aware of the
possibilities of logic. Only later, with the efforts of
Frege, Schrbder, and Lewis, did the propositional interpre-
tation take complete symbolic form. But in the transition,
logic, as we shall see, moves permanently away from the
purely mathematical form.

In his attempt to provide for a dual-interpretation
for his calculus, Boole arrives at some interesting results.
Some of these are indeed awkward, but others are genuinely
fruitful. Consider, for instance, his claim that ”every
proposition expresses a relation among elements. Those
elements may either be thin s, in which case the prOposition
belongs to that class which logicians call categorical; or
they may be gpgpg'pp events represented by elementary
propositions, in which case the connecting proposition
belongs to that division which logicians term conditional

or hypothetical."l"2 This constitutes Boole's attempt to

 

uzBoole, op. cit., ”Sketch of a Theory and Method
of Probabilities, Founded Upon the Calculus of Logic,"
p. 1&5.

31

separate term logic from propositional logic. It is of some
interest that he made no attempt to provide a symbolism
adequate to the form of the conditional, while the
categorical is readily symbolized as 'x=vy'. He does, however,
make an attempt to symbolize conditionals within the calculus,
maintaining that “the conditional proposition 'If it rains,
it hails' is expressed by the equation x=vy in which v is

the representative of time partially indefinite.'u3 Boole is
thus greatly expanding the interpretation of 'v'. In the
case of the categorical, 'vy' means “some y” so that 'x=vy'
means, ”all x is identical with a part of the class of y's.”
This use of 'v' is difficult enough to follow. As was

noted earlier, its status as a quantifier is suspect because
Boole forces it into the role of an elective. In the case

of the conditional (and hence the propositional interpre-
tation), 'v' is being used as a device to limit the span of
time through which 'y', in 'vy', is to be considered. Thus,
roughly, 'vy'means ”sometimes y'. Hence, in the case of the
conditional example noted above, 'x=vy' is to be interpreted
as I'raining is identical with a time-span of hailing." Boole
is thus trying to carry over his analysis to cover prOpo-
sitions. Perhaps the only defensible interpretation of
'x=vy' would be one in which 'x' was allowed to range over

time spans also. Thus such an expression might be interpreted

 

“3lbid., p. 1&8.

32

as 'times when it rains = some times when it hails.’ Boole,
however, suggests no such interpretation. It is not surprising
that Boole is remembered for the class interpretation of his
calculus, and not for the propositional interpretation.
Nonetheless, to picture Boole as having been concerned only
with classes of objects is not enough, as should now be clear.
As we mentioned earlier, Boole was an experimenter, and
deserves to be remembered for more than the most finished
aspects of his work.

In committing himself to the dual interpretation of
the calculus, Boole arrives at other strange conclusions
regarding time. For instance, whereas in the term (i.e.,
categorical) interpretation of the value of 'l' is the
universe, in the hypothetical interpretation, the value of
the same symbol is eternity.

In a more modern vein, a difficulty presents itself
concerning all of Boole's attempts to use various symbols
to range over time-spans: in order to make such an
interpretation work, it now seems that such "quantifiers"
should be used to extend over gpgpp-time spans, or durations,
rather than over time spans alone. Using '1' to stand for
eternity would then be acceptable, provided that by eternity
is understood all space-time spans.

Yet another peculiarity is Boole's use of 'y=l' to
mean, for instance, ”it rains.” Here, '1' is being used
to mean the True. Presumably '1' meaning eternity, and '1'

meaning the True arewnot entirely separable interpretations,

33

for Boole makes no attempt to distinguish between them.
The interesting features of such an equation as 'y=l' are
two: First, Boole apparently regards 'y=l' as being
significant because of his collapsing of all uses of 'to be'
into '=' in the calculus. Then assuming that he regards
'it is raining is true,' and 'it is raining' as being the
same statement, we can read 'y=l' as 'it is raining is true.'
The other interesting feature of the equation is that it
shows the peculiarity of the algebra with which Boole was
working. As was mentioned at the beginning of the chapter,
it was a two-valued algebra. Hence, either x=l, or x=0,
i.e., x is true, or x is false, which amounts to an algebraic
interpretation of the law of the excluded middle.

By the time Boole wrote his L213 pf Thou ht, the
dual interpretation of the algebra was fairly well worked
out. He no longer expresses the time interpretation of
symbols when the subject concerns propositions. In fact, he
drops any attempt to work out the rough areas with regard
to symbolization. He says merely that "Logic is conversant
with two kinds of relation, relations among things, and
relations among facts. But as facts are expressed by
propositions, the latter species of relations may, at least
for the purposes of logic, be resolved into a relation among

propositions.'uu He apparently still felt that the dual

 

huBoole, op. cit., A2 Investigation pf the Laws pf
Thou ht, p. 7.

3&

interpretation was tenable, but had given up trying to work
out the specifics.

The third phase of Boole's logic, which is of minor
importance in relation to the other two, is Boole's final
assessment of the scape of logic. This assessment of logic
is not as significant for how it is reflected in his logic,
as it is in the way in which it presupposes events which
were yet to take place in logic. In a way, this phase
constitutes Boole's last attempt to work out many of the
difficulties which have already been considered. Also, this
phase cannot be completely separated from the other two,
for it is a consequence of both.

Toward the end of his life, Boole became more and
more aware of the nature of the symbols used in the calculus.
In a paper written after 1855, Boole first maintains that
there is a twofold sense of the term 'logic' by saying that
”...the word logic in its primal sense means the science of
the laws of thought as expressedfiih the calculué]. But there
is a secondary and narrow sense of the term logic, according
to which it is the science of the laws of thought as expressed
by the terms of ordinary language.'u5 In saying this, he
recognizes that perhaps some of the functions of language
are not adequately expressible in the calculus. But what is

more, Boole sought to provide a place withinx the scheme of

 

hsBoole, Op. cit., “LOgic and Reasoning,“ p. 212.

35

logic for mathematics itself, from which the apparatus
for the calculus was borrowed.

Instead of leaving the term 'logic' in the uncertain
position of having more than one interpretation, Boole
instead proposed that a completely new sense of that term
be recognized. This new sense Boole speaks of as 'higher
logic.‘ By way of explanation, he says that "the general
principles of 10gic considered as the science of all thought

“6 can only be apprehended through

that admits of expression,
the particular manifestations of them which are founded in
the logic of class, the logic of number, and other sciences
into which it may be developed."l"7 Thus Boole sought to
bring all formal sciences under the heading of logic. He
recognized the similarities which exist between these
disciplines, and apparently thought that since logic is the
science of the laws of thought, that it should take into
account all disciplines displaying similarity to the logic
of class, i.e., all of the formal sciences.

Whether he was directly motivated to adopt such a
position by any recognition that logic would have to be

broadened in order to account for both terms and propo-

sitions is not clear, for beyond such statements as the one

 

6
Which amounts to a definition of 'higher logic.’

“7Ibid., p. 226.

36

just quoted, he says little else. In an untitled paper he

reiterates his position, saying that,

”In its highest conception...logic might be
said to be the philosophy of all thought which
is expressible by signs, whatever the origin

of that thought, whatever the nature of those
signs may be. Nor is this conception either
vague or unreal. There is a philosophy of signs
which governs and explains all their particular
uses and applications...The perfect idea of
logic is not that of a mere system of rules,
but a philosOphy from which, as from a common
stem, all sciences whose method is deductive
are developed, and with which they all stand

in vital connection."l'8

Such pronouncements as these sound full of promise, and

one wonders what would have become of them had Boole had

the opportunity to work them out in detail. Unfortunately,
before he had such an opportunity, he died. It is difficult
to say what he might have done, and we shall not attempt to
do so. It will be sufficient to note that within Boole's
own thoughts on logic are contained many of the seeds of

later developments, developments which have brought logic

to its present state.

 

uaRoyal Society Manuscripts, No. C. 57, quoted in
Boole, George, Studies lp_Logic and Probabiiigy, R. Rhees
(ed.)(London: Watts and Company, 1952), p. l&.

37

&. CONCLUSION.

We have seen the contributions which Boole made,
and also the shortcomings to which his theory was liable.
It should at once be apparent that his work was truly
original and that he justly deserves a place as one of
the major figures in the development of modern logic.

It will become more and more apparent as we proceed
through the remaining chapters that Boole also deserves
recognition as one who shaped our thought £222: logic, for
much of what was discussed in this chapter served as

kindling for later efforts in the theory of logic.

CHAPTER III
GOTTLOB FREGE
1. INTRODUCTION.

While Boole got mathematical logic started on its
way by generalizing on the range of applicability of
certain notions which had been previously considered to be
purely mathematical, Gottlob Frege (18&8-1925) is the man
most responsible for the present shape of the subject. He
accomplished this incredible feat by making a small number
of crucial innovations in the structure of logic, and by
making demands for rigor in the performance of logical
operations. He demanded rigor, for example, in proof-
procedure.1 Such rigor had been lacking in the past. Also,
he demanded that the components of calculi be explicitly
stated. This led him further to suggest that what we now
call the primitive basis of any system should be capable of
being easily stated, i.e., it should be as sparse as possible
without injuring the system itself. Had Frege simply stopped
at this point, his contribution to logic would have been
great enough. But besides such general innovations, he also

introduced a profound complexity into logic by introducing

1Frege, Gottlob, The Foundations Lf Arithmetic (a
translation of Die Grundlagen der Arithmetik, Breslau, 188&)
J. L. Austin (trans. )(Oxford: Basil Blackwell, 1950), g l.

38

39

a sizeable number of new notions -- notions sometimes borrc

from mathematics, like that of a function, and at other

times freshly created, like the concept of quantification.

Another of the major distinguishing features of Frege
thought is his espousal of what has come to be known as the

logicist thesis. He thought, that is, that arithmetic cou]

ultimately be reduced to logic, via the concept of number.

He constructed his logic for the main purpose of providing

such a basis for mathematics. How well he (and later

Russell and Whitehead) succeeded in establishing the logici

thesis is still being debated, usually around the question

as to whether or not set theory, which is needed by the

logicists in the performance of their reduction, is to be

cons idered a part of logic.

The casual student of logic is often only vaguely

aware of the scope of Frege's contributions.

Indeed, he
is

in general best remembered today for the unusual notatic
which he introduced, and for the distinction which he

introduced between sense and reference. His full theory of

logic, however, reaches far beYond these concernS, and they
can be understood only in terms of the full theory.

Our main interest will be with Frege's theory of

1
031C -- his position on philosophical problems which are

In
031; germane to logic itself. Here again, Frege proves to

b
e the master of the subject. Frege's theory of logic is

a.
a flull and complex as Boole's is sparse. Where Boole was

r
e1“ ctant to assert, or to commit himself to more than he

140

absolutely had to, Frege goes out of his way in explaining
the philosophical groundwork of logic. Indeed, Frege was
a. truly philosOphical logician -- concerned not only with
the science itself, but also with such problems as the
accurate description of the subject-matter of logic. Here
again, he comes into sharp contrast with Boole. Boole's
theory of logic has to be coaxed out of his writings and
put together piece by piece. In dealing with Frege, we are
so completely and continually immersed in theory that in
Pla. cos it is quite difficult to separate the theory of the
Sci once from the science itself. While Boole clung tenacio
to neutrality on issues where decisions were not absolutely
nec essary, Frege is far removed from neutrality. This fact
m‘kes Frege's theory more. interesting, but at the same time
leave, it more open to criticism.
We shall discuss first a few relevant aspects of the
1031 co-philosophical enviornment in which Frege Operated, f
understanding this environment is essential for a proper
understanding of his own theory. Without paying attention
to context, Frege's position appears more radical than it

ac t:‘-la.lly was.

2
‘ FREGE 's INTELLECTUAL ENVIRONMENT.

From the time of his first published work, Frege was
a
"Pe of several major positions held by individuals workin
b
0th- in logic, and in mathematics. Much of Frege's own

13081-"::ion was formulated as a reaction to this intellectual

bl

enviornment. 0n the one hand, there were the formalists,
such as J. Thomae, and Heine. There were also the ”main-
stream", or popular logicians of the day. These individuals
were mostly of a thickly psychologico-idealistic bent,

going to great lengths to base logic on what now appear to
be impenetrably obscure mentalistic concepts. Also, there
was the empiricist position; here Frege chose to come to
grips with the work of one of his recent predecessors,

J. S. Hill. Each of these positions had a great influence

2 and although they were more centered in mathe-

on Frege,
matics than in logic, Frege's theory of logic is a direct
reaction to them. We shall consider a brief description of
these positions, together with Frege‘s reactions to them.
Let us first consider the position held by the
psychologist-logicians. B. Erdman; for instance, maintained
that ”...psychology teaches with certainty that the objects
of memory and imagination, as well as those of morbid
hallucinatory and delusive ideation, are ideal in their

nature.... Ideal as well is the whole realm of mathematical

ideas properly so called, from the number series down to the

 

2It is not known, however, how familiar Frege may
have been with the algebra of logic. Venn alludes to this,
when, in his Symbolic Lagic (1st ed., 1881) he remarks,
concerning Frege's system, that hetiregélconstructed his
logic ”in entire ignorance that anything of the kind had
been done before.” (§!mb01i¢ L0 10. po #15.)

“2

objects of mechanics."3 Also, ”objects are things ideated,”u
and ”according to its origin, the ideated divides on the one
hand into objects of sense-perception and of self-consciousness,
and ontthe other hand into original and derived."5 Taking
such statements into account, it becomes apparent that the
'psychologists' with which Frege was concerned were in fact
idealists, believing that that which is real is somehow ideal.
When these individuals speak of logic as being concerned with
the “laws of thought,“ they intend by that expression the
laws of all that is, in an underived sense. But such a
position is quite unlike Boole's conception of the laws of
thought. Boole thought that the mind was separate from the
external world. He spoke of logic as being concerned with
the laws of thought in the sense that derivatively, logic
is £132 concerned.with extra-mental entities. But Erdmann
maintained that all the objects there are are mental entities,
so that the laws of thought are immediately concerned with
everything, since everything occurs within thought.

Frege's general objection to any position which
maintains that logic is basically psychological is that the
laws of logic are too immutable to be grounded in something

which appeared to him to be as subject to variation as is

 

3Erdmantenno, Logik (Halle, A.S.: Max Niemeyer,
1892), I, p. 85.

uIbid., p. 81.

51b1d., p. 38.

#3

thought. He says, for instance, that ”I understand by 'laws
of logic' not psychological laws of takings-to-be-true, but
laws of truth....if being true....is independent of being
acknowledged by somebody or other, then the laws of truth
are not psychological laws; they are boundary stones set in
an eternal foundation, which our thought can overflow, but

6

can never displace.” Later, he says, ”one could scarcely
falsify the sense of the word 'true' more mischievously than

by including in it a reference to the subjects who judge."7

This aspect of Frege's attack on the idealist

logicians, while forthright, is carried on somewhat uncritical-
ly. What he seems to say is that idealism just cannot be

true —- that there 2233 be a world of extra-mental entities,
and that the idealist blot out such distinctions as subject

8 His arguments are, however, somewhat incon-

and predicate.
elusive. He gives no support for his contention that that
which is mental is too variable to provide a ground for logic.
Perhaps the reason why Frege was reluctant to provide

elaborate arguments was that he thought the position

represented by Erdman1to be radical enough that all that

 

6Frege, Gottlob, The Basic Laws of Arithmetic (a
translation of parts of Grundgesetze der —Arithmetix7,
Montgomery Furth (trans. ed. )TBerkeley: University of
California Press, 196R), p. 13.

7Ib1d. , p. 1“.

8On this point, cf. ibid., pp. 17 ff.

an

was needed was to bring it to light, and allow it to fall to
pieces of its own accord. For whatever reason, Frege simply
asserts and reasserts that idealism as such cannot provide
an explanatory basis for mathematics and logic.

At one point, however, Frege presents a more telling
argument against all those who maintain that logic deals
with the laws of thought. He says, for instance, that "I
take it as a sure sign of a mistake if logic has need of
metaphysics and psychology -- sciences that require their
own logical first principles.'9 That is, he took all
mentalistic notions as presupposing logic; hence they could
not be based on any such laws of thought. Logic simply
could not, according to Frege, be carried on in mentalistic
terms, for they were derivative.

Is there any sense in talking about the laws of
thought? According to Frege, there is, but only a derivative
sense. He says, ”...what is fatal is the double meaning of
the word 'law'. In one sense a law asserts what is; in
another it prescribes what ought to be. Only in the latter
sense can the laws of logic be called 'laws of thought': so
far as they stipulate the way in which one ought to think."10

Thus he accords the laws of thought as laws of logic the place

 

91bid., p. 18.

1°Ib1d., p. 12.

hS

of paradigms, or normative rules, a position quite different
from that accorded to them by either Erdmannor by Boole.

The formalism with which Frege was concerned was not
the same as the later formalism propounded by Hilbert, but
was of the more radical type propounded by Heine and Thomae.
The central tenet of this type of formalism, according to
Frege, was that according to the formalist, ”Arithmetic is
concerned only with the rules governing the manipulation of
arithmetical signs, not, however, with the reference of the
signs.'11 Signs, according to the formalists, were devoid
of content -- all of their significance was supposedly
derived from the roles they played as they were governed by
various rules. Frege, in fact, includes a definitive
quotation from Thomae in which Thomae says:

The formal conception of arithmetic accepts

more modest limitations than does the logical

conception. It does not ask what numbers are

and what they do, but rather what is demanded

of them in arithmetic. For the formalist,

arithmetic is a game with signs, which are

called empty. That means they have no other

content (in the calculating game) than they
are assigned by their behavior with respect

 

11Frege, Gottlob, Grundgestze der Arithmetic (Jena,
1893-1903)(Hildesheim: Georg Olms Verlagsbuchhandlung, 1962),
II, a 880

h6

to certain rules of combination (rules of the
game). (From Elementare Theorie Egr’éggly-
tischen Functionen £$22£ Complexen Verander-
lichen.)12
This constitutes one of the clearer statements of the
formalist position. To be sure, there were minor differences
amongst formalists; Heine, for example, took the less radical
position of maintaining that numbers are merely signs (i.e.,
numerals), while Thomae takes the more extreme view
maintaining that the question as to what numbers 353 has no
place in mathematics whatever. But aside from such differ-
ences, the central feature of the position remained the idea
that signs have no significance by themselves, but are
assigned a significance according to the rules by which they
are bound.13
Frege argued against this position from several
directions. On one hand, he maintained that creative
definition, i.e., anything but the barest nominal definition,

is out of the question. He says, for instance, that "even

the mathematician cannot create things at will, any more than

 

12Quoted in ibid., g 86; translated in Frege, Gottlob,
Translations from the Philosophical Writings of Gottlob
Frege, P. Gesch and M. Black (trans. ed. )(Oxford: Basil
Blackwell, 1960), p. 182.

13Thus the difference between the formalists and the
idealists becomes clear; while the idealists thought that
logic is about something, i.e., the world as revealed by the
conceptions of the mind, the formalists thought the expres-
sions of logic to be empty, or devoid of content. Such
expressions gain meaning only by way of the rules which
govern them.

In.

the geographer can; he too can only discover what is there
and give it a name.'1u Here Frege was reacting to the idea,
which seemed to him implicit in the formalist position, that
one could begin with any sign, including conventional signs
like 'l,’ assign it meaning, and have it operate as if it
were still a conventional sign in all of the appropriate
contexts. Also, it seemed to Frege that the formalists were
bound to the position that within mathematics itself, apart
from its instances of application, the rules themselves were
“as arbitrary as those of chess.'15 Frege remarks also that
Thomae had attempted to contrast the rules of arithmetic and
the rules of chess, maintaining that the rules of chess were
more arbitrary than those of mathematics. But this could
only be the case, Frege maintained, if the application of
mathematics was being considered.

Regarding the game-liek nature of arithmetic pro-
pounded by the formalists, Frege says, ”let us try to make
the nature of formal arithmetic more precise. The obvious
question is 'how does it differ from a mere game?‘ Thomae
answers by alluding to the services it could render to
natural science. The reason can only be that the numberical

signs have reference and the chess pieces have not. There

 

luFrege, op. cit., The Foundations of Arithmetic, I 96.

15Frege, op. cit., Grundgestze der Arithemtik, g 85.

#8

is no ether ground for attributing a higher value to arithmetic
than to chess."16

This leads directly to another of Frege's criticisms
of Formalism: that it confuses numbers with numerals. Frege
maintained that the formalists actually assigned the
characteristics of numbers to numerals -- such numerals
supposedly being “eternal manifestations of the rules which

govern them.'17

He pointed out, for example, that signs would
be of no use whatever if “they did not serve the purpose of
signifying the same thing repeatedly in different contexts,
while making evident that the same thing was meant."18 In
effect, Frege's criticism is an attempt to exhibit formalism
attempting to pull itself up by its own bootstraps -- the
formalists cannot do without reference and meaning, and yet
they try to accomplish everything while yielding nothing.

In fact, maintains Frege, what the formalists do
accomplish is the smuggling in of meaning, where there should
be none, if the formalist position were taken strictly.19

Also, since the formalist rules for mathematics are concerned

with numbers generally, it appeared to Frege that it would be

 

161212;, e 90. Cf. Translations from the Philosophi-
cal Writings of Gottlob Fre e, p. 185.

 

17Ibid., g 96.

181bid., g 99.

192222;; g 110 ff. Cf. Translations from the Philo-
s0phica1 Writings of Gottlob Fre e, pp. 205 ff.

1+9

difficult to exhibit the special nature of such a number as
0 without again sneaking in meaning where it really should
not be.20

Our discussion of Frege's remarks on formalism must
of necessity remain incomplete; he spends roughly half of
the second volume of the Grundgesetze in detailed criticism
of the formalist position, and we will not be able to trace
all of these arguments. What we have done is to give his
criticism in outline, in order to provide a rough picture of
the influences which shaped his later thought.

Yet another facet of Frege's thought comes to light
in connection with his criticism of empiricism. Frege
objected to the idea of asserting any necessary connection
to hold between numbers and observable things. It seemed
to Frege that Mill had fallen into just such a pattern of
error; in fact, Mill's position seemed rather foolish to
Frege. In order better to understand Frege's criticism,
let us first briefly examine Mill's position with regard to
the formulation of the notion of number.

Concerning the definition of number, Mill remarks
that,

"the fact asserted in the definition of number

is a physical fact. Each of the numbers two,

three, four, etc., denotes physical phenomena,
and connotes a physical property of those

 

2°Ib1d., g 113.

50

phenomena. Two, for instance, denotes all
pairs of things, and twelve all dozens of
things, connoting what makes them pairs or
dozens; and that which makes them so is some-
thing physical; since it cannot be denied that
two apples are physically distinguishable from
three apples...they are a different visible
and tangible phenomenon."

In the Grundlage, Frege embarks upon a caustic criticism
of Mill's position. After remarking that Mill had the sound

sense to base mathematics on defintions, he remarks that

"...this spark of sound sense is no sooner

lit than extinguished, thanks to his precon-
ception that all knowledge is empirical. He
informs us, in fact, that these definitions
are not definitions in the logical sense;

not only do they fix the meaning of a term,
but they also assert along with it an observed
matter of fact. But what in the world can be
the observed fact...which is asserted in the
definition of the number 777,861+?"22

The mistake which Mill was making in such passages
seemed obvious to Frege: he tried to connect numbers to

observable things by definition, thereby forcing a necessary

 

connection between the realm of empirical fact, and the

realm of mathematical objects. But besides such difficulties,
Frege also recognized that Mill was incorrectly trying to
equate the meaning of '+' with physical addition. Here

again, Mill stumbles over his own empiricism. As Frege puts

 

21Mill, John Stuart, A System 2f Lo ic, (London:
Longmans, 1843), p. #00.

22Frege, op. cit., The Foundations 2g Arithmetic,
99-

51

it, “Mill understands the symbol + in such a way that it will
serve to express the relation between the parts of a physical
body or of a heap and the whole body or heap; but such is
not the sense of that symbol."23

If Frege's assessment of Mill's position is correct,
such criticisms of Mill as those just mentioned are for the
most part sound. However, it is often difficult accurately
to determine just what position Mill was trying to defend
with respect to the definition of number. Frege, with typical
literalness, picks just those passages from Mill's work which
are ripe for criticism. If Millvwere held to just these
passages, Frege's criticisms would be devastating to his
position. But in other passages Mill seems to hold the
weaker (and presumably more defensible) position that we
come to 522! numbers by the association of numberals with
observable things. Frege takes no notice of such passages,
and hence shows little charity.

Such,then, was the intellectual atmosphere in which
Frege found himself immersed. The period was characterized
by the presence of philosOphical extremists of the various
sorts just described. These individuals were not extremists
because of the sort of views which they held, but because

of the lengths to which they went in defending them. It

 

23Ibid.

52

was during this period, for example, that extreme idealism
was enjoying some popularity in Germany.

Much of Frege's work is a reaction to these extremes.
He felt that each position had its own clear area of failure.
Further, if sense was to be made of such generic notions as
that of number, and the relationship between logic and
mathematics clearly explained, a completely new approach
would have to be developed. With this in mind, we shall now
proceed to an examination of a few relevant portions of
Frege's logic, and then to a discussion of his theory of

logic.

3. SOME BASIC ASPECTS OF FREGE'S LOGIC.

Frege developed his logic as a groundwork for
mathematics, and not as a discipline in and of itself. His
claim was that the laws of arithmetic are analytic in the
sense that they can be strictly (i.e., by definition)
derived from logic. As he says, "Arithmetic thus becomes
simply a development of logic, and every prOposition of
arithmetic a law of logic, albeit a derivative one. To
apply arithmetic to the physical sciences is to bring logic
to bear on observed facts; calculation becomes deduction.'2u

Everything which appears in his logic is there because it

is necessary for the development of relations, etc. which

 

2“Ib1d., g 87.

53

are needed to define the notion of number, and the operations
of arithmetic. Thus to explain why something appears in his
logic is to explain how it functions in the chain of reason-
ing from the basic propositions (axioms) to the desired
operations necessary for the transition to arithmetic.

In some respects, Frege's theory of logic is more
easily separated from his logic than was Boole's from his
logic because Frege was determined to state the fundamental
principles of his logic explicitly. As a result, what
belongs to his logic is easily rec0gnizab1e as such. But
Frege‘s theory of logic thoroughly pervades his logic, in
the sense that very little of his logic can be understood
without understanding most of the theory with which it is
associated. Hence, the treatment of his logic which follows
will be confined to a discussion of just the primitive basis
of his Begriffsschrift, for this is all we shall need for
the discussion of his theory.

Frege's logic appears in the form of what he calls a

 

Begriffsschrift,25 alternatively translated as 'idography,‘
or 'concept-script.'26 The particular rendering of the

begriffsschrift with which we shall be concerned is contained

in the early portions of Frege's later work, Grundgesetze der

 

 

25Not to be confused with The ngriffsschrift, Frege's
early pamphlet, which contains the first rough presentation
of his logic.

26Neither of the later renderings are totally adequate,
as they do not present the meaning of 'begriffsschrift' in
an enlightening manner.

 

5h

Arithmetik. It is in this work that his logic is most fully
developed.

With regard to the vocabulary of the logic, there
are several basic types of expressions to be considered.

First, as is well known, Frege used the sign 'f-' in
constructing the basic sort of expression in the system, the
assertion. Used to the left of further components, this sign
indicates that the truth-value named by the expression which
follows it is being pointed out by the individual producing
the sign.

The sign '-' placed to the left of the name of a
truth-value indicates that the content expressed by the
expression which follows it is being formulated, but is not
being asserted.

To the right of a content stroke, where no quanti-
fication is involved, there can appear either a Greek capital,
which serves as a place-holder for a particular statement,
i.e., as a statement-constant, or a Roman italic, which
serves as a free variable, and is associated with a group of
possible statement-substituends. Put together, such an
expression as '[~"' forms a judgment. Frege says, "the
presentation in Begriffsschrift of a judgment by use of the
sign 't—‘ I call a proposition gf’gggriffsschrift or briefly

a proposition."27

 

27Frege, op; cit., The Basic Laws 2: Arithmetic, g 38.

55

Another innovation which Frege introduced into logic
was the use of symbols such as 'E ' and ' S' as argument
place-markers in function-expressions. Introducing such
symbols allowed Frege to use an expression to name a function
without there being the necessity of introducing an argument
expression for the function. Were this not done, functions
could only be spoken of by using an example of the use of a
function-expression, i.e., the name of a 22122 of that
function. Thus, for example, '( +h)=6' names a function
without need of regard for its various possible arguments.

Assertions of the form 'f1:%brw are particularly
interesting in Frege's system, for this expression is the
judgment that it is not the case that 'A' is the True, and
"" is the False. While Frege's manner of formulating the
Philonian conditional is well known to most students of
logic, few realize the full complexity of Frege's formulation
of it. Let us consider an example of Fregean conditional:
reading from left to right, we have first "' which is the
assertion sign. Next, the horizontal segment between the
assertion sign and the junction of the vertical line segment
to the antecedent, stands for the content of the whole
conditional. The vertical line segment connecting the
antecedent with the consequent is call the ”conditional
stroke,“ indicating the nature of the expression, while the
two horizontal segments, which immediately precede 'ZX' and

't“ are content lines for the two expressions which follow.

56

Also, it must be remembered that the expression
'I-L—A', is as much a function-sign as is '¢é'. Consider
Frege's remark on this point, that ”...I introduce the
fucntion of two arguments 'Ft;_ ' by stipulating that its
value shall be the False if the True be taken as g-argument,
and any object other than The True be taken as t-argument,
and that in all other cases the value of the function shall
be the True."28

In Eggriffsschrift, negation is handled in terms of
a negation stroke attaching to the content-stroke of a
proposition, in the following manner: 'rw-FV. This amounts
to the assertion that '(‘ is not the case.‘29 Here again
the symbolism is perhaps more complex than might first
appear, for included in the expression mentioned above, there
are two content-strokes. The horizontal segment to the left
of the negation sign expresses the content of the negated
argument, while the segment to the right of the negation
stroke expresses the content of '[".

With the use of both the conditional sign and the
negation sign it becomes simple to express conjunction and

(inclusive) disjunction reSpectively as follows: 'h‘E X',

and 'rtr-A'.

281bid., p. 51.

 

29Frege, Gottlob, ngriffsschrift (translation of
monograph by same name, Halle, 1879), in Translations from
the Philosophical Writings of Gottlob Frege, P. Gesch and
I.Black (trans. ed.)(0xford?—Basil Blackwell, 1950), g 7.

 

57

The next feature of the symbolism of Frege's logic
which we shall consider is by all accounts one of the few
contributions to logic which have singlehandedly altered
the entire scope and conception of the subject. What I
have in mind is Frege's use of what we now call quantifiers.
Considering the pronounced effect which this innovation was
to have on later developments, its introduction was simple
enough; Frege says, "'w—Qh)‘ is to denote the True if for
every argument the value of the function (P(§) is the True,

30

and otherwise is to denote the False.“ The scape of the
gothic letter stands over the concavity. After detailing
many rules governing binding by quantifiers, etc., Frege
goes on to mention that propositions such as 'Nfl-a2=l'
mean that "there is at least one square root of 1."31 'Thus
he makes a provision for defining existential quantification
in terms of universal quantification.

Three other sorts of signs of general interest are

the following: the Greek smooth-breathing over a vowel

indicates the range of values of the function which follows.

 

30Earlier, in The Begriffsschrift, Frege gives the
following formulation of generality: '

”NJ—9(a)

this signifies the judgment that the function is a fact
whatever we take its argument to be." This formulation is
less desirable for purposes of illustration because it lacks
the clarity found in the later formulation. Ibid., p. 11.

311b1d.

58

Thus, 'é(€2- 6)' indicates the range of values of the
function ({2— g). The significance of this sign will be
discussed in the section on theory. Second, the sign '\ t'
is used to indicate the function for the definite article.
Finally, the sign ' f-' is used to assert a definition.

There are several other sorts of signs which appear
in the vocabulary of Frege's logic. Most of these are
connected with the establishment of such relations as
'successor of' etc., which are used in the definition of
number. Since they are not basic constituents in his logic,
and since they bear only very indirectly on the theory
discussed later, we shall not deal with them here.

Frege provides no rules for well-formedness, as such.
That is, he does not provide any sort of a recursive
definition of well-formedness which can be used mechanically
to determine whether or not an expression is well-formed.
Instead, he provides prototypical examples of expressions
throughout the text, apparently presuming that any expres-
sion which resembled one of these would be significant.32
He does, however, provide long lists of rules governing

33

the transition from one expression to another.

 

32Here again it must be remembered that Frege was
not constructing a system for general use. He was merely
getting enough together to provide a basis for the definition
of number. Hence, it is not surprising that he apparently
did not think of making all the rules of the system explicit.

33Frege, op. cit., The Basic Laws 2: Arithmetic,
p. 105.

59

We shall touch only briefly on the axioms which
Frege uses in his system, as they are of relatively minor
importance in the discussion of theory to follow. They are
of interest mainly by comparison with various axiom-sets
employed today. The seven axioms (“Basic Laws," as Frege

calls them), with variation; are as follows:

(1) a
r—t:::a (I g 18)
(2) ‘ f(a)
r—‘——\EJ——f(a) (11a 5 20)
(3) Mflmfln
PEMMNISH (IIb g 25)
(LP) Sc—kfktufl)
f(b)
3(a=b) (III g 20)

(sn—j— J‘ .)=(-—-b)

 

 

 

 

f‘ *1 ‘3‘):(_r-—b) (IV a 18)
(6) k—(e’ f(€)=a’3(a))=(—\y-f(a)=3 (a))
(v 5 20)
‘3
(7) |-——a=\€(a= 6) (VI 3 1M3“

The interpretation of most of these is simple enough.
Perhaps the third and fourth axioms may seem unfamiliar, but
this is simply due to Frege's use of ‘P and a to quantify
over functions and produce second level functions.

Perhaps the one theoretically significant feature

of these axioms is brought out by Furth, when he says that

 

Shlbid.

6O

care must be taken in attempting to translate these expressions

into more modern notation. As an example of why such care
must be taken, he says,

...the range of variables in Basic Laws I and IV

[Nos. 1 and 5, above] is not confined to truth

values, but extends over all objects whatever;

accordingly the constants eligible to replace

them are not merely sentences (names of truth

values) but names of any objects. Thus, for

example,
"the Parthenon~9(2+2=b-9the Parthenon)"

(assuming the constituent names to be (denoting)

proper names of the language) would be a

legitimate instance of Basic Law 1.35
It seems initally doubtful that such an interpretation is
in fact possible. We shall, however, postpone further
discussion of the point until we take up the discussion of
Frege's notion of concept and object, in the section on
theory.

Frege employs four basic rules of inference. These
correspond to Modus-Ponens, Transposition, Hypothetical
Syllogism, and a rule for the concatenation of antecedents
in conditional expressions.

While he does not specifically recognize a separate
set of theorems which are deducible from the axioms, Frege

comes close by recognizing that whatever is derivable from

the basic laws will also be a law. He says at one point,

 

351b1d., translator's introduction, p. x.

--1

 

61

”Now that we have become acquainted with the Basic Laws and
the methods of inference which are to be employed, it is
time to derive from them laws we shall be using later, so
as to exhibit at the same time the style of calculation."36
He then goes on to produce many proofs of derived laws which
were to be used later in dealing with mathematics.

We have now dealt, albeit briefly, with the
essential features of Frege's logic. Such aspects as have
been discussed were readily intelligible because they are
translatable into patterns of interpretation in use today.
But of course Frege had no such scheme of interpretation on
which to rely. Thus in order to insure the significance of
this mass of symbols, Frege developed a thorough, and very
complex theory of logic. In dealing with this theory, we
shall see how Frege conceived of the subject-matter of logic,
to what sorts of things his position committed him, and how

we gain knowledge of such things.
h. FREGE'S THEORY OF LOGIC.

Frege's theory of logic constitutes a well worked-
out groundwork for his system of logic. In fact, as Furth
points out, ”Frege's explanation of the primitive basis of
his system of logic, and particularly of the primitive

symbolism, is undertaken in terms of a deeply thought out

 

361bid., p. 105.

62

semantical interpretation, which in turn embodies an entire
theory of language."37 Frege's theory of logic is designed
primarily to aid in the accommodation of logic to mathematics,
but the significance of the theory is much more general.

We shall begin this discussion with an indication of
Frege's general attitude toward the theory of logic.

Frege's theory is strongly realistic. In fact, his
ontology is populated by many varieties of immutable,
abstract, only partly knowable things. As Wells remarks,
”Frege's ontology is concerned with abstract entities such
as functions, senses and ranges etc. Many ontologists would
regard these as beings of reason (333i: rationis). Frege
takes a more extreme realistic view of them."38

Frege's extreme realism, however, does not appear
to be based on any mysterious underlying philosOphical
convictions. Rather, it is the natural outcome of his rather
common-sensical commitment to four ideas: (1) he believed
that there is an objective reality which is independent of
human knowledge; (2) he believed that much, but not all of
this reality is accessible to human knowledge; (3) he believed
that all knowledge is of timeless, immutable objective

truths; and (h) he believed that not only the natural

 

371212;, translator's introduction, pp. vi-vii.

38Wells, Rulon, "Frege's Ontology," Review 2:
Meta h sics (New Haven: Philosophy Education Society, Inc.,

1950’51 ’ p. 51”..

63

sciences, but logic and mathematics have objective truths
as their subject-matter.39 With these points in mind, let
us turn to the body of the theory.
Frege thought the subject-matter of logic to be truth.
His pronouncements on this point indicate his straightforward
and uncomplicated approach to the question of subject-matter.
For instance, he says, "All sciences have truth as their
goal; but logic is concerned with it in quite a different
way from this. It has much the same relation to truth as
physics has to heat or weight. To discover truths is the
task of all sciences; it falls to logic to discern the laws
of truth."“0 It is only a short step for Frege from this
conception of subject-matter to the rules of language:
“rules for asserting, thinking, judging, inferring, follow
from the laws of truth."l‘1
But aside from such pronouncements, Frege says little
else which directly bears on questions concerning the subject-
matter of logic. The next step in completely specifying the
subject-matter would, in this case, consist of Fregds
analysis of truth itself. But on this point, he pleads

justifiable ignorance: "...every...attempt to define truth

collapses. For in a definition, certain characteristics

 

39On this point, of. ibid., p. 562.

qurege, Gottlob, "The Thought” (ca. l90h), P. E. B.
Jourdain (trans.), Mind (Edinburgh: Thomas Nelson and Son,
Ltd.’ 1956), va, NO. 259’ p0 2890

b11b1d.

6%

would have to be stated. Aniin any particular case the
question would arise whether it were true that the character-
istics were present. 50 one goes round in a circle.
Consequently, it is probable that the content of the word
'true' is unique and indefinable.”u2

In the passage just quoted we see a bit of reasoning
which is characteristically Fregean. Time and time again we
will notice that Frege is quite willing to talk about
fundamental notions, while at the same time, he maintains
that we are not in a position fully to understand these same
notions. Frege held the view that there was a kind of
epistemological barrier which prevents our fully grasping
basic notions. In fact, at several points, he asks his
reader not to “begrudge him a pinch of salt" when he talks
of fundamentals.

Why, then, it might be asked, if Frege knew that
such notions as truth were not fully accessible to the
human intellect, did he bother with them at all? It would
surely seem that if sense could not be made of the notion of
truth, no sense could be made of an account of the nature of
the subject-matter of logic in which it was employed. The

answer to such a question relies on one of Frege's basic

 

uzIt is interesting to note in passing, that Frege's
account of the indefinability of "true” parallels, nearly
word for word, G. E. Moore's later account of the
indefinability of 'good'. Cf. ibid., p. 291.

65

intellectual traits mentioned earlier: he believed that our
knowledge is of eternal truths which are not mind-dependent.
The notion of truth being the subject-matter of logic is no
construction of the mind, but is discovered by it. Frege
is claiming that we can gain 3223 knowledge of truth, and
that all we can gain will be of help in understanding logic.
Let us consider an example of how Frege puts the
partially understandable notion of truth to work. At one
point, he says, ”I understand by 'laws of logic' not
psychological laws of takings-to-be-true, but laws of truth.
If it is true that I am writing this in my chamber on the
13th of July, 1893...then it remains true even if all men
subsequently take it to be false. If being true is thus
independent of being acknowledged by somebody or other, then
the laws of truth are not psychological laws; they are
boundary stones set in an eternal foundation, which our

1&3

thought can overflow, but never displace.“ Frege is in
effect saying that while we cannot grasp the nature of truth,
we know enough about it to know that its laws, i.e., the

laws of logic, have an extra-mental ground, since we seem

to be able inter-subjectively to judge the truth or falsity
of an assertion.

Truth is not the only fundamental notion to receive

this sort of treatment by Frege, although it is perhaps more

 

 

66

general in significance than any of the others (and as a
result, he says relatively little about it). We will meet
this “partial analysis“ again in the discussion of such
notions as function, object, range, concept, relation, and
others. All such notions are, according to Frege, so
completely fundamental that we can only catch glimpses of
their natures. Concerning them, we must be satisfied with
something less than perfection.

The laws of logic, then, are only partially accessible
to us. Frege maintains that "the questions why and with what
right we acknowledge a law of logic to be true, logic can
only answer by reducing it to another law of logic. Where
that is not possible, logic can give no answer."ub Frege
is thus maintaining that not only is knowledge of the
subject-matter of logic necessarily incomplete, but also
that complete knowledge of the basic constituents of logic
is not forthcoming. Frege quite probably believed that the
“rock bottom," atomic constituents of knowledge, and amongst
them the most fundamental laws which support logic, are
themselves outside the scope of human knowledge. Hence the
laws of logic must be in a sense self-supporting.

Although this tepic will be dealt with again later,
while on the topic of truth as the subject-matter of logic,

and the relationship between truth and the laws of logic, it

 

uhlbid., p. 15.

 

67

should be noted that Frege posited the existence of something
called the True, which serves as the referent for all true
statements, and the False, which serves as the referent for
all false statements. In this manner, Frege reified truth-
values.“5

From looking at just the notion of truth alone, it
becomes apparent that there are two questions with which we
shall have to deal rather fully in order to grasp Frege's
theory of logic. The first is, for Frege, what sorts of
things exist?, and the second, how do we know what sorts of
things exist?

Let us take the more simple of the two questions
first, i.e., Frege's answer to the question, how do we know
what sorts of things exist? Frege's answer is brief enough
to be almost overly simple. He says, “If we want to emerge
from the subjective at all, we must conceive of knowledge
as an activity which does not create what is known but grasps

#6

what is already there." This constitutes Frege's only

firm pronouncement on the question of knowledge (although

 

u5The True is, however, not to be confused with Frege's
conception of truth conceived of as the subject of logic.
This latter notion was employed by Frege only in the latest
of his writings, and is used in a vague, non-technical sense.
Thus the laws of truth are nothing more than the laws which
govern the relation between statements and the constituents
of the world which they indicate. ”The True,“ on the other
hand, is a technical expression used to name a special kind
of object to which all true statements refer.

p. 23.

.4 /_

 

68

he repeats it often). However, although it is brief, this
pronouncement is of the greatest importance. Frege is saying
that the mind is totally passive, save for the sole activity
of naming. This is, the mind does not have the power to
construct knowledge, but is quite like a tabula £33g_onto
which knowledge is transcribed.

Frege holds this position concerning knowledge so
rigidly, that, for instance, he maintains that definition can
only be construed as the most simple sort of nominal
definition. On this point, he says, ”It is important that
we make clear at this point what definition is and what can
be attained by means of it. It seems frequently to be
credited with creative power; but all it accomplishes is
that something is marked out in sharp relief and designated
by a name.'h7

Frege's position with regard to knowledge is trouble-
some to this extent: if we do indeed act as spectators in
the world of knowing, and merely "grasp what is already
there,“ then how is it that such things as the True, etc.,
do not seem more familiar to us than they in fact do? That
is, how are we to decide what exists and what does not
exist? In Frege's language, what governs the grasping of

that-which-is-already-there? Without a solution to this

 

“71b1d., p. 11.

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59

problem no ontological dispute could possibly be settled,

for in such a case, there would be no means available to
adjudicate disparate claims concerning the existence of
anything. Before we wind our way through Frege's ontology,
let us briefly consider how he himself decides what is actual
and what is not.

Frege is of no direct help on the problem just
mentioned. He tells us how he conceives of knowledge, and
what sorts of things there are, but he does not tell us
what credentials a thing must have in order to be counted as
actual. But while he does not directly address himself to
this question, nonetheless a very plausible solution lies
in full view. Frege's ontology incorporates mainly those
sorts of things which correspond one-to-one with the various
major kinds of expressions that occur in his formal language.
This suggests that the test used to determine whether or
not a thing is actual was whether or not it is one of a kind
which corresponds to a particular kind of essential expression
in the language of 10gic. Frege arrived at his ontology
largely by looking to what language suggested might be
actual, and then he reified such things.

Such a conclusion might invite the criticism that
after all, Frege was not so narrowly concerned with language
that he would allow it to dictate what should be considered
to be actual. In answer, it might be pointed out that Frege
is no mere slave to the needs of language. For example, one

of Frege's contributions was the recognition that mysterious

A [—2

7O

"variable entities” need not be posited in order to correspond
to variables. This is an example in which the correspondence
mentioned above breaks down. Senses, also, do not participate
in a direct correspondence to a unique sort of expression
(although senses are essentially language-related entities).
But the fact that Frege was not blindly led by language to
adOpt his ontological views does not imply that his ontology
was not forged with a strong eye to the needs of language.
A fair appraisal of the situation seems to be that while he
did not demand a strict correspondence between each sort of
expression and a unique sort of thing, nevertheless, he did
rely heavily on what appeared to him to be the demands of
language. Frege likely thought that such notions as that
of function, range of valueg etc., could only make sense if
something definite and actual corresponded to each of them.
After all, there was no provision made for the mind's
creating "convenient fictions” to account for such notions.
Since it could only uncover what was indeed already there,
such parts of language as function—expressions could only
serve as evidence that another sort of thing had been
discovered: functions. What made sense in language had to
correspond to something actual, for it must be remembered
that Frege allowed no appeal to the mind as a ground for
the making of such sense.

Thus in each case, Frege's ontology responds to the
needs of his language and is in fact largely a product of

those needs. It does not appear that there is a basis for

71

the opposite claim, that his language is constructed on a
prior ontological foundation. For in any case, his ontology
never appears as well worked-out as does his language.
Therefore, language seems to hold the place of priority.

To see just how much the ontology associated with
Frege's logic is dependent on language, let us now consider
the main parts of his ontology.

Since Frege's ontology is complex, we shall use a
variation of an outline of his ontology provided by Wells.

All things fall into one of the following categories:

A. Objects
l. ordinary denotations
a. truth-values
b. ranges
0. function correlates
d. other things
2. ordinary senses

B. Functions

1. functions all of whose values are
truth-values
a. with one argument (concepts)
b. with two or more arguments

(relations)

2. functions not all of whose arguments
are truth-values
a. with one argument
b. with two or more arguments

#8

We shall first consider the major distinction between

objects and functions, as it is central to Frege's ontology.

 

uaWells, OE. Cite, pe SUB.

72

For Frege, an object is that which a function becomes,
when that function takes an argument and issues in a value.
Notice immediately that he thought of both functions and
objects alike, not as parts of language, but as entities of
a sort which correspond respectively to function-expressions
and to names. After explaining objects in terms of functions,
he maintains that a function is something which becomes an
object, when it itself is completed with an argument (which
of course would itself be an object or a function). While
this account might at first appear to be a case of circular
definition, it is really not definition at all. Frege is
simply explaining each of the two notions by appeal to a
mutual relevance between them. From this point, he goes on
to build an explanation of each of the notions separately.

Concerning objects, Frege says, "I regard a regular
definition Ef objecté] as impossible, since we have here
something too simple to admit of logical analysis. It is
only possible to indicate what is meant. Here I can only
say briefly: an object is anything that is not a function,
so that an expression does not contain any empty place."l‘9
While saying what an object is, Frege has included a very

helpful remark, a remark which states a sufficient condition

for the recognition of an object: an object is something

 

9Frege, Gottlob, "Function and Concept,"1Translations
from the PhilosOphical Writings 2£_Gottlob Frege, P. Gesch
and M. Black (trans. ed:)(Oxford: Basil Blackwell, 1960),

p. 32.

 

73

which corresponds to an expression which has no completely
indeterminate place. On the other hand, functions are those
things which correspond to incomplete expressions. "The
function," as Frege says, ”by itself must be called incomplete,
in need of supplementation, or 'unsaturated'."50

Since understanding how Frege arrives at the
distinction between functions and objects is so crucial to
understanding how he arrives at the rest of his ontology,
let us consider a restatement of the point just made. Frege
is claiming that there is one difference between functions
and objects which is basic; functions are unsaturated or
incomplete, while objects are saturated, or complete. He
does not explain what constitues ”being saturated,” or
”being unsaturated"; at best, he relies on a prior under-
standing of the notion of an argument (which is itself
either an object or a function) for an explanation of
saturatedness. But while all this seems highly suspect,
Frege in effect exonerates himself by l) maintaining that

what he has offered is not in fact a definition of either

 

'function' or 'object,’ for no definition is possible,

since both notions are so basic; and 2) by adding substance
to his talk 22223 the dichotomy, by showing how things which
fall under the one heading or the other can be recognized.

He is using the fact that functions correspond to incomplete

 

5°Ib1d.

7h

expressions to shed light on the more basic notion of
saturatedness.

Frege's frequent remark that functions are named by
incomplete expressions gives further credence to the point
made earlier: there is a strong tie between the parts of
language which Frege recognizes, and the kinds of things
which he countenances in his ontology. In this case, in
fact, much of the significance which attaches to the notion
of a function derives from the very fact that a function
corresponds to a certain kind of expression.

Let us consider an example of what Frege calls a
function. '2.§ 3+ g' denotes a function (counting 'E ' as
simply a place-marker), i.e., it denotes that which becomes
complete when it is combined with an argument, which would
be denoted by some other expression substituted for each
occurrence of 'g', above. When, for instance, the function
2.§3+ g is combined with the argument 1, the result is an
object: a value of the function in question (in this case,
3).

Putting aside for the moment discussion of such sorts
of objects as values, ranges, etc., there are several other
features of functions which must be considered.

First, there is no way to tell which part of a
complete expression such as '2.13+l' denotes a function, and
which part or parts denote arguments. Virtually 222 part of
the expression can be removed in such a manner that what

remains is a function-expression. But, as Wells remarks,

75

"given an expression and the function it denotes, one can
ascertain the arguments; given the expression and the
arguments, one can determine the function."51

Second, there are two major kinds of functions, those
all of whose values are truth-values, i.e., those the
expressions for which, when completed, refer to the True
or the False, and those whose values are not truth-values.
The function §2=lb is an example of the former sort, while
the function denoted by the expression 'mother offi', is an
example of the latter sort.

Further, there are two sorts of functions all of whose
values are truth-values; concepts are functions which take
only one argument, and relations are functions taking two
or more arguments. Concerning concepts, Frege says, "the
concept (as I understand the word) is predicative (it is,
in fact, the reference of a grammatical predicate)."52
Further, he says, "I call the concepts under which an object
falls its properties."53

The method which Frege employes in developing his

theory of logic should now be obvious. He begins with highly

 

51Wells, pp; cit., p. Shh.

52Frege, Gottlob, ”On Concept and Object,” Trans-
lations from £23 Philosophical Writings 2: Gottlob Frege,
P. Gesch and M. Black (trans. ed.)(0xford: Basil Blackwell,
1960), P0 “30

531b1d.

76

suggestive, but highly metaphorical assertions concerning
fundamental components of the theory, and then proceeds, by
giving partial descriptions of these components, until a good
deal of sense can be made of the notions in question. In the
case of the notion of a function, he began with the property
of being unsaturated, which is suggestive but quite Opaque,
and then proceeded to unravel the notion so that what began
as obscure is connected to what is understandable. In this
way, more sense can be made of both.

While Frege leads us toward an understanding of what
he meant by ‘function,’ there remains a great deal of
obscurity connected with the notion. All that began as
obscure remains essentially obscure, i.e., we have no better
an understanding of the nature of unsaturatedness, although
we now know more of conditions which are necessary and
sufficient for the recognition of things which have the
property. Frege is aware of this surd of obscurity; he
maintains, as has already been pointed out, that complete
understanding of these notions is impossible. He has in
effect built a workable edifice on a mysterious, but not
necessarily shaky foundation.

We have now covered the function-half of the main
division in Frege's ontology. Let us next consider the
various sorts of objects which he recognizes.

In Frege's view, there are two main sorts of objects,
senses and denotations. This distinction is widely known,

since it has been employed recently by such individuals as

 

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77

Church and Carnap. The article in which Frege brings the
distinction to light ("On Sense and Denotation") has become
something of a classic in analytic philosophy. In view of
the wide familiarity attending to the distinction, we shall
look only briefly at the central aspects of it.

Frege holds there to be three components in the
meaning of an expression. As he says, ”the reference of a
proper name is the object itself which we designate by its
means; the idea which we have in that case is wholly
subjective; in between lies the sense, which is indeed no
longer subjective like the idea, but is yet not the object

5h

itself.” The distinction between sense and reference
arose in the following manner: Frege recognized that some
affirmations of identity of the form 'A is identical with B'
seemed to have more significance than merely asserting that
A was identical with itself. Thus, “the morning star is
identical with the evening star” seemed to mean more than
just 'the morning star is identical with the morning star.’
In order to account for this difference, he maintained that
there are two major aspect of meaning -- the object referred
to, i.e., Venus, and that component which accounts for the

distinction between the meaning of 'the morning star,‘ and

'the evening star.‘ Furthermore, he says that the sense of

 

5“Frege, Gottlob, ”On Sense and Reference,” Trans-
lations from £32 Philosophical Writings 2f Gottlob Frege,
P. Gesch and M. Black (trans. ed.)(0xford: Basil Blackwell,
1960), Pa 600

78

an expression is every bit as objective (i.e., not mind-
dependent) as is its reference. Almost as an afterthought,
Frege also recognized that there is often a purely private
aspect to meaning -- an aspect which, unlike sense, is not
available for public scrutiny. In the case of terms he
called this "the idea."

Thus Frege accounts for identity statements of the
form 'A is identical with B' by saying that what is being
asserted is that 'A' and 'B' have distinct senses associated
with the same denotation.

Frege further says that every expression having a
denotation has a sense, but not conversely. Thus expressions
which only purport to refer have sense, but no denotations.
Also, while to each sense there corresponds at most one
denotation, for each denotation, there can be many senses.

Frege extends his discussion of sense and reference
by maintaining that for statements, the reference is either
the True, or the False, and the sense is a thought: "with-
out wishing to give a definition, I call a thought something
for which the question of truth arises. So I ascribe what
is false to a thought just as much as what is true. So I
can say: the thought is the sense of the sentence.... The
thought, in itself immaterial, clothes itself in the material

garment of a sentence and thereby becomes comprehensible to

79

us.'55 In this way Frege accounts for the variation in
meaning between all true statements ; while they all have the
same reference, each has its own sense. ”In the reference
of the sentence, i.e., the True or the False, all that is
specific is obliterated. We can never be concerned only
with the reference of a sentence; but again the mere thought
yields no knowledge, but only the thought together with its
reference, i.e., its truth-value."56

By treating statements in a fashion closely parallel
to that of names, Frege accomplishes quite a good deal.
According to Furth, treating sentences as complete names
”...works a formidable simplification in the semantical
theory governing language, on several fronts. Frege desires
to express laws of propositional logic by using variables,
and these variables should be appr0priately replaceable by

57

sentences." Furthermore, not only is the relationship

 

55Frege, 22; cit., ”The Thought," p. 292. or. also
Frege, 22; cit., ”On Sense and Reference,” p. 65. The
concept of truth used with reSpect to the sense of a sentence
is identical with the concept of reality. Thus to say that
a sentence has true sense is to say that the state of
affairs which corresponds to the statement is actual. This
use of the concept of truth must not be confused with the
concept of the True, which as we have seen is a distinct
object to which Ell statements which have true sense, refer.
Put differently, two statements may have two different true
senses to which they correspond, but 333 true, the statements
would refer to the same object, the True.

56Ibid.

translator's introduction, p. xii.

80

between names and sentences parallel, but also interlocking.
Frege puts this aspect of the relationship as follows: ”I
have in fact transferred the relation between the parts and
the whole of the sentence to its reference, by calling the
reference of a word part of the reference of the sentence,
if the word itself is a part of the sentence."58 In this
fashion, the sense-reference distinction can be seen to
pervade the various levels of language.

There are, however, important questions which attend
the sense-reference distinction, but which are external to
the distinction itself. One such problem concerns Frege's
reason for drawing such a distinction. Another concerns the
relevance of such a distinction to questions concerning
Frege's ontology, since the distinction seems so obviously
a semantical one.

The solution to the first problem is obvious enough.
As we have seen, he wished to use the distinction to clear
up difficulties concerning the role of identity. But a
further, less recognized reason is that he drew it in order
to be able conveniently to set considerations of sense aside
once he began the develOpment of the Begriffsschrift in his
Grundesetze. He wished to develop his logic in such a way
that it would be a logic of denotations, and not of senses.

As Furth points out, his purpose in setting up the distinction

58Frege, op. cit., ”On Sense and Reference," p. 65.

81

was partly negative: to get rid of one side of it.59 This
interpretation seems amply borne out by the noticeable lack
of interest displayed in the subject of sense in works, and

most noticeably the Grundesetze, which appeared after "On

 

Sense and Reference.”

The answers to the question as to whether the sense
and reference distinction is only semantical appears to be
that it was as much an ontological distinction as a
semantical one. That is, it cuts across linguistically-
determined boundaries. Such would be the case in instances
in which the sense of one expression serves as the reference
of another (as, for example, in the ease 'the sense of the
word ---'). As Wells says, "...the sense-denotation distinc-
tion is not merely semantical; it is ontological as well.
Whatever is a sense of some expressions is also a denotation
of some (possible) expression; but there are denotations
which, if treated as senses, have no correspondent

6O

denotation." As a matter of fact, there are many
complexities which Frege introduces into his discussion of
sense and reference which are often ignored in commentaries.
For instance, Frege maintains that the reference of a

statement in indirect discourse is its own sense, and that

such a statement names neither the True, nor the False;

 

translator's introduction, p. xviii.

6oWells, op. cit., p. 555.

82

likewise, he maintains that the indirect reference of a word
is its sense.

But no matter how the sense-denotation distinction
cuts across the distinction between various sorts of
expressions, every object is a potential component of a
sense, or a potential denotation for some exPression.

We have now only to consider the various sorts of
ordinary denotations in order to complete our discussion of
Frege's ontology.

Among the main sorts of denotations are, first,
truth-values, which were mentioned at the beginning of this
section. The clearest account of the status of truth-values
is given by Frege when he says, "a statement contains no
empty place, and therefore we must regard what it stands for
as an object. But what a statement stands for is a truth-
value. Thus the two truth-values are objects."61 Of all of
the components of Frege's ontology the True and the False are
the most difficult to understand. Virtually all that we
know about them is that they stand to statements as objects
named stand to names, and that all statements which corre-
spond to actual states of affairs refer to the True, and all
statements which correspond to states of affairs which are
not actual refer to the False. The contention that Frege

used the demands of language to determine what he thought

61Frege, op. cit., "Function and Concept", p. 32.

83

existed, once again seems to be supported; Frege is claiming
that the True and the False are objects, and supporting his
claim by pointing out that that which refers to it are
complete, i.e., are statements.

The only other interesting sort of ordinary denotation
is what Frege calls a range of values for a function. As
was mentioned in Section II, above, Frege uses the expres-
sion 'é’,(€_+7)' to stand for the range of values of the
function {+7. The modern student of logic can easily
comprehend the notion of a range by reference to sets, in
this manner: The range of a function is a set of ordered
pairs, one member of each pair being an argument for that
function, and the other in each case being the value of that
function which results from the function taking that
particular argument. Indeed, such an account closely
parallels Frege's understanding of the notion of a range,
but with one important exception; for Frege, the Greek
smooth breathing is a primitive, and in no case should it
be thought of as being defined in terms of classes. This
difference, however, is a mere oddity, and provides no
genuine difficulty for the understanding of his concept of
the range of a function.

Another oddity concerning Frege's conception of ranges
is that anything at all can serve as an argument to some
functions (i.e., functions which were not restricted to truth-
Vfllues). On this point, Frege says, "Not merely numbers,

bUt objects in general, are not admissable as argumenté]; and

8b

here persons must assuredly be counted as objects. The two
truth-values have already been introduced as possible values
of a function; we must go further and admit objects without
restriction as possible values of a function."62 It is

this liberality which led to Russell's discovery of a
paradox in Frege's axioms. Specifically, the breadth of

the range of possible values for axiom V, listed above

(p. 59) was so broad that it led to the now famous paradox.63

5. CONCLUSION.

We have now seen most of the important aspects of
Frege's theory of logic; there remains the task of drawing
many of these aspects together.

In a way, Frege has already told us how he conceives
of the subject-matter of logic. It is clear from the
foregoing discussion, for instance, that he considered logic
to be about something more than just language alone. To be
sure, he was well aware both of problems concerning the
structure of language, and of the relation holding between
language and the world. Such considerations, however, did
not prevent him from pursuing questions concerning the nature
of those things-in-the-world which the language of logic is

about.

62
Frege, op. cit., ”Function and Concept", p. 31.

63For the details of the paradox, cf. The Basic
'8 f Arithmetic, p. 127.

‘—

85

It is clear also that Frege would not agree that logic
is exclusively about the workings of the mind. To be sure,
the mind has its place in the scheme of things, but Frege
considered the operations of the mind to be too variable to
provide the subject-matter of logic.

The primary subject-matter of logic in Frege's
estimation was the extra-linguistic world of logical
entities which we have been considering. While he considered
the world to be pOpulated by the ordinary group of medium-
sized objects with which we are all familiar, he also
considered there to be such things as functions, the True,
the False, thoughts, concepts, senses, ideas, etc., which
we have already discussed. These are all things posited in
order to provide pp existent, extra-ligguistic, extra-mental,
logical 2332 £25 232 world.

But how, it might be asked, does this account square
with Frege's own claim that logic is about truth? The two
accounts square nicely. When he says that logic has no
(particular truths with which to deal, but deals with the
conditions for truth, he is in effect pointing to the world
of logical entities just mentioned. That is, it is these
entities which provide for the possibility of there being
particular truths. It is the laws governing these entities
which comprise logic.

Frege is not the only logician with whom we shall
deal who holds that logic-has the extra-linguistic world

as its primary subject-matter. We shall see that Church,

86

Whitehead, Lewis, and to some extent, Wittgenstein go beyond
the bounds of language and the mind to explain the workings
of logic. In particular, we shall look to Church's theory

of logic as a working out of many of the most useful aspects
of Frege's theory. Most importantly, the two logicians share
the same concept of the subject-matter of logic; both consider
logic to be about things in the world. We shall postpone
further discussion of this position until the conclusion of
this work, where it will be compared with a view of a dif-
ferent stripe, when the theories of Church and Quine are
compared.

By way of a modest evaluation of Frege's own theory,
perhaps we should begin with the most obvious objection to
it. As Wells says, "many philosophers today would
spontaneously object E0 Frege's ontologfl on the ground of
parsimony. Frege sets up his ontology to make logic and
mathematics intelligible; these philOSphers would maintain
that he has posited more entities than is necessary for the
purpose.”6u Such a criticism, however, seems empty at the
outset. Frege wanted to present logic as being the basis of
mathematics in as clear a manner as possible. He did not give
a great deal of thought to how little he might commit him-
self to, and get away with. Frege's desire was apparently
t0 provide as broad an ontological base as possible on which

t0 construct logic.

6uWells, OEe Cite, p. 560.

87

While it is pointless to criticize Frege for being
somewhat too generous in stocking his ontology, and while
it is true that he succeeded in constructing a massive
theory, still much of his theory has been forgotten, or at
least set aside. Much of this negative reaction is a direct
result of the reaction against many types of platonism
during the last several decades. During this period, it has
become unfashionable, for instance, to talk of functions as
things (presumably abstract) referred to by function-
expressions. Without attempting to decide whether or not
functions exist, many logicians began talking of functions
as kinds of expressions, thereby simply avoiding many of the
philosophical problems which accompany logic.

Besides the reaction to realism, some of Frege's
ideas proved to be too difficult to adOpt. His notion that
all true statements have the same object as a referent, i.e.,
the True, is a good case in point. The difficulty with this
notion, and others like it, is not just that it is abstract.
Rather, it has proved to be easier to assign the difference
in meaning between two statements to their respective

referents, and not to what might be construed as their

5

 

senses (i.e., their thoughts).6 Treating the reference of

 

65Care must be taken, however, in making such a
statement. Some later authors (most notably Carnap) have
attempted, by various means, to preserve the idea that all
true statements have the same denotation. Thus Frege's
conception is not as antiquated as it may appear.

88

statements in such a way as Frege did seems contrived, for
in the case of terms, the difference in meaning is accounted
for in terms of difference in reference. Why this should
not carry over to the treatment of statements is unclear.
From his own pronouncements it seems that Frege had trouble
accounting for the truth statements. Apparently he provided
too powerful a place for it in the semantic relation, and in
doing so, forced that relation into difficulty.

Even though Frege's theory of logic is not without
its difficulties, it still stands as a landmark. It is one
of the few cases in which a real innovator in logic saw the
necessity of making the philosOphical ground of his system
explicit. As we shall see, much of Frege's theory (and even
more of his vocabulary) has been borrowed by later logicians.
But too often such borrowings have been uncritical; too
frequently assumptions are made to lie quietly, and logic

is carried on as if it had no theory.

CHAPTER IV
CHARLES SANDERS PEIRCEl
1. INTRODUCTION.

Logic played a great part in the work of Charles
Sanders Peirce (1839-l9lh). ”If we had to describe Peirce's
vocation in a single word, the most adequate would be
'logician.‘ That was the title he preferred himself; and
certainly, the study of logic was his grand passion which
overshadowed all others."2 The fruits of this interest
were many significant additions to logic, and to the theory
of logic. Unfortunately, many of these advances lay hidden

until well after they could exert a useful influence on the

 

1The following chapter constitutes only an outline of
the theory of logic of C. S. Peirce. While no work on the
theory of logic during the period under consideration could
possibly be complete without a consideration of Peirce's work,
no single chapter could adequately deal with it. Parts of
Peirce's theory of logic are infused into virtually every
part of his philosophy. While in the case of Boole, and
that of Frege, we have observed some points of connection
between logic and, say, metaphysics, the connections have
always been tenuous enough that sense could be made of the
theories without any extensive knowledge of particular
metaphysical or epistemological doctrines. In Peirce's
case, however, we are confronted with a theory which is
inseparable from such fields as ethics, metaphysics, etc.
A full treatment of Peirce's theory would demand that we
first acquaint ourselves with broad areas of his philosophy.
Because such treatment is impossible in a limited space,
this chapter should be treated as a recipe for further study
in Peirce's theory.

2Goudge, Thomas A., The Thought of C. S. Peirce
(Toronto: University of Toronto Press, 1950),— p. 111.

 

89

90

development of logic. Perice published only a fraction of
his writings. As Kneale says, "Unfortunately Peirce was like
Leibniz, not only in his originality as a logician, but also
in his constitutional inability to finish the many projects
he conceived."3 The result of this lack of published
material was that most of his efforts went unnoticed by those
who were nearer the ”mainstream" of the development of logic.
Thus, for example, while Russell knew Frege's work quite
well, and to a large degree shaped his own work around it,

he apparently knew little of Peirce's work, in that he makes

no mention of it in the Principia and only passing mention

 

in Principles.

Some of Peirce's work became available through Lewis'
treatment of it in his Survey 23 Symbolic Logic, published
in 1918, but it was not until late in the 1920's that the
task of publishing his papers was begun. Even today, many
of his papers are not generally available; knowledge of
Peirce's contribution is still far from being complete.

In the next section, we shall look briefly at some of
Peirce's more significant advances in formal logic. This
being completed, we shall turn our attention to a discussion

of his theory of logic.

 

3Kneale, William and Kneale, Martha, The Deve10pment
22 Logic (Oxford: Oxford University Press, 196T), p.fih27.

 

 

91
2. PEIRCE'S LOGIC.

As we shall see in the next section, Peirce's
conception of logic was much broader than Boole's, or Frege's.
Indeed, the kind of logic with which these latter two were
concerned forms only a part of logic, in Peirce's view. It
is as if the logic of Frege and Boole could be mapped onto
Peirce's 10gic. Nonetheless, Peirce was quite active in this
narrower area, i.e., with the vigorous formal logic like that
of Frege and Boole, and it is with this area that we shall
concern ourselves here.

While Peirce's most productive period in logic was
between 1879 and 188U, the years in which he was in
residence at Johns Hopkins University, the full scope of his
dealings in the subject range from 1867 to 1902. During
that time Peirce made such remarkable advances as l) the
introduction of inclusive disjunction into Boolean algebra;
2) the introduction of quantifiers in a manner closely
resembling, but nonetheless independent of, Frege's use of
quantifiers; 3) the introduction of what is now called prenex
normal form, and full conjunctive and disjunctive normal
form; h) the introduction of a tabular method for determining

combinations of truth and falsity of statements which

“Although Jevons recognized the same Operation in
his Principles _o_f; Science. It is not clear which came
ffidrst, but it does not matter; both were genuine instances
0f discovery.

92

closely resembles the truth-tables now in use; 5) the
introduction of a logic of relations; 6) the introduction of
the class-inclusion relation; 7) the introduction of a
system of graphs which serves many of the functions of a
natural deduction system.

In the remainder of this section, we shall discuss
each of these developments by discussing the most important
of Peirce's works in logic, in their chronological order.5

In 1867, Peirce wrote two articles, which (as Roberts
points out (p. 15)) were part of a series of articles which
appeared in the Proceediggs of £23 American Academy 23 Arts
322 Sciences. In the first of these, entitled ”On an
Improvement in Boole's Calculus of Logic,"6 Peirce introduces
the Operation of inclusive disjunction, mentioned above.

The introduction is made in the following manner:
Let the sign of equality with a comma

beneath it eXpress numerical identity...let

a+,b denote all the individuals contained

under a and b together. The operation here

performed will differ from arithmetical
addition in two respects: first, that it

 

5The outline of this section has in part been adapted
from Roberts, D. D., The Existential Graphs of Charles S.
Peirce (typescript)(Urbana, University of Illinois Library,
. Many of the reference used in this section were
Suggested by Roberts' work, especially Ch. II, Sec. B.

6Peirce, Charles Sanders, The Collected Pa ers of
(”larles Sanders Peirce, C. Hartshorne and P. Weiss (eds. )
Cambridge: Harvard University Press, 1931—1935), Vol. 3,
99.. l-l9. Hereafter all footnote references to this work
V111 read as follows, giving as form the volume and para-
graph respectively: CP, 3.1-19.

93

has reference to identity, not to equality,

and second, that what is common to a and b is

not taken into account §wice over, as it

would be in arithmetic.

In the second of these two papers, entitled ”Upon the
Logic of Mathematics,"8 Peirce first uses the sign '2:' to
denote the logical sum, as opposed to arithmetical summation,
although his full use of quantifiers did not come until
later.

In a paper entitled "Grounds of Validity of the Laws
of Logic: Further Consequences of Four Incapacities,"9
Peirce foreshadows the recognition of semantical paradoxes,
which were not explicitly recognized as such until the late
1920's by Ramsey. Of Peirce's discussion of the semantical
paradoxes, Bochenski says, ”former endeavors to solve the
problem of semantic antinomies...seem to have fallen into
complete oblivion in the time of the 'classical' decadence.
Nor did the mathematical logicians know anything about them
until Rustow, with the sole exception of Peirce, who had
read Paul of Venice and had given a subtle commentary on

him in this respect...."10

70?, 3.3f.

8cp, 3.2o-uu.

91868, CP, 5.318 ff.
loBochenski, I. M., A History of Formal Lo ic, Ivo

Thomas (trans.)(South Bend3—Notre Dame—University Press,
1961), p. 387.

9b

In an 1870 paper entitled "Description of a Notation
for the Logic of Relatives, resulting from an Amplification
of the Conceptions of Boole's Calculus of Logic,” Peirce
introduces class-inclusion in a systematic fashion,
employing the sign "<' for the relation in question. Lewis
says of this innovation, ”the relation of 'inclusion in' ...
appears for the first time in the rDescription of a Notation
for the Logic of Relatives.‘ Aside from its treatment of
relative terms and the use of the 'arithmetical' relations,
the monograph gives the laws of the logic of classes almost
identically as they stand in the algebra of logic today."11

Peirce provided for the process of reduction to both
full disjunctive normal form and its dual, the full
conjunctive normal form in a paper entitled "On the Algebra
of Logic."12 As Church points out,13 these forms were not
given their present names until 1922, when Heinrich Behmann
used the term 'disjunktive normalform.’ Nonetheless, it is
clear that Peirce foreShadowed the develOpment of normal
forms: ”Both Eull normal forma were given in 1880 by

Peirce. Schr3der and Peirce state the normal forms for the

 

11Lewis, Clarence Irving, A Survey 2£,S bolic Logic

(Berkeley: University of California Press, 1918 , p. 83.

121880, cp, 3.15u—251.

13Church, Alonzo, Introduction £2,MathematicalLogic
(Princeton: Princeton University Press, 1956), p. 186 n.

 

 

95

class calculus, but extension to the prOpositional calculus

1U

is immediate.”

In another manuscript (untitled)15 written about
1880, Peirce discovered what is now known as the dual of the
stroke function. In this paper, "Peirce sketched a Boolean
Algebra in which the variables took statements rather than
nouns as substituends. In this remarkable fragment
unpublished until 1933, Peirce also anticipated by some
thirty years Sheffer's discovery of the fact that all Boolean

functions could be defined in terms of the single primitive

'neither--nor--' of joint denial."16 In the article in

question, Peirce himself says:

...I have thought that it might be curious
to see the notation in which the number of
signs should be reduced to a minimum; and
with this view I have constructed the
following. The apparatus of the Boolean
calculus consists of the signs =, >Wnot
used by Boole, but necessary to express
the particular prOpositions), +, -, x, l,
O. In place of these seven signs, I
prOposed to use a single one.

I begin with the description of the
notation for conditional or ”secondary"
propositions. The different letters signify
prOpositions. Any one proposition written
down by itself is considered to be asserted.

1“Ibid.

15

16
Berry, George D. W., "Peirce's Contributions to

the Logic of Statements and Quantifiers," Studies in the
..Philosophy of Charles Sanders Peirce, P. P. Weiner, and
I"EZ'A'KIric H. flung, (eds.HCambridge: Harvard University
Press, 1952), p. 151;.

GP, b.12-2O.

96

Thus,

A
means the proposition A is true. Two
propositions written in a pair are con-
sidered to be both denied. Thus,

AB
means that the propositions A and B are
both false; and

AA
means that A is false. ...

17

Few realize the full extent of the similarity between this
passage and the passage in an article by H. M Sheffer
(quoted below, p. 193) in which he quite independently
sets down the joint-denial function.

Peirce gives his clearest statement of his logic of
relatives, as well as a full account of his use of
quantifiers, in a work entitled ”The Logic of Relatives,"
which appeared as a part of Studies 12 Lo ic, 2y Members

2: the Johns Hopkins University.18

Of the developments in
this paper, Kneale says, “what Peirce created was a symbolism
adequate for the whole of logic and identical in syntax with
the system now in use. He himself called his invention the

1
General Algebra of Logic; ...” 9

 

17CP, n.12-15.
180p, 3.328-358.

19Kneale, op. cit., p. #31.

97

As an indication of how Peirce introduced quantifiers,

consider the following statement, made in a later paper:20

In En algebra which Peirce is proposing]
every proposition consists of two parts, its
quantifiers and its Boolian. Its Boolian
consists of a number of relatives united by
non-relative multiplication and aggregation.
... To the left of the Boolian are written
the quantifiers. Each of these is a’n’or a

with one of the indices written subjacent
to it to signify that in the Boolian every
object in the universe is to be imagined
substituted successively for that index and
the non-relative product (if the quantifier
is‘“’) or the aggregate (if the quantifier
is Z) of the results taken. Thus

114.231,, =(11142t12v413 etc.).
(1.2119122‘P‘623 etc.) etc.,

will mean everything loves something. But
Zj’m’bu =111.¢21. L31. etc.+' .‘(:12.
"(’22. 132. etc. \P'13"23' etc. ‘P etc.

will mean something is loved by all things.22

21

One of the most original papers by Peirce is one

entitled “On the Algebra of Logic: A Contribution to the

 

20Also entitled ”The Logic of Relatives,” (CP,
3.U56-552) although it is essentially a review of some
of SchrOder's work, and hence is fairly unlike the
earlier article of the same title.

21
Peirce defined "Lfiyb' as follows: “I write
l4’b calling this the operation of non-relative addition,

or more accurately, of aggregation." (CP, 3.U95:7

220R, 3.u98-502.

98

PhilosOphy of Notation."23

In this paper, written in 1885,
is found the precursor of modern truth-tabular technique, a
reiteration of quantification theory, a method for reduction
to prenex normal form, a modern defintion of identity, the
”Peirce-Dedekind“ defintion of an infinite class, and a
justification of the use of material implication in formal
logic.

With regard to Peirce's use of two truth-values, and
to his use of truth-tables, Church maintains that "The
explicit use of two truth-values appears for the first time
in @e paper mentioned abovej."2u Peirce introduces truth-
values in the following manner:

Now an assertion concerning the value

of a quantity either admits as possible or

else excludes each of the values v and f.

Thus, v and f form the set objects each

connected with one oniy of n objects,

admission and exclusion. Hence there are,

nm, or 22, or 5; different possible

assertions cogcerning the value of any
quantity, x.2

 

Peirce follows this passage with an actual graph to depict
the possible arrangement of values. While this graph bears

little resemblance to those in use today, one which he uses

 

23oF. 3.359-u03.

2”Church, Op. cit., p. 25.

25'v' and 'f' are used by Peirce to stand for truth
and falsity, respectively.

26CP, 4.212 ff.

99

later in the same passage bears a great deal of such
similarity.

With regard to the introduction of prenex normal
form, Church points out that the "use of the prenex normal
form was introduced by C. S. Peirce, although in a different
terminology and notation. Peirce uses the term 'Boolian'27
for what we here, following Principia Mathematica, call the
matrix, and speaks of the prefix as ‘Quantifier' or
'quantifiers."'28

We have now provided ourselves with the barest
indication of some of Peirce's more outstanding contribu-
tions to the develOpment of logic. We have not, for instance,
even touched on his unpublished and unfinished book entitled
Grand Logic (parts of which, unfortunately, are scattered
throughout the Collected Papers). From this work, one can
glimpse the fruits of Peirce's great desire for systemati-
zation. But to pursue fully the many details of his logic,
while interesting, would involve us in an unmanageably large
task.

I Before moving on to a discussion of Peirce's theory
of logic, a few comparisons with Frege are in order.

First, and perhaps most importantly, while Frege's

logic arose from mathematics in order to form a basis for

 

27Cf. quote on p. 98.

28
Church, Op. cit., p. 292.

100

mathematics, Peirce's logic had broader origins. Peirce's
logic originated from many inter-connected points; from his
knowledge of mathematics, to be sure, but also from his
knowledge of Aristotelian logic. It arises from the demands
of his epistemology (as we shall see), and from his desire

to formalize such sciences as -crystalography. As a
logician, Peirce was in many ways a broader intellect than
Frege. Frege, as we have seen, was influenced by individuals
who were mostly contemporary with him, and were also
mathematicians.

Second, it should be now apparent that both Frege and
Peirce were genuine innovators and systematizers. Their
discoveries are remarkably similar, especially since they
had no knowledge of each other's work. In the case of the
introduction of quantifiers alone, for example, not only are
their additions to quantification-theory similar, they were
also made at very nearly the same time.

This temporal proximity with which Frege and Peirce
made their discoveries points to something further; it is
much more than coincidence that their similar contributions
were made at the same time. In fact, logic was, at that time,
at a stage of development at which such developments as those
made by the authors in question were appropriate. That is,
the foundations for such advances had been laid. All that
were needed was intellects of the magnitude of Frege and

Peirce.

101
3. PEIRCE'S REMARKS ON ALTERNATIVE THEORIES OF LOGIC.

As we saw in the last chapter, Frege's theory of
logic was developed largely as an alternative to various
other theories with which he had come in contact. To a great
extent, this is also true of Peirce's theory. Once again
the only difference is that the sphere of influence on
Peirce was more extensive than that which affected Frege.
It is especially interesting to note, however, that in
several cases, the aversions of the two logicians coincide.
This is the case with regard to their attitude toward, for
example, idealist logicians such as Erdmann, etc.

In order to make this comparison between Frege and
Peirce clear, and in order to provide a foundation for the
discussion of the central features of Peirce's own theory,
we shall now turn to a discussion of Peirce's reaction to
several other schools of thought in the theory of logic.

Peirce apparently ”...felt obliged to consider and
give sufficient arguments for rejecting certain of the
classical theories of logic which found defenders in his own
day."29 In several places in his writings,30 he comments
on more than a dozen different theories. Some Of these, while

important to Peirce, have grown obscure enough in the

 

29Feib1eman, James K., Ag Introduction to Peirce's
Ehilosophy (New Orleans: The Hauser Press, 19h37, p. 81.

30Mainly at CP, 2.15 ff.

102

intervening years that they require no separate discussion
here. There are several, however, which require special
attention, and we shall now deal with these.

Perhaps the most complex, and at the same time more
interesting, of Peirce's attitudes concerning other theories
deals with the relation between logic and psychology.
Concerning this relation, Peirce says at one point that "my
principles absolutely debar me from making the least use of
psychology in logic."31 In making such a statement he was
arguing against any position which seemed to confuse the two
disciplines. He was 223, however, suggesting that
psychological concerns are totally irrelevant in logic, but
that logic itself is essentially non-psychological.

In taking up the matter of the nature of the relation
between psychology and logic, Peirce dealt with what he
called the "German logicians.” These included Sigwart,
Wundt, and Erdmann, whose doctrines were discussed in the
last chapter. It should be pointed out again, however, that
when these individuals are accused of failing to make a
distinction between psychology and logic, that this charge
is correct. Erdmann's Lo ik,32 for instance, is really a
text-book in idealism of a heavily psychological sort. In

this massive work one finds little of what we are at all used

 

31C?, 5.157.

Erdmann Benno Logik Halle A. S.: M. Nieme er
1892). , ’___g___( 9 y 9

103

to thinking of as logic.33 Thus, when Peirce's criticisms
of the ”German school" seem to be aimed at a theory of logic
which no individual has in fact held, one must remember just
how “psychological” these logicians in fact were.

The reason that Peirce found any mixture of logic and
psychology unacceptable was that "the psychological question
is what processes the mind goes through. But the logical
question is whether the conclusion that will be reached, by
applying this or that maxim, will or will not accord with the
£323."3u Furthermore, I'...psychologica1 theory substantially
ends where consciously controlled thought begins, with which
alone logic has even an indirect connection."35 Peirce's
claim, that is, is that strictly speaking, essentially
psychological concerns are irrelevant for logic, although
there is a connection of a contingent sort which holds

between them.

 

33As a sample of the kind of subject-matter dealt

with in Erdmann's book, consider the following passage
selected at random:

To the extent that a mass of perceptions
presents the same thing as have previous stimuli
and the excitations triggered by them, it
reproduces the memory-traces that stem from the
sameness of previous stimuli and amagamates

with them to form the object of the

apperceived idea. (Logik, Vol. I, p. #2)

 

 

Nearly all of the rest of the work resembles this passage.

3“or, 5.85.

350?, 2.63.

10h

As to which of these two disciplines, logic or
psychology, is more fundamental than the other, Peirce says
that ”Psychology must depend in its beginnings upon logic,
in order to be psychology and to avoid being largely logical
analysis. If then logic is to depend upon psychology in its
turn, the two sciences, left without any support whatever,
are liable to roll in one slough of error and confusion."36

When Peirce speaks of psychology, he thinks of it in
a fairly broad sense. In his criticism of various theories
of logic, he actually makes a three—fold distinction among
theories which are in some manner connected with psychology.
On the one hand, he deals with Erdmann and Wundt, who are
indeed idealists, but who claim to derive logic from what
they call ”epistemology." Secondly, he deals with those who
confuse logic with the 2233 of psychology.37 Such
individuals claim, according to Peirce, that we are shown
the truths of logic by observing how we think. In criticiz-
ing this view, Peirce reveals a good deal about his own
attitude toward the relevance of psychology for logic:
”Logic is not the science of how we do think; but, in such
a sense as it can be said to deal with thinking at all, it

only determines how we ought to think; nor how we ought to

36C?, 2.51.

37Cf., CP, 2.52.

105

think in conformity with usage, but how we ought to think in
order to think what is true."38

Peirce's criticism of Mill also comes under the
heading of his discussion of the relation between psychology
and logic. In particular, Peirce says, "it is J. 8. Mill who
insists that how we ought to think can be ascertained in no
other way than by reflection upon those psychological laws
which teach us how we must needs think.”39 Peirce objects to
this, maintaining that if in a given case, Mill holds that

”there is pp other reason to be given for thinking in a

 

given way than barely that the thinker is under complusion

so to think, is he not applying that Criterion of
Inconceivability against which we have heard him fulminate in
his finest style?"uo Against Mill's arguments concerning
compulsion and thought, in which he describes how we are
”forced" to draw some conclusions and-not others, Peirce
further says, "the evident truth is, that psychology never
does prove a compulsion of thought of an absolute definitive
kind for conscious operations of the mind."u1 Peirce then
concludes his criticism of Mill by maintaining that he cannot
understand how Mill can support such a position as that

outlined above.

 

381b1d.

39cp, 2.u7.

uoIbid.

“lop, 2.u9.

106

Peirce's criticism of Mill is not so much interesting
for the insights it provides into logical or psychological
controversies (for the issues with which Peirce was
concerned have to dowwith Mill's associational psychology
which has few proponents today), as it is of interest with
regard to the difference between Peirce's criticism of Mill
and Frege's criticism of Mill. In particular, what disturbs
Peirce in Mill's work does not seem to bother Frege, and
zips-versa. Peirce is totally undisturbed by Mill's
empiricism in his theory of logic, and Frege is just as
unconcerned with his psychology. On the other hand, Frege
was strongly opposed to his empiricism, and Peirce to his
psychology.

While against any view which confused logic with
psychology, or which tried to ground all or part of logic
in psychology, Peirce was well aware that there was still
some sort of connection between psychology and logic. He
says at one point, for instance, that "formal logic must not
be too purely formal; it must represent a fact of psychology,
or else it is in danger of degenerating into a mathematical
recreation."n2 What fact of psychology logic represents will
become clear when we discuss the heart of Peirce's theory;
it is enough now to simply note that Peirce had in mind a

view in which logic had to do with certain psychological

 

107

states of affairs. To hold this is, however, perfectly
consonant with the view that psychology and logic are
completely separate sciences.

Besides this commentary on the relation of psycholog
to logic, Peirce discusses several other theories. Among
these is the view that "the goodness and badness of reasonings
is not merely indicated by, but is constituted and composed
of the satisfaction and dissatisfaction, reSpectively, of a
certain logical feeling, or taste, within us."L,3 Peirce
does not offer a formal criticism of this view, in that he
offers no explicit suggestions as to what might be wrong with
it. It is clear, however, that Peirce strongly disapproves
of such a view. Consider, for example, his claim that "Judged
by English standards...Sigwart's teaching is calculated to
undermine the vigor of reasoning, by a sort of phagedenic
ulceration. So it would seem 3 riori; and g posteriori the
impression made upon me by young reasoners who have been
most diligent students of Sigwart is that of debility and
helplessness in thought.”uu Peirce apparently thought that
such a position, once eXposed to plain view, would die a

natural death.

 

108

The next view which Peirce considers is that put forth
by Gratry.“5 This view revolves around the idea that logical
principles are to be somehow based on “direct individual
experiences."u6 But Peirce claims that “since the principles
of logic are general, only a mystical experience could give
them."“7 Peirce goes on to maintain that Gratry did in fact
hold such a view, in that he claimed that induction is the
process of proceeding from the finite to the infinite. Once
again, Peirce's remarks about this position do not constitute
a direct criticism, although his aversion to any such
position is clear from the way in which he describes the
position itself.

Another position which Peirce considers is the claim
that the principles of logic are to be founded on the light
of reason. He says, for example, ”...the Aristotelians, who
compose the majority of the more minute logicians, appeal
directly to the light of reason, or to self-evidence as the
support of the principles of logic."u8 Before proceeding to
a criticism of this view, Peirce first maintains that
Aristotle himself did not really believe this. Peirce

criticizes the view in question by maintaining that self

“5A Priest of the Sorbonne in the mid 19th century,
and author of £3 Lo 1 ue, (Paris, 1855).

“6cr., CP, 2.21.

“71bid.

”BOP, 2.27.

109

evidence requires some final justification itself, since it
requires a retrogression "to a first demonstration reposing

1+9

upon an indemonstrable premiss," and to call upon, for
instance, God, to bolster such self-evidence is to "call
that in question which they intended to prove, since the
prodfs, themselves, call for the same light to make them

50

evident." Once again, Peirce's criticism is not a strong
one. The brevity with which he treats many of these positions
makes it seem that he did not consider them to be a serious
threat to the position which he himself maintained. Many
of his criticisms seem to be in the nature of a simple
cataloging of various historical positions.

One of the more interesting positions which he
considers is that in which logic is held to be founded
"upon a philosophical basis.” Peirce does not associate this
view with anyone in particular; it is likely that he intended
it as another criticism of the idealist logicians, with
whom he had previously dealt. With regard to this view,
Peirce says, ”to me, it seems that a metaphysics not founded
on the science of logic is of all branches of scientific

inquiry the most shaky and insecure, and altogether unfit

for the support of so important a subject as logic, which

 

“90?, 2.28.

Soap, 2.29.

110

is, in its turn, to be used as the support of the exact
sciences and their deepest and nicest questions."5

In connection with this view, Peirce once again takes
steps to vindicate Aristotle, whose logic is indeed
apparently based on metaphysics. But, says Peirce,
"Aristotle evidently bases the metaphysics upon a grammatico-
logical analysis of the Greek sentence."52

While Peirce considers several other views, they are
of little enough consequence to need no discussion here. But
before proceeding on to a discussion of his own theory, there
are two considerations concerning his criticism of other
views worth mentioning. First, as is by now obvious, Peirce
was intimately familiar with virtually every major theory
of logic which had ever been seriously held. This fact alone
makes Peirce quite unique. As has been mentioned previously,
Peirce's intellectual background was much broader than any
other logician of his time. Having so much more to draw
upon, his own theory of logic can be seen to be all the more
subtle. Secondly, Peirce offered no strenous objection to
any of the other theories which he considered. Instead,
after considering them, and offering only modest criticisms,
luathengoes on to develop his own view as an alternative to

all of these other views. As we examine Peirce's own theory,

 

51CP, 2.36.

520p, 2.37.

111

our task will be in part to determine the ways in which his

theory is in fact superior to all of those which he mentions.
h. PEIRCE'S THEORY OF LOGIC.

Peirce considered logic to be £22 science pf 3312:
controlled, deliberate thought. How he comes to this view,
and what is involved in it will occupy our attention in this
section.

Actually, Peirce's theory of logic is broad enough
to require some initial structuring. Accordingly, besides
the definition given above, and its immediate explanation,
we must consider the following questions: 1) What is the
place of logic among the various sciences?; 2) What does
logic depend upon? That is, what are the fundamental
principles which logic assumes? 3) What does the definition
of logic imply? That is, how are we to unravel the definition?
Such structuring is necessary in dealing with Peirce's
theory simply because his theory is quite unlike any of the
others with which we shall deal.

Peirce is one of the three logicians in the entire
history of logic who incorporates lOgic into an entire
world-view (the other two being Leibniz and Aristotle).
Since we cannot deal with the entirety of this world-view,
our discussion will of necessity be incomplete. We can only
hope to come to gripes with the general outline of the

theory.

112

Peirce's classification of sciences53 proceeds as
follows: All science, he claims,is either A. Science of
discovery, B. Science of review, or C. Practical science. A
science of review would be one dealing with ordering the
results of science of discovery, while by practical science,
Peirce means ”all such well-recognized science now ipfiactu,
as pedagogics, gold beating, etiquette, pigeon-fancying..."5h

Sciences of discovery fall under three headings: A.
mathematics; B. philosophy; C. idioscopy. According to
Peirce the distinction between these three is that

Mathematics studies what is and what is

not logically possible, without making itself

responsible for its actual existence. Philoso-

phy is positive science, in the sense of

discovering what really is true; but it limits

itself to so much of truth as can be inferred

from common experience. Idiosc0py embraces

all the Special sciences, which are principally
occupied with the accumulation of new facts.

Continuing with the expanding classification, Peirce
Vdivides philosophy into three branches: A. phenomenology;
B. normative science; C. metaphysics. Phenomenology can be
initially understood as that science which deals with the
most pervasive or universal features of the common experience

which is mentioned in the quotation above. Philosophy rests

on experience, and it is phenomenology which describes this

 

53cr., CP, 1.180 ff.

Shop, 1.243.

550?, 1.18u.

113

experience. Metaphysics, on the other hand, "is the science

of reality."56 Furthermore, ”its business is to study the

most general features of reality and real objects."57 It is

a science whose ”...main principles must be settled before

very much progress can be gained either in psychics or in

8
physics.'5

claims

Continuing with this expanding classification, Peirce

that

Normative science has three widely separated
divisions: i. Esthetics; 11. Ethics;
111. Logic.

... ...Ethics, or the science of right and
wrong, must appeal to Esthetics for an aid in
determining the summum bonum. It is the theory
of self controlled, or deliberate, conduct.
Logic is the theory of self-controlled, or
deliberate, thought; and as such, must appeal
to ethics for its principles. It also depends
upon phenomenology and upon mathematics.
Although performed by means of signs, logic
may be regarded as the science of the general
laws of signs. It has three branches: 1,
Speculative Grammar, or the general theory of
the nature and meanings of signs, whether they
be icons, indices, or symbols; 2, Critic, which
classifies arguments and determines the
validity and degree of force of each kind; 3,
Methodeutic, which studies the methods that
ought to be pursued in the investigation,o£n
the exposition, and in the application of
truth. Each division depends on that which
precedes it.59

560?, 5.121.
570?, 6.6.
580?, 6.u.

59CP, 1.191.

11h

Here at once, we not only have logic placed in relation to
all the other sciences, but we also have an outline of the
presuppositions on which logic rests, together with points
which follow from the definition given. In particular, we
shall first discuss how it is that logic appeals to ethics,
to phenomenology, and to mathematics. We shall then turn to
a discussion of the parts of logic as it is regarded as the
theory of signs, i.e., to speculative grammar, critic, and
methodeutic.

With regard to the relation between ethics and logic,
part of Peirce's thinking on this matter should already be
clear; for when he maintains that logic is not about how we
22 think, but about how we 22523 to think, the evaluational
aspect of logic becomes quite clear. In the definition of
logic already quoted, it is the aspect of self-control, and
deliberateness which bears a relation to ethics. The reason
for this, as we have already observed, is that Peirce defines
ethics as the science of self-controlled conduct. Burks
claims that ”as a pragmatist,(§eirc§}held thinking to be a
kind of conduct, from which it follows that 'self-controlled
thinking Eeasonina is a special case of self—controlled

6O

conduct." Thus, since ethics studies self-controlled

(Bonduct in general, logic has to rest on ethics.

6OCP, 5.533. Burks, Arthur, "Peirce's Conception
of Logic as a Normative Science," Philosophical Review
(Ithaca: The PhiIOSOphical Review, 1933), No. 52,
PP. 191-192.

115

While at some places it appears that Peirce wants to
make logic a branch of ethics?1 his attitude seems basically
to be that the two sciences are in fact separate, but very
closely analogous. He claims at one point that "the
righteous man is the man who controls his passions.... A
logical reasoner is a reasoner who exercises great self-
control in his intellectual operations."62 At another place,
he claims that ”logical norms...correspond to moral laws."63
That is, he seems to be maintaining that logic resembles
ethics in its use of normative principles. Thus when he says
that logic appeals to ethics for its principles this should
not be construed as meaning that logic therefore becomes
merely a part of ethics.

Since we shall not delve further into Peirce's
conception of the nature of ethics, what has already been
said on the topic of the relationship between logic and
ethics will have to suffice. The relationship is clear as
long as it is remembered that Peirce conceived of ethics as
the study Of the deliberate and conscious modificationOOf
habits. Since it studies controlled conduct, it is closely
related to logic.

The question as to how logic depends on phenomenology

is somewhat more complicated than the same question concerning

 

61Cf., CP, 1.611; 1.573.

62CP, 1.608.

63oF, 1.609.

116

ethics, for his views on phenomenology are a good deal more
difficult to understand than are his views on ethics. Peirce
conceived of phenomenology as the science dealing with the
most pervasive and universal features of experience. Goudge
explains the relation between logic and phenomenology by
saying that for Peirce, ”phenomenology too, must have
something to contribute‘Ep logié], inasmuch as the categories
it discovers, if they are genuinely universal, will
inevitably find exemplification in the pattern of human
reasoning. Peirce relates that they did indeed serve him

as a key wherewith to 'unlock many a secret' in the domain

of logic.”6u Thus, insofar as the fabric of experience
influences the fabric of reasoning, the science of reasoning
depends on the science of experience.

While we shall not deal in detail with Peirce's
phenomenology, we should at least discuss enough of it to
make the relation between Peirce's doctrine of the
categories and his account of reasoning a bit clearer.

The doctrine of the categories, which comprises
Peirce's phenomenology65 takes the categories to be the
basic types of phenomenal experience. Peirce delineates, in
one of the best-known parts of his writing, three categories.

The categories are held to be universal in the sense that

 

6uGOUdge, OE. Cite, p. 1130

65Cf., CP, 1.280.

117

each phenomenon is related to each of the categories, even
though one of them may be more prominent than another in a
given phenomenon.

The category of firstness may be tentatively described66
as the immediate element in all experience. It is, to
borrow Quine's phrase, the ”fanciless medium of unvarnished
news."67 The difficulty, according to Peirce, is that because
of its immediacy, firstness cannot be described. He suggests
that we can arrive at a notion of it through a suitable
process of abstraction, from such notionras that of
particular qualities. Peirce himself explains firstness
best, when he says,

It cannot be articulately thought; assert it,

and it has already lost its characteristic

innocence; for assertion always implies a

denial of something else. Stop to think of

it, and it has flown! What the world was to

Adam on the day he opened his eyes to it,

before he had drawn any distinctions, or had

become conscious of his own exiggence--that

is first, present, immediate...

While secondness, like firstness, cannot be described but

only alluded to, it appears as something more familiar to

 

66It should be understood that the following

description of the categories does not constitute an attempt
completely to describe Peirce's theory. What is given is a
mere outline.

67Quine, Willard Van Orman, Word and Object (New York:
John Wiley and Sons, 1960), p. 2.

580p, 1.357.

118

us. This category is ”one which the rough and tumble of life
renders most familiarly prominent. We are continually
bumping up against hard fact."69 As Goudge says, the world
of experience is ”not merely qualitatively such-and-such; it
is also ineluctably before us, an actuality, hip 3£.pggg.
The here-ness and now-ness are not reducible in any way to
a matter of spatio-temporal location. Nor can they be
reconstrued in terms of quality. Together they constitute
an ultimate feature of phenomena, -best designated by the
word‘gggg.”7o

Understanding the category of thirdness is of the
greatest importance, for it is with this category that logic
is most closely concerned. As Opposed to qualitativeness
and factuality, thirdness is the generality and lawfulness
which govern things. It is thirdness, for instance, which
accounts for our ability to predict future events on the
basis of observed cases. ”Phenomenological inquiry, then,
finds the most prominent illustrations of Thirdness to be
generality, law, meaning, representation, mediation,
continuity, triplicity, and thought or inference."71 Logic

is concerned with thought, and thought is one of the things

which arise "from the examination of the manifold of

 

69CP, 1.320.
70G0Udge, 02¢ Cite, p. 880

711bid., p. 93.

119

experiencé], and are observable features of it. This point
will only be clear if we remember that for Peirce observation
is a process which includes a judgmental as well as a non-
judgmental factor.”72 Since Peirce thinks of reasoning in
terms of the category of thirdness, how logic rests on
phenomenology becomes clear. Furthermore, there is another
tie between logic and phenomenology; the category of thirdness
can also be thought of as accounting for the presence of
purpose, since purpose is one of the uniting features of
experience. Hence, the normative character of logic also
rests, in part, on phenomenology. As Burks says, Peirce
"...thought that the view that logic analyses reasonings does
make it normative. This is because he conducted his analysis
of reasoning by means of his category of Purpose or
Thirdness."73

Logic depends on the third area mentioned above, i.e.,
mathematics, in a somewhat different manner than it does on
the other two disciplines already discussed. We have already
seen how Peirce accounts for the difference between
mathematics and logic; mathematics studies that which is
logically possible, while logic (as a part of philosOphy),
studies that which is really true. There is, however, an

important and strong relation between them; mathematics, in

 

72
73

Ibid.

Burks, OE. Cite, p. 189 no

120

Peirce's view, is the practice of necessary or deductive
reasoning. Logic, on the other hand, is the theory of such
reasoning. In this sense, logic is about mathematics. Thus,
”in contrast to mathematics, which reasons deductively,
deductive logic studies deductive reasoning or arguments....
Thus logic as a whole analyses arguments to discover their

.7“

nature. Peirce himself says that the business of logic is

the ”analysis and theory of reasoning, but not the practice
of it.”75 Buchler remarks that Peirce says that ”an interest
in the essential elements of mathematical hypothesis and of
deductive processes 'in their intellectual pedigress and in
their conceptual affiliations with ideas met elsewhere...is
the logical interest, p35 excellence.”76
We have now seen how logic, as the theory of self-
controlled deliberate thought, rests on ethics, phenomenology,
and mathematics. We now must discuss, in outline, the nature
of logic itself. The points to be covered are first, the
transition from talk of thought to talk of language, and then
the divisions of the subject, construed in linguistic terms.
The key to the transition from talk of thought to talk

of signs is Peirce's contention that all thought is performed

1.
7 Ibid., p. 188.

75

760?, b.320. Buchler, Justus, "Peirce's Theory of
LOgic,' Journal pf PhilOSOphy (New York: The Journal of
Philosophy, Inc., 1939), Vol. xxxvi, p. 21h.

CP, b.13h.

121

77

by means of signs. Therefore, according to Peirce, logic
can be considered to be the science of the general laws of
signs. As he says, logic is ”the science of the necessary
laws of thought, or, still better (thought always taking
place by means of signs), it is general semeiotic, treating
not merely of truth, but also of the general conditions of
signs being signs."78 In conceiving of logic so broadly,
Peirce thought it to be of advantage to conceive of two
senses of logic. The one just expressed in the quotation
above is the broader sense, while ”in its narrower sense, it
is the science of the necessary conditions of the attainment
of truth.'79 This narrow sense is actually the traditional
sense, and encompasses the logic of those individuals with
whom we have already dealt. But ”...between logic in Ehe
narro€]sense and logic in the broader sense there is no
conflict, for the simple reason that the former...becomes a
branch of the latter. In this broader sense logic is nothing
else than the general theory of signs, which Peirce calls,
after Locke, 'semiotic'.”

To be sure, Peirce's conception of semiotic did not

contain the same classifications (developed by Carnap) that

 

77Cf., CP, 1.191.
78cp, 1.uuu.
79Ibid.

80Buchler, op. cit., p. 197.

122

we now attach to that notion, viz. syntax, semantics, and
pragmatics. Nonetheless, even though his classification is
not as sharp as ours, it still covers the same territory.
Peirce also makes three distinctions within semiotic:
Speculative grammar, critic, and methodeutic (or speculative
rhetoric). Speculative grammar is like both semantics and
syntax, in that it deals with the nature and meaning of
signs. Critic is like semantics, but is restricted to concerns
having to do with the nature and validity of arguments.
Methodeutic resembles Carnap's conception of methodology,
although it also in part resembles pragmatics.81

For the first time, then, we can see the full sc0pe
of Peirce's conception of logic. Logic for him was truly the
entire theory of signs. As we shall see, all the other
cOnceptions of logic with which 13 have been dealing fall
under the heading of critic. But for Peirce, formal (i.e.,
mathematical) logic was only one area (and a fairly narrow
one at that) amongst a broad spectrum of concerns with signs.

What remains for us, then, is a discussion of these
three areas which actually constitute logic for Peirce.

This being done, we will have covered, though in outline,

all of his theory.

 

81Although care should be taken when speaking of the
present-day classification; the three fields are still not
definable with perfect precision.

82Actually, Peirce's full theory of signs is extremely
complex, besides being fascinating. Since discussion of this
theory would lead us far afield, we must once again resort to
outline.

123

Speculative grammar is best understood as Peirce's
theory of lingusitic meaning. As he says, it is ”the
doctrine of the general conditions of symbols and other signs
having the significant character."83 Thus, "... the division
of all signs into terms, sentences, and inferences belongs
to speculative grammar.”8u

In general, a sign (called a 'representamen' by
Peirce) is "anything which determines something else (its
interpretant) to refer to an object to which itself refers
(its object)."85 This conception of the function of signs
differs radically from, say, Frege's conception. Peirce is
one of the few thinkers to realize that in the relationnof
referring, the individual doing £23 referring plays an
integral role. While Frege was content, as is even now the
custom, to talk simply about the relation between signs and
things, Peirce's conception of signs unfolded in the following
manner: first, every sign stands for an object independent
of itself. This is called its object. Furthermore, ”it
stands for an object to somebody (or something) in whom it
arouses a more developed sign, the interpretant. And,

finally, a sign stands for an object to an interpretant in

 

8309, 2.93.

BuBuchler, op. cit., p. 199.

850?, 2.203.

12b

some respect, that is, it represents the 'common characters'
of the object, and in this respect is called the ground."86

In point of fact, Peirce's theory lacks the simplicity
displayed by current accounts of how signs function. He
allows, for example, for signs which are in no way concrete,
i.e., for those which are of a quite different sort than
those which can be Observed. Nonetheless, the effect of his
theory is the inclusion Of pragmatic considerations in the
reference relation. Peirce is aiming, that is, at the broad-
est theory Of signs possible.87

Thus when he says that speculative grammar "has for
its task to ascertain what must be true of the representamen
used by every scientific intelligence in order that they
E.e., the representamerflmay embody any meanin ,"88 he means
that it must account not only for the relation of the sign
to its object, but must also take its user into account.

Critic (or Critical Logic) "is concerned with the
types and degree of validity of inferences. Its classification

of inferences is effected by laying down rules that define

8
and govern them." 9 That is, critical logic includes

 

86Feibleman, OEe Cite, pe 89e

87Hence, Peirce's speculative grammar also resembles
Carnap's concept of pragmatics; this again serves to show
that the relation between Peirce's classification and
Carnap's classification cannot be strongly drawn.

88CP, 2.229.

8gBuchler, ibid.

125

essentially all of what we have heretofore called 'logic.'
More particularly, it includes all of the advances made in
formal logic by Peirce, which were discussed earlier. In this
sense, all of ”ordinary" logic is subsumed by Peirce's

broader conception of the subject. Peirce realized how
limited other logics were in relation to his own; on several
occasions, he denounces ”mere formalism” which amounts, in

his view, to the mere construction of calculi. Peirce singled
critic out for special consideration by identifying the

narrow sense of logic with it. In this case also, he displays
his awareness that his conception of logic subsumes the

logics of both of his best-known contemporaries (i.e., Frege
and Russell), and most of his predecessors.

Under the heading of critic, Peirce also provided a
complete analysis of the structure of the syllogism. In
particular, he discusses at length the role which terms play
in syllogisms, i.e., how they function in extension and in
intension. Next he analyses propositions, pointing out how
propositions may express existing states of affairs or as
expressing possibility. In this dichotomy, he uses Peter of
Spain's classification of £3 inesse and 222.2222: Further-
more, Peirce points out how hypothetical prOpositions embrace
all others.

At one point, Peirce calls speculative rhetoric

0
(methodeutic) the "highest and most living branch of logic."9

 

9°CP, 2.333.

126

Further, he says that speculative rhetoric "would treat of
the formal conditions of the force of symbols."91 What
Peirce has in mind is a study of the logical aspects of the
employment of signs -- a study not unlike present-day
pragmatics, (but also, as was mentioned earlier, like
methodology). In fact, the two areas of concern are
particularly alike in that Peirce, while wishing to take the
notions of purpose and of the function of the interpreter of
signs into account in his logic, was still quite intent on
avoiding the introduction of any purely psychological concerns
into logic. For instance, in his definition of a sign,
Peirce takes into account the interpretant, as we have
already seen. But the interpretant is not simply the
interpreter, i.e., the person involved, but the effect pf £23
sign on the interpreter. The interpretant, that is, is,
roughly speaking, a sign produced in the interpreter.
Speculative rhetoric, then, while concerned with the use,
i.e., the force of signs, is still an analysis of signs.
Speculative rhetoric (which Peirce also calls methodeutic)
bears, in this sense, a resemblance to a broadly conceived
system of methodology; it is concerned, as Peirce says,

with the methods for the investigation, exposition, and

application of truth.

 

910?, 1.559.

127

We have now completed the outline of Peirce's theory
of logic. In order to add meat to the bones which have been
provided, one would have to complete the discussion of
Peirce's theory of signs, besides dealing fully with his
ethics, phenomenology, and theory of mathematics. If this
were done, one would have the full scope of Peirce's theory
of logic at his command. As it is, we have been able only to
mark out the relative proportions of his theory in such a

way that they can be added to at leisure.
5. CONCLUSION.

Peirce's theory of logic fits into the pattern of the
development of the theory of logic in an odd manner. No one
else whom we have discussed, or whom we shall discuss,
constructs his logic in order to meet philosoPhy at so many
places. Logic for Peirce shades off into philOSOphy in
virtually every direction in which philosophy seems to be
headed; into ethics, esthetics, phenomenology, epistemology,
analytic philosophy, not to mention such extra-philosophical
areas as natural science, etc. We have seen how he relates
logic to each of these fields in such a way as actually to
gain something Of itself from each of them.

To be sure, Frege provided enough of a theory so that
we might be able to construct the rest of a purely
philosophical system from his attitudes concerning aspects

0f logic. But again, logic appears to be free-standing for

128

Frege. For him it only required enough explanation of an
extra-logical sort to prevent peOple from misinterpreting
the structure and significance of his system. Peirce's
attitude, on the contrary, is that a free-standing logic is
empty and quite useless; it must be firmly grounded in every
area which it touches.

The breadth of Peirce's intellect should now be
apparent; he was not only fully aware of the work in logic
being done by some of his contemporaries, but he was also
fully aware of the use to which logic might be put (e.g., in
analyzing communication, etc.), and of the limitationsof

logic.

CHAPTER V
PRINCIPIA MATHEMATICA, I
1. INTRODUCTION.

Any discussion of the theory of logic associated with
Principia Mathematica must run afoul of two difficulties.
First, the Principia is not noted for the theory of logic
which is contained largely in its introductions. The
massive system of logic Of the Principia so overshadows
questions of interpretation, that difficulties in interpreta-
tion are usually simply overlooked. Many of these
difficulties have been resolved in the works of later writers,
including the later works of the authors themselves. But
even where difficulties in interpretation persist, they are
usually of such a nature that they can be held in abeyance
while one works with the logic itself. Questions on matters
of theory have had relatively little effect on the general
interpretation of the Principia.

The second major difficulty is that Alfred North
Whitehead (1861-l9h7) and Bertrand Russell (1872-) took
conscious steps to avoid enmeshing themselves in controversies
concerning the theory of their logic. They attempted to keep

the Principia phiIOSOphically antiseptic. In this reSpect,

 

it differs from the rest of the work of both authors. The

£EE$2212122.2£ Mathematics, for instance, is concerned to a

 

large degree with theory, although most of the discussion

129

130

falls within the realm of mathematics. After the Principia,
of course, both authors went on to distinguish themselves in
their respective endeavors.

To say that Russell and Whitehead paid little
attention to theory is not to say that they were uninterested
in questions of interpretation. Topics in the theory of
logic, according to the authors, simply had no place in the
task which they had set for themselves. The task, of course,
was the reduction of mathematics (including geometry) to
logic, and the treatment of the antinomies which had been
exposed earlier by, among others, Russell. Accordingly, an
attempt was made in the Principia to adopt a disinterested
view of the theory of logic, to shelve such problems in order
to deal more adequately with the central task. There is
little direct reference to this attitude of disinterest by
Russell and Whitehead, although at one point they maintain
that ”We have...avoided both controversy and general
philosophy, and made our statements dogmatic in form."

In the face of all this, the pressing question seems
to be, why worry about the theory Of logic of the Principia
at all? In the light of the Sparseness of theory, and the
authors' disinterest, it seems inappropriate to delve into

such theory. Perhaps attention to, for example, Russell's

 

1Russell, Bertrand and Whitehead, Alfred North,
Principia Mathematica (Cambridge: Cambridge University Press,
1927), p. v. Hereafter all footnote references to this work
will read as follows giving page number after the initials:
PM, pe ve

131

later writings on the subject would be more rewarding for
Russell went on to hold some interesting and controversial
positions in the theory of logic. But again, our task here
is to evaluate critically those theories of logic which were
associated with the greatest advances 12.12512 during the
last century.

To disregard the theory of logic of the Principia
would be unfortunate in many respects. Although the theory
is sparse, it is of vast importance. The logic of the
Principia had such an overwhelming influence on subsequent
developments in logic, that the theory associated with it
permeated nearly all logic developed subsequent to its
publication. For example, students even today are introduced
to what is called the "prOpositiona1” calculus. This
expression is a direct desendant from the Principia and its
theory. There is very little recognition, however, of what
the notion of a proposition involves, and even less Of what
the authors of the Principia intended by it. As we proceed,
it should be kept in mind that we are dealing with the theory
of the Principia largely in order to clarify such notions.

The theory of logic of the Principia can be neatly
divided into two parts: that which is contained in the
introduction to the first edition, and that contained in the
second edition. The difference in point of view between the
editions is striking; Whitehead's influence is pronounced
in the first edition, but is nonexistent in the second.

Russell wrote the introduction to the second edition after

132

he had come under the considerable influence of Ludwig
Wittgenstein. The effect is a total re-direction in the
theory of logic associated with the Principia.

The present chapter will deal exclusively with the
theory of logic of the first edition. The next chapter will
deal with the theory of the second edition, and Wittgenstein's

philosophy of logic.

2. THE BACKGROUND OF PRINCIPIA MATHEMATICA.

Before taking up questions concerning the theory
associated with the Principia, it will be helpful to discuss
its background and lineage.

Just as Frege's logic was in no sense a direct product
of the logic of his best known predecessor, Boole, the logic
of the Principia cannot be traced back either to Frege or to
any other single source. Indeed, few realize that the recent
history of formal logic is so discontinuous; Boole had little
influence on Frege. Even though Russell and Whitehead pay
homage to Frege, he had little direct influence on the

authors of the Principia. This is evidencaiby the disregard

 

shown by Whitehead and Russell for the advances made in
various aSpects of theory by Frege. Whereas Frege maintained
a sharp distinction between language and extra-linguistic
fact, the same relation is blurred in the Principia in
several important areas. Also, Frege maintained that
entities corresponding to variables do not exist, and that

only ranges of possible arguments correspond to them. In the

133

Principia, it is not at all clear that such variable entities

 

are not countenanced; on one interpretation of the concept
of a propositional function, variable values are a definite
possibility.

Such discontinuity is not totally undesirable; though
it is true that many of the advances made earlier had to be
re-discovered, often very laboriously, many of the obvious
shortcomings inherent in the works of earlier writers were
not surrepitiously adopted by those who came later. In the
case of Frege, for example, it is well that he did not simply
inherit Boole's logic, but began instead with completely
new foundational ideas. Also, had Russell and Whitehead
simply adopted, say, Frege's set of axioms, they would surely
have been at a distinct disadvantage with respect to the
position which they eventually adOpted.

There is a distinct parallel between Boole, Frege,
and Russell-Whitehead; they were all influenced by the
current trends in mathematics in their own days, and by the
most influential of the mathematicians. In Boole's case, there
was the great influence Of the earlier algebrattists; in
Frege's, of the formalists; in the case of Russell-Whitehead
the mathematicians who most influenced their logic were, by
their own admission, v. Staudt, Pasch, Pieri, Veblen, and
most notably, Cantor and Peano.

In this discussion of the first edition of the

__rincipia, attention will be confined to the contributions

 

130

of Cantor and Peano, and to the influence of these two men
on Whitehead and Russell.

Regarding Cantor, the authors of the Principia state
that ”in arithmetic and the theory of series, our whole work
is based on that of Georg Cantor."2 Indeed, Cantor's
influence on Whitehead and Russell's work in mathematics is
very evident, as will be seen below.

Cantor's best known mathematical work was done in the
theory of sets. Cantor defines 'set' as follows: "By an
'aggregate' (Menge) we are to understand any collection into
a whole (zuzammenfassen zu einem ganzen) M of definite and
separate objects of m of our intuition or our thought.

These objects are called the 'elements' of M."3 The theory
which revolves around this definition is in some respects
like Boole's algebra, but while Boole managed to restrict
his analysis of classes to a small number of operations, and
maintained the constructivity of his system, Cantor branched
out into some extremely rarified areas of the study of
classes. In particular, he exposed some very peculiar
properties of sets (and accordingly was held up to a good

deal of ridicule).

2PM, Viiie

3Cantor, Georg, Contributions to the Founding of the
Transfinite Numbers, P. E. B. Jourdain (ed. )(New York:
DOver Publishing Company, 1960), p. 85. (A reprint of the
first edition, 1915.)

135

One such discovery was that "the set of rational
integers 1, 2, 3, ... contains precisely as many members as
the 'infinitely more inclusive' set of all algebraic
numbers.“+ Another example of the same property is associated
with the proof that the power (which is Cantor's term for
cardinal number) of the set of positive integers is the same
as the power of the set of squares of positive integers, even
though the latter set contains only some of the members of the
former set.5 Eventually, infinite sets were defined as those
which stand in a one-to-one correspondence with a proper
sub-set of themselves.

Another important advance made by Cantor was the
recognition that for any set S, 1(5, which is the set of all

sub-sets of S (otherwise known as its power set), has a

 

greater cardinal than S. This is known as "Cantor's
Theorem.” The importance of this theorem lies in the fact
that even transfinite numbers obey this rule. Thus, for
example, it was shown that 2c is greater than c for any
cardinal c, whether finite or trnasfinite.6 As Kneale points
out, this follows as an immediate corollary from Cantor's

7

theorem.

 

“Bell, E. T., Men of Mathematics (New York: Simon
and Schuster, 1937), p. 565.

50f., Kneale, William and Kneale, Martha, The Develop-
ment of Logic (Oxford: Oxford University Press, 1961), p.—EDO.

Remembering that for Cantor, cardinal numbers are sets.

7Cf., Kneale, op. cit., p. “U2.

136

Still other contributions made by Cantor were the
proofs that transcendentals are infinitely more numerous than
positive rational integers, and that the points in a line-
segment, and those in a line are equal in number. Another
contribution came in a roundabout way. As Bell says,

With the opening of the new century Cantor's

work gradually came to be accepted as a

fundamental contribution to all mathematics

and particularly to the foundations of

analysis. But unfortunately for the theory

itself the paradoxes and antinomies which

still infect it began to appear simultane—

ously. These may in the end be the

greatest contribution which Cantor'g theory

is destined to make to mathematics.

The paradox which Cantor discovered was remarkably simple:
"Let us suppose the S is the set of all sets. By his own
theorem about power sets ((57%. But since ‘QS is a set of
sets (namely the set of all sub-sets of S), it must be part
of the set of all sets, that is, of 5. And from this it
follows that figs-5', which is contradictory to the result

we have just obtained."9 Bell's claim that the discovery

of this antinomy may have been Cantor's greatest achievement,
must be taken with a grain of salt. It is not the discovery

of the antinomy as such that is remarkable; it is the’

pressing of the limits of set theory to such a point that

 

88811, 020 Cite, p0 5710

9Knea1e, op. cit., p. 652.

137

such defects became apparent, that stands as Cantor's
monument. His achievement was the placing of mathematics on
a more stable foundation.

Such advances made by Cantor had a pronounced effect

on the authors of the Principia. In the Principles 2:

 

Mathematics, Russell, for instance, in parts II, III, and
IV, concerns himself with topics taken directly from Cantor's
work. -In the Principia itself, nearly all of the work done
in analysis and number theory rests squarely on Cantor's
achievements. Where Russell and Whitehead appear to differ
with Cantor in method or style, this is due to the influence
of Peano, who, through the novel notation he introduced, was
able to present Cantor's achievements in a much more
economical fashion.

There is yet another influence which Cantor may have
had on Russell and Whitehead, although they make no Specific
mention of it. This is Cantor's idea that sets are extra-
mental realities as Opposed to being mere constructions of
the mind. This attitude is reflected, for instance, in his
definition of 'set,’ given above. He there speaks of sets
as collections of things graSped either by our senses or our
minds. Cantor believed that he was discovering the existence
of these various mathematical entities, and not simply
”constructing" them. Consider, as another example, Cantor's
definition of 'cardinal number': "We will call by the name
'power' or 'cardinal number' of @eaM the general concept

which, by means of our active faculty of thought, arises

138

from the aggregate M when we make abstraction of the nature

of its various elements and of the order in which they are

given."10 Here again, Cantor's position that he is discovering

(by means of rational activity), rather than merely construct-

ing is apparent. He believed that numbers (even transfinite

ones) exist as surely as do the objects of perception, and

that therefore, it is only appropriate to speak of discovery.

As Kneale says, ”like all great mathematicians, Cantor thought

of his work as the discovery of laws not made by man. He was,

he said, only a faithful scribe with no claim to merit

except for his style and the economy of his exposition."ll
Even though Cantor held to a realistic position with

regard to the existence of sets, it would be difficult to

hold that any realism connected with the first edition of

the Principia is due wholly to him. Yet surely Cantor must

 

be counted as an important part of the total enviornment
which gave rise to the theory of logic of the Principia.
While it is tempting to speculate that so-and-so had such-
and-such effect on the authors of the Principia, in most
cases such speculation is futile. The complex intertwining
of the developments of the last two decades of the 19th
century make it impossible to say with certainty that it was

only Cantor who was responsible for the mathematics of the

 

10Cantor, op. cit., p. 86.

11Kneale, op. cit., p. “#2.

139

Principia, or only Frege or Peano who was reSponsible for
its logic. To adopt such a position is to miss the point.
There is no single line of influence running from either

Frege, or Cantor, or anyone else to the Principia.

 

Another major influence on the Principia was the

 

works of G. Peano. Before dealing with his contributions,
however, we shall briefly discuss some apparent incongruities
concerning the contributions of Frege and Peano as they are

accounted for in the Principia.

 

At one point, the authors of the Principia maintain
that "In all questions of logical analysis, our chief debt
is to Frege. Where we differ from him, it is largely because
the contradictions showed that he ... had allowed some error
to creep into his premisses."12 But on the preceding page,
it is maintained that ”[Eeano'élgreat merit consists not so
much in his definite logical discoveries nor in the details
of his notations (excellent as both are), as in the fact that
he first showed 22! symbolic logic pgg_£p pg freed £322.l£§
undue obsession El22.£22 forms p£ ordinary algebra, and
thereby made it a suitable instrument for research."13 These
statements seem odd to us now, because there is a widespread
desire to credit Frege, and not Peano, with freeing logic

from any dependence on algebra. Over the intervening years,

 

12PM, vii.

13PM, viii: italics mine.

1&0

it has indeed become apparent that Frege deserves the honor

bestowed on Peano by the authors of the Principia. However,

 

Whitehead and Russell should not be judged too harshly for
misplacing the laurels. In the first place, Frege and Peano
were working at the same time on the project of grounding
mathematics in logic (1893 and 1894 respectively), and while
Frege had an edge on Peano in that the Begriffsschrift
appeared some ten years earlier than any of Peano's work,
this made little difference, for Frege was generally known
through his Grundgeset__z__e_, and not through the Begriffsschrift.
In the second place, Whitehead and Russell themselves were
working in very close temporal proximity to both Peano and
Frege, and can therefore hardly be blamed for misjudging the
relative contributions of these two men. Also, in a sense,
the comment concerning Peano made above is correct; while
Peano was not the first to ”free" logic from algebra, he did
in fact make it a suitable instrument for research by
introducing a clear notation.

Some of Peano's specific contributions are as follows:
1) he first made the distinction between real and apparent
variables, i.e., between bound and free variables; 2) he
used a distinct symbol for class-membership. His use of the
symbol '6;' for this purpose differs slightly from current

usage, in that he incorporated it into such expressions as

 

1&1

y
14 3) he

'xEPx', meaning, "the class of x's such that Px.";
employed a well-defined system of punctuation in complex
expressions. This punctuation involved the first use of a
dot-notation;15 h) he distinguished class membership and
class inclusion; 5) he first used the 'j' symbol for definite
descriptions; 6) he employed a separate existential
quantifier (although Peirce had done so previously).

Peano's other distinction was his attempted grounding
of mathematics on a set of nine postulates. "Four of these
are truths about equality, but the other five are the
following special postulates:

1) 1 is a number.

2) The successor of any number is a number.

3) No two numbers have the same successor.

1:) 1E in later formulationa is not the
successor of any number.

5) Any property which belbngs to l and also
to the successor of any number which has
it belongs to all numbers.

Many who credit Peano with being the first explicitly to

state the groundwork of mathematics have these postulates in

mind. Nidditch, for example, says, "before Peano (1858-1932),

 

1“Nidditch, P. H., The Development pf Mathematical
Logic (London: Routledge and Keegan Paul, 1962), p. 75.

15Actually, this amounted to no real advance over
Frege's system, for in it, there was no need for separate
punctuation. The configuration of the basic function-signs
provided a punctuation-free notation.

16Kneale, OE. Cite, p. “730

1&2

however, no one made use of the logic of statements for
making clear the arguments of everyday mathematics, and so
viewing logic as an instrument for getting clear and tight

«17

reasoning in such mathematics. Since Nidditch was
presumably equally aware of Frege's contributions, the
explanation for such a statement seems again to be that
Peano's advances in the field of notation got him credit

which he should at least share with Frege.

Peano's lasting effect on Principia Mathematica is
apparent at many places in the work. The most obvious
contribution occurs in the area of symbolism. Whitehead was
so enthused over Peano's notational innovations (which
Russell had altered at various points), that he said, "I
believe that the invention of the Peano and Russell symbolism

... forms an epoch in mathematical reasoning."18 As we shall
see, the great bulk of symbolism used in the Principia may
be traced back to Peano.

Peano's other contributions to the Principia are
found in the sections concerned with mathematics. His work
served as a catalyst for earlier developments in mathematics,

again largely through the novel notation which he introduced.

To be sure, some aspects of the development of mathematics

 

17Nidditch, op. cit., p. 73.

18"On Cardinal Numbers," p. 367, quoted from Quine,
Willard Van Orman, “Whitehead and the Rise of Modern Logic,"
The Philosophy of A. N. Whitehead, P. A. Schilpp (ed. )(New
York. Tudor Publishing Company, 19Ul), p. 138.

 

 

 

1&3

in the Principia diverge markedly from Peano's own treatment;
Quine, for instance, points out that for Peano, the notion
'cardinal number of a' could only be defined in the context
of an assertion of equality between two numbers, i.e., 'Num a

= Num b.' The version adopted in the Principia, after

 

Whitehead had become acquainted with Frege's work, is such
that 'Num a' can be defined in isolation.19 But such extreme
diversion is rare; much more often Peano's mathematics is
accepted by Whitehead and Russell ip_£p£2.

The two other sources of influence on the Principia

 

which are worth mentinning, are those exerted by the earlier
writings of the authors themselves. As we proceed through

the discussion of the theory of logic of the Principia,
occasional reference to those works will be seen to be
helpful. Consequently most of the discussion of them will

be left until it becomes relevant. However, some introductory
remarks concerning Russell's Principles pf Mathematics, and
Whitehead's Universal Algebra are in order.

Whitehead's Universal Algebra is usually considered

 

to be simply the last word, quite literally, in algebraic
logic. Actually, only one part, Book II, is directly
concerned with logic in the Boolean tradition. Book I deals
with some very general notions concerning algebra without

regard to interpretation. In this part, Whitehead develOps

 

19cr., PM, p. 100 ff. Cf. also, Quine, op. cit.,
p. 156.

luh

a theory of equivalence to cover all possible interpretations
of the system of algebra. After Book II, various special
topics in algebra are considered, such as higher-order
algebras. Taken as a whole, the book is in no sense simply
the finishing touch on the algebra of logic; it may be more
aptly described as the bridge from algebraic logicto the

richer logic of the Principia. In this connection, consider

 

the following innovations to be found in the work. First,
Whitehead introduced what amounts to a separate existential
quantifier, to denote the existence of the membership of
some class or other. As Quine says, "Whitehead introduced
a curious quasi-term '3. ', which he thought of as‘a
modification of 'l' to the following effect: xii, like x1
[Ehe union of the universe-class 1, and any class x is x;
but the use of the notation 'xjj' is understood as implying
incidentally the further information that something belongs
to x."20 Also, in Universal Algebra, Whitbhead develops a
total reinterpretation of classical logic by assigning a
prOpositional interpretation to the variables which it
employs. Thus 'xy' is used to stand for the conjunction of
x andy, x{y for the disjunction of the two and -x for the
denial of x. Concerning such reinterpretation, it should

be remembered that Boole came close to exposing the fact

that class-symbols can be thus reinterpreted, but at best,

 

”Wine, _2_0 - _ci_.t-. pp. 133-131».

1&5

his develOpment is sketchy. Whitehead's treatment, on the
other hand, is explicit and very systematic.

Besides such interesting developments in logic,
Universal Algebra contains some fundamental ideas which are

*

carried over to the Principia. For instance, even in this

 

work, it is clear that Whitehead takes propositions to be
extra-linguistic things, ige., roughly what sentences name.
Thus, 'x=y' is true under his account, just in case the
statements replacing 'x' and 'y' name the same proposition.
This extra-linguistic interpretation of the notion of
proposition provides difficulty in interpreting certain

parts of the Principia, as we shall see.

 

Throughout Universal Algebra, there are many other

 

indications that Whitehead was in no way interested in a
purely linguistic interpretation of the nature of logic.
Consider, for example, his definition of 'manifold' (i.e.,
'class,' or 'set'): "consider any number of things
possessing any common property. That property may be
possessed by different things in different modes: let each
separate mode in which the property is possessed be called
an element. The aggregate of all such elements is called
the manifold of the prOperty."21 The fact that Whitehead
adopted such an attitude toward questions of interpretation

will be seen to be of importance later in the discussion.

 

21Whitehead, Alfred North, A Treatise 23 Universal
Algebra with Applications (Cambridge: Cambridge University
Press, 1898), p. 317.

 

1&6

Russell's earlier work was also a great influence on
the Principia. While we shall discuss such work in detail
when it becomes relevant to do so, some preliminary remarks

are in order here. In the preface to the Principia, it is

 

remarked that ”the present work was originally intended by

us to be comprised in a second volume of The Principles 23

Mathematics."22 The authors go on to remark that all that

 

kept them from doing this was that the complexity of the
project made the work so large that separate publication
became mandatory.

The purpose of the Principia, and that of Russell's
Principles are in part the same. In Principles, Russell
states that “the present work has two main objects. One of
these is the proof that all pure mathematics deals exclusively
with concepts definable in terms of a very small number of
fundamental logical concepts.... The other object of the
work...is the explanation of the fundamental concepts which

23

mathematics accepts as indefinable.” The first of these

two purposes is shared with the Principia, the other is not.

 

The Principia is indeed less philoSOphical than Principles

 

in the sense that more attention is devoted in the latter
work to questions concerning interpretation, to questions

concerning the nature of various mathematical notions, and

 

22
PM, p. v.

23Russell, Bertrand, The Principles pi Mathematics
(London: Allen and Unwin, 1903), p. xv.

 

1&7

to questions which now fall mainly into the domain of the
philosophy of science, such as concerning the nature of
space, time, etc.

Since such interpretative work had already been done,
it is possible that Russell and Whitehead, when they wrote

the Principia chose, for that reason, largely to avoid such

 

areasof inquiry. But as we shall see, in most cases where
difficulty arises concerning interpretation in the Principia,
referring back to Principles provides little help. Many

confusions arising in Principles are carried whole and

 

undiminished to the Principia.

The influences on the development of the Principia
which we have considered in this section are only a few of
the more important ones. Nonetheless, it should now be
clear that developments leading up to the publishing of the

Principia form an intricate panorama of developments both

 

in logic, in the theory of logic, and in mathematics.
In the next section, we shall consider a few aspects

of the logic of the Principia. The points discussed will

 

be of help in the discussion of theory which follows.

3. SOME ASPECTS OF THE LOGIC OF THE PRINCIPIA.

Russell and Whitehead had three purposes in mind in
developing the logic of the Principia. First, according to
the authors, "it aims at effecting the greatest possible
analysis of the ideas with which it deals ... and at

diminishing to the utmost the number of undefined ideas and

1&8

undemonstrated propositions from which it starts." Second,
”...it is framed with a view to the perfectly precise
expression, in its symbols, of mathematical propositions."
Third, ”the system is Specially framed to solve the paradoxes
which, in recent years, have troubled students of symbolic
logic and the theory of aggregates.”2u In this section, we
shall look briefly at the logic which they developed in
order to attain these goals. In so doing, we shall
temporarily set aside all questions of interpretation. All
that will be attempted here is the presentation of those
aspects of the logic itself which have a bearing on later
questions of interpretation, and the presentation of such
symbolism as may be of help in the next section.

One further preliminary question must be considered.
Just how much of the totality of the Principia should be
counted as a part of logic? At issue is whether or not to
include set theory as a part of logic. Frege maintained that,
since mathematics could be reduced to logic, no real
distinction could be held to hold between the two fields.
In this vein, Whitehead and Russell held that ”considered as
a formal calculus, mathematical logic has three analogous
branches, namely (1) the calculus of propositions, (2) the

calculus of classes, (3) the calculus of relations."25 In

zuPM, p. 10

25PM, p. 88.

 

1&9

this sense, set theory should be considered to be a proper
part of logic. But this sense of 'logic' is the widest
possible, since it includes not only that which has been
traditionally considered to be logic, but also that which
was supposed to be immediately derivable from it. Whitehead
and Russell recognize the distinction between logic
considered this broadly, and what is ordinarily treated as
logic. They maintain, for instance, that

"symbolic logic is often regarded as

consisting of two coordinate parts, the

theory of classes, and the theory of propo-

sitions. But from our point of view, these

parts are not coordinate; for in the theory

of classes we deduce one proposition from

another by means of principles belonging to

the theory of propositions, whereas in the

theory of propositions we nowhere require

the theory of classes. Hence, in a

deductive system, the theory of propositions
necessarily precedes the theory of classes."

26

In this section, we shall follow this notion of logic as
the theory of propositions, and set aside any consideration
of issues in the theory of sets, even though such division
may be in principle unwarranted.

The logic of the Principia is built upon what Russell
and Whitehead call ”primitive ideas," and ”primitive
propositions.” As they put it, "following Peano, we shall

call the undefined ideas and undemonstrated propositions of

 

26PM, p. 90.

 

150

the system primitive ideas and primitive prOpositions

 

 

respectively, The primitive ideas are explained by means of
descriptions."27 In a more modern cast, the distinction
between primitive ideas and primitive prOpositions amounts
to the distinction between the axioms of the system
(primitive propositions) and the undefined expressions of
the system (primitive ideas). But this is only a rough
characterization, for as we shall see, some of the primitive
propositions themselves might be better treated as primitive
ideas.

In general, it is difficult to place the various

basic notions of the Principia into one of the four

 

categories which Church calls the "primitive baSis" of a
system (i.e., vocabulary, formation rules, rules of inference,
and axioms).28 As in the case of Frege, the difficulty

stems from the lack of recognition of some basic distinctions
and features of the system, such as the distinction between
syntax and semantics, and the distinction between use and
mention (though this applies less in the case of Frege than

in the case of the Principia).

 

27PM, p. 91. On this point, cf., Peano, Formulaire,
1897, I, 62-63, and ibid., 1901, p. 6. According to Russell
(Brinciples, p. 27) Peano, in his later works, no longer
eYBTTETny_distinguished primitive ideas as such.

28Church, Alonzo, Introduction £2 Mathematical Logic
(Princeton: Princeton University Press, 1956), p. 107.

 

151

Our task here, then, will be to go through the list
of these primitive notions, to see how they are symbolically
represented in the system, and to discuss those questions of
interpretation which will not be dealt with in the next
section.

In the body of the Principia, there are ten primitive

 

ideas listed as such. Six of these appear in *1, three in
*9, and one in *12.

The first primitive idea, and perhaps the most
important, in that it will be seen to be the cause of a
great many interpretational difficulties, is that of the
elementagy proposition. Elementary propositions can be
understood for the moment as those parts of the system which
are in some way wholly determinate. The pressing question
concerning them is whether they are merely simply subject-
predicate statements (with presumably singular terms
serving as subjects), or whether they are what correspond to
such statements (in the same way that that which is named
correspond to its name). The letters 'p,' 'q,' 'r,' and 's'
are used in the system symbolically to represent elementary
propositions.29 Besides these, any proposition formed

exclusively from elementary propositions by means of

 

29The closest Whitehead and Russell come to
specifying the complete vocabulary of the first part of
the work can be found on p. 5 of the introduction.

152

30

truth-functions will also be an elementary proposition.
This inclusion appears to be due to the fact that propositions
so formed will be determinate in the same sense as those
propositions corresponding to the single propositional
letters.31
The second sort of primitive idea is that of an
elementary prppositional function. Perhaps the easiest way
in which to understand this notion (though by no means the
clearest) is as that which is partially indeterminate and
when made determinate, becomes an elementary proposition.
How this notion is to be understood depends entirely on the
decision as to what elementary propositions are, for
whatever they turn out to be, elementary propositional
functions will be similar enough in Elflg that they can
become elementary propositions when properly "fulfilled."

Any decision as to what functions are, then, will have to

await the decision as to what propositions are.

 

30Although Whitehead and Russell do not include the
concept of an elementary truth-function as a primitive idea
of the system proper. They introduce the concept informally
(introduction, p. 8), but in the system itself, all that
they recognize are two specific truth-functions, negation
and disjunction. Our use of the expression is one of
convenience, and care must be taken lest an historically
inaccurate interpretation be adopted.

31However, the comment on p. 91 of the Principia
that combinations of elementary propositions by negation,
disjunction and conjunction are also elementary propo-
sitions, is an informal one, and is formally acknowledged
only in primitive propositions such as *l.7.7l.72.

153

There are two ways in which a propositional function
may be symbolized. In the first case, "if p is an
undetermined elementary prOposition, 'not-p' is an elementary
prOpositional function."32 In the second case, so-called
function letters, 'f,' 'g,' '¢,' “'f,' '1,‘ 'g,' and 'F',
are used; they correspond to elementary prOpositional
functions whenever the function symbolized takes an argument
or arguments which produce elementary propositions.

With regard to function expressions, two cases need
special recognition. An expression such as '¢x,' where 'x'
is unbound, is said ambiguously to denote a proposition.

That is, '¢x' denotes a proposition from its reange; which
one it denotes is determined when 'x' is replaced with a
suitable substituend, i.e., with an individual constant.

The other expression to be considered is '¢;.' The
sign 'lh' appearing over a variable is used to show that the
function has that place associated with it, and that no other
significance is to be attached to the variable presently
occurring in that place. In this way, this sort of sign
serves exactly the same purpose as does Frege's sign 'g'

as in '¢E', i.e., it is merely a place marker, allowing

 

32PM, p. 92. Apparently, Russell and Whitehead
intended 'p,' 'q,' 'r,' etc. in the sense in which they were
held to stand for elementary propositions, to be constants
corresponding to some (undetermined) proposition. In
speaking of elementary propositional functions, however,
'p' is being used as a propositional variable.

151»

the function-expression to stand alone as the name of a
function, with no regard for the arguments which it might
take.

These two categories of primitive ideas, elementary
propositions and elementary prOpositional functions, are by
far the most important in the sense of being most central to
the system. The crucial questions concerning the theory of

logic of the Principia revolve around these notions.

 

Accordingly, the brief treatment given them here will be
fully expanded later. The other primitive ideas, however,
while not of such direct importance, throw additional light
on the theory of logic of the Principia.

The third primitive idea is assertion, which is
thought of in the same manner as Frege had earlier, and is
symbolized in roughly the same manner, 'f’.P,' where p is
the asserted proposition. As in the case of Frege, this
notion is perhaps the least clear of all of the primitive
ideas. In the introduction, it is claimed that ”The sign
'f-,' called the 'assertion—sign,' means that what follows
is asserted.... In ordinary written language a sentence
contained between full stops denotes an asserted prOposition,
and if it is false, the book is in error.33 The sign 'hr'

prefixed to a proposition serves this same purpose in our

 

33Which is not always true, since some statements
between full stops may be being held up for consideration
instead of flat assertion.

155

symbolism."Bu The difficulty with such a notion is that the
dot-notation by itself can be thought of as performing the
same function as the punctuation system of the English
language. That is, statementhood can be determined as well
with dots in the Principia notation, as it can with periods
in English. But there is one difference between formal
systems and ordinary language which Whitehead and Russell
never mention explicitly, but which definitely plays a role
in the use of the assertion-sign. This is the fact that
assertions in logic are made in an essentially contextless
manner,iae., there is no 3 priori way in which to make the
subtle distinction between an asserted proposition and one
which is merely being "looked into." The authors of the

Principia may well have had this lack of context in mind

 

when they incorporated the use of the assertion-sign into

the system. For instance, they hold that one of its uses is
to insure that a sub-componenet of a complete, asserted
proposition will not necessarily be taken to be asserted
itself. Questions concerning this form of the fallacy of
division rarely arise in expressions in English, for context
alone is usally enough to show that statement sub-components
of complex statements need not be thought of as being asserted

just because the entire statement is.

 

3“PM, p. 8.

 

156

Such an interpretation of the use of the assertion-
sign does not, however, rid the notion of all difficulty.
Specifically, the distinction between assertion and mere
consideration is not a very clear one to begin with. This
lack of clarity is transmitted from ordinary language to

the distinction as it is employed in the Principia. The

 

distinction remains largely an intuitive one. Since
assertion is listed as a primitive idea, the authors are
under no duress completely to explain what is meant by it.
But unlike the other primitive ideas, there is relatively
little initial extra-systematic significance attached to it;
thus the use of assertion often seems superfluous in the
system.

Thus the difficulty surrounding the notion of
assertion concerns its place in the system itself. While
Whitehead and Russell make no linguistic-metalinguistic
distinction, it is clear that assertion does not function
at the same level as do the other primitive ideas. For
instance, the assertion-sign in the expression 'f-.P' can
perhaps be best thought of as a metalinguistic predicate
ranging over expressions of the object-language. In general,

the authors of the Principia make no systematic distinction

 

between levels of language. This gives rise to serious

difficulties in dealing with questions of interpretation.
Assertion of a prOpositional function which is

another type of assertion, is eXplained as follows: "Let

(bx be a propositional function whose argument is x; then

157

we may assert ¢x without assigning a value to x.... This
is only legitimate if, however, the ambiguity may be
determined, the result will be true."35
The fifth and sixth primitive ideas are negation and
disjunction, respectively. While these ideas are subsumed
under the idea of an elementary propositional function
mentioned earlier, Whitehead and Russell wished to single
these two ideas out for special attention. Faced with the
decision as to whether they should be defined or included as
further primitive ideas, they simply chose the latter. 1
Under the heading "The Fundamental Functions of
Propositions," it is maintained that "There are four special
cases which are of fundamental importance, since all the
aggregations of subordinate prOpositions into one complex
proposition which occur in the sequel are formed out of them
step by step."36 Then the claim is made that not all of
these functions are independent since some of them can be
defined in terms of others. Finally, negation and disjunction
are chosen as primitive, and-the others are defined in terms
of them. The reason for choosing them was that "simplicity
of primitive ideas and symmetry of treatment seem to be gained

by taking Ehesel two functions as primitive ideas."37 In

 

35PM, pp. 92‘930

36PM, p. 60

37Ibid.

158

explaining these two functions, Whitehead and Russell
employ what has become the standard truth-functional

definition, which is well enough known not to require
repeating here.

One further interesting point concerning the use of
truth functions in the Principia is that they were employed
to the exclusion of intensional, i.e., modal functions. It
seemed to Whitehead and Russell that the mathematics which
they were trying to evolve had no need for intensional
notions. As they put it, the fact that they would consider
only truth functions of prOpositions "is closely connected
with a characteristic of mathematics, namely, that
mathematics is always concerned with extensions rather than
intensions. The connection, if not now obvious, will become
more so when we have considered the theory of classes and
relations."38 Actually, Whitehead had no other aversion to
modal operators; he made wide use of modal notions in
Universal Algebra.

The next two primitive ideas occur in *9. One is the
notion of an individual. The description of an individual
given in that chapter is that "...x is 'individual' if x is
neither a prOposition nor a function."39 Further, "Such

objects will be constituents of propositions or functions,

 

38PM, p. 8.

39PM, p. 132.

 

159

and will be genuine constituents, in the sense that they do
not disappear on analysis, as (for example) classes do, or

phrases of the form 'the so-and-so.'"L’O

But such pronounce-
ments are of little help until we can decide what a proposition
is, and what constitutes a propositional function. For the
present, the only other point concerning individuals which
is of interest is that within the system, they correspond to
the letters, 'a,' 'b,' 'c,' 'x,' 'y,' 'z,' and 'w.' Questions
concerning interpretation will be taken up again in the next
section.

The last sort of primitive idea which we shall
consider is that of a first-order prOposition which is
symbolized either as '(x) 9x,‘ or as '(3 x)Qx.' Whitehead
and Russell go about eXplaining the significance of first-
order functions in a number of different ways, although all
of them are tightly interconnected. At one point they say
that "in the present number, we introduce two new primitive
ideas, which may be expressed as 'q>x is always true' and
'¢x is sometimes true..."'L"1 Another way in which first-
order functions are explained can be found in the introduction
where it is maintained that corresponding to any prOpositional

function there is a range of values which satisfy ("saturate,"

in Frege's language) the function, i.e., make it into a

 

”0PM, p. 51.

”1PM, p. 127.

160

proposition.“2 According to Russell and Whitehead, an
assertion to the effect that all prOpositions of the range
are true will be symbolized by '(x)Qx.'

In introducing first-order functions, we are presented
with yet another way in which to understand elementary
propositions. In *9 it is held that "in a prOposition of
either of the two forms (£)NJL£’ (§L£)°5L19 the 3 is called

an apparent variable. A prOposition which contains no

 

apparent variable will be called 'elementaryf and a function,
all whose values are elementary propositions, is called an
elementary function."L’3

There is another point concerning first-order
propositions which will be relevant in the discussion of
theory. According to Russell and Whitehead, since elementary
functions and first-order functions are of different levels,
then when negation and disjunction are applied to them, they
will have different meanings in each case. Thus the negation
of (x)¢x is held to be (a x)~Qx. In *9, the authors claim
that negation and disjunction must be thought of as having
been re-defined for first-order functions. Many of the
propositions of *9 are thus analogous of propositions of

*1, and were constructed for the purpose of such re-definition.

 

ung., PM, p. 15.

“BPNI, p. 1270

161

The other class of primitive notions consists of a
set of 19 primitive prOpositions, i.e., those prOpositions
which are assumed without proof. However, it must not be
supposed that this set of prOpositions correSponds to what
we now call a set of axioms for the system. Actually, there
are indeed axioms of the system, and those which might
profitably be thought of as being meta-axioms.

The first group of primitive propositions, the
axioms of the system, are nine in number. Five of these
(*l.2.3.&.5.6) are listed in the opening chapter, and are
those which are usually considered to be :22 axioms of the

Principia. Unfortunately, it is incorrect to think of the

 

system in this way, for there are four other axioms, all of

which are very important. They are:

*9.1 F:¢x.3x(3z).¢z Pp
*9.11 \-:¢xv¢x.3.(az).¢z Pp
*12.11 \-:(3§):Qx.'£x.5:x Pp
*12.11 H(35):Q(x.y).§x,y.§:(x,y) Pp

The last two are variations of the Axiom of Reducibility,
which will be discussed later. While full treatment of these
axioms is beyond the scope of this discussion, it is worth
noticing that they are axioms because they form the basic
building-blocks of the system. All that is subsequently
derived is derived from them. In short, these propositions
are genuine axioms, as they are expressions of the system,

and are those which are assumed without proof.

162

On the other hand, the rest of the primitive
propositions are statements about certain characteristics of
the system. Thus, for example,

*1.1. Anything implied by a true elementary

proposition is true. Pp.

*l.ll. When Qx can be asserted, where x is a

real variable, and Qx DYX can be

asserted, where x is a real variable,

then ‘rx can be asserted, where x is

a real variable. Pp.

*9.12. What is implied by a true premiss is

true. Pp.
The first two of these propositions correspond to what we
now call rules of inference, while the third resembles a rule
of conditionalization. In each case, a certain property of
the system is being exposed. In each case, the axiomatic
nature of the propositions is preserved, in that the presence
of the prOperty in question is simply estipulated. In this
way, these propositions are similar to metatheorems which
are assumed without proof. Such an analogy, however, cannot
be drawn very strongly. It might be supposed, for instance,
that the propositions in question must operate at a meta-
level, for the language used to indicate them is ordinary
English, and hence, is outside of the system itself. Such
as assumption, however, is too hasty. Whitehead and Russell
made no stipulation to the effect that English could not be
used within the system. They apparently felt perfectly at

ease in stating *1.1 above in the same list which included

purely symbolic axioms. All that can be claimed is that

163

these different sorts of propositions are doing different
jobs in the system, and that from our modern point of view,
those which have to do with characterizations of the system
might better be thought of as meta-propositions.

We have now covered some of the basic features of the
logic of the Principia. To cover all such features would
require much more space than can be spared. Should need for
any further discussion of points in logic arise, they will be

dealt with in the next section.

&. THE THEORY OF LOGIC OF THE PRINCIPIA MATHEMATICA.

In dealing with the theory of logic of the Principia

Mathematica, one point must be kept well in mind: We are not

 

seeking a way in which the subject-matter of the Principia
222 be consistently interpreted. That goal has already been
attained; those who later adopted the logic of the Principia
were usually quite content to take the path of least
resistance in questions of interpretation, and merely assumed
that the authors were in fact dealing with language, i.e.,
that their logic was basically about linguistic entities,
such as statements. Our task, on the other hand, will be to
analyze Russell and Whitehead's ppp_approach to questions of
interpretation. It is only in this way that we can talk of
the theory of lOgic of the Principia.

There are a number of serious difficulties which
must be met in the discussion to follow. The authors'

distaste for coming to grips with questions of interpretation

16&

within the confines of the Principia has its effect in the
lack of rigor to be found in the introduction. Often there
is little attempt made to dwell on those points which must
have seemed troublesome or doubtful. Many such statements
are indeed dogmatic, as the authors maintained they would be.

Also, the interpretation given to the system is not
altogether consistent. The basic notions of proposition and
propositional function, for instance, receive at least three
different treatments (in the opening section of the
introduction, in the section of the introduction devoted to
the theory of types, and in the introduction to *1). These
accounts diverge widely on fundamental issues.

Therefore, our task will consist not only in describing
the theory of logic of the Principia, but actually re-
constructing it, i.e., in fitting together those disparate
pieces into an intelligible whole.

The basic question which confronts us in this
discussion of theory is, what are prOpositions as they are
countenanced in the Principia? The rest of the interpretation
of the subject-matter of the Principia depends directly on
an answer to it. Once we have decided what to count as
propositions, we will be able to determine what sorts of
things prepositional functions are. Once both these notions
are explained, the interpretation of the rest of the system
will be seen to easily follow.

With respect to the question as to what constitutes a

proposition, there appear to be two distinct answers possible.

165

On the one hand, 'proposition' can be taken to be merely a
synonym for 'statement,‘ i.e., a sentence with an associated
meaning. On the other hand, a proposition can be taken to
be that which a statement is about, i.e., that which stands
in the same relation to a statement as that which is named
stands to its name.uu There are several variations on this
theme, but they all revolve around the distinction between
propositions as states of affairs generally, or propositions
as linguistic entities.

But now consider the following statement: "By an
'elementary' prOposition we mean one which does not involve
any variables, or, in other language, one which does not
involve such words as 'all,' 'some,' 'the' or equivalents
for such words. A proposition such as 'this is red,‘ where
'this' is something given in sensation, will be e1ementary."u5
Is this a definition of 'proposition' in the linguistic
sense, or the non-linguistic sense? If read quickly, the
answer would undoubtedly be that propositions, as here
defined, are simply statements, and elementary propositions
are those with singular terms as subjects. Such an answer

will not, however, stand up under close scrutiny. From the

first sentence of the above quotation it seems that

 

qun this distinction, cf., Church, Alonzo,
"Propositions and Sentences," The Problem pi Universals
(South Bend: Notre Dame University Press, 1956).

“5PM, p. 91.

166

propositions must be linguistic, for since the presence of
words in propositions is strongly implied, it would seem to
demand that we consider propositions as being the sort of
things which contain words, i.e., as statements. But the
second sentence does not support such a conclusion. It may
not seem obvious that this statement is not about linguistic
entities. But consider that the use of double quotation

marks in the Principia (and in many other works of the day)

 

did not necessarily mean that what fell between them was

being mentioned.“6 There is no simple answer to the question
as to what sorts of things propositions are. The definition
given is ambiguous; it could mean that elementary propositions
are statements in which no quantifier-words appear. On the
other hand, it could mean that propositions are what such

statements are about, i.e., states of affairs which correspond

to such statements.

 

u6Actually, the use of double quotes in the Principia
is rather complicated, in the sense that they perform several
fairly distinct tasks. In the first place, they are used on
some occasions as indicators of instances of mention. But
also they are used to point up the substantiality of the thing
being referred to 2x the expression in quotes. That is, they
form, together with the expression enclosed within them, a
sort of artificial substantive expression, a "that" clause set
off by quotes. For example, consider the following remark:
"...When we judge (say) "this is red," what occurs is a rela-
tion of three terms, the mind, and "this," and red. On the
other hand, when we perceive "the redness of this," there is
a relation of two terms, namely the mind and the complex
object ”the redness of this."" (PM, pp. &3—&&.) Here we have
a good example of how quotes are used to draw attention to
the things correSponding to the linguistic eXpressions occur-
ring between them. Although the first use of quotes in the
statement is an instance of mention, the others, and partic-
ularly the last, are surely used to point out non-linguistic

objects.

167

The Principia is teeming with statements of the sort

 

quoted above. Often, ambiguity makes straightforward
interpretation impossible. Such statements £32 be interpreted
as being merely about language. The effect of such an
interpretation forces the representation of Whitehead and
Russell as thinking that logic is ultimately about language.
We shall see, however, that they considered the subject-
matter of logic to be propositions (and higher-level objects)
in the broad sense, i.e., as states of affairs which
correspond to statements, and that their lapses into talking
as if propositions were fundamentally linguistic, are the
result of a lack of care in keeping talk about statements
separated from talk about what they are about.

There is another reason for getting clear about the
notion of a proposition», Some of the fundamental expressions

of the Principia are concerned directly with propositions.

 

Consider, for instance, the following primitive proposition:
”*l.1 Anything implied by a true elementary proposition is
true. Pp."u7 How are we to understand this assertion if we
cannot determine what sort of thing a proposition is? If
propositions are simply expressions, then it appears that
*1.1 is as much a part of the system of the Principia as are
the purely symbolic eXpressions occurring in it. The only

<iifference then would be that those expresSions couched in

 

“7PM, p. 9b.

168

symbols would exhibit the form of prOpositions, while such
expressions as *l.l would be about such propositions.

Thus far, we have considered one ambiguous definition
of 'elementary proposition.' There are many occasions on
which it seems that Russell and Whitehead genuinely believe
that propositions are basically linguistic entities. At one
point, for instance, they remark that ”a proposition which
contains no apparent variable is called 'elementary."'u'8
There can be no doubt that what is being talked about is a
kind of expression, for it is clearly maintained that
variables are linguistic in nature;u9 hence on this account
elementary propositions must be considered to be linguistic
in nature.

At other places, it seems equally clear that
propositions are to be taken as being extra-linguistic in
nature. For instance, the authors hold that "It follows from
the above theory that a 'proposition' in the sense in which
a proposition is supposed to be £22 object of a judgment, is
a false abstraction, because a judgment has several objects,
not one.” It is then claimed that "Owing to the plurality of

the objects of a single judgment, it follows that what we

call a "proposition" (in the sense in which this is

 

“8PM, p. 127.

ugAlthough it is interesting to note that values of
variables are taken to be "any set of entities, prOpositions,
functions, classes or relations, according to circumstances."

PM, p. &.

169

distinguished from the phrase expressing it) is not a single
entity at all. That is to say, the phrase which eXpresses

a proposition is what we call an "incomplete" symbol."50 This
is the clearest indication that Russell and Whitehead
understood propositions to be objects indicated by the
subjects of statements, and not statements themselves.

But is such a statement evidence enough for the claim
that Russell and Whitehead thought of propositions as being
the subject-matter of statements? Surely it is enough to
support the contention that they were well aware of such an
interpretation. But it also seems to be enough to support
the claim that basically, they thought of propositions in the
broad (i.e., realistic) sense, and that apparent remarks to
the contrary are mere linguistic confusions on their part
of the notions of proposition and statement. If this claim
can be borne out, then the entire realm of notions in which
'propositions play a part (e.g., prOpositional functions, etc.)
will have to be interpretable as being compatible in nature
-with prOpositions as states of affairs. PrOpositional
functions, for instance, will have to be interpreted as the
.E£2§:2£ thing whibh can become a prOposition, (i.e., a state
of affairs) under the interpretation being considered.

A plausible solution to this problem of interpretation

is that even though pronouncements to the effect that

 

50PM, p. &&.

170

propositions are basically extra-linguistic are infrequent,

nonetheless the authors of the Principia thought that the

 

subject-matter of logic was not simply language. Under this
interpretation, on those occasions in which it appears that
propositions are basically linguistic, in point of fact,
'proposition' in such cases is being used to mean what we
would ordinarily mean by 'statement.‘

The theory of logic of the Principia, with specific
regard to propositions, then, rests on the belief that While
the basic subject-matter of logic consists of states of
affairs, dealing with such states of affairs is often best
implemented by talking about the statements which indicate
them. In this fashion, there are two levels operative in the

logic of the Principia, that which is concerned with

 

statements, and (often implicitly) that which is concerned
with the things which correspond to such statements.

We can see a strong resemblance to the theory of
Frege's logic in such a theory: both display the desire on
the part of their authors to ground logic in extra-mental,
extra-linguistic fact, while at the same time both theories
hold that it is economical, and often clearer to avoid
direct reference to the fundamental subject-matter of logic,
and to concentrate on the language which corresponds to such
subject-matter.

Most commentators on the Principia on the other hand,

 

have ignored the interest shown in it in treating propositions

as states of affairs and have concentrated only on the purely

 

-_ '_ “1

171

linguistic interpretation.51 The work 232 be successfully
interpreted in this fashion, but such an interpretation
misrepresents Whitehead and Russell's intent.

The rest of this section will be devoted to working
out the theory outlined above.

First, if 'proposition' can be interpreted as meaning
a state of affairs, then properties ascribed to propositions
should be interpretable as characteristics which propositions
construed in this way could possess. Such is the case with
the notion of truth.

The first remark made in the Principia concerning
truth is that "the 'truth-value' of a proposition is truth
if it is true, and falsehood if it is false."52 In a
footnote to this statement, it is remarked that "this phrase
is due to Frege."53 In fact, the statement only looks as
though Frege might have been responsible for it. In
particular, in this statement it is not clear that truth and
falsehood are being understood as objects in the way Frege

thought of them. Russell and Whitehead's statement is

 

SlOn this account, cf., Hilbert, David and Ackermann,
“Wilhelm, Principles of Mathematical Logic (New York: Chelsea
IHiblishing Co., 19507: p. 28. I; translation of Grundzuge Der
'fheoretischon Logik (Berlin: Julius Springer Verlag, 1928):T:
Nidditch, pp; cit., p. 77, and Jflrgensen, Jorgen, A Treatise
of Formal Logic (New York: Russell and Russell, 1962 ,
£53. 231 ff, (first published in Denmark, 1931). This will
serve as a sample of prevalent attitudes, past and present,
‘to the interpretation of the Principia.

 

52PM, p. 7.

53Ibid.

172

ambiguous from another point of view also, for by considering
it alone, it is impossible to tell whether they intend truth

and falsehood to be properties of "linguistic” propositions,

or of propositions construed in the realistic sense outlined

above.

These ambiguities are simply left unresolved. In
Section II of the introduction, the subject of truth is taken
up again, in connection with the discussion of the theory of
types. There it is held that "in fact...we may define truth
...as consisting in the fact that there is a complex
corresponding to the discursive thought which is the judg-
ment.”54 It is also remarked that "when the judgment is £322,
in the case of the kind of judgments we have been considering,
there is a corresponding complex of the objects of the
judgment alone."55 In neither case is it possible to tell
whether by 'truth' is intended a property of statements or a
property of the states of affairs which correSpond to them.
The pertinent question seems to be, does truth consist in
the actuality of the state of affairs corresponding to the
statement, or does it consist in the statement pointing out
that state of affairs?

In general, the notion of truth in the Principia is

treated in such a way that both statements and states of

 

5u'PM, p. &3.

55PM, p. nu.

In

LA

 

173

affairs are taken into account; the stress is laid on the
correppondence between actual states of affairs, and
statements which indicate them. The notion of truth, then,
is essentially neutral with reSpect to the dischotomy between
statements and the states of affairs which they indicate.
While such neutrality does not actively support the
contention that propositions can be understood as states of
affairs, still it does not impede such an interpretation by
restricting the application of truth to linguistic entities.
The subtleties in the treatment of truth in the

Principia might easily be overlooked. If only the first

 

remark quoted above concerning truth were taken into account,
it might be supposed that it was a purely lingusitic property
of presumably linguistic propositions. As we have seen,
truth should be interpreted as involving states of affairs
also, and hence as allowing the broader interpretation of

the notion of proposition.

The next notion to be taken into account is that of
the propositional function. To support the claim that Russell
and Whitehead held the subject-matter of logic to be states
of affairs indicated by statements, propositional functions
must be interpretable in a fashion analogous to that of
propositions, for propositional functions become prOpositions
when suitable replacements are made.

But consider the first mention of propositional
function: "Let ¢>x be a statement containing a variable x

and such that it becomes a proposition when x is given a

171+

fixed determined meaning. The $>x is called a 'propositional
function'; it is not a preposition, since owing to the

56 In

ambiguity of x it really makes no assertion at all."
£233 instance the relation being discussed is that which
obtains between an incomplete or "open" statement and a
completed one. There is simply no other manner in which

such a claim could be interpreted. Under this interpretation,
with regard to a simple subject-predicate statement such as
'John is dead,‘ we might regard the associated propositional
function as 'x is dead'; when it is completed, it becomes a
statement.

Such an interpretation is straightforward, and fits
in well with the conception of propositions as statements.
The remaining question is whether an interpretation of the
notion of propositional function can be found, by which they
are conceived as being of the same level as propositions 322
states of affairs, i.e., whether they can be thought of as
being the sort of thing which becomes a state of affairs.

As in the case of propositions, propositional
functions also can be interpreted in two ways. Consider, for
instance, the claim that "a function Em, a prOpositional
functioé], in fact, is not a definite object, which could be

or not be a man; it is a mere ambiguity awaiting determination."57

 

56PM, p. 1U.

57PM, p. as.

175

Again, "A 'property of x' may be defined as a propositional
function satisfied by x."58 In both cases, propositional
functions are taken to be essentially extra-linguistic.
Under this interpretation, propositional functions are very
much like Fregian functions, i.e., they are attributes.
Functions, then, are what become propositions. Where
propositions are considered to be states of affairs which
can be indicated by statements, propositional functions are
incomplete entities. When completed, they become states of
affairs.

Which of the two interpretations of 'propositional
function' should we take as the one intended by Russell and
Whitehead? The answer is clear; to take either to the
exclusion of the other would serve only to exclude a
significant part of the theory of logic of the Principia.
The interpretation of the notion of a propositional function
conforms to that of a proposition; Russell and Whitehead
recognize functions as sorts of basically extra-linguistic
things, but they also recognize what we call open sentences,
and use them to talk of such things. Unfortunately, they
use the same name for both. As Quine puts it, "The phrase
'propositional function' is used ambiguously in Principia

Mathematica; sometimes it means an open sentence and sometimes

 

it means an attribute. Russell's no-class theory uses

 

58PM, p. 166

176

propositional functions in this second sense as values of
bound variables; so nothing can be claimed for the theory
beyond a reduction of certain universals to others, classes
to attributes."59 The authors of the Principia seem well
aware of this ambiguity; at one point they maintain that
"the question as to the nature of a function is by no means
an easy one. It would seem, however, that the essential

"60 Of course

characteristic of a function is ambiguity.
this statement is true of propositional functions under
either interpretation, for ambiguity can easily be construed
as linguistic ambiguity (of a variable) or as a sort of
material ambiguity, i.e., as unsaturatedness, or
indeterminateness of a state of affairs.

There is one point concerning propositions and
propositional functions which might be misunderstood in such
a way as to provide some difficulty for interpretation. In
particular, it has been remarked that 'QQ' stands for a
propositional function, and '¢a,' ' Qb,‘ etc. (or '(x)¢x'
etc.) stands for a proposition. But it is not clear just
what '¢x' stands for. The difficulty which could arise is
that if 'Q?‘ names a function, and '¢a' a proposition, it
might seem that '¢x' could be used to name some sort of

single variable prOposition. While some passages in the

 

59Quine, Willard Van Orman, From a Lovical Point 2£
View (Cambridge: Harvard University Press, 1953), p. 122.

60PM, p. 39.

177

early part of the introduction are unclear enough to make
this view appear to be a likely interpretation,61 the
difficulty is eventually resolved, when it is held that
"when what we assert contains a real variable Es in '¢xfl ,
we are asserting a wholly undetermined one ofmall the
propositions that result from giving various values to the
variables."62 From this statement it is clear that '?x'
stands for no particular proposition, i.e., it stands for a
proposition, but which preposition is left undetermined.
Thus there is no need to countenance variable states of
affairs.

Taking stock of the theory of the Principia as it
has been thus far developed, the position most representative
of the authors' intent is that they conceived of logic as
being basically concerned with propositions in the broad
sense outlined above. They believed logic to be concerned
with states of affairs and with those things which serve as
components of states of affairs. They did not conceive of
logic so narrowly as some have been led to believe; their

concern ranged far beyond considerations of language. But

 

élThe difficulty is this: it is remarked (pp. lU-lS)

that $3: is an ambiguous value of @x. Now since Russell
and Whitehead make no distinction between substituend and
value, one might consider (fix to be a value in the same sense
that, say, Qa, 0b, etc. are; if carried through, this would
mean that like Qa, Qx would be a definite single,
'proposition, but also a variable one -— an "ambiguous" state
of affairs.which was not a function.

62131“, p. 180

178

while their attention was ultimately focused Ion the broad
interpretation of the subject-matter of logic, they realized
the value of the economy to be gained in restricting large
portions of the discussion contained in the introduction to
talk about language. That is, while they realized that
prepositional functions were more than open sentences, they
found it useful to deal with them by talking about the sort
of language used to name such functions, i.e., open sentences.
This theory of logic looks very much like the one held
by Frege. Both theories accept the existence of such things
as functions etc., which are different from function-
expressions. Both use talk about language as a convenient
medium for getting at such entities. But a difference
suggests itself at this point. Frege used what he found in
language as a guide in deciding what did, and what did not,
exist. Existence for him was bound to language in that the
needs of language suggested a sort of theory which served as
a basis for making existence claims concerning such things
astruth-values and functions. From reading just Principia

Mathematica, it might appear that Russell and Whitehead were

 

doing the same thing; letting language be the guide to
existence. But judging from other works, and especially

those of Whitehead, such is not the case at all. In fact,
Whitehead's metaphysics goes far beyond the meager information
which the structure of language provides for the description
of the nature of propositions. For Whitehead, propositions

were ”matter of Fact in Potential Determination, 23 Impure

179

Potentials for the Specific Determination of Matters of Fact,

63

2E Theories.” At another place he says,"A proposition is
a new kind of entity. It is hybrid between pure
potentialities and actualities."6u There are many other such
passages all of which indicate that Whitehead had a much
stronger metaphysical base for assuming the existence of
propositions than had Frege. Frege's assumption was one of
expedience; he could not imagine the various sorts of
expressions which he used as being related to no sort of
specific thing in the world, so he simply assumed that
corresponding to each sort of expression, there existed a
sort of thing. Whitehead's reasoning, on the other hand,
indicates that his conception of propositions was not
primarily motivated by logical concerns, but rather by
metaphysical ones. He apparently saw propositions as
constituents of the universe, and as having no necessary
connection with language, with specific regard to their
existence.

The last notion to be dealt with is that of an
individual. Individuals are spoken of in the following
manner: "...we will use such letters as a, b, c, x, y, z, w,

to denote objects which are neither propositions nor functions.

 

63Whitehead, Alfred North, Process and Reality
(New York: Macmillan and Company, 19297, p. 32.

6“Ibid., p. 282.

180

65

Such objects we shall call individuals." The only other
remark made concerning individuals is that "such objects will
be constituents of propositions or functions, and will be
penuine constituents, in the sense that they do not

disappear on analysis, as (for example) classes do, or

“.66

phrases of the form 'the so-and-so. Concerning this
notion, there are two important considerations which must
be taken into account.

First, the notion of an individual as presented
conforms to the rest of the theory of logic described
earlier. That is, individuals are conceived of as objects
which have linguistic correlates which can be replaced with
the list of letters given above. Here again, then, the
basic concern is broader than that of dealing solely with
language. And again, analysis shows that the discussion is
being carried on at two levels, one linguistic, and one
primarily non-linguistic.

Also, by defining individuals as being those things
which are not propositions or prOpositional functions,
Russell and Whitehead have in effect set up a trichotomous
division of the subject-matter of the Principia. Not only

does the notion of individuals form a homogeneous extension

of the theory of the Principia, it in effect closes it.

 

65PM, p. 51.

66Ibid.

181

But we are not through with the notion of an individual
quite so easily. There remains the question as to whether
there is a way in which individuals might be described other
than in the negative fashion already mentioned. Unfortunately,

the Principia is of little help, for all that is said about

 

individuals is contained in the quotation given above. The
only way to understand the notion of an individual as it is
used in the Principia is by elimination; that is, by
understanding first what propositions are, and then
understanding what constitutes propositional functions. But
this amounts to nothing more than following the negative
definition given in *9, which has been quoted above. There
is no reference in the Principia to an independent means for
theoretically describing individuals. That is, Russell and
Whitehead give no indication of how one might begin, say,
with individuals and functions, and show how propositions
might be defined in terms of them.

This lack of definite positive manner for describing
the sort of things which correspond to the individual-symbols

in the Principia is simply a result of the notion of an

 

individual being taken as a primitive idea. But the treatment
of this notion differs from the treatment of the notion of
Ixropositions and propositional functions in that while these
latter two were not defined within the system, still the
auithors apparently felt under some constraint to offer the
'basic points of an essentially philOSOphical description of

tfliem (although again, it was not necessary for them to do so,

182

for both of these notions are undefined parts of the system).
But with regard to individuals, even the rudiments of a
description are not forthcoming; knowledge of individuals
must, if the inquiry is restricted to the Principia, be
abstracted from knowledge of propositions.

But simply because they gave no full description of
individuals does not mean that Russell and Whitehead were
not convinced that none was possible. Their position on this
point might easily be mistaken, owing to remarks made in *12
to the effect that only the relatively lowest type of
discourse need be taken into account in determining the types
of other expressions occurring in that context. In
particular, they say that "it is unnecessary, in practice,
to know what objects belong to the lowest type, or even
whether the lowest type of variable occurring in a given
context is that of individuals or some other. For in practice
only the relative types of variables are relevant."67 But
this relativization of type theory is not done at the
expense of dispensing with the idea that certain kinds of
things are unchangeably individual. Careful attention to the
quoted passage shows that even though types can be relativized,
still the genuinely lowest level will be that of (presumably

irreducible) individuals.

 

67PM, p. 161.

183

Also, even though no full explanation of individuals
is given, this does not mean that the authors had no further
ideas on what individuals might be like. In 1911, during the
publication of the first edition of the Principia, Russell
gave a Presidential address to the Aristotelian Society in
which he outlined his views on the nature of particulars.
Since he had not yet come under the influence of Wittgenstein,
his views had not yet significantly changed from those which
he held during the writing of the Principia. The paper in
question consists of a discussion of the relation between
universals and particulars, and -would take too much space
for a full treatment here. Russell's main point, however,
was that there were two sorts of things, universals, and
"...Particulars, which enter into complexes only as the
subjects of predicates or terms of relations, and, if they
belong to the world of which we have experience, exist in
time, cannot occupy more than one place at one time in the

68

space in which they belong." This is at best a rough

indication of the way in which Russell conceived of the

nature of individuals. But while it is rough, this description
at least indicates that the authors of the Principia thought
of individuals as being more than just what is left after

'propositions and functions are abstracted from the world.

 

8
Russell, Bertrand, Logic and Knowledge, Essays
1901-1950, E. R. Marsh (ed.)(London: George Allen and
Ufnvin, Ltd., 1956), p. 12h.

18h

Perhaps the best explanation of why no theoretical
explanation of the concept of an individual is given in the

Principia itself is that when it was written, neither author

 

took the concept of an individual to correspond to a separate
(metaphysical) category of entities. Whitehead took several
different kinds of things to be individuals. Among these
are prehensions, actual occasions, nexus and societies.
Since positive characterizations do not stretch beyond the
confines of single categories, no such characterization was
attempted.

In Russell's case, the situation is largely similar.

He thoughtof particulars as ranging over a number of distinct

 

categories. Hence he too had good reason for not attempting
a positive characterization of individuals.

This brings us to the end of our discussion of the
various sorts of things countenanced in the theory of logic

of the Principia. Virtually all other topics of a theoretical

 

nature discussed in the Principia are developed in terms of
the trichotomous distinction between propositions,
propositional functions, and individuals. As an important
example of this latter sort of development, let us consider
the treatment of the notion of implication.

Implication is defined as a relation among prOpositions,
in the following manner: "When a proposition q follows from

a proposition p, so that if p is true, q must also be true,

185

we say that p implies q."69 Further, it is maintained that
"The essential property that we require of implication is
this: 'What is implied by a true prOposition is true.' It
is in virtue of this property that implication yields
proofs.”70 Here, then, we have a theoretical notion which is
definable totally in terms of the more basic theoretical
notions mentioned above, together with the consent of truth,
which has also been discussed. To understand implication as
it is formulated above, one must rely entirely on the
knowledge of the basic constituents of the theory of logic
of the Principia.

Finally, there is one last area of concern which must
be mentioned in connection with any discussion of theory.
This is the formulation of the theory of types in the

Principia. While a full discussion of the theory falls

 

outside the scope of our topic, there are some relevant
points which must be made.

The theory of types was formulated by Russell and
Whitehead to avoid the paradoxes which had beset logic since
the time of Cantor and Frege. The theory set up in the

Principia does this in an overly restrictive manner, as was

 

later pointed out by Ramsey, when he developed the simple

theory of types in the mid- 1920's.

 

69PM, p. 9n.

7OIbid.

186

The essentials of the theory as presented in the

Principia are as follows: a type is defined as the range of

 

significance of a propositional function (i.e., all of the
arguments which that function can actually take, the test of
this being whether the propositional expression formed from
the function-eXpression and an argument-expression actually
makes a meaningful assertion). The hierarchy of types is
apparently71 conceived as being built from functions of
individuals, functions of functions, functions of functions
of functions, etc. But within each type there occur orders,
generated in the following fashion: The matrix associated
with'some value of a propositional function is defined as
being the value in question, with none of its arguments
generalized (i.e., none of its variables quantified).
Individuals are of the lowest order. Any function whose
matric involves no higher order variable than individual
variables is a first-order function (assuming that it
actually does involve such an expression as argument) and so
on.

The significance of ordering functions in the manner
just sketched is that an expression of type one might be of
32y_order. Consider the following example: If a particular

function involves two arguments, a first-order expression and

 

71The hierarchy of types per 32 is not generated
within the Principia, but merely assumed. On the other hand,
the hierarchy of orders of types is fully described.

 

187

an individual-expression, then the eXpression is of order
two and type two. But if the first-order argument is then
bound, this forces the function to take as arguments only
the values of the individual-expression, which is its only
genuine argument. We now have a type one function (taking
only individuals as arguments), but it still remains an
order 2E2 expression, for its order is determined by the
order of its matrix, which is of order two. In this way,
type one expressions can be of any order.

This complex mechanism was set up to insure that
significant generalizations must be those which are restricted
to the orders of some definite type (although for practical
purposes, allowance was made for some typically ambiguous
expressions).

The relevance of this ramified theory of types to the
theory of logic under consideration is this: since the theory
of types was set up to eliminate certain kinds of arguments
and expressions, the question arises as to whether the
authors thought that propositions, functions, and individuals
conceived in the realistic sense (as states of affairs etc.)
also conformed to the theory of types. That is, is the theory
simply an artificial constraint on language, or does it
actually describe what they took to be the hierarchy of
entities of all sorts?

Few realize that Russell and Whitehead thought that
they were describing the way things are, and were not merely

sprucing up language. There are two bits of evidence which

188

confirm this. The only such evidence which occurs in the

Principia occurs in *12, when it is maintained that "the

 

division of objects into types is necessitated by the
vicious—circle fallacies which otherwise arise. These
fallacies show that there must be no totalities which, if
legitimate, would contain members defined in terms of them-

"72 Presumably, when the authors said that they are

selves.
ordering ob‘ects, they meant this in a broad enough sense to
take into account objects other than linguistic objects. The
other piece of evidence that their concern with the theory

of types was broader than just a linguistic one comes from
Whitehead's later work. This is his idea that the hierarchy
of eternal objects (which can be construed -very roughly as
being propositional functions in the broad sense) conforms to
the theory of types. That is, the hierarchy, as he describes

it is perfectly consonant with the theory of types of the

Principia. While a discussion of this hierarchy would carry

 

us far afield, consider just one remark made by Whitehead:
”There is not, however, one entity which is merely the class
of all eternal objects. For if we conceive any class of
eternal objects, there are additional eternal objects which
73

presupposes that class but do not belong to it." This is

just a glimpse of how Whitehead conceived of eternal objects

 

72PM, p. 161.

73Whitehead, op. cit., Process and Reality, p. 73.

189

as conforming to the theory of types, but it is enough to

show that he considered the theory to be an accurate description
of the way in which things are. A more extensive discussion

of the place of types in Whitehead's metaphysics can be found

in the chapter entitled "Abstraction,” in his Science and the

Modern World.

 

Thus the theory of types is perfectly consonant with
the theory of logic being considered, for its subject-matter
(propositions, etc.), even when conceived in the broadened
fashion discussed above, still allow the theory of types to
make sense as a description.

The discussion of the theory of logic of the first

edition of the Principia is now complete; we have discussed

 

not only the basic theory, but those topics which follow
from such basic ideas.

The chapter to follow deals with the radical changes
made in the second edition of the Principia, together with
some relevant aspects of the phi1050phy of Wittgenstein, as
they bear on the discussion of the theory of logic of the

Principia. The final assessment of the theory of the

 

Principia will be made at the end of that chapter, as we

 

will then be in a position to compare the two theories.

CHAPTER VI
PRINCIPIA MATHEMATICA, II
1. INTRODUCTION.

In 1927, the second edition of Principia Mathematica
was published under the supervision of Russell. While he
made no attempt in this edition to make it appear that
Whitehead was not responsible for the changes in it, such
was nonetheless the case. The position maintained in the
second edition is in no way due to Whitehead, for while
Russell was expounding atomism and the theory of
extensionality, Whitehead was busy at the other end of the
philosophical spectrum, writing Process :22 Reality.
Whithhead's distress over the fact that no mention was made
of his non-participation in the new edition is indicated

by the following note which he sent to the editor of Mind:

The great labour of supervising the second edition
of the Principia Mathematica has been solely under-
taken by Mr. Bertrand Russell. All the new matter
in that edition is due to him, unless it shall be
otherwise expressly stated. It is also conven-
ient to take this opportunity of stating that the
portions of the first edition -- also reprinted in
the second edition -- which correspond to this

new matter were due to Mr. Russell, my own share
in those parts being confined to discussion and
final concurrence. The only exception is in
respect to *10, which preceded the corresponding
articles. I had been under the impression that

190

191

a general statement to this effect was to appear

in the first volume of the second edition.1
It is no longer the Principia of Russell and Whitehead with
which we are dealing. This fact is unfortunate, for much of
the novelty of the first edition was due to the interplay of
the separate concerns of the two authors.

There are two main areas of difference between the
editions of the Principia. In the first place, there are
purely formal differences occasioned by various discoveries
in logic during the years between the publication of the two
editions. The other area of change has to do with a profound
reorientation to questions of interpretation. This change is
due almost entirely to the influence of Ludwig Wittgenstein
(1889-1951) on Russell.

This chapter will be devoted, then, to a brief
account of the logical differences between the two editions,
to the positions held by Wittgenstein which influenced

Russell, and to Russell's reaction to them in the new edition.

2. CHANGES IN THE LOGIC 0F PRINCIPIA MATHEMATICA.

In a way, the second edition of the Principia is an
entirely different work, although there is little evidence
of it beyond the introduction. The main difference is the

introduction of the primitive idea 'p and q are incompatible'

 

1Mind, (Edinburgh: Thomas Nelson and Son, Ltd., 1926),
ns. Vol. 35, January, p. 130.

192

for the two fundamental functions of prOpositions employed
in the first edition, negation and disjunction. Russell
exPlained that he had left the text unchanged (despite the
replacements) because ”any alteration of the prOpositions
would have entailed alteration in the references, which
would have meant a very great labour."2 Even if the new
primitive function had been introduced into the main part of
the work, there still would not have been a very noticeable
difference in it; for once the new function had been introduced,
the more familiar connectives would have been defined in
terms of it. In this way, the proofs would have remained
essentially the same. Nonetheless, the very foundation of
the system had been changed. Let us look briefly at how
this change took place.

In 1913, H. M. Sheffer published a paper entitled “A
Set of Five Independent Postulates for Boolean Algebras,
with Application to Logical Constants."3 As the title
indicates, the primary purpose of the paper was to provide
a set of postulates for Boolean algebra. This set, as

Sheffer says, ”differs from the previous sets(1.e., those

 

2Russell, Bertrand and Whitehead, Alfred North,
Principia Mathematica (Cambridge: Cambridge University
Press, 1927), p. xiii. Hereafter all footnote references
to this work will read as follows giving page number after
the initials: PM, p. xiii.

3Sheffer, H. H., ”A Set of Five Independent Postulates
for Boolean Algebras, with Application to Logical Constants,”
Transactions of the American Mathematical Society, Vol. 1h,
1913, pp. E81189. —

193

of Huntington and Schr3de£l(l) in the small number of
postulates, and (2) in the fact that the set contains no
existence postulate for z, u, or a.'h In the discussion of
this postulate set, Sheffer employs the sign "' to stand
for a certain relation obtaining between elements of a given
set. In setting up the postulate-set, he first assumes the
existence of a class K, and a binary K-rule of combinationl.
Then he lists properties of K and of I, some of which are:

2. Whenever a and b are K-elements, a|b is a
K-element.
5

Def. a'=a{a

h. Whenever a, b, and the indicated
combinations of a and b are K-elements,

al(b\b')=a'. 5

Nowhere in the introductory portion of the paper is the stroke
truth-functionally defined, although it is obvious by the way
in which it is employed that the function has all of the
truth-functional characteristics with which it was later
associated.

In the second portion of the paper, under the heading
”application to Primitive logical constants,I Sheffer notes

that

 

“Which are special elements which correspond to such
Boolean symbols as '1' and '0' employed in the postulate sets
already mentioned. Sheffer, op. cit., p. #82.

5Where 'a" is presumaldy the complement of 'a'.

6Sheffer, op. cit.

19¢

On these two primitive ideas Eegation and
disjunctiofilin view of the following interpre-
tation of K and |, our set 1-5 @f postulates]
has an important bearing. For if K is the
class of all propositions of a given logical
type, then whenever p and q are two propositions
of this type, p|q may be interpreted as the.
proposition neither 2 22£_35 in other words,

i has the properties of the logical constant
neither-22E. This logical constant we may
symbolize by‘A, and for obvious reasons we may
name rejection.

Theorem 1. I i2_anz list 23 primitive ideas

£2; lggig both negation and disjunction 2E2

primitive, EEEX.EEZ.2£ replaced bz_£hg single

primitive idea rejection.7
In this way Sheffer explicitly recognized the relation between
the stroke and the primitive functions employed in the
Principia.

Also, besides recognizing Sheffer for introducing the
stroke, Russell also credited him with having developed a
new method in logic. As he puts it, “...a new and very
powerful method in mathematical logic has been invented by
Dr. H. M. Sheffer. This method, however, would demand a
complete re-writing of Principia Mathematica."8 No further
mention of such innovations is made in the Princi ia, and
the method to which Russell referred was never to be seen

in print.9

 

71bid., p. 481.

8P“, p. xv.

9Probably what Russell had in mind was an unpublished
paper of Sheffer's, entitled "The General Theory of Notational
Relativity.” (Typescript, Cambridge, Massachusetts, 1921).

195

The other main innovation of a strictly logical
nature in the second edition of the Principia is the
replacement of the primitive propositions *1.2.3.h.5.6 by
a single axiom using only the stroke. This was done by '
J. Nicod (“A Reduction of the Number of Primitive Pr0positions
of Logic,“ Proceedings of the Cambridge Philosophical Society,

Vol. xix). The single axiom is as follows:

 

This paper, as Sheffer points out, is based on the work done
in several of his earlier papers, all of which.appeared in
the Bulletin 2£'£h£.American Mathematical Society. Some

of these papers are: "Total Determinations of Deductive
Systems with special reference to the Algebra of Logic,"
(March, 1910); "Superpostulates: Introduction to the Science
of Deductive Systems," (November, 1913); ”Mutually Prime
Postulates,“ (March, 1916).

In the paper under consideration, Sheffer says,

In a volume entitled Analytic Knowledge,
which the writer hopes to publish in the near
future, a new method which may be characterized
as a sort of Prolegomenon to Every Future
Postulate Set -- is developed in detail, and
is then applied to the solution of awnumber
of fundamental problems in logic, mathematics,

and Mengenlehre. (p. 3)

The heart of the paper is indicated by the following
statement:

Deductive systems, it is well known, may
be determined by means of postulate sets in
various ways. ... May there not be, then, a
set of 'superpostulates,‘ of which Hilbert's,
Veblen's, Huntington's and other postulate
sets are special cases? There is. And, as a
matter of fact, the 'invariant' of these
postulate sets turns out to be of an extra-
ordinarily simple character. (p. 2)

The system which Sheffer develops is much too complex to be
dealt with here. Perhaps these statements, however, give
some indication of the object of Russell's enthusiasm.

 

196

{pink-)3 | EHNHB I {(slq)|((pl8)l(p8))3] .

Russell noted that this axiom, together with the rule of

10 could do all that the six axioms

inference p, p\(q\r) :.r,
of the first edition, together with modus ponens, and
substitution could do.

Concerning these innovations Russell remarks that
"From this E.e., the innovations there follows a great
simplification in the building up of molecular propositions
and matricies; ..."11 In a way, he is of course correct;
for using the stroke and Nicod's axiom, together with the
modified rule of inference, a great economy in the primitive
basis of the system is effected. But as Kneale says, ”it
can scarcely be said that the reduction achieved by Nicod is
a simplification which makes the theory easier to grasp."12
That is, while such innovations make the primitive basis
(quantitatively) more simple, the development of the system
is in like proportion complicated. If Russell had actually
employed Nicod's axiom in the body of the work, without
defining the other connectives and establishing derived

rules, the logic of the Principia would have become

unbearably complex.

 

1°'\' is here defined as non-conjunction, not as its
dual, non-disjunction.

11
PM, p. xiii.

12Kneale, William and Kneale, Martha, The Development
f Logic (Oxford: Oxford University Press, 1931), p. 527.

197

The only other extra-theoretical change which should
be noted is a slight but important change in terminology.

In the first edition no differentiation is made between kinds
of elementary propositions;they are simply defined as being
those which involve no variables. It was apparent from the
beginning, however, that there is a difference between kinds
of elementary propositions. On the one hand, there are
those such as Russell's example of “this is red” where the
subject is something ”given in experience.” This was held
to be the most fundamental sort of prOposition. On the
other hand, propositions constructed out of these by the use
of negation and disjunction were also classed as elementary,
for they too did not involve variables.

In the second edition, this difference is made
explicit. The first sort mentioned above, i.e., the “immediate”
variety, are now called "atomic“ propositions. Russell says,
"atomic propositions may be defined negatively as prOpositions
containing no parts that are propositions, and not containing
the notions 'all' or 'some.‘ Thus, 'This is red,’ 'this is
earlier than that,‘ are atomic propositions."13 On the other
hand, propositions built up from atomic propositions by
means of the stroke are now called "molecular propositions."
Together, the two sorts are called elementary prOpositions.

As Russell put it, ”elementary propositions are atomic

 

13PM, p. xv.

198

prOpositions together with all that can be generated from
thepuby means of the stroke applied any finite number of
times. This is a definite assemblage of propositions.”1u

We have now covered the important changes made between
editions, which are of a purely logical nature. Were these
the only changes occurring in the second edition, they would
hardly warrant consideration in a separate chapter. Of far
more consequence are the changes yet to be dealt with;

changes in interpretation gave the second edition an

entirely different character from that of the first.

3. A. INFLUENCES ON THE THEORY OF LOGIC OF THE SECOND

EDITION OF PRINCIPIA MATHEMATICA; INTRODUCTION.

While the theory of logic of the first edition of the
Principia was difficult to reconstruct, due to the authors'
reluctance to delve fully into questions of interpretation,
the theory associated with the second edition is just as
obscure, due mainly to changes in Russell's handling of the
concept of a proposition. In the course of the introduction
and appendices to the second edition, Russell manages to
change completely the theory of logic of the Principia.

We have already alluded to the reason why Russell
changed his mind on many questions of interpretation; between~

the two editions he had come under the influence of his pupil,

 

lap“, Po xv110

 

199

Ludwig Wittgenstein. Wittgenstein's work during this period
was done mainly in the theory of logic, his best known work
of the period being the Tractatus Logico-Philosophicus which
deals largely with questions concerning the interpretation
of logic.

Russell's revision of the second edition of the
Principia, while not a simple restatement of Wittgenstein's
position, is nonetheless perfectly consonant with its main
points. Where their positions diverge, the difference is
usually of minor importance.

The remainder of this section will be structured
as follows: We shall first briefly discuss Russell's theory,
to provide a basis for comparison with Wittgenstein's theory.
We shall then consider Wittgenstein's theory itself.

The second edition of the Principia presents a more
conservative view on questions of theory than did the first
edition. That is, Russell attempts to rid the work of some
major notions countenanced in the first edition. For example,
while provision was made in the first edition for countenanc-
ing functions as a sort of thing (i.e., as properties), in
the second edition Russell adopts a sort of logical atomism
by maintaining that propositional functions occur only through
their values. While we shall look at this claim in some
detail later, it can be tentatively taken to mean that
functions are no longer considered to be a separate kind of
thing. Russell wishes to maintain that all that exists are

individuals (i.e., values of predicative functions), while

200

functions themselves serve only as ways of explaining the
similarities among various individuals.

Another major change between editions of the Principia
is Russell's contention that functions of propositions are
always truth functions. According to this position, such
propositions as those of the form 'A believes p' are not
considered to be molecular propositions involving the function
9 believe39. Thus, if replacements for 'A believes p' are
to be propositions at all, they must be of an entirely
different sort than those which are truth functional
molecular propositions. In the first edition, reference is
made to the fact that "such functions as 'A believes p' are
not excluded from our i.e., Russell and Whiteheadflgfl
consideration, and are included in the scope of any general
propositions we make about functions; but the particular
functions of propositions which we shall have occasion to
construct or to consider explicitly are all truth-functions."15
The change between editions, then, is that while 'A believes
p' was considered to be a function of propositions in the
first edition, although not a truth-function, it is not
considered to be a function of propositions at all, in the
second edition.

Both of the changes just mentioned reveal a basic

alteration in the theory of logic of the Principia. Both

 

15PM, I). 80

201

of the changes are, by Russell's own admission, taken from
Wittgenstein. The reason Russell gives for adopting them

is that by doing so, he might avoid the necessity of
adapting the axiom of reducibility, which he considered to
be unacceptable as an axiom. We shall look into this matter
again in a later section.

The only other major theoretical change between
editions is that in the second edition, the notion of a
proposition is markedly altered. It is no longer possible
to tell with certainty in the second edition, what sort of
a thing a proposition is, or even what the alternative
interpretations might be. Here again, the main influence
on Russell was Wittgenstein.

In order to understand these changes fully, we shall

now turn to Wittgenstein's own theory.

3. B. INFLUENCES ON THE SECOND EDITION OF PRINCIPIA
MATHEMATICA; THE PHILOSOPHY OF LOGIC OF LUDWIG

WITTGENSTEIN.

While Ludwig Wittgenstein was not a logician in the
same sense as were the system-builders whom we have been
considering, his work has had a marked influence on
subsequent developments in the subject. Since we are
directly concerning ourselves only with those individuals
who have made the greatest strides in logic, we shall not
consider Wittgenstein in his own right. But since his

(indirect) influence on the subject has been so great,

202

neither can we neglect his work. Hence in this section we
shall briefly discuss the major features of Wittgenstein's
theory of logic, so that we may better understand Russell's
later theory.

Wittgenstein's best-known work is his Tractatus Logica-
Philosophicus.16 It is a work concerned with topics
concerning the relation between formal aspects of language
and the world at large. The Tractatus is an unusual work in
many ways; perhaps its most striking feature is its strange
format. Wittgenstein arranged the work into an extended
series of terse assertions, and assigned a number-value to
each assertion (or in some cases to short paragraphs which
dealt with a single point). The assignments were made in
order that the reader could see which points followed which,
other points, which points were intended to elucidate (i.e.,
to be subservient to) others, and in general, to exhibit the
entire structure of the discussion. While this format is a
great help in understanding Wittgenstein's intent, the terse
character of the assertions he makes in it render the work
obscure in many places.

While nearly all of the Tractatus bears on problems
in the theory of logic, we shall be able to deal only with
those areas of major importance to Wittgenstein's own theory,

and to Russell' theory. We shall first examine Wittgenstein's

 

16First published in 1921, in Wilhelm 0stwald‘s
Annalen der Naturphilosophie.

203

general theory of logic, and then some of the specific points
which he develops from it.

Logic, according to Wittgenstein, “is not a body of
doctrine, but a mirror image of the world."17 On the same
point, he says also that ”the prOpositions of logic describe
the scaffolding of the world, or rather they represent it.
They have no subject-matter."18 Everything else Wittgenstein
says about logic serves to explain what he means by
characterizing logic in this way. From these statements, it
is apparent that Wittgenstein thought of logic as being quite
different from any other discipline. But the highly meta-
phorical nature of the language he uses in such assertions
makes it impossible, without additional discussion, to
understand just what he thought logic to be. There are
three areas of his theory which need detailed attention: the
nature of the world, the nature of language, and the nature
of the “mirroring” relation which holds between the two.

First, we shall consider Wittgenstein's conception of
the nature of the world which serves as the subject-matter
of logic. In doing this, we shall first discuss his general

ontology. Using this discussion as a basis, we shall then

 

17Wittgenstein, Ludwig, Tractatus Lo ico-Philosophicus,
D. F. Pears and B. F. McGuinness (trans.) London: Routledge
and Keegan Paul, 1961), 6.13. (First German edition, 1921.
First English edition, 1922.) Hereafter all footnote
references to this work will read as follows listing Wittgen-
stein's own numbering system after the initials: TLP, 6.13.

18"Sie .handeln' von nichts." TLP, 6.13.

20h

consider those specific features of his ontology which bear
most directly on considerations in the theory of logic. In
particular, we shall deal with the status of form in
Wittgenstein's theory.

Second, we shall be concerned with Wittgenstein's
conception of the nature of language. In dealing with this
topic, we shall consider his conception of the nature of
propositions, and what about them that makes them capable
of mirroring the world.

The third area of concern will be with the notion of
"mirroring" itself. That is, we shall try to discover whether
Wittgenstein considered mirroring to be strict formal
isomorphism between language and extra-linguistic fact, or
some looser sort of parallelism between the forms of
statements and the forms of the objects which serve as their
subject-matter.

First, let us consider Wittgenstein's general
ontology.

Wittgenstein conceived of the world as being basically
composed of atomic objects. These combine to form states of
affairs (in alternative terminology, “atomic facts”). These
atomic facts are divided further into the class of those
which exist~ (which Wittgenstein calls 'facts') and the class
of those which do not exist. Wittgenstein's ontological
hierarchy is simple and rigid; facts may grow more and more
complex, but they are always basically combinations of

objects. Furthermore, ”The world is the totality of

205

facts..."19 In order to better understand the hierarchy,
let us consider each level separately.

Objects, according to Wittgenstein, are simple;
”Objects make up the substance of the world. That is why

'20 Furthermore, they are the basic

they cannot be composite.
constituents of the world, in that they are the basic
constituent in all states of affairs. As Wittgenstein
says, ”substance i.e., objecta is what subsists independently
of what is the case."21 Also, ”Objects, the unalterable,
and the subsistent are one and the same."22

While Wittgenstein makes it clear that he considers
objects to be fundamental, he does not attempt a description
of the nature of objects. He goes on to characterize them
in several ways (saying, for example, that space, time and
color are forms of objects), but he does not discuss such
questions as e.g., their ontological status (i.e., whether
they form a separate category, etc.). In fact, it is likely
that he leaves the discussion of objects open-ended for a
definite purpose. For instance, Wittgenstein uses the

concept of an object to cover a variety of entities. As

one commentator says, "I have counted at least a dozen kinds

 

19TLP, 1.1.

20TLP, 2.02.

21TLP, 2.02b.

22TLP, 2.027.

 

206

of entities which may be suggestedEy the term 'object3 by
23

reading the Tractatus....” He goes on to say that among
these are, for instance, sense-data, besides various sorts
of extra-sensory things. From this it seems clear that one
of the reasons why Wittgenstein did not attempt a full
discussion of objects was that the concept cut across several
categories. That is, he did not conceive of the notion as
marking out its own peculiar sort of entity. The term
'object' served only to group several different kinds of
things which shared the feature of being basic in the world.
Wittgenstein's treatment of objects resembles Frege's
treatment of the same concept; both single objects out by
allowing language to determine what is, and what is not, an
object. That is, both seem to hold that objects are those
things which correspond to the subjects in subject-predicate
statements.2h Classifying objects in this way has the effect
of singling out those things which (like Aristotle's primary
substances) can be held to be the basic constituents of the
world, and also the effect of cutting across many different

ontological categories, as was mentioned above.

 

2 . - . -
3Maslow, Alexander, A Study ig,Wittgenstein's
Tractatus (Berkeley: University of California Press, 1961),

p. 9.

h
2 Thus, says Wittgenstein, "A name means an object"
(TLP, 3.203) and, "In a proposition a name is the

representation of an object.” TLP, 3.22.

207

On the next level of Wittgenstein's ontology, he
considers an atomic fact to be ”...a combination of
objects.'25 He does not attempt to give an explanation of
the nature of combination in general. When he does consider
the topic, he makes obscure assertions, such as ”In a state
of affairs E.e.~, atomic facB objects fit into one another
like the links of a chain."26 Since he does not go on to
unravel such assertions, we are largely left to our own
devices in attempting an explanation of combination. However,
since he takes the totality of facts to comprise the world,
it seems likely that he conceived of the concept of
combination in the broadest possible sense. In this way, any
relation which holds between objects wwould be a combination.
There is one further feature of the concept of combination
of objects into atomic facts which we shall discuss after
finishing the general discussion of Wittgenstein's ontology;
this is his idea that the form of an object is the manner
in which it enters into states of affairs.

The last major level of Wittgenstein's ontology is
that of a fact. As he says, ”what is the case -- a fact --
is the existence of states of affairs."27 That is, he

differentiates between those states of affairs (i.e., atomic

 

25TLP, 2.01.

26TLP, 2.03.

27TLP, 2.

208

facts) which are actual (i.e., exist) and those which are

not actual. As Anscombe says, “...to the question 'What is

a fact?‘ we must answer: ”It is nothing but the existence

of atomic facts.‘ ...And to the question: 'Is there such a

thing as a negative fact?‘ we must answer: 'That is only the

non-existence of atomic facts.’ Thus the notion of a fact is

supposed to be explained to us by means of an atomic fact,

or elementary situation."28 As we noted earlier, it is the

totality of facts for Wittgenstein that constitute the world.
The foregoing overview of Wittgenstein's ontology

provides us with a base from which to discuss the concept of

form, which relates to objects and states of affairs. The

concept of form is of central importance in that when he

speaks of logical propositions mirroring the ”scaffolding"

of the world, Wittgenstein presumably understands “scaffolding"

to mean ”form.“ He says, for instance, that I'propositions

222! the logical form of reality. They display it."29
In general, Wittgenstein considers form to be "the

possibility of structure."30 With specific regard to objects,

he says, "the possibility of its [i.e., the object'é]

31

occurring in states of affairs is the form of an object."

 

28Anscombe, G. E. M., A__n_ Introduction to Wittgenstein' s
Tractatus (London: Hutchinson —University Library, 1959), p. 30.

29TLP, 0.121.
3°TLP, 2.033.

31TLP, 2.0101.

209

It is his view, that is, that at every level, from the most
simple things to the most complex, it ig‘fggg EEEEE.$2.£EE
determiner 2£_ggg things £23 combine :ith one-another. Form
determines how objects may combine with each other into states
of affairs. Moreover, ”The structure[1.e., forélof a fact
consists of structures of states of affairs."32
We shall see later that form-in-the-world, as Wittgenstein
conceives of it, is mirrored by the form of the propositions
(which are linguistic). These correspond to the states of
affairs having the form. Wittgenstein attempts in his way to
build a strong parallelism between the logical features of
language, and the formal features of the world.
Before dealing in detail with the linguistic side of
the mirroring relation, there are a few further points
concerning Wittgenstein's ontology worth noting. For instance,
there are some difficulties in interpreting Wittgenstein's
ontology. On the one hand he appears to maintain that all
that really exists are objects, and that states of affairs,
which are accumulations of objects, are nothing superadded
to the objects which they contain. This difficulty pervades
the interpretation of the system. On the one hand, he holds
to a strict atomism. On the other hand, he countenances

form, which serves as a connecting factor between objects.

The difficulty consists in attempting to establish the

 

32TLP, 2.033.

210

ontological status of such ”connecting“ notions. Whether
form exists in its own right remains an open question. As
we shall see, there are difficulties which parallel these
in intepreting Wittgenstein's theory of language.

In order better to understand the concept of form,
and to deal with the second major part of Wittgenstein'a
theory of logic, we shall now consider this theory of language.

The heart of Wittgenstein's theory of language is his
concept of a proposition. For him, propositions are
basically linguistic entities. He says, “the simplest kind
of prOposition, an elementary proposition, asserts the
existence of a state of affairs[§.e., an atomic fact-J."33
Then he says, "an elementary prOposition consists of names.
It is a nexus, a concatenation of names.'3u From such
assertions as these, it is clear that Wittgensteinw thought
of propositions as linguistic entities.

Wittgenstein's concept of a prOposition strongly
resembles our present concept of a statement. Consider, for
example, his remark that "I call the sign with which we
express a thought a propositional- sign -- and a prOposition
is a propositional sign in its projective relation to the

world."35 Thus a propositional sign is a sentence, a

 

33

3“TH, 9.22.

TLP, 0.22.

35TLP, 3.12.

211

syntactical configuration of letters obeying certain
grammatical rules. A proposition, on the other hand, is a
statement, a sentence with an associated meaning. This is
borne out in such a way that “elements of the prOpositional
sign correspond to the objects of the thought."36 In this
way, the names which are a part of the prepositional sign
correspond to the objects which the author of the propositional
sign had in mind: "the configuration of objects in a
situation corresponds to the configuration of simple signs
in a propositional sign.'37

We noted earlier that the ontological status of that
which provided for the combination of objects into states of
affairs is questionable. A similar difficulty now arises in
connection with the discussion of propositions. The status
of the linguistic "glue," i.e., that which combines terms
into statements is also questionable. Wittgenstein's position
may indeed be perfectly consistent on this point, in that he
may have in mind heaps of objects corresponding to simple
linear concatenations of names. While consistent, however,
difficulties still haunt such a view. What, for example,
of statements such as 'Grendel is a beast?‘ What we are to
count as being a name in such a statement is not clear. If

the statement is a simple concatenation of names, than 'a

 

36TLP, 3.2.

37TLP, 3.21.

212

beast' must be a name. But of what it is a name we are not
told. If held strictly to the idea that statements are
concatenations of names, we would also have to make provision
'is' as a name of some sort (which Russell in fact tried to
do in some of his earlier work).

Another interesting feature of Wittgenstein's theory
of propositions is his claim that "all propositions are
results of truth operations on elementary propositinns."38
Further, he says that ”a proposition is an expression of
agreement and disagreement with truth-possibilities of

39 To deal with this thesis more

elementary propositions."
adequately, let us first look at Wittgenstein's use of truth
tables. This will provide us with a better idea of what he
means by 'truth-possibilities.'

In introducing truth-tables, he first says that
"Truth-possibilities of elementary prOpositions mean
possibilities of existence and non-existence of states of
affairs."l‘0 Then he says, ”We can represent truth-
possibilities by schemata of the following kind ('T‘ means

'true,' 'F' means 'false'; the rows of 'T's' and 'F's' under

the row of elementary propositions symbolize their truth-

 

38TLP, 5.3.

39TLp, 0.0.

“OTLP, 0.3.

213

possibilities in a way that can easily be understood); ... ."hl

By maintaining that all propositions are products of truth
operations, he is held to the view that any proposition is
in principle capable of being represented by a truth-table,
i.e., that there is always a way of computing the truth-
possibilities of the proposition in a purely mechanical
fashion.

In establishing such a position, there are three
separate cases to be considered. First, the most obvious
case of truth-functional combination concerns propositional
connectives. In this case, Wittgenstein is maintaining that
statements of a form such as 'p:>q' are truth—functional
compounds of the truth-possibilities of their constituent
propositions. He explicitly deals with this case, by
providing truth-tables for the sixteen binary propositional
connectives.

The second case is more difficult. In order to
establish that all propositions are results of truth-
combinations of elementary propositions, Wittgenstein must
provide for quantified statements. He attempts this by
holding that such propositions can be reduced to series of

elementary propositions. As Black says, “Wittgenstein's

 

thhere then follows three standard truth-tables.
Wittgenstein's use of them, while not a genuine innovation,
in that Peirce had invented them,is still regarded as the
starting point of their current use. TLP, 9.31.

z‘2c1‘., TLP, 5.101.

211:

conception of the relation of general propositions to
elementary propositions is hard to grasp. It seems certain
that he wanted to construe general propositions as conjunctions
and disjunctions of elementary propositions."u3 On this
account, an expression such as '(x)FxD(3x)Fx' would be an
abbreviation for an expression of the form 'Fa.Fb.Fc. ...
ZDFa v Fb v Fc v ... .' That is, it occurred to Wittgenstein
that universal quantification could be reduced to an unending
series of conjuncts, and existential quantification to a
similar series of disjuncts. As Ramsey says,
Mr. Wittgenstein has perceived that, if we

acceptfliij account of truth-functions as expres-

sing agreement and disagreementw with truth-

possibilities, there is no reason why the argu-

ments to a truth-function should not be infinite

in number. As no previous writer has considered

truth-functions as capable of more than a finite

number of arguments, this is a most important

innovation. Of course if the arguments are

infinite in number they cannot all be enumerated

and written down separately; but there is no

need for us to enumerate them if we can

determine them in any other waza as we can by

using propositional functions.
Wittgenstein (and Ramsey) believed that he had found a

simple procedure for reducing quantified expressions to

truth-functional compounds. It was not until later that

Church proved that no method was generally available for

 

u3Black, Max, A Companion to Witt enstein' s 'Tractatus'
(Ithaca: Cornell University Press, 19 , p. 281.

uuRamsey, F. P., The Foundations of Mathematics
(London: Routledge and Keegan Paul, 19317— p. 7.

2 1 5 7,» :3

testing whether, say, such an expression is a tautology.

But while this is the case, there are still several important
solvable cases of this "decision problem.‘I For our purposes,
perhaps the most important of these is Herbrand's proof that
there is such a decision procedure for the class of
"Well-formed formulas having a prenex normal form in which
the matrix satisfies the condition of being a disjunction of
elementary parts and negations of elementary parts or
equivalent by laws of the propositional calculus to such a
disjunction.'u5 While Wittgenstein's attempt to resolve all
quantified expressions into propositional expressions was
doomed to failure in general, his procedure still holds for
many cases; it was (though unknown to him) a partial

success.

The third class of expressions with which Wittgenstein
dealt in attempting to establish that all propositions are
the results of truth operations on elementary prOpositions
concerns intensional propositions. In the case of an
expression such as ‘A believes p,' Wittgenstein was faced
with a choice: either it could be truth-functionally
analyzed, or not considered to be ”proposition at all. His
resolution of the problem was to maintain that "'A believes
that p,' 'A had the thought p,‘ and 'A says p' are of the

form '“P' says p': and this does not involve a correlation

 

uSChurch, Alonzo, Introduction 32 Mathematical Logic
(Princeton: Princeton Univdrsity Press, 1956), p. 256.

testing whether, say, such an expression is a tautology.

But while this is the case, there are still several important
solvable cases of this “decision problem.I For our purposes,
perhaps the most important of these is Herbrand's proof that
there is such a decision procedure for the class of
”Well-formed formulas having a prenex normal form in which
the matrix satisfies the condition of being a disjunction of
elementary parts and negations of elementary parts or
equivalent by laws of the propositional calculus to such a
disjunction.""’5 While Wittgenstein's attempt to resolve all
quantified expressions into propositional expressions was
doomed to failure in general, his procedure still holds for
many cases; it was (though unknown to him) a partial

success.

The third class of expressions with which Wittgenstein
dealt in attempting to establish that all propositions are
the results of truth operations on elementary prOpositions
concerns intensional propositions. In the case of an
expression such as ‘A believes p,' Wittgenstein was faced
with a choice: either it could be truth-functionally
analyzed, or not considered to be .,~.proposition at all. His
resolution of the problem was to maintain that “'A believes
that p,' 'A had the thought p,' and 'A says p' are of the

form '“P' says p': and this does not involve a correlation

 

uSChurch, Alonzo, Introduction £2_Mathematical Logic
(Princeton: Princeton University Press, 1956), p. 256.

 

217

of a fact with an object, but rather the correlation of facts
byneans of the correlation of objects.""‘6 He exempts such
expressions, that is, from the ordinary role of propositions.
What he appears to be saying is what Frege maintained
concerning such cases of indirect reference, that the
reference in such cases is what would ordinarily be the sense
of the sentence.

Another major feature of Wittgenstein's theory of
prOpositions is his claim that all propositions of logic
are tautological. This leads directly to the related claim
that such propositions have no content, i.e., they have no
subject-matter.

He introduced the notion of tautology and
contradiction by saying that

among the possible groups of truth-conditions

there are two extreme cases.

In one of these cases the proposition is

true for all the truth-possibilities of the

elementary propositions. We say that the

truth-conditions are tautological.

In the second case the proposition is

false for all the truth-possibilities: the
truth-conditions are contradictory.“7

 

Wittgenstein's claim is that not only are all propositions
truth-functional, but also that the propositions of logic

form a limiting case.

 

“6Tip, 5.502.

“7Tip, 0.06.

218

After defining 'tautology' and 'contradiction,‘ he
goes on to say that ”prOpositions show what they say:
tautologies and contradictions show that they say nothing...
(For example, I know nothing about the weather when I know
that it is either raining or not reaining.)"L‘8 Further, he
says, "It is the peculiar mark of logical prOpositions that
one can recognize that they are true from the symbol alone,
and this fact contains itself in the whole philosophy of
logic. And so too it is a very important fact that the truth
or falsity of non-logical propositions cannot be recognized
from the propositions alone."""9 Thus being tautological and
being devoid of empirical meaning are two sides of the same
coin.

The difficulty confronting the view that all logical
propositions are tautologies is a special case of the
difficulty with the view that all propositions are truth-
functional: there is simply no general method for
mechanically determining the tautologousness of quantified
statements.

In holding that logical propositions are vacuous,
Wittgenstein was also led to hold that there are no ”logical
objects.“ That is, he maintained that there are no objects

which uniquely correspond to logical expressions. Part of

“BTLP, 0.061.

“9TLP, 6.12.

219

the reason for his saying that ”...there are no 'logical
objects' or 'logical constants' (in Frege's or Russell's
sense)"50 is that ”the results of truth-operations on truth-
functions are always identical whenever theywre one and the
same truth-function of elementary propositions."51 This
amounts to saying that logical connectives do not name
anything in the world, since they are inter—definable. If,
on this account '.' and 'eu' were really distinct from '13,‘
i.e., if they named distinct relations, then they should not
be capable of being exhaustively defined in terms of one-
another.

We have now discussed two major areas in Wittgenstein's
theory of logic, his ontology and his theory of propositions.
We must now pass on to the third major part of his theory,
to the question as to what connection there is between the
propositions of logic and the things in the world. Since we
have seen that he considered logical propositions to be
tautologies, and since he believed that there are no uniquely
logical objects, we are in a position to ask whether he
thought there to be 321 connection between logic and the
world of extra-logical fact.

Wittgenstein thought that while logical propositions

have no subject-matter, i.e., while there is no type of

 

SOTLP, 5.0.

51TLP, 5.01.

220

entity which uniquely correspond to logical propositions,
still they could be said to be about the world. He describes
the connection between such propositions and the world as
follows: ”The logical propositions presuppose that names
have meaning and elementary prOpositions sense; and that is
their connexion with the world. It is clear that something
about the world must be indicated by the fact that certain
combinations of symbols - whose essence involves the
possession of a determinate character - are tautologies."52
The first part of this remark is clear enough; the purely
formal aspect of a logical proposition corresponds to no
object. The connection between such a proposition and the
world depends on what Quins calls the 'vacuous' parts of the
proposition.

It is the second part of the claim made above with
which Wittgenstein attempts to point out the difference
between logical prOpositions and all others: logical
propositions exhibit purely formal properties in such a way
that they ”show“ the formal properties of language and of the
world. As he remarks, ”The fact that the prOpositions of
logic are tautologies 322:3 the formal - logical properties
of language and the world."53 But what does such ”showing“

involve? Wittgenstein does not tell us. It is not clear

 

52TLP, 6.120.

53TLP, 6.12.

221

whether he intended to maintain that there was some strict
identity between symbols and what they symbolize, or whether
he wanted to claim that there was some sort of parallel
between logic and the world. Although this seems to be the
more likely of the two alternatives, just what form this
parallel is to take is not made clear. Wittgenstein never
goes beyond his metaphorical use of the expression 'mirrors,‘
in his explanation of the relationship between logic and the
world.

A suggestion which may help in clarifying
Wittgenstein's claim that logical propositions mirror the
structure of the world is as follows: mirroring might be
considered to be an analogical relationship obtaining between
logical propositions and the world in such a way that the
structure of a state of affairs is analogous to the form of
the statement which, when interpreted, could be used to refer
to it. It is possible to catalogue states of affairs on
purely formal grounds by cataloguing the statements which
refer to them on analogous formal grounds.5h Thus, states
of affairs which have only one logical subject are analogous
to statements which have only one logical subject, and so on
for two, etc. This interpretation seems to be *borne out

by Wittgenstein's claim that ”the configuration of objects in

 

ShFor an example of this, cf., Leonard, Henry S.,
The Principles 2f Right Reason (New York: Henry Holt and Co.,
19577, Unit 16. """

222

a situation correspond to the configuration of simple signs
in a propositional sign."55 The analogy seems plausible in
relatively simple cases, but is much more difficult to

56

comprehend in more complex ones. Furthermore, the principle
behind such analogy is no clearer than the notion of mirroring
itself. It is likely that Wittgenstein was aware of such
difficulty, which may explain the lack of clarity.

However, the fact that no clear statement of the
nature of the relation between the form of statements and the
formal constituents of a state of affairs is provided by
Wittgenstein does not mean that none can be given. In fact,
as we shall see in the concluding chapter of this work,
Wittgenstein's line of reasoning concerning this aspect of
the nature of logic, as well as his line of reasoning in the
case of other aspects, is extremely promising.

We have now come to the end of our discussion of
Wittgenstein's theory of logic, although we shall return to
various points already discussed in our discussion of Russell,
which follows. In conclusion, there are several points

concerning Wittgenstein's contribution worth noting. First,

Wittgenstein had the distinction between syntax and semantics

 

55TLP, 3.21.
56
Such complex cases would be, for example, those of
the form 's is P' in which the predicate is intensional. In
such cases, there is no simple one-to-one correspondence
between parts of the state of affairs and the parts of the
statement.

223

well in mind in constructing his theory. The nature of logic,
on this account, depends on this distinction, in that it is
the syntactical features of language which mirror the formal
features of the world and thereby constitute logic. In fact,
Carnap, who first fully and systematically developed the
notion of syntax, says that ”It was Wittgenstein who first
exhibited the close connection between the logic of science
(or 'philosOphy' as he calls it) and syntax. In particular,
he made clear the formal nature of logic and emphasized the
fact that the rules and proofs of syntax should have no

57 Second, although

reference to the meaning of symbols....'
he made such distinctions as that between formal and non-
formal, he was able (as few have since been) to provide a

balanced account of the roles played by both sides of the

distinction in explaining the nature of logic.

0. RUSSELL'S THEORY OF LOGIC IN THE SECOND EDITION OF

PRINCIPIA MATHEMATICA.

We have already briefly discussed the main points in
Russell's theory. They are again, 1) the change in the
conception of the nature of propositions, 2) the idea that
propositions only occur in truth-functional combination, and

3) the idea that propositional functions occur through their

 

57Carnap, Rudolf, The Logical S ntax 2f Language
(London: Routledge and Keegan Paul, 1937), p. 282.

22“

values. We shall now examine each of these ideas more
closely, largely in the light of our discussion of
Wittgenstein.

With regard to prOpositions, Russell says that ”we must,
to begin with, distinguish between a proposition as a fact
and a proposition as a vehicle of truth and falsehood."58
This distinction is basic to his account of propositions,
for they are ultimately explained in terms of it. The
distinction can be understood for the moment in the following
manner: propositions as facts are sentences; they are sets
of marks obeying certain syntactical rules, but with no
particular meaning attaching to them. PrOpositions as
vehicles of truth and falsity, on the other hand, are more
like statements. By his making such a distinction, it is
clear that Russell intended the concept of a prOposition to
cover two different sorts of things.

After making the distinction, Russell remarks that
he will take up the question of propositions as vehicles of
truth and falsity first, and then return to prOpositions as,
facts. Of the former sort, he says: '

When we say that truth or falsehood is, for

logic, the essential characteristic of prOpo-

sitions, we must not be misunderstood. It does

not matter, for mathematical logic, what

constitutes truth or falsehood; all that matters
is that they divide propositions into two

 

58PM, I). 6600

225

classes according to certain rules. Let us take
a set of marks

X1, x2, see X2n-l, x2“.
Let us put, as unexplained assertions,

T(X2m+1) (“1‘") 9

F(x2m) (Ini- n).

Let us further introduce the symbol xr x5, and
assume

T(xr|xs) if F(xr) or F(xs);
F(xr|xs) if T(xr) and T(xs).

Assume further that, if p, q, s are any one

of the x's or any combination of them by means
of the stroke, the above rules are to apply

to p|q, etc., and further we are to have:

1*{pd(plp)3 .
T £p3q. D .s\qap\s3 ,

where ”pzbq' means 'p\(q\q).' Further: given
T{p\(q|r)3 and T(p), we are to have T(r).

Taking the above as more conventional
rules, all the logic of molecular proposi-
tions follows, replacing '}-.p" by IT(p),n59

Russell believed that by performing such replacement, he had

shown that we need not worry about the nature of prOpositions

when doing logic: ”Thus from the formal point of view it is

irrelevant what constitutes truth or falsehood: all that

matters is that propositions are divided into two classes

according to certain rules. It does not matter what

 

59Ibid.

226

propositions are, so long as we are content to regard our
primitive propositions as defining hypotheses, not as
truths."60 Russell then goes on parenthetically to admit
that philosophically the procedure just discussed may indeed
have non-formal presuppositions. These, he says, make no
difference to the discussion at hand.

Russell's claim that for propositional logic, we do
not need to know what propositions are, but only that they
are true or false, and further, his claim that we need to
know nothing about truth or falsity, shows a marked change
in attitude in the philosophy of logic, and a drastic change
from the theory of the first edition of the Principia. In
particular, Russell is greatly restricting the conception
of logic employed in the first edition.

Russell's final position with regard to propositions
is that regarded as vehicles of truth and falsehood, they
are “particular occurrences," and considered factually, they
are series of ”similar occurrences.” He is maintaining that
the former sort of propositions are in fact statements
considered as tokens (i.e., particular instances of

statements), while considered factually, they are the

 

60Ibid., pp. 660-1. This statement in particular shows
the radical nature of the change between editions of the
Principia. While the primitive prOpositions of the first
edition were held to be simply true, now Russell wishes to
hold that the primitive propositions are implicit definitions.
This is a good example of the very different attitude toward
questions of theory which Russell developed between 1915
and 19250

227

sentence-types out of which propositions as statement-tokens

are constructed. Russell's own explanation is that

“when we say ”'Socrates' occurs in the proposition
'Socrates is Greek," we are taking the prOpo-
sition factually. Taken in this way, it is a
class of series, and 'Socrates' is another class
of series.... The particular 'Socrates' that
occur at the beginning of our sentence does

not occur in the proposition 'Socrates is

Greek'; what is true is that another particular
closely resembling it occurs in the proposition.
It is therefore absolutely essential to all such
statements to take words and propositions as
classes of similar occurrences, not as single
occurrences. But when we assert a proposition,
the single occurrence is all that is relevant.
When I assert "Socrates is Greek,” the particular
occurrences of the words have meaning, and the
assertion is made by the particular occurrence

of that sentence...”

The shift between the two editions is clear; instead
of having to decide whether propositions are statements or
the states of affairs to which they correspond, as in the
first edition, we must now decide whether propositions are
statements, complete with meanings, or sentences, constructed
without regard to meanings. The effect of the shift is to
move logic away from an immediate concern with the world to
which language corresponds, to a position in which logic is
conceived as having no immediate relevance for the world.
Russell's attitude in the second edition is one of
conservatism, a pulling back of the frontiers of logic. The

full effect of such a shift will be seen later, in the final

 

61PM, pp. 660-5.

228

chapter of this work, after we have discussed others who
shared this kind of attitude toward logic. One observation,
however, is appropriate at this point. Russell's shrinking
of the boundaries of logic was possible only after the
system had been fully constructed. The origination of the
system was spearheaded by the broader interpretation,
discussed in the previous chapter.

The other two theoretical innovations of the second
edition, the idea that functions of propositions are always
truth-functions, and the idea that a function can only occur
in a proposition through its values, are introduced by
Russell in an attempt to circumvent the necessity for
recognizing the axiom of reducibility. In the second edition,
Russell observed that the only merit of the axiom of
reducibility is that it does what it was intended to do. That
is, it is pragmatically acceptable, but is acceptable from
no other standpoint.

The axiom of reducibility was adopted in the first
place as a device to permit reference to totalities of
propositional functions of a given type from within that
given type. Kneale Provides an interesting example of such
a totality. Remembering that a real number is a cut among
the rational numbers, or more precisely the lower set defined
by such a cut, Kneale says, “from this it follows that a set
of real numbers is a set of sets of rational knumbers. Now
if S is a non-empty set of real numbers with an upper bound,

the least upper bound of S is supposed to be a real number

229

which is also the union or logical sum of all the real numbers
in S, i.e., which has for its members all those rational
numbers which are members of any of the members of S."62 He
goes on to say that this looks very much like a violation of
the ramified theory of types, since the set of rational
numbers which constitutes a real number, being infinite,
"cannot be specified except by mention of a propositional
function which all the members satisfy, and the functional
expression which is supposed to specify the least upper bound
of an infinite set of real numbers must apparently involve
reference to the totality of propositional functions that
specify real numbers, including ppgp function 22322.£E
expresses."63 The axiom of reducibility makes such expressions
legitimate. What it says is that in orders of a given type,
for every prOpositional function there is a predicative
function which is equivalent to it. The propositional
functions specifying real numbers remain segregated into
orders, but there is no need to fear over-generalization,
for, for every prOpositional function specifying some real
number, there will be a lowest order function ranging over
the same number, which is equivalent to it.

Russell's attempt to circumvent the need for this

axiom is two-sided. He holds, in effect, that functions need

 

2Kneale, op. cit., p. 662.

63Ibid.

230

no "reduction” to functions of individuals, because it is
only in this way that functions can actually occur. This
amounts also to an adoption of Wittgenstein's atomism. 0n
the other hand, he maintained that his would not be overly
restrictive, since all functions could be seen to fit this
extensional pattern. It might be said in Russell's initial
defense, that he realized that there were difficulties with
such an extreme view. He says, for instance, that ”it i.e.,
Wittgenstein's view that functions of propositions are
always truth-functions, and that functions only occur through
value§]involves the consequence that all functions of functions
are extensional. ...We are not prepared to assert that this
theory is certainly right, but it has seemed worthwhile to
work out its consequences...."6u
Functions, in the view under consideration, cannot
occur as the arguments of other functions. As Russell says,
"There is no logical matrix of the form f” ¢22). The only
matricies in which $22 is the only argument are those
containing ¢a, Ob, QC, ..., where a, b, c, ... are

65

constants...” He then goes on to maintain that matrices
so formed are still not logical matricies, for they were not
generated by replacing prOpositional letters in propositional

expressions by function-expressions. This amounts to

 

60
65

PM, pa Xiv.

PM, p. xxxi.

231

maintaining that functions are nothing more than the sum of
their values, which in turn closely resembles Wittgenstein's
view that objects are the only things there are.

It is quite easy to see how the problem of the axiom
of reducibility has been circumvented: there is no longer
any need to establish an equivalence between functions of
higher orders and predicative functions for it is being
maintained that the higher order functions are nothing 2252
than their values at the lowest level. As Russell says,

"we may define a function $3; as that kind of similarity
between propositions which exists when one results from the
other by the substitution of one individual for another."66
In no case, however, does he admit that functions have
existence separate from their values.

While such a notion does get at the difficulties
surrounding the lack of obviousness of the axiom of
reducibility, it is like removing a diseased tooth by going
through the back of the patient's head; the job gets done,
but the patient is eviscerated in the process. The ideas
under consideration are radical areas of divergence from the
theory of the first edition. As in the case of Wittgenstein,
the status of functions is never made perfectly clear; for
although Russell, like Wittgenstein, apparently wants to

collapse ontology into the basement of individuals, both

 

66PM, p. xxx.

232

still talk as if functions were nonetheless something, but
neither provides a final answer.

Russell attempts to defend the thesis that functions
occur only through values by maintaining that functions of
propositions are always truth-functions. This position is
now known as the Thesis of Extensionality. The heart of the
thesis is the assumption that utterances made in an
"intensional“ language, i.e., one containing such indirect
expressions as those of the form 'A believes p,' can be
exhaustively reduced to an extensional language. Actually,
this characterization is only appropriate to Russell, for in
Wittgenstein's system, statements containing instances of
indirect reference were simply left in limbo; all that he
definitely maintained was that whatever actually :33 a
function of propositions was in fact a truth-function.
Russell, on the other hand, attempts to show how such
statements as 'A asserts p' can be formulated in the Principia
notation in such a way that it becomes obvious that 'p' is
not occurring truth-functionally. Taking the case in which
'p' in the statement form 'A asserts p' is 'Socrates is
Greek,‘ Russell says,

thus a person who asserts "Socrates is Greek"

is a person who makes, in rapid succession,

three noises, of which the first is a member

of the class 'socrates,“ the second a member

of the class "is,” and the third a member of

the class ”Greek.” This series of events is

part of the series of events which constitutes

the person. If A is the series of events con-

stituting the person,oCis the class of noises
”Socrates," lathe class "is,” and‘d‘ the class

233

“Greek," then A asserts that 'Socrates is
Greek! is (omitting the rapidity of succession)

(3 X.Y.z).x£<1.y€fl.z€‘6‘.x ‘YU x; z w
yj,z(;A.

It is obvious that this is not a f nction of

p as p occurs in a truth-function. 7
While there is something suspicious about this example, it
is not until Russell analyses the form 'A believes p' that
the nature of his actions becomes apparent. In this example,
belief is simply reduced to a relation between thoughts:
”...we shall say that when a man believes 'Socrates is Greek'
he has simultaneously two thoughts, one of which 'means'
Socrates while the other 'means' Greek, and these two
thoughts are related in the way we call 'predication."‘68
After getting over this hurdle, the rest of the example is
straightforward, for what Russell has done is simply to 2252
beliefs $232 objects in such a way that "A believes p“
becomes purely extensional in nature. That is, Russell is
maintaining not simply that intensional statements are
translatable into an extensional language, but that their
intensionality is only apparent in the first place.69 The
Thesis of Extensionality in Russell's case appears to be the

simple assumption that any assertion can be taken to be an

assertion about objects.

 

67PM, p. 6610

68PM, p. 662.
69On this point, cf. Carnap, 0p. cit., p. 206.

230

Whether Russell realized the radical nature of the
position which he was propounding is not clear, for he leaves
it almost totally undefended. In fact, it was not until
Carnap's Logical Syntax 2: Language that the thesis was to
find a coherent statement.7o Carnap makes the thesis
plausible by maintaining that all that is being attempted is
the translation of intensional contexts into extensional
ones. His formulation has found little success in the
ensuing years; Russell's formulation has found none at all.

The second edition of Principia Mathematica has every
appearance OB being an experiment in theory, carried out by
Russell. His drastic revisions are left unsupported. Further,
his borrowings from Wittgenstein leave him in an awkward
position, for it is not clear how much of the rest of
Wittgenstein's own theory he is willing to accept.
Wittgenstein's position, as we have seen, depended not merely
on the adoption of atomism (which Russell also adapted), but
also on a definite set of ideas concerning the nature of form
as the subject-”matter” of logic. Russell simply avoids any
reference to questions concerning such subject-matter. The
absence of such reference further complicates the interpretation
of Russell's position, for while the first edition provided
at least a partial indication of the nature of the subject-

matter of logic (i.e., propositions considered as states of

 

7OCfe’ ibide’ s 67 - g 71; g 7“ - g 810

235

affairs), Russell's shrinking of the boundaries of the
notion of a proposition removes even the last glimmer of a

possible interpretation.
5. CONCLUSION TO THE DISCUSSION OF PRINCIPIA MATHEMATICA.71

Having finished the discussion of both editions of
Principia Mathematica, some concluding remarks concerning
the theory of logic which it contains are now in order.

The lessons to be learned from the logic of Principia
Mathematica are many. It is a masterful example of a fully
complete system of mathematical logic; it was the first
system to use a really workable notation; it had an explicitly
stated primitive basis. The subsequent developments in logic
have been largely constructed on the foundation of the
Principia. Logicians openly recognize their debt to the
Principia. The theory of logic presented in it presents a
somewhat different picture. The question we must consider is,
what are the lessons to be learned from the theory of logic
of the Principia?

Compared with Frege's theory of legic, the theory
presented in the Principia is sparse; in some areas, it is
at best fragmentary. Where Frege built a detailed ontolOgy
for the sole purpose of leaving no doubt as to how his logic

was to be interpreted, Russell and Whitehead are content to

 

71This is to serve as a conclusion for both Principia
Mathematica chapters.

236

provide only the most necessary points of theory. Their
characterization of the nature of propositions provides
only enough for the reader better to understand how
prOpositions function within the system; how they relate to
the world, or exactly what they are, are questions which
receive only passing mention. Thus much of our discussion
has been concerned with adding meat to the bones of theory
provided in the Principia. We have in large part inferentially
expanded what Whitehead and Russell eXplcitly provided. In
neither edition did the authors attempt to present their
views on the nature of logic; Frege, on the other hand,
carefully considered such questions as "is logic a part of
psychology?” and ”is logic a science?“ Furthermore, the
theory of logic of the Principia really adds little to
previous theories; Frege's and Peirce's theories of
propositions were much more explicit, and in some respects
more defensible.

Does all of this mean that Russell and Whitehead were
inferior theoreticians? Does it mean that they were simply
not interested in questions of theory? In both cases the
answer is, absolutely no! The lesson to be learned from the
shape of the theory of the Principia concerns the state of
the develOpment of logic 23.3p3£.£ipp, and in no way concerns
the authors. Specifically, it should now be apparent that
logic, in the time of Russell and Whitehead, had attained a
level of sophistication in which $5.!gp simply pp longer

necessary £2 surround logic with 2 massive theory which

237

served 52 connect $2.!iEEHEEEE familiar objects. Logic was
showing its mathematical nature; like mathematics, the
abstract notions with which it dealt were becoming familiar
enough so that people working with such notions no longer
needed to have such abstractions broken into little pieces,
to be digested bit by bit.

In Frege's day, people had to be shown how logic
differed from mathematics. The broader sweep of the
variables of logic was quite new, and presumably very difficult
to grasp. By 1910 the necessity of leading people by the hand
through logic had passed; logic had prOgressed to the point
at which its umbilical cord to interpretation could be severed.

The consequence of this situation is good for logic,
but less than good for philosophy, especially that part
concerned with the nature and interpretation of logic; for as
soon as logic was separated from the necessity of being
accompanied by a full-fledged theory, logicians began
treating theory as a weak sister, to be tolerated, but not
catered to. Philosophers, on the other hand, went on to
concern themselves with formal questions, but for mainly
different reasons than to provide logic with a comprehensive
theory. Thus analytic philosOphy has grown up around such
topics as meaning criteria, and criteria for explanation.
While these topics border on the theory of logic, little
attempt was made to make any firm connection.

Fortunately, there have been several notable

exceptions to this pattern. These individuals are both

238

logicians and philosophers; it is they who are most concerned
with questions concerning the connection between the two
fields, and it is with them that we shall deal in the rest

of this work.

CHAPTER VII
C. I. LEWIS
1. INTRODUCTION.

In this chapter, we shall be concerned with the
theory of logic of C. I. Lewis (1883-1962). Both Lewis'
logic and his theory of logic differ in important respects
from those already considered. His logic differs in that he
uses a different sort of implication relation in his system,
in addition to the familiar material implication relations
employed by Frege, Russell and Whitehead. Another
distinguishing feature of his logic is his use of modal
notions as an integral part of his system of implication.

Lewis' theory of logic differs from others in that
his concept of logic arises directly from his theory of the
nature of inference. Moreover, his theory of logic is
pragmatically oriented; in his view, different logics are
constructed out of various logical facts, which are
conceptual meanings-in-the-use-of—language. The logician's
task is one of deciding which of the vast array of logics
best fits his needs. His decision, according to Lewis, will
be a pragmatic one.

In discussing Lewis' logic and his theory of logic,
it should be kept in mind that he was working outside of the
logicist tradition. While he realized that logic and

mathematics share many characteristics, and that in its

239

200

simpler forms, logic can mirror, say, algebra, yet his
interest in logic had a broader base than did the interests
of those we have already considered (with the obvious exception
of Peirce). Logic for Lewis was not primarily an instrument
on which to build mathematics, but was rather a science to be
studied in its own right. Much of his work aims at extending
logic to its full limit, giving special regard to no
particular application. The divergences we shall note
between Lewis' work and the works of many of his predecessors
are a direct result of these different interests. Consider,
for example, Russell and Whitehead's remark that.they excluded
intensional notions from consideration in the Principia
because mathematics was basically extensional, and since they
would not include something which they did not need for the
grounding of mathematics, they would not include such
intensional concepts. Since Lewis has no such limited
applicational aim, he considers such contexts to be a natural
part of his total enterprise.

We shall now consider Lewis' logic, and then go on to

consider his theory of logic.
2. A. LEWIS' LOGIC; INTRODUCTION.

We shall discuss several areas in Lewis' logic; first,
we shall discuss the general development of his logic, covering
many aspects which are important, but which nevertheless have
received little attention in the past. Second, we shall

discuss Lewis' theory of strict implication. In this

201

discussion, we shall consider the defects which Lewis found
inherent in the system of the Principia, and the devices he
introduced to correct these defects. Also, we shall discuss
the modal logic which he introduced to facilitate the building
of his system of strict implication.

In dividing Lewis' logic into these categories, two
words of caution are necessary. First, anyone who is familiar
with Lewis' work will realize that the areas mentioned above
are not perfectly separable; there will of necessity be some
overlap in our discussion. Second, while we shall discuss
Lewis' theory of logic fully in the subsequent sections, it
is impossible to exclude all theoretical concerns from the
discussion of his logic. For Lewis, logic and the theory of
logic go largely hand-in-hand; he himself makes little attempt
to separate them. Let us now proceed to consider each of

the areas mentioned above.
2. B. THE GENERAL DEVELOPMENT OF LEWIS' LOGIC.

Lewis is best remembered for two works in logic, his

Survey pg Symbolic Lo ic, published in 1918, and his Sypbolic

1
Lo ic, which he co-authored with C. H. Langford. Sypbolic

 

1Although by the authors' own assertion, most of the
material with which we shall deal is due to Lewis. They say,
”Chapters I-VIII and Appendix II have been written by
Mr. Lewis; Chapters IX-XIII and Appendix I, by Mr. Langford.
Each of us has given the other such assistance as he might;
but final responsibility for the contents of the chapters is
as indicated."

202

Lpglp_differs from the earlier work mainly in its inclusion
of material which did not appear in the earlier work.2
Otherwise, nearly all of what is done in the earlier work is
carried over to (and expanded in) the later work. This being
the case, we shall concentrate on the later work, since it is
the fuller development.

In Symbolic Lo ic, Lewis develops the subject inaa
most interesting fashion: he begins with the simplest kinds
of lOgic and builds the more complex forms of logic. He first
deals with the Boole-Schr6der algebra and its extensions in
general term-logic, then with propositional logic, next with
functional logic, and finally with the logic of strict
implication (first propositional contexts, and then briefly
in quantificational contexts). Finally, in a well-known
appendix, he recounts his full system of implication, listing
the axioms presented both from the Survey, plus those presented
later in Symbolic ngip, As we shall see, his reason for
developing logic in that way was simply that by constructing
the traditional forms of formal logic, he was in a position
then to criticize them and to build his own system around
such criticisms.

The novelty in the manner in which he constructs logic

is that he uses each level of logic to serve as a basis for

 

2Thus in Symbolic Logic, the system of strict
implication is greatly extended, amny valued systems are
considered, and the appendices were added (which give further
detail to and concept of strict implication). None of

these were included in Survey pg Symbolic Logic.

203

the succeeding levels. The clearest instance of this is
found in the shift from the calculus of classes to the logic
of propositions. Lewis reinterprets certain features of the
algebra of logic to cover propositions, and then begins to
develop propositional logic on that basis. The procedure he
employs is similar to one in accordance with which many parts
of mathematics have grown: they are both like a building,
each floor of which is slightly larger than the one preceding.
Thus, each successive level in logic forms a base for the one
which follows, even though its successor is more extensive
and encompassing than itself.

In a way, Lewis' procedure is like Boole's; Boole,
it will be remembered, wished to reinterpret his calculus to
cover propositions. In his case, however, it did not work,
simply because he was unwilling to go beyond the limits of
the calculus in constructing the prOpositional interpretation,
Lewis, on the other hand, carries Boole's program to its
furthest extent. After making the transition from the
algebra to propositional logic, he extends logic to cover much
more than the algebra alone had covered. In the same way,
Lewis believed that his system of strict implication over-
shadowed the Principia-type logic of prOpositions, including,

and going far beyond it.

200
2. C. THE LOGIC 0F CLASSES, AND THE LOGIC OF TERMS.

The first chapter of Symbolic Lpgig is devoted to an
historical account of logic. The next two chapters are
devoted to a systematic presentation of Boole-SchrSder logic.
While chapter two is supposedly devoted to a logic of classes,
and chapter three to a logic of terms, Lewis confuses classes
with terms-in-extension. He begins chapter two with the
claim that he will first present the algebra as a calculus
of classes, and then go on to present other interpretations
of it (among these being the logic of terms). However,
immediately after making this division, he begins speaking of
classes as if they are invariably associated with terms.

This weakness, however, is of little consequence for the work
as a whole; the important distinction is between extension
and intension, and how it holds at various levels of logical
complexity.

The class-logic which Lewis presents is a very simple
one, which includes the following components: first, there
are four ”assumed ideas”:

(1) Any term of the system, a, b, c, etc.,

will represent some class of things,
named by some 10gical term.

(2) a x b represents what is common to a and b;
the class composed of those things which
are members of a and members of b, both....

(3) The negative of a will be written -a. This

represents the class of all things which are
not members of the class a; ...

205

(0) The unique class 'nothing' will be
presented by 0....

Next, there are three definitions:

1.01 l = -0 Def.
1.02 a + b = -(-a x -b) Def.
1.03 aCb is equivalent to a x b = a Def.

Finally, there are six postulates for the system:

1.1 a x a = a.

1.2 a x b = b x a.

1.3 a x (b x c) = (a x b) x c.
1.0 a x 0 = 0.

1.5 If a x -b = 0, then aCb.

From this basis, Lewis goes on to develop many of the properties
of classes, which are standard enough not to require repetition
here.

Of particular interest for our purposes, however, is
that while Lewis presents the Boole-Schr3der algebra as a
calculus of classes, he is fully aware that it can be given
other interpretations. He says, for instance, that ”in
dealing with the logic of terms in propositions, either of two

interpretations may frequently be chosen: the prOposition

 

3Lewis, Clarence Irving, and Langford, Cooper Harold,
Symbolic Lo ic (New York: Dover Publishing Company, 1959),
pp. 27-29. First edition, 1932.)

\\

 

206

may be taken as asserting a relation between the concepts which
the terms connote, or between the classes which the terms
denote."’4 Lewis is fully aware, that is, of the possibility
of giving the system an intensional interpretation: ”the laws
governing the relations of concepts constitute the logic of
the connotation or intension of terms; those governing the
relations of classes constitute the logic of the denotation
or extension of terms.'5

While Lewis recognized the possibility of giving the
logic of terms an intensional interpretation, he does not
develop such an interpretation. In chapter two, Lewis is
concerned to present the logic of terms from the most widely
held points of view. Thus in this chapter he maintains that
he will deal exclusively with an extensional interpretation
”To avoid confusion...."6 His decision at this point was
based on heuristic considerations, rather than on
considerations of relative importance. Later, in chapter
three, Lewis goes on to maintain that the extensional
interpretation of term logic is superior due to the wider
area of applicability of term logic so interpreted.7 These

two accounts do not conflict, however. Lewis is simply

 

Ibid., p. 27.

 

H
0‘
p-
D.
e

 

7On this point, of. below, p.207 ff.

20?

moving from a description of how the calculus had been
interpreted, to his own critical consideration of it.

Lewis, of course, is not the first logician with whom
we have dealt who chooses not to deal with intensional
contexts, after recognizing that such interpretations are
possible.8 It will be remembered, for instance, that after
recognizing "sense," Frege chose not to build his logic around
it, but instead built it around denotation.9 Likewise, in
the introduction to Principia Mathematica, Russell and Whitehead
recognize intensional contexts, but do not go on to provide
for them in their system. In both cases the reason for the
exclusion is the same: neither Frege, nor Russell and
Whitehead needed an intensional interpretation for the task
at hand, which was the grounding of mathematics in logic.

Lewis' reason for excluding intensional notions from
term logic, however, are somewhat different; he maintains
(in Chapter III of Symbolic ngig) that extensional logic
provides the broadest possible expanse of application with

the least amount of complication. He believes that such

 

81t must be kept strictly in mind that we are here
discussing only Lewis' refusal, in Chapter II, to elaborate
intensional interpretation in the context of term logic; his
assessment of the significance of intensional concepts in
other contexts will be discussed later.

It was not, in fact, until Church wrote "A Formu-
lation of the Logic of Sense and Denotation' that both
facets were systematically taken into account. This article
is found in Structure Method 52g Meaning, P. Henle, H. Kallen,
and S. Langer (eds.)(New York: The Liberal Arts Press, 1951).

208

complication would be introduced by the introduction of
intensional notions. Let us look briefly at how he arrives
at this view.

Lewis maintains that ”the laws of intension and those
of extension are analogous, but they do not apply to the same
terms in the same way: the relations of a given set of terms
in intension may not be parallel to their relations in

10 He maintains, for instance, that some

extension.”
statements might be taken to be true in intension while taken
to be false in extension. Thus, "'some illegal acts are
moral' might be true in intension, meaning that, as the law
stands, it is conceivable that a moral person might find
himself in a situation requiring him to break it; but it
might be false if it meant, in extension, that certain deeds
actually done are at once moral and illegal."11
Since such variation is possible, Lewis maintains
that a logic which would be adequate for dealing both with
intensional and extensional interpretations would have to be
such that there would be some way of separating the two
interpretations. If no such method were provided, confusions
could arise between the two interpretations. "Thus a logic

which should be adequate to deal with all the relations of

terms which figure in and affect inference, would have to

 

10
Lewis, op. cit., p. 09.

11
Ibid.

209

represent their relations of intension differently from their
relations of extension and such a logic must include not only
the laws of intension and the laws of extension but also the
laws connecting the two."12

Even though the Boole-Schroder algebra can be
13

interpreted as being extensional or intensional, Lewis
maintains that it is still unsatisfactory: ”the Boole—
Schro.der algebra is inadequate Es a complete representation
of the logic of terms, though it is completely applicable to
the relations of extension, and likewise completely applicable
to the relations of intension.'1u That is, since on Lewis!
view, the algebra is not adequate as a device for showing the
relations between the two interpretations, it is in that
respect inadequate; if left ambiguous as to interpretation,
various mistakes in reasoning can occur.15
Since Lewis wishes neither to leave the algebra
ambiguous with respect to interpretation, nor to provide the

extra apparatus which would render it a "complete” logic of

terms,16 he is forced to adopt either one interpretation or

 

12Ibid., p. 68.

13For the essentials of the partial intensional
interpretation he does give, cf., ibid., p. 66.

1“Ibid., p. 69.

150“ this point, Cf. ibid., Po 500

16But, he claims, it would not be difficult to make
the algebra a complete logic of terms, saying that 'it is
not a difficult matter, and awaits only the attention of

250

the other, when presenting the algebra. As we have seen, he
chose the extensional interpretation. On the one hand, he
claimed that whenever a universal prOposition is true in
intension, it holds in extension also. Then he says that

”for this and certain other reasons, the logic of terms in
extension is applicable to a much wider range of actual
inference than is the logic of intension."17 0n the other hand,
he claims that "as a matter of fact interpreting the
traditional modes in intension does not help them a bit. If
the interpretation is completely and consistently intensional,
precisely the same forms of inference which are fallacious in
extension will be fallacious in intension also. The logic of
the intension of terms is a slightly complicated matter, and
perhaps of no great importance."18 For both of these reasons,
Lewis decided to give the algebra a strictly extensional
interpretation.

While this discussion of Lewis' attitude towards the
interpretation of the algebra may seem unnecessarily detailed,
yet it will be of importance when we deal with questions of
interpretation of more complex forms of logic, in subsequent

sections.

 

some student who will bring to bear upon the subject the
exact methods now available." Ibid., p. 69.

17Ibide, p. 500

laIbid. , p. 66.

251

2. D. LEWIS' LOGIC; THE TRUTH-FUNCTIONAL LOGIC 0F

PROPOSITIONS.

The second main portion of Lewis' logic deals with
the logic of propositions and propositional functions, His
view concerning the interpretation of propositional logic
differs markedly from his view concerning the interpretation
of the algebra. While in dealing with the algebra, he
maintains that an intensional interpretation is not of much
significance, we shall see in this and subsequent sections
that his attitude toward that interpretation of propositional
logic was just the opposite. To arrive at such a position,
Lewis begins by setting up the logic of propositions in a
purely extensional (i.e., truth-functional)19 manner, by
transforming the operations of class logic into prOpositional
terms.

After formulating the logic of propositions in this
way, Lewis goes on to point out that such an interpretation
is unsatisfactory. Finally, he formulates a logic based on
strict implication, a relation which he considers to be
intensional. Thus this initial presentation of the logic of
propositions has the negative purpose of providing a ground

for criticism. Nonetheless, we shall present this treatment

 

19In general, when speaking of propositions in
extension, Lewis means that they are taken as classes of
truth-p0551b111tYe Cf. ibide, pa 78 and pe 87o

252

in outline in order to be able to pinpoint what Lewis
considers the source of difficulty to be.

As was mentioned, Lewis was not the first to develop
the logic of propositions by reinterpreting the logic of
classes; Boole, and later Whitehead (in Universal Algebra),
expanded the algebra to cover propositions. Lewis'
distinction in reviving this procedure was that he had the
advantage of having all of the advances in propositional logic
made since the time of Peano at his disposal. The logic
which he was developing was more complete than were the
logics of either Boole or Whitehead.

Lewis begins his development of the truth-functional
logic of propositions by stating that “the Boole-Schr3der
algebra applies to assertable statements as well as to
classes. ...That is to say, a logic of classes can be

extended to assertable statements by applying it to the

20 He

classes of cases in which such statements are true.'
goes on to say that the algebra can be applied to propositions
by assigning the following interpretations to the constituents
of the algebra:

(1) a, b, c, etc., represent statements

in extension: the classes of instances in
which the statements are true.

 

2°Ibid., p. 78.

253

(2) a x b represents the joint assertion
of a and b; the class of cases in which a and
b are both true.

(3) -a represents the contradictory of
a, or "a is false”; the class of cases in which
a is not true.

(0) a + b represents "a is true or b is
true”; the class of cases in which at least
one of the two, a and b, is true.

(5) 0 represents the null class of cases,
so that a==0 symbolizes "a is true in no case“
or “a is always false."

(6) Consonantly, a = 1 symbolizes ”a is
true in every case” or ”a is always true.'

(7) aC—b will mean "All cases in which a
is true are cases in which b is true“ or “If a
is true, then b is true.'
(8) a = b will mean “the cases in which a
is true are identically the cases in which b is
true" or "a is true when b is true, and false
when b is false.'21
Besides these principles, Lewis introduces another, which
he considers to be peculiar to this sort of prOpositional
logic, by claiming that ”...there is no difference, for
propositions, between being true and being always true;
2
between being false and being always false.” 2 Because of
this, he says that "thus there is one further principle,

...which is required in order to restrict the terms of the

algebra to statements which are prOpositions; namely, 'for

 

21Ibid., pp. 78-79.

22Ibid., p. 79.

250

every element a, either a = 0 or a = 1."'23 By adopting this
principle, Lewis claims that we have ”a calculus of propositions
in extension,“ which he calls the Two-Valued Algebra.

In setting up the logic of propositions in this way,
Lewis carefully points up its extensional nature. He says,
for instance, that "p, q, r, etc., do not really represent
prOpositions--that is, different meaningful assertions--but
only the truth or falsity of these, the class of cases (1 or
0) in which they are true.”2u Again, thereason Why Lewis
dealt with the extensional nature of the logic of propositions
was largely in order to point out its shortcomings. He uses
his earlier description of the truth-functional logic of
prOpositions as a basis for building his own system, which
he maintains is free of difficulty. We shall treat this
matter in detail later.

After providing the nine rules for reinterpreting the
algebra of terms, which are given above, Lewis next provides
a symbolism suitable to the new interpretation. 'qu'
becomes 'p;>q,' 'p+q' becomes 'p v q,' and 'p = q' becomes
'p 5 q.‘ Thus he translates many of the theorems of term
logic into theorems of propositional logic by providing
suitable replacements for the symbolism of term logic. Thus,

for example,

 

23Ibid.

2“Ibid.

255

1.2 a x b = b x a.
becomes,
102 pqueqpe

Because such propositions are quite standard, we shall not
consider then in detail here.

Besides the straightforward translations from term
logic to the logic of propositions, Lewis formulates another
set of propositions, whidihe claims cannot be obtained from
term logic without the help of the additional two principles
which hold only for propositional logic, which were mentioned

above. He formulates these principles in the following manner:

7.1 (p=0)

“JP

7.11 p _ (p=1)
This new set of propositions includes several which will be

of great importance later. They are,

7.0 p.:.q=P

7.01 ~p- mopaq

7.5 PQ-D-Paq qua-qop

7.51 updq.:.p:>q diq.:.q=P 25
He holds that the properties which these propositions exhibit
are peculiar. It seemed odd to him that a false proposition,
by merely being false, implied any proposition, that a true
proposition was implied by any proposition, that any two true

propositions implied each other, and likewise for any two

 

251bid., p. 86.

256

false propositions. But all of these properties are the
result of the extensional, or two-valued interpretation. At
the end of his development of the logic of propositions,
Lewis gives a preliminary summary of why he considers tuch

properties to be odd:

The fact that almost all developments of
symbolic logic are based upon this relation of
extension or material implication--a relation of
truth-values, not of content or logical
significance-~is due to their having been built
up, gradually, on the foundation laid by Boole;
that is, upon a calculus originally devised to
deal with the relations of classes. There is
nothing more esoteric, behind these peculiar
properties of material implication, than t is
somewhat unfortunate historical accident.

The rest of our treatment of Lewis' logic will
constitute a brief sketch of how he sought to overcome the

effects of this "unfortunate historical accident."27

 

26Ibid., p. 92.

27Besides the account of propositional logic just
discussed, Lewis also provides an account of quantificational
logic, which he also builds up by reinterpreting term logic.
The only interesting feature of his presentation is that he
employs Wittgenstein's device of translating (x)Fx into
Fa.Fb.Fc..., and (3 x)Fx into Fa v Fb v Fc ... . He main?-
tainskthat "the procedure here set forth is so convenient,
and so well exemplifies the connection between the calculus
of propositions and the calculus of ro ositional functions,
that we shall adopt it, in spite of his theoretical inadequacy."
Ibid., p. 92. Since besides this, his presentation of
quantificational logic is quite standard, we shall pass over
it to the consideration of his system of strict implication.

257
2. E. DIFFICULTIES WITH THE CONCEPT OF MATERIAL IMPLICATION.

To this point, we have seen Lewis carefully build up
truth-functional logic, presumably as he thought it had been
built up, by the reinterpretation of the logic ofciasses.
Furthermore, we have seen him maintain that the interpretation
of this whole develOpment is accidental, an historical quirk.

Lewis, however, maintains more than just that all of
this development is an unfortunate accident; he claims that
it has had the undesirable effect of giving rise to an
inadequate conception of implication, .i.e., to the concept
of truth-functionally determined material implication as it
was conceived by Russell and Whitehead. Lewis' main point
is that ”the logic of propositions should be so developed
that the usual meaning of 'implies,' which is intensional,
should be included. Although the relation of classes a<;b
is often important when the corresponding relation of intension
does not hold, the relation of material implication is seldom,
if ever, of any logical importance unless the usual intenstional
relation of implication also holds."28 That is, he thinks
that material implication does not capture the ordinary
meaning of 'implies': “The relation of material implication,

which figures in most logistic calculi of propositions, does

 

28Ibid., p. 120.

258

not accord with Ehausual meaning of 'implies."'29 We now
see Lewis' change of attitude with regard to questions of

intension versus extension; the extensional interpretation,
while apprOpriate as the more useful interpretation of the

algebra of logic, $3 not enough for prOpositional logic.

 

Thus a greater amount of attention must be paid to questions
concerning intensional notions.

Material implication leads, according to Lewis, to
“paradoxes,” such as “'a false proposition imples every
proposition' and 'a true proposition is implied by any'

... ."30 Besides these paradoxes, Lewis thinks it quite
peculiar that, concerning any two propositions chosen at
random, “the chance that the first...will materially imply

the second is 3/0. The chance that the second will materially
imply the first is 3/0. The chance that either will
materially imply the other is 1/2. And the chance that
neither will materially imply the other is 0."31

In each of these cases, Lewis is criticizing those
prOperties of the logic of propositions which stem from the
extensional interpretation, discussed above. Because the

truth-functional definition of material implication gives

 

29Lewis tentatively characterizes the ”usual” meaning
of 'implies' by stating that 'p implies q' will be synonymous
with 'q is deducible from p.' We shall, however, deal with
this matter more fully later. Ibid., p. 122.

3°Ibid.

31Ibid., p. 105.

259

rise to the ”odd properties,“ Lewis concluded that material
implication did not capture the real meaning of 'implies.'
Regarding Lewis' discomfort with the notion of material
implication, two points should be remembered. First, Lewis'
criticisms are aimed at what 21823 be considered to be an
instance of a bad choice of a name. As Quine says, "it is
doubtful that Lewis would ever havelgpnstructed his own theory
of implication if Whitehead and Russell, who followed Frege
in defending Philo of Megara's version of 'If p then q' as
'Not (p and not q),' had not made the mistake of calling the
Philonian construction 'material implication' instead of the
material conditional."32 Thus while Lewis believaithe oddity
to lie in the circumstance that implication had beenfbrced
into an extensional interpretation, as the result of an
historical accident, Quine is claiming that perhaps the
difficulty might lie in the fact that the material conditional
was erroneously named 'implieation.‘33 But such considerations

are of only passing interest, for whether or not the difficulty

 

32Quine, Willard Van Orman, Word and Object (New York:
John Wiley and Sons, 1960), p. 196.

33Russell and Whitehead were aware that their use of
'implies' was not the only one. Consider, for instance, their
comment that ”The meaning to be given to implication in what
follows may at first sight appear somewhat artificial; but
although there are other legitimate meanings, the one here
adopted is very much more convenient for our purposes than
any of its rivals." Russell, Bertrand, and Whitehead,
Alfred North, Principia Mathematica (Cambridge: Cambridge
University Press, 1927), p. 90. They were not, like Lewis,
after the ”usual” meaning of 'implies.'

260

which gave rise to Lewis' revision of the concept of
implication is a matter of the meaning of words does not
matter. What is important is that Lewis saw what he
considered to be an inadequacy, and proceeded to rectify it.

The second point worth noting is that concerning the
paradoxes, “the paradox lies only in this: some strange
consequences of the principles(§f material implicatiofi] show
that the meaning to be attached to material implication, is
not the meaning one had intended to translate into symbols.'3u
This is simply another way of stating what has already been
established. However, this point will be of importance when
we deal with some apparently odd properties inherent in

Lewis' own conception of implication.
2. F. THE LOGIC OF STRICT IMPLICATION.

Under the heading ”The Logistic Calculus of Unanalyzed
PrOpositions," Lewis sets out to "...develop a calculus based
upon a meaning of 'implies' such that 'p implies q' will be
synonymous with 'q is deducible from p."35 Further, he says
that 'it is entirely possible so to develOp the calculus of
propositions that it accords with the usual meaning of

'implies' and includes the relation of consistency with its

 

3L‘Feys, Robert, “Carnap on Modalities” The Philosophy
p£_Rudolf Carnap, P. A. Schilpp, (ed)(LaSa11e, Illinois: The
Open Court Publishing Company, 1963), p. 290.

35Lewis, op. cit., p. 122.

261

ordinary properties."36 The new implication relation which

Lewis introduces is called 'strict implication.‘

The Calculus of Strict Implication, a calculus of

propositions built around the notion of strict implication,

is composed of the following constituents:

First, the system contains the following five primitive,

undefined ideas:

1.
2.
3.
he

5.

Propositions: p, q, r, etc.

Negation: ~p. ...

Legical product: pq, or p.q. ...
Self-consistency or possibility: 0p. This
may be read "p is self-consistent" or "p is
possible" or ”it is possible that p be true.“
As will appear later, 0p is equivalent to
"It is false that p implies its own negation,”
and O (pq) is equivalent to "p and q are
COUSIStente” eee

Logical equivalence: p = q. 37

Next, Lewis includes the following three definitions,

which he casts in the form of strict equivalences:

11.01 p v q. = . ~(~pvq)
11.02 p-iq. = .~¢(pvq) 38
11.03 p = q. = 3 p-iq-q-H’

Next, he provides an initial list of seven postulates,

remarking that others will be added later. For reasons of

economy, we shall list all of the axioms which he uses in all

of the variant systems he constructs. In Appendix II of

 

36

Ibid., pp. 122-3.

371bid., p. 123.

38Ibid., p. 120.

262

Symbolic nglp, Lewis provides the following complete list of
postulates. Those on the left are theaxioms taken from the
system developed in Survey (which he provides mainly for
purposes of comparison); those on the right constitute the

full set from Symbolic Logic:

 

Ale pq.-3.qp Bl. pCI-‘aOQP

943- qpo-aop 82- qp.-a.p

A3. p.-a.pp BB. p-aopp

A0. P(qr).-§.q(Pr) B0. (pq)r.-9.p(qr)
A5. p-a~(~ p) BS. p-9~(~’p)

A60 p-aq.q-9r:-%.p-€r 86. p-Bq.q-5r:-9.p—3r
A7. ~Op-3‘vp B7. psp-‘JqS-aoq

A8. p-aq.-s.~oq—‘a~0p 88. 0(pq)-30p

B9. (3 p-qH ~(p-?q). ”(p-36m)”

Finally, Lewis provides four transformation rules; a
rule for uniform substitution, a rule for substitution of
strict equivalents, a rule of adjunction, and what he calls
"inference,” which he formulates in the following manner:
"If p has been asserted and p-Bq is asserted, then q may be

00

asserted.”

 

39Ibid., p. 093. One of the more curious features of
the primitive basis of the system of strict implication is
Lewis' use of an existential quantifier in axiom B9. No
provision had been made in the primitive ideas, or in the
defined ideas, for this operator. Kneale maintained that
Lewis introduced the axiom to insure that the system could
not be considered to be a mere notiational variant for the
system of material implication, in which 'Op' would be
equivalent to 'p,' and 'p~3q' would be taken as a typograph—
ical variant for 'p:3q.' (Cf. Kneale, William, and Kneale,
Martha, The Develppment 23 Logic (Oxford: Oxford University
Press, 1961), p. 552.) Since the quantifier appears nowhere
else in the system, this may suggest that Lewis added it as
an afterthought.

 

hoLewis, op. cit., p. 126.

263

We will not, of course, be able to examine even a small
part of the development of the system of strict implication.
Taking this description of the primitive basis of the system
as a guide, we shall only examine those features of the system
and its interpretation which will be of significance in the
discussion of theory which follows.

By far the greatest area of difficulty which surrounds
the system of strict implication concerns the interpretation
of the key notions involved in it. What these difficulties
are, and how Lewis attempts to interpret various aspects of
the system, will be of interest in preparing for the
discussion of theory.

But before proceeding to a consideration: of Lewis'
remarks concerning interpretation, one point of possible
confusion must be clarified. Because Lewis, like Russell
and Whitehead,.did not employ an object-language meta-
language distinction, some confusion has arisen concerning the
nature of the propositions which serve as the relata of the
strict implication relation. That is, because he employed
no device within the system by which he could establish.
systematically the nature of the relata (and hence the kind
of relation involved in strict implication), some have been
led to believe that, in fact, Lewis' modal logic deals only
with non-linguistic propositions. Thus, for instance,

R. M. Martin, in speaking of modal logic, says that "formu-

lations of such a logic seem always to involve variables

260

ranging over a kind of entity called pgopositions. But what
these entities are remains rather obscure."l‘1 On the contrary,
Lewis regards propositions quite differently. He says, for
example, "A proposition is any expression which is either
true or false.""‘2 Again, in defining strict implication, he
says, '...'p implies q' is to mean 'it is false that it is
possible that p should be true and q false' or 'the statement
'p is true and q is false' is not selfconsistent."u3 While
we shall discuss such questions fully later, it should be
kept in mind that in all questions of interpretation, Lewis
held to this basic conception of the linguistic nature of
propositions. As we shall see, however, there is a discrepancy
between what Lewis pgyp on the topic of the nature of
propositions, and what the system of strict implication
commits 212.329 on the same question. We shall also discuss
how Lewis himself might have accounted for this discrepancy.
Returning to Lewis' interpretation of the system of
strict implication, it must first be noted that when he
maintained that 'p-{q' meant that it was not possible that p
be true and q false, he understood this in such a way that

the truth or falsity of 'p-fiq' could not be totally determined

on the basis of the truth or falsity of p and of q. When

 

thartin, Richard M., ”Does Modal Logic Rest upon a
Mistake?" in Philosophical Studies, XIV 8-11, 1963.

uzLGWiS, OE. Cite, p. 900

uaIbide, p. 12“.

265

Lewis set about avoiding the pitfalls which he saw inherent
in the truth-functional logic of the Principia, one of his
methods was to get rid of the truth-functional method as the
3212 determiner of the truth or falsity of molecular expression.
That is, ”...it is Lewis' main interest to maintain that the
deducibility of conclusion from premise in a valid argument
depends on the holding between them of a relation called
strict implication and that this relation cannot be defined
truth-functionally."hu As Lewis himself says, "the strict
relations, p-gq, and p=q..., are not...truth-functions; that
p-gq or p=q holds cannot generally be determined by merely
knowing whether p and q are true or false.”u5

To emphasize the point concerning the divergence
between the strict and material system of implication,

Lewis provides the following table:

pq paq pdq
1 1 1 undetermined
1 0 O 0
0 1 l undetermined
0 O l undetermined

In other words, 'p-dq' is false if p is true and q false, and
is otherwise undetermined. However, just because the
modalities incorporated into the system are not truth-

functionally determinable does not mean that none of the

 

uuKneale, OE. Cite, p. 5530

usLewis, op. cit., p. 107.

 

CODS‘

both

that

dete

tru1

fr01

He

trr
0t1
or

B}

 

266

constituents of the system are. Since the system contains
both modal and non-modal operators, it contains both those
that are, and those that are not truth-functionally
determinable.u6

While Lewis rids his logic of total dependence on the
truth-functional interpretation, he does not radically depart
from established patterns of interpretation in other respects.
He claims, for instance, that "the laws of the relation of
strict implication, ...are tautologies, ...but they are not
truth-value tautologies.”u7 Also, he states that ”there are
other tautologies than those of truth-value, because there
are other principles of division, and hence other ways of
exhausting the possibilities, than by reference to alternatives
of truth-valued”8 That is, while he thought the laws of
logic to be tautologies, he wished to extend the concept of
tautology beyond the narrow truth—value interpretation.
Accordingly, a tautologyfbr Lewis is ”...a principle whose
truth is demonstrable by exhaustion of some set of alternative
possibilitiesfl'u9 Thus, 'pfiq' holds when 'p‘aq' is a
tautology. Lewis further explains his conception of

tautology by saying that while every tautology has to be

capable of being stated by reference to truth and falsity,

 

h60n this point, cf. ibid., p. 200.
07

Ibid., p. 213.

“81bid., p. 200.

“91bid.

267

the principle of division used for separation need not itself
be based on truth and falsity.50

For an explanation of what Lewis considered to be
involved in the nature of 32y principle used for the
determination of tautologies, we shall have to have a preview
of a bit of his theory of logic. In particular, he says
that ”the tautology of any law of logic is merely a special
case of the general principle that what is true by definition
cannot conceivably be false: Thus any logical principle is
tautological in the sense that it is an analytic proposition.“51
This is the foundation of Lewis' interpretation of the system
of strict implication. However, since we have yet to discuss
other points concerning the logic of strict implication, and
since further discussion concerning the nature of the laws
of logic would carry us into the realm of theory, we shall
postpone further discussion along these lines until later.

Before proceeding, however, two points should be
noted. First, Lewis' conception of tautologies as being
analytic statements has given rise to a great amount of
further work along these lines. While we cannot deal with
such work here, it might be pointed out that the best known
of the subsequent works on this topic is found in Carnap's

Meaning and Necessity. This work has engendered many further

 

Sone, ibid., p. 2&0 no

Sllbida, p. 2110

268

advances in modal logic. The second point is that Lewis
himself illustrates how other principles than truth-functional
ones might be used to determine tautologies. He does this by
setting up alternative systems of many-valued logics, which
he thought to be different in kind from the standard two-
valued systems.

With regard to the relation between strict implication
and material implication, Lewis points out some interesting
facts. Many of the theorems in the system of strict
implication hold both for strict and for material implication,
But there are a few crucial ones which do not hold for both.
For instance,

~(p~q)o-%-p-%q
cannot be proved in the system. Lewis says "this is a main
distinction between strict implication and material
implication. On this point the properties of strict
implication are in accord with the usual meaning of 'implies,'
while those of material implication are not."52 On the
other hand

p-aqo—i-qu
does hold in the system. As Lewis points out, strict
implication is both narrower and stronger than is material

implication.

 

521bid., p. 130.

269

More importantly, perhaps, neither

p.-3.q-§p, nor

~po-aop-iq
is provable in the system. Both of these are theorems in the
system of material implication, but not in the system of
strict implication. This difference may seem to have the
effect of circumventing the paradoxes of material implication.
Such however, is not quite true. It turns out that
19.70 ~0p.-$.p-3q and
19.75 ...osopo—qu-Bp
are both theorems of the system. The first states that an
impossible proposition implies any prOposition, while the
second states that a necessarily true Pr0position is implied
by any proposition. Although this consequence has disturbed

53

many of Lewis' successors, and has been used as a tool by

 

53As a result, both Church and Sobocinski have develop-
ed modal systems which avoid these paradoxes of strict impli-
cation. (Cf. Church, Alonzo, “The Weak Theory of Implication,“
Kontrolliertes Denken, Untersuchngen zum Lpgikkalkul und zur
Lo ik der Einzelwissenschaften (Munich: Karl Alber Verlag,

1951), pp. 22- 37. Cf. also Sobocinski, Boleslaw, "Axiomati-

zation of a Partial System of Three-Value Calculus of
Propositions," _’l_‘_he Journal of Computing System (St. Paul:
Institute of Applied Logic), Vol. 1, no. 1, pp. 23- 50. ).

The presence of the paradoxes of strict implication in
Lewis' system has also given rise to some highly original work
by Anderson and Belnap. They claim that the presence of the
paradoxes in Lewis' system is as objectionable as the presence
of the paradoxes of material implication in the Principia.
Their system is an attempt to rid the calculus of entailment
of the paradoxes altogether. (Cf. Anderson, Alan Ross, and
Belnap, Nuel D., ”The Pure Calculus of Entailment,' Journal
pg Sypbolic Lo ic, Vol. XXVII, pp. 19-51.

 

 

270

critics of modal logic, nonetheless, Lewis himself was not
disturbed by it. He calls the two theorems ”paradoxes of
strict implication," but goes on to say that they are
“inescapable consequences of logical principles which are in
everyday use. They are paradoxical only in the sense of being
commonly overlooked, because we seldom draw inference from a
self-contradictory prOposition.”5u While the same might be
said of the paradoxes of material implication, apparently
Lewis did not consider the case to be parallel. The most
plausible reason for his lack of concern is that since on his
account strict implication is closer to the usual meaning of
'implies,' what follows from it will be genuinely unavoidable,
whereas what follows from material implication, insofar as
material implication does not perfectly conform to the usual
meaning of 'implies,' is to be held suspect.

There are many other areas of great importance with
regard to Lewis' logic and its interpretation which we shall
not be able to cover. Among these perhaps the most prominent
issue concerns the interpretation of the modalities which
Lewis employs. We have not, for instance, discussed the
alternative systems which Lewis constructs from the full axiom
set mentioned above. These five systems by including or
excluding various axioms, bestow varying degrees of strictness

on the interpretation of the constituent modalities. Thus,

 

SuLewis, pp; cit., p. 175.

271

for instance, one of these systems, 85, has been provei to be
equivalent to a variant of the Boole-Schrgder algebra, in
which, as Kneale points out, the modal sign '0' is
introduced by the rules:
Op
Op

Such a tidy interpretation is not possible in the case of the

1 if and only if p f 0,

0 if and only if p = 0. 55

other four systems. In fact, the variation in possible
interpretaion prompted Lewis to admit that "prevailing good
use in logical inference -- the practice in mathematical
deduction, for example -- is not sufficiently precise and
self-conscious to determine clearly which of these five
systemlexpresses the acceptable Principles of deduction."56
We will not be able to deal either with such problems
as the interpretation of iterated modalities, and the problem
of interpreting modalities as they occur in quantificational
contexts. Our purpose in presenting this glimpse of Lewis'
logic has been simply to provide us with enough background

to make the following discussion of theory somewhat more

understandable.

 

55Cf. Kneale, pp; cit., p. 552. Regarding 55, it
might also be pointed out that it has been proved complete
(by Kripke, Saul A., "A Completeness Theorem in Modal Logic,”
Journal pi Symbolic Lo ic, Vol. XXIV, pp. 1-10.). A complete
ternary connective has also been discovered for SS (by
Massey, Gerald J., ”The Theory of Truth-Tabular Connectives,
both Truth-Functional and Modal,” Journal 22 Symbolic Lo ic,
forthcoming.).

56Lew13, OEe Cite, ppe 501-2e

272

The main point which must be kept in mind with regard
to Lewis' logic is simply that he was never interested in
attempting to replace the extensional logic of the Pringipia
with a different sort of logic, but rather to show that there
are alternative sorts of implication which might better serve
the logician's needs than does material implication. We have
seen that Lewis felt perfectly at home with extensional logic,
as long as it was accorded its proper place among all other
logics. Thus he holds to the extensional interpretation of
term logic, and retains material implication as a part of the
system of strict implication. Lewis' logic is a broader
system of logic, a logic which he thought to be adequate for
the expression of different forms of implication.

Our discussion of Lewis' theory of logic will center
on such topics as the reasons which he provides for the idea

that various logics can support various forms of inference.
3. LEWIS' THEORY OF LOGIC.

A statement such as ”...the subject-matter of symbolic
logic is merely lpglp -- the principles which govern the
validity of inference,"57 ordinarily would appear to be fair
game for any paradox-hunter. As it turns out, however, not
only is the statement not paradoxical, it serves as a guide

to Lewis' entire theory of logic. Much of this section

 

57Ibid., p. 3.

273

will be devoted to an explanation of the meaning of this
statement.

In order better to come to grips with Lewis' theory
of logic, we shall consider the full sc0pe of his theory, as
he first develops it in Surve , as well as the later
development in Symbolic Logic. Dealing with his theory in
this way has the advantage of allowing us to examine the
entire process of development of his theory. As we shall see,
the foundations which are laid in Survey are carried over,
largely intact, and systematically develOped in Symbolic Loaic.

In Surve , Lewis maintains that ”Symbolic Logic is the
development of the most general principles of rational
procedure, in ideographic symbols, and in a form which exhibits
the connection of these principles one with another."58 Here
we see Lewis with a theoretical foot in two centuries; there
is still the tendency to rely on the mentalistic concept of
the ”laws of thought" when speaking of the subject-matter of
logic, while at the same time, there is the explicit
recognition of the necessity of setting up rigorous formal
systems in the linguistic medium of logic. Lewis later draps
all talk of the subject-matter of logic being principles of
rational procedure, but only because in the more refined

theory of symbolic logic, he finds no need for such concepts;

 

58Lewis, Clarence Irving, A Survey 23 S mbolic Logic
(Berkeley: University of California Press, 1918 , pl 1,

270

his later theory goes far beyond a pure mentalistic conception
of the subject-matter of logic.

Lewis nowhere, in fact, seems strongly committed to the
mentalistic conception of the subject-matter of logic. After
introducing it, he provides no further discussion of what might
be involved in the notion of "the principles of rational
procedure." His uncritical use of such notions probably is
an indication only of the fact that Lewis was employing them
in an historically engrained way. That is, Lewis may well
have been simply following historical guidelines in accepting
these notions.

Had Lewis carried through with the conception of the
subject-matter of logic as being the principles of rational
procedure, there is every indication that his later work in
epistemology *would have provided a strong basis for it. His

theory of the structure of the mind in Mind and the World

 

92222: for instance, would be used to provide a basis for
such a theory of logic.

There is yet another feature of the Survey definition
of symbolic logic, given above, which is of interest. When
Lewis speaks of the medium of logic as being composed of
"ideographic symbols,” he is using the eXpression in the same
way as Peirce had used it, when he coined the term. As we
proceed, more of Peirce's influence on Lewis will become
apparent; the pragmatic aspect of Lewis' theory, as well as
his use of certain technical expressions, was heavily

influenced by Peirce's thought.

275

Even as early as the writing of Surve , Lewis was
interested in maintaining that symbolic logic was just a part
of logic in general: "the presentation of the subject—matter
of logic i.e.[Ehe general principles of rational proceduré]
in...mathematical form59 constitutes what we mean by symbolic
logic."60 The difference between symbolic logic and other
sorts of logic amounts to a difference in the forms of
symbolism employed. This can be the only area of genuine
difference, for, as Lewis admits, all forms of logic have the
same subject-matter. He also recognizes, however, that while
symbolic logic is separable from traditional logic, the
separation is by no means perfect. As he says, “...the
really distinguishing mark of symbolic logic is the
approximation to a certain form, regarded as ideal."61 The
form in question, i.e., the mathematical structure, can only
be approximated by symbolic logic, ”...hence the difficulty
in drawing any hard and fast line between symbolic logic and
other logic.”62 The only firm distinction which Lewis

eventually draws is between symbolic logic and the algebras

 

59Such form being comprised of three aspects: “(1)
the use of ideograms instead of phonograms of ordinary
language; (2) the deductive method...; and (3) the use of
variables... ." Ibid., p. 2.

60Ibid.
61Ibid.

62Ibid.

276

of logic. This distinction is carried over to Symbolic Lo ic,
as we have seen. But even in this case the distinction stands
because the forms of the two sorts of logic have been
separated enough to make the distinction useful. Their
subject-matter remains the same.

While Lewis was interested in pricturing symbolic
logic by reference to the notion that 1) its subject-matter
is rational or reflective procedure in general; 2) its
medium is ideographic symbolism; 3) it employs variables of
fixed range; and 0) it is developed deductively, he contrasts
this sort of lOgic with what he calls '10gistic.‘ He maintained
that is would be ill-advised to call logic '10gistic,‘ "...
because 'logistic' is commonly used to denote symbolic logic
together with the application of its methods to other
symbolic procedures. Logistic may be defined as 322 science

which deals with types pf order 33 such."63 He goes on to

 

maintain that while logistic may use principles of symbolic

logic, ”...still a science of order in general does not

necessarily presuppose, or begin with, symbolic logic."6u
In the last chapter of Surve , Lewis provides a

detailed discussion of his conception of logistic. In this

discussion, however, he does not carry through with the notion

 

63Ibid., p. 3.

60
Ibid.

277

65

that logistic is the science of order, but instead maintains
that logistic comprises all formal systems, i.e., all systems
which incorporate the use of the principles of symbolic logic.
Thus he says, "'Logistic' may be taken to denote any
develOpment of scientific matter which is expressed

exclusively in ideographic language and uses predominately

(in the ideal case, exclusively) the operations of symbolic
logic."66 On this account Euclidian geometry becomes a part

of logistic simply because of its deductive structure.

In a stricter sense, however, Lewis finally takes
'logistic' to mean any discipline which can be reduced to logic.
Any branch of mathematics which can be so reduced, for example,
becomes a part of logistic. Thus he says, “...gll rigorously
deductive mathematics gets its principles of operation from
logic; logistic gets its principles of operation from symbolic
logic. Thus logistic, or the logistic develOpment of
mathematics, is a name for abstract mathematics, the logic
Operations of whose development are represented in the
ideographic symbols of symbolic logic."67

While Lewis later drOpped all reference to logistic

as a separate science, going so far as to exclude the portions

 

65A1though this meaning of the term has the longer
history; in his Dictionary 2; Philosophy article entitled
”Logistic,“ Church points out that as early as the 17th
century, 'logistic' was used to cover all sorts of calculation.

 

66Lewis, op. cit., 5 Survey p£ Symbolic Lo ic, p. 200.
671bid., p. 103.

278

of Survey which dealt with the subject from the post-war
reprint of the book, nonetheless he retained the strict
conception of logistic noted above, but under the heading

”logistic method.” In Symbolic Logic, Lewis uses the concept

 

of 'logistic system' to mean any system which is developed
from an explicitly prescribed primitive basis, and used the
term 'logistic method' to refer to that procedure by which
various parts of, for instance, mathematics can be built
upon a purely logical base. Thus he claims that Principia
Mathematica is a good instance of the employment of the
logistic method, especially since it begins with the logic of
propositions and then proceeds to construct other sorts of
logic and mathematics, all in one continuous development.
While such topics as Lewis' conception of logistic are
interesting, they shed little light on the main portions of
his theory of logic. For this reason, we shall now consider
the remaining portions of the theory.

The theory of logic which Lewis presents in Surve ,
while being incomplete in the sense that such central notions
as "the general principles of rational procedure" are not
fully developed, nonetheless resembles the theory presented
in Symbolic Logic in many important respects. His insistence
on a strong role for language in logic, for instance, is
carried over from the earlier work to the later one. The
emphasis on language and its properties is made even stronger

in the later work, as we shall see.

279

Lewis' theory of logic, as the full version of it is
presented in Symbolic Lo ic, revolves around the following
considerations: first, he conceives of logic as the canon
of inference. Inference is an operation performed on
prOpositions. At this point, however, two issues relating
to Lewis' theory of logic must be separated. We must
differentiate between what Lewis ggyg that the subject-
matter of logic is, and what his system of logic actually
commits ElE.£2'

On the one hand, it is clear that Lewis thought that
when he talked about propositions, that these were basically
linguistic. When he defined 'proposition,' as we have seen,
he asserts that they are linguistic. Furthermore, every
statement he makes concerning the nature of propositions
supports this view.

On the other hand, however, there are certain definite
aspects of his logic which will simply not permit such an
interpretation. For instance, in the cases of nested
modalities such as 'p-$(p-§p),' to hold that their relata
are statements would force one to the position that the
statement 'p' cuts across three type-levels (as would be
required by the eXpression in question).

In sum, Lewis thought that logic was about statements
in intension, while his system can best be interpreted as
being about intensions themselves.

One way in which Lewis could have attempted to

extricate himself from such difficulty would be by holding

280

that the substituends for the variables connected by the
symbol "6' are of the form 'that p.' In this way, 'P'
remains a statement-variable, but the entire expression
indicates an intensional entity. Thus 'P-%Q' should be read,
'the proposition that P strictly implies the proposition that
Q.‘ Understood in this way, the difficulties surrounding the
decision as to whether logic is about language or extra-
linguistic entities vanishes, for at one and the same time,
both the statement and its meaning are taken into account.
Such a move would have the effect of incorporating meta-
linguistic devices into the object—language in order to
better indicate the full sc0pe of the subject-matter of
logic.68

Second, Lewis does not consider it to be necessary to
maintain that logic describes, or has any similar direct
contact with the extra-linguistic world. The meanings which
implication relates are conceptual meanings, as we shall see.

Third, logic is not only about propositions, but is
also about their relations. The relations which statements
bear to one another are fixed and unalterable; but which of
these relations are chosen for incorporation into systems of

logic is indeed a matter of choice, especially since not all

 

68Furthermore, such a mode of interpretation might
be profitably employed in describing the subject-matter of
the first edition Of the Principia. There also, there was
no attempt to systematically separate linguistic from
non-linguistic concerns.

281

logical relations are a part of logic. Thus we choose those
logical relations which best serve our needs in the
construction of systems of logic. This conventional aspect

of logic carries over both to the idea that analytic statements
(i.e., tautologies) are products of choice among logical

facts, and to the idea that there are many different relations
which can be used to support inference, and we must choose
among them. The remainder of this section will be devoted to

a detailed discussion of these points.

In dealing with the question as to what constitutes
logic, Leiws' basic position, taken in Symbolic Lo ic, is that
”...what is usually called 'logic' -- meaning the canon of
inference -- represents simply a selection of relationstéf
prOpositions and/or termi], with their laws, which are found
useful for certain purposes, notably those commonly connoted
by 'deduction.'”69

Furthermore, Lewis denies there being any close
connection between logic and the world. In a revealing
passage, he says

...There have always been those who take

logical truth to state some peculiar and

miraculous property of reality or the universe,

and thus fall into a state of mystic wonderment

about nothing. The facts which the principles

of logic state are simply facts of our own

meanings in the use of language: they have
nothing to do with any character of reality,

 

69Lewis, op. cit., Symbolic Lo ic, p. 255.

282

unless of reality as exhibited in human
language-habits. The universe can 'be what it
likes'; it cannot make a definition false; and
it cannot exhibit what is logically
inconceivable, for the simple reason that

logical conCeption exhausts the possibilities.70

In limiting logic in this manner, Lewis does not suggest that
the language which logic is about has nothing to do with
reality, or that the relations which statements bear to one
another have nothing to do with reality. His suggestion is
that there is no necessity for logic, in itself, to go beyond
the bounds of ”the facts of our own meanings in the use of
language.”

To this point, Lewis' position has been described as
holding that logic is about language and the relations between
the parts of language which can give rise to inference. Now
we must notice another central feature of the theory, which
is closely related to this point, namely, that the relations
which prOpositions bear to one another are not products of
convention, but how they are employed lp’a matter of convention.
Lewis says that "whether a particular relation is included or
omitted in a 'system' is a matter of choice. Systems are
man-made, even in that sense in which relations and the
truth about them are not. ...the justification of one's

procedure, in this re5pect, is purely pragmatic....”71

 

70Lewis, op. cit., p. 212.

71Ibid., p. 256.

283

The immutable facts which logic is about are, for
instance, relations such as (in the most familiar case) truth-
functional relations, but also include modal or intensional
relations. Lewis is maintaining that we discover logical facts
and relations, such as the various sorts of implication,
rather than construct them. But how we use these facts and
relations is another matter. While we can do nothing to alter
these fundamental relations, we can do something about how
they are put to use. It is at this point that the pragmatic
aspect of his theory becomes apparent. After deciding how we
want a system of logic to function, we choose the appropriate
logical facts to incorporate into it. Thus, in building a
system, "if consistency and independence of propositions
shouldn't be important to us we might choose the system of
Material Implication as the canon of our inferences."72 Or
again, if one was constructing an intuitionist system, then
perhaps the law of the excluded middle might be drOpped.

Lewis is simply saying that the facts are there to be used
or not used, and that the decision is up to the logician.

In putting the same idea in a slightly different way,
Lewis says that "we could not arrange facts in a certain
order if the facts did not have certain relations. But facts
do not arrange themselves; and quite generally the relations

which facts actually have allow an almost unlimited variety

 

72Ibid., p. 260.

280

of different orderings."73 Or again, he says that the ”facts
summarized in a system are not created by the system --
inference maps facts, but doesn't create the geography of
them.”7u

There are a few immediate conclusions which may be
drawn from all of this. In the first place, from Lewis'
talk of choosing among logical facts, one might be tempted to
draw a comparison between Lewis' theory and Wittgenstein's
theory. Both talk of logical atoms from which we choose in
the construction of logic. However, while Wittgenstein's
facts were states of affairs, Lewis' ”atoms” are facts-of-
meanings-in-the-use-of-language. For Lewis, the choice
process was not one of our selecting among external facts for
the components of the system of logic, but actually a

selection among 10 ics, which differ in the conceptual

components which they include. The "meanings" among which we

 

choose are conceptually determined; the evidence of our choice
among these meanings occurs in the language which we use.
The basis for choice among logics is pragmatic; we choose

the conceptual components which will "do the job."75

 

73Ibid., p. 251.

7uIbid.

75For a full description of the mechanics of this
choice process, cf. Lewis, Clarence Irving, Mind and the
World Order (New York. Dover Publishing Company, 1956),
pp. 230- 270. (First published in 1929. )

285

At another level, however, the comparison between
Lewis and Wittgenstein is of some interest. One of the
advantages of Wittgenstein's theory of logic was that it
suggested that there is a vast array of logical facts which
could be incorporated into systems of logic. Lewis made this
notion even more explicit by claiming that the field of logical
fact (i.e., facts of meanings in the use of language) just
cannot be exhausted by any system of logic. He says, for
instance, that "logical order is immensely wider than
traditionally observed relationships."76 At another point,
he says that "just as the weight of every grain of sand on
the seashore is a physical fact, though no part of 'physics,’
so the truthszbout the relations of propositions are all of
them logical facts, andthe multiplicity and variety of such
facts is beyond all imagining; but to be a logical fact in
this sense does not mean to be a part of 'logic."'77 With
such a wide range of facts from which to choose, Lewis'
theory might be described as a pragmatic "accommodation"
theory. We can accommodate logic, that is, to our needs by
choosing the proper logic from the world of conceptual fact.

There is yet another conclusion which may be drawn at
this point. Lewis' theory of logic is an extension of his

epistemology in certain significant reapects. As an example

 

76Lewis, op. cit., §ymbolic Logic, p. 253.

77Ibid.

286

of this, consider his theory of categories in 2122.222.£EE
World Order. This theory, too, can be called a pragmatic
"accommodation" theory, for in it there is rec0gnized both an
element which we cannot change (viz. the qualia, or elementary
phenomena), and also the ability to incorporate these elements
into our conceptual scheme, as the outcome of essentially
pragmatic decision-making procedures. In this way, Lewis'
theory of logic ”fits” a large part of his epistemology. The
reason for this resemblance, moreover, is not difficult to
discover: Lewis, as we have seen, was interested in insuring
that there be a range of choice in the selection of a relation
for use in supporting inference. While he thought that such
indeterminancy was desirable, he was also interested in
providing a firm foundation for this field of choice.

This conventionalism together with an underlying
basis of unalterable fact, which is the central theme of
Lewis' theory, is also strongly reflected in his general
theory of language. Consider, for instance, this passage

from Analysis 2: Knowledge and Valuation:

 

The use of linguistic symbols is indeed determined
by convention and alterable at will. Also what
classifications are to be made, and by what
criteria, and how these classifications shall be
represented, are matters of decision. Insistence
on these facts is sound. Nevertheless such
conventionalism would put the emphasis in the
wrong place. Decision as to what meanings shall
be entertained, or how those attended to shall

be represented, can in no wise affect the relations
which these meanings themselves have or fail to
have. Meanings are not equivalent because
definitions are accepted: definitive statements
are to be accepted because, or if, they equate

287

expressions whose eguivalence of intensional

meaning is a fact.7
Here we see essentially the same position which we have been
discussing, but in a much wider context.

After Lewis sets up this theory of pragmatic
conventionalism in the manner in which it has been described,
he goes no farther in developing either the ontological
implications of his theory or a full theory of reference.
But while he does not pursue such matters, there are still
other ways in which light can be shed on his theory.
Specifically, we shall discuss some of his further thoughts
on the topic of inference, and on the nature of logical
truth. Both of these topics will be of help in providing a
better understanding of the theory as it has been presented.

Lewis' main point concerning the nature of inference

is that ”Inference remains an operation even when implication

 

is a relation of the system."79 Such being the case, inference
does not have the immutable character displayed by logical
facts. It is alterable. As Lewis says, ”We make an inference
upon observation of a certain relation between facts. Whether
the facts have that relation or not we do not determine. But

whether or not we shall be observant 2f just this particular

 

 

78Lewis, Clarence Irving, An Analysis of Knowledge
and Valuation (LaSalle, Illinois: The Open Court Publishing

Company, 1935), p. 97.

 

79Lewis, op. cit., Symbolic Logic, p. 258.

288

relation of the facts and whether we shall make that relation

£22 232l3‘2£.22£ inferences are things which we do determine."80

 

Furthermore, inference is external to the system of
facts which support it. As Lewis says, "The logistic rule
which controls or allows the operation of inference reveals
its externality to the facts within the system itself by being
incapable of expression in the symbols of the system. The
system may be discovered or delimited in terms of inference:
nevertheless the fact of the inference is nothing in the

"81 Lewis thinks of inference as being our way of

system.
using the (conceptual) facts before us.

While it seems that Lewis holds inference to be a
mental operation, he cuts his discussion of it off before
taking up such issues directly. Perhaps he would consider
inference to be a conceptual category, ordering other
conceptual components (i.e., logical facts).

With regard to Lewis' conception of logical truth, he
claims that not only can logic be thought of as being the
canon of inference, but also ”...as that subject which comprises

all principles the statement of which is tautological."82

In our discussion of Lewis' logic, we have already seen that

 

BOIbid.

811b1d.

821bid., p. 235.

289

he claimed that there are alternative ways of conceiving of
tautologies (see above, p. 266). We must now discuss how his
position on the nature of tautologies accords with the rest
of the theory.

Lewis claims that "...any logical principle...is
tautological in the sense that it is an analytic proposition.
The only truth which logic requires, or can state, is that
which is contained in our own conceptual meanings -- that
which our language or our symbolism represents.”83 Lewis'
idea of the nature of tautologies strongly reflects the same
sort of conventionalism we have seen in other contexts
already. Here he is simply maintaining that the truth of any
law of logic ‘3..eXplicates, or follows from, a meaning which
has been assigned, and requires nothing in particular about
the universe or the facts of nature.”8u

In maintaining such a view, he is attempting to set
the laws of logic apart from other sorts of scientific laws,
in that they are about nothing. "...there are only certain
analytic propositions, eXplicative of 'logical' meanings,
and these serve as the 'principles' which thought or

inference which involves these meanings must, in consistency,

adhere to.”85

 

8
3Ibid., p. 211.

8L‘Ibid.

851bid.

290

There is one difficulty which haunts this view. From
this account it seems that the very logical facts which were
earlier held to be immutable are those which Lewis is now
claiming are capable of being established by definition. It
is not immediately clear how Lewis would propose to handle
this difficulty. It may be the case, however, that he intends
the logical meanings which are established by definition to be,
like inference, outside of the system of logical facts which
such meanings would employ. In this way, the system of facts
could remain immutable relative to the system, and the logical

truths of the system could still be held to be conventional.

u. CONCLUSION.

Lewis' theory of logic is centered around his major
concern in logic: the exhibition of alternative methods of
conceiving of the basis of inference. In providing for these
alternatives, Lewis relies heavily on the conceptualist point
of view; he considers the facts of logic to be facts of
conceptual meaning. In his view, there are many of these
facts, and alternative logics are composed of alternative sets
of them. In this way, how inference is conceived of in a
system depends on which conceptual facts are accounted for
in that system.

The choice of logical fact, in Lewis' view, is
reflected in the use of language. For different purposes,

different facts are chosen, and different languages employed.

291

Lewis considers logic to be about statements-as-
they-carry-conceptual-meaning. His linguistic orientation,
while not as sevenaas Russell's, nonetheless pervades his
theory. But as we have seen, his idea that logic is about
statements-in-intension is not consistent with the
requirements of his system.of logic. The system demands that
logic be about intensions (i.e., the conceptual meanings

themselves), rather than about statements-in-intension.

CHAPTER VIII

CONCLUSION: CHURCH, QUINE, AND THE FUTURE

1. INTRODUCTION.

We shall conclude this study by considering the wide
disparity between the theories of logic of two well-known
currently active logicians, how that disparity helps to draw
together our findings concerning all of the earlier theories
discussed, and how it points to future problems in the theory
of logic. The theme of this discussion will be that while
the disparities between the theories of logic which we have
considered are quite genuine, nonetheless there are some
areas of fundamental agreement among these theories. We shall
maintain, furthermore, that these areas of agreement could
serve as the basis for further attempts in the construction
of a truly acceptable theory of logic.

We have saved the theories of Willard Van Orman Quine
(1908-) and Alonzo Church (1903-) until now for two reasons:
first, because there is no clearer example of the wide dis-
parity which can exist between theories of logic; and second,
because neither of the theories will need so much discussion
that the main points of the conclusion might be obscured.
Quine's theory strongly resembles Russell's later theory.
There is also a similarity between his theory and the theory
which Lewis explicitly asserted. However, the theory which
better fits Lewis' system of logic, as we saw earlier, is of

292

293

a sort which is opposed to a position such as Quine's. Thus
the two theories are highly articulated versions of the
theories with which we have already dealt. While we have
already set down the particulars of the major theories of
logic in some detail, nevertheless, by considering Quine and
Church, one may put the contrasts in sharper focus, see also
the similarities, and perhaps draw some conclusions for the

future.

2. CHURCH'S LOGIC; QUINE'S LOGIC.

In opposition to our method of operation in other
chapters, we shall not attempt a discussion of the basic points
of the logics of the two logicians in question. The great
bulk of their work presents itself as sOphisticated versions
of a Principia-type logic, both incorporating such further
advances as explicit metalanguages and as dealing eXplicitly
with problems in decision theory. This is not to say, however,
that their logics are just reiterations of the Principia; the
sophistication and subtlety of these works exceeds that of the

Principia. The point is, rather, that the tools which Quine

 

and Church employ in the construction of their logics bear a
basic resemblance to those used in the Principia. Furthermore,
the technical advances over earlier works in logic, and
contrasts between their works would provide important material
for a history of logic, but it is larggly irrelevant to our

concern with the development of the theory of logic.

29“

Hence, we shall only briefly discuss the achievements
of each of these men in the field of logic and then move on
to discuss some aspects of the relationship between their
contributions to the theory of logic.

Quine's most widely-recognized contribution to logic
is his rigorous treatment of the logicist development of

mathematics in Mathematical Lo ic, first published in 19h0.

 

Among other advances made in that work are the following:
first, he added rigor to the system by the inclusion of such
devices as a quotation-system whereby quotes could be
explicitly framed in the metalanguage (he calls this “quasi-
quotation”), and by pointing out the difference between use
and mention. He also provided rules for the iteration of
operators and predicates in molecular expressions. Second,
he set up rules governing the binding of variables, and the
transformation of quantificational formulae in such a way as
to avoid the pitfalls which would otherwise attend such
operations.

Of a more substantive nature, Quine develops an
alternative to the theory of types, which he says has

unnatural and inconvenient consequences.1 His alternative

 

1In particular, he says that because the theory of types
”...allows a class to have members only of uniform type, the
universal class V gives way to an infinite series of quasi-
universal classes, one for each type. The negation -x ceases
to comprise all nonmembers of x, and comes to comprise only
those nonmembers of x which are next lower in type than x."
Quine, Willard Van Orman, From a Logical Point of View
(Cambridge: Harvard University Press, 1953), p. 91.

295

revolves around his conception of the “stratification" of
formulas. A formula is stratified "...if it is possible to
put numerals for its variables (the same numeral for all
occurrences of the same variable) in such a way that '6'
comes to be flanked always by consecutive ascending numbers
('ne1+1').~2

As it stands, this replacement for type-theory is still
analogous to typeatheory. Quine goes on, however, to suggest
a way in which stratification can be used to avoid the
shortcomings of type-theory. He says, "Whereas the theory of
types avoids the contradictions by excluding unstratified
formulas from the language altogether, we might gain the same
end by continuing to countenance unstratified formulas but
simply limiting EBB. If 'x' does not occur in Q,
(a x)(y)((y€x)§ Q) is a theorem.3 explicitly to stratified
formulas. Under this method we abandon the hierarchy of types,
and think of the variables as unrestricted in range."3

There are many other areas in which Quine makes
advances over the logics which were written between 1915 and
19h0. His treatment of definite descriptions, for instance,
is somewhat more simple than was Russell's,“ although it

remains on the same philosophical footing.

 

2Quine, Willard Van Orman, Mathematical Logic
(Cambridge: Harvard University Press, 19h0), p. 157.

3Quine, op. cit., From 2 Logical Point 2: View, p. 92.

“In fact, Quine's theory of definite descriptions
resembles Frege's theory, more than it does Russell's.

296

In general, Quine's contribution to logic has come
mainly in the form of a tightening up of logical concepts
and procedures, mainly through his systematic presentation
of the logicist program.

Church's best known work was done in the mid-1930's,
in the area of decision theory. In a 1935 paper entitled "An
Unsolvable Problem of Elementary Number Theory,” he develops
what has come to be known as ”Church's Thesis,” that the
”functions which can be computed by a finite algorithm are
precisely the recursive functions, and ...that an explicit
unsolvable problem can be given."5 Even better known is his
paper entitled ”A Note on the Entscheidungsproblem," in
which Church shows that there is no algorithm to test the
validity of a formula of first-order logic (or more
specifically, that there is no “algorithm which can test a
formula to determine whether or not it can be derived from
the rules laid down by Hilbert and Ackermann.")6

Besides these significant contributions, Church has
made contributions to such disparate fields as combinatory
logic, and to the logic of sense and denotation, in which he
attempts to systematize Frege's concept of "sense“ by

constructing an intensional logic.

 

5Davis, Martin (ed), The Undecidable (Hewlett, New York:
The Raven Press, 1965), p. 88.

6Ibid., p. 108.

297

Church's interest in logic covers a broader range of
topics than does Quine's. Church, like Lewis, develops his
logic outside the strict confines of the logicist program.
Thus, his most popular work, Introduction :2 Mathematical
Logic covers the various orders of calculi in more detail
than does Quine. For example, he provides several alternative
propositional calculi, and two first-order calculi. He does
not present a straightforward development leading directly
to the definition of number.

For our purposes, however, we must neglect the dif-
ferences between Church's logic and Quinés, and concentrate
on their similarities. For they are basically similar not only
one to another, but to the earlier systems that we have already
discussed. They use many of the same concepts (’proposition,"
”prOpositional function," "theorem,” etc.) and the same
operations (rules of inference, etc.). In short, the mechanics
and furniture of the two systems of logic bear a fundamental
and unmistakable resemblance to one another.

In itself, such similarity is not surprising; after all,
these logics have been developed from many of the same roots,
and for largely the same purposes. What is more surprising
is the diSparity among the theories which accompany these
systems of logic. While there is an underlying stability to
logic itself (regardless of the ultimate directions which
various systems take), the underlying theories that we have

discussed seem to show no such general agreement. But even

298

this fact should not be too surprising. Like any other science
(and especially any other formal science), it is not unusual
for there to be wide discrepancies in underlying theory, while
there is broad agreement on the mechanics of the science.

What is genuinely surprising is the lack of interest
in attempting to find some pervasive unity for all theories
of logic, principles which stand to the various theories of
logic as the fundamental similarities in logic stand to the
alternate systems of logic.

The heart of these concluding remarks will be the
contention that even though such theories as those of Quihe
and Church are in fundamental disagreement on the question of
its location and nature, such theories are in agreement that
logic is fundamentally about some type of £252. The final
suggestions will center around the idea that perhaps the best
direction for future work in the theory of logic will be
concerned with the question of the nature of form in general,
and the relation which such form bears specifically to
linguistic form.

To complete this study, we shall now discuss the

essentials of the theories of logic of Quine and Church.

3. THE THEORIES OF LOGIC OF QUINE AND CHURCH.

The basic divergence between the theories of Quine and
Church is apparent from the following statements; first,

Quine:

299

”logic, like any other science, has as its
business the pursuit of truth. What are true
are certain statements; and the pursuit of
truth is the endeavor to sort out the true
statements from the others, which are false.

... statements will constitute not merely the
medium of this book (as of most), but the
primary subject-matter.‘8

... the truths of mathematics treat explicitly
of abstract non-linguistic things, e.g., numbers
and functions, whereas the truths of logic, in

a reasonably limited sense of the word 'logic,‘
have no such entities as Specific subject-
matter. This is an important difference.”

Next, consider these remarks, made by Church:

Traditionally, (formal) logic is concerned with

the analysis of sentences or of propositions

and of proof with attention to the form in

abstraction from the matter.10

... a proposition, as we use the term, is an

abstract object of the same general category

as a class, a number, or a function.
Quine thinks that logic has no extra-linguistic objects as
its subject-matter, while Church thinks it does. Herein lies
the difference between the two. Let us look briefly at how

each of the two arrives at his position.

 

7Quine, Willard Van Orman, Methods 22 Logic (New York:
Holt Rinehard and Winston, 1950), p. xi.

81b1do, p0 xv1o

9Ibid., pp. xvi-xvii.

10Church, Alonzo, Introduction to Mathematical Logic
(Princeton: Princeton University Press, 1956), p. 1.

11

 

Ibid., p. 26.

300

Church's theory of logic is a refined version of Frege's
theory of logic. The main points of Frege's theory are,
however, carried over intact to Church's theory. Church
maintains, for example, that terms haveroth sense and
reference,12 and that statements also function as names, in
that they too have both sense and reference.

Church builds his theory in the following fashion:
first, as we have seen, he claims that in logic we are
interested in form as opposed to matter. In order better to
study these formal features (of both linguistic and non-
linguistic things) he suggests that a formalized language be
constructed, i.e., one which would reverse the tendency of
the natural languages to obscure formal structures in language.
He then points out that there will be some features of the
natural languages which will become a part of the formalized
language, such as proper names.13

Church's theory of proper names is, as he himself
asserts, due to Frege. Thus he claims that names can have a

denotation, i.e., what the name names, and also that names J

 

12For an explanation of these notions, cf. Frege
chapter, pp. 77 ff. We will explain only those notions
which are either original with Church, or are altered
enough by him to require discussion.

13Here lies, in fact, one of the important differences
between Quine's theory and Church's theory: Church employs
an artificial language simply to overcome the inadequacies
of ordinary language, while Quine seems to hold that artifical
languages are to be constructed for their own sake.

301

have a sense. Church claims that the sense.of a name "...is
what is grasped when one understands a name...”1u
There are a few slight differences between Church's
account of proper names and Frege's. Church places more
emphasis on theidea that the sense of a name determinesits
denotation, than does Frege. Church, in fact, calls this
determination of denotation by sense "being a concept” of
the denotation.15 Here we see also a shift in vocabulary:
for Frege, a concept was a function with one argument-place,
all of its values being truth-values; while Church's use of
the term ”...is a departure from Frege's terminology. Though

\

not identical with Carnap's use of concept in recent

16

publications, it is closely related to it... . It also

agrees well with Russell's use of class-concept in The

Principles pf Mathematicsl7... ."18

 

 

1“Church, op. cit., p. 6.

1
50f. ibid.

16The term 'concept' will be used here as a common
designation for properties, relations, and similar entities...
it is not to be understood in a mental sense...but ratherElle
something objective that is found in nature and that is
eXpressed in language by a designator of nonsentential
form. Carnap, Rudolph, Meaning and Necessity (Chicago:
University of Chicago Press, l9b7T: p. 21.

17Russell uses 'class concept' to stand for a class-
defining characteristic. Cf. Russell, Bertrand, Principles
pf Mathematics (London: Allen and Unwin, 1903), g 69.

 

 

1.8 .
ChurCh, OE. Cite, P. 6 no

302

Another subtle shift between Frege and Church concerns
the general notion of a function. Frege, it will be recalled,
considered a function to be a kind of incomplete entity,
which becomes an object when it results in a value of itself
by taking an argument. Church, on the other hand, claims
that a (one-valued singulary) function is "...an operation
which, when applied to something as argument, yields a certain
thing as the value of the function £25 that argument."19 The
shift here is subtle, and perhaps of only passing interest;
Church is placing more emphasis on functions-in—relation-to-
arguments although both Frege and Church consider functions
to be non-linguistic, to be the worldly correspondent of
expressions which contain free variables.

Church, like Frege, sees great utility in considering
statements to be names. In adopting such a position, he comes
into clearest contrast with Quine. Church begins by stating
that ”In order to give an account of the meaning of sentences,
we shall adopt a theory due to Frege according to which
sentences are names of a certain kind.... An important
advantage of regarding sentences as names is that all the
ideas and explanations of Ehe theory of name-a can be taken
over at once and applied to sentences, and related matters,

as a special case."20 He then goes on to posit the existence

 

19
20

Ibid., p. 15.

Ibid., p. 2b.

303

of two objects, one of which (truth) serves as the denotation
of all true sentences, and the other (falsehood) which serves
as the denotation of all false sentences. In reifying the
truth-values, Church follows Frege exactly.

With regard to the sense of a statement, which Frege
calls Gedanke, Church instead uses the term 'proposition.'21
This is a fortunate shift in terminology, for it brings
Frege's conception of the sense of a sentence more into line
with one of the current accepted usages of 'proposition.‘ It
will be remembered that, for Frege, the sense of a sentence
is the state of affairs which corresponds to the statement.
Frege's term, which is usually transtaled 'thought' conveys
the mistaken impression that senses of sentences are really
mental entities, which neither Church nor Frege wishes to
hold.

In developing the mechanics of the sense and reference
of statement, Church makes some clarifications of Frege's
theory. As in the case of terms, he maintains that the sense
of a statement determines its referent, i.e., its truth-value.
As he says, ”according to our usage, every proposition
determines or is a concept of (or, as we shall also say, has)

some truth-value."22

 

21
Frege, it will be remembered, used the term

'proposition' to refer to asserted egpressions of his
begriffsschrift.

22Church, op. cit., p. 27.

30h

Another important shift between Frege and Church occurs
when Church maintains that "A variable whose range is the two
truth-values...is called a prOpositional variable. We shall
not have occasion to use variables whose values are
propositions, but we would suggest the term intensional

propositional variable for these."23 That is, while Frege

 

did not even recognize the possibility of constructing a
logic of sense, Church has provided a place for it in his
theory.

These then, are the basic features of Church's theory.
Perhaps the most significant feature of his theory (for our
purposesk‘is that Church is quite willing to posit the
existence of such entities as truth—functions, where he feels
that his theory demandsthem. Furthermore, he is convinced that
logic is about these abstract objects. Like Frege, he
conceives of a world of things which correspond to linguistic
structures in the same way that theories of mathematics posit
things which correspond to mathematical expressions. He
allows himself the convenience of approaching his system of
logic as the scientist would a theory; that is, his ontology
is shaped to the needs of the system.

Quine, on the other hand, approaches the theory of
logic with an external ontological criterion: instead of

allowing the system of logic itself to be a guide to

 

231bid., pp. 27-28.

305

existence, he is initially unwilling to be led into positing
the existence of such objects as truth-values and senses. He
brings this negative criterion £2 his theory and constructs
his theory in such a way as explicitly to avoid the necessity
for countenancing Church's abstract objects. Quine's negative
attitude toward realism in the theory of logic is apparent
from such remarks as ”Most of the logicians...who discourse
freely of attributes, propositions, or logical modalities
betray failure to appreciate that they thereby imply a meta-
physical position which they themselves would scarcely condone.”2u

At various times, Quine has adopted two different, but
related, theories of the subject-matter of logic. 0n the one
hand, he has claimed that the question as to the subject-
matter of logic is best left Open, and that we should deal
with language, because by so doing we can set aside more basic
questions as to subject-matter. On the other hand, he has
maintained that logic is simply about language, with no other
questions left open.

Regarding the first Of these two positions, consider
Quine's statement that

Just what Ehej subject matter of mathematical

logig is, it is not easy to say; the usual

characterizations of logic as "the science of
necessary inference,“ ”the science of forms,"

 

2“Quine, Op. cit., From 3 Logical Point pf View,
p. 1570

306

etc., are scarcely informative enough to be

taken as answers.
But if we shift our attention from

subject matter to vocabulary, it is easy to

draw a superficial distinction between truths

of logic and true statements of other kinds.2
In making such statements, he is maintaining that questions
concerning the ultimate subject-matter of logic may be put
aside; that all problems concerning the subject-matter of
logic may be reduced to problems of language.

More Often, however, Quine appears to take the
narrower position that considering language as the subject-
matter of logic is not merely a convenience, but that logic
‘_2 ultimately about aspects of language. He says, for example,
that "mathematical logic at its most elementary levels deals
with statements, or declarative sentences, and with ways of
compounding them into further statements."26 Again, he says
that ”logic seeks to systematize, as simply as possibla,the
rules for moving from truths to truths..."27 Further, he
claims that ”laws of logical inference refer to recurrences
of sentences, on the assumption that a sentence true in one

occurrence will be true in the next."28

 

25

261bid., p. 11.

Quine, op. cit., Mathematical Logic, p. l.

27

Quine, op. cit., From 3 Logical Point pf View,
p. 165.

28Quine, Willard Van Orman, Word and Object (New York:
John Wiley and Sons, 1960), p. 227.

307

Quine's theory of logic is a refined version of the
linguistically-oriented theories with which we have already
dealt. It is a more explicit reiteration of the position
taken by Russell, in the second edition of the Principia.
Russell, it will be remembered, changed the conception of a
proposition adopted in the first edition of the Principia—
(which was essentially ambiguous as to the status of
propositions), to the idea that propositions were statements.
Quine adopts this position, and builds his theory on the
basis of it.

Besides the explicit statements by Quine such as we
have already seen, more of his theory of logic emerges from
his general philosophy of language, contained in such works
as 123$.222 Object. We shall not attempt to cover his entire
theory, as it would not serve our present purposes. Yet there
is one aspect of it which is of importance in this discussion.
This is Quine's idea that the truths of logic are not
immutable.

In stating that ”...the laws Of mathematics and logic
are true simply by virtue of our conceptual scheme,"29 Quine
adopts a position which bears a strong resemblance to Lewis'
contention that we can alter our conceptual scheme in order

to take various logical facts into account. Quine's position

 

29Quine, op. cit., Methods pf Lo ic, p. xi.

308

is slightly different from Lewis' position, and depends
directly on his conception of the structure of language.

Quine pictures language as a field of interconnecting
statements,30 with Observation statements at the periphery,
and those more and more removed from experience toward the
inside of the field. Adjustments (in truth-value) can be
made anywhere in the field, although those statements which
are more central inthe field (such as the laws of logic) are
more resistent to revision than those more directly connected
with experience. That is, those connected more directly with
experience can be altered more readily inthe face of a
recalcitrant experience, while modifications are made at the
center only ”if it is found that essential simplifications
will ensue. There have been suggestions Eor exampla stimulated
largely by quandaries of modern physics, that we revise the
true-false dichotomy of current logic... ."31 The general
resemblance of this position to Lewis' position is obvious;
both consider shifts in our conceptual scheme which affect
logic to be based on pragmatic considerations.

Furthermore, Quine's position that logical truths have
a conceptual basis is in this respect like Lewis' position,

and yet it does no harm to his idea that logic deals with

 

301.e., they are interdependent for their meaning.
For a more complete exposition of this doctrine, of. "Two
Dogmas of Empiricism,” in Quine, From 3 Logical Point pf
View.

31Quine, Op. cit., Methbds 23 Lo ic, p. xiv.

309

language. While our conceptual scheme is the arbiter of what
is, and what is not logically true, it is still the case that

it is statements which are logical truths, and it is still

 

them with which logic is concerned.

Before closing our discussion of Quine's theory, the
contrast between his theory and the theory associated with
Lewis' logic should be noted. As we have seen on several
occaSions, the position which Lewis explicitly adopts with
regard to the nature ofthe subject-matter of logic does not
agree with the theoretical requirements of his logic itself.
Thus while his stated position resembles Quine's position,
i.e., that logic deals primarily with statements, the theory
actually associated with his logic is Opposed to Quine's
theory. That is, Lewis' use of the symbol "5' seems to
demand that the relata of the sign be intensions rather than
statements-in-intension.

Quine, however, understands Lewis' system in such a
way that "3' would be a statement-connective. He says that
”Lewis, Smith, and others have undertaken systematic revision
of 'D' with a view to preserving just the properties
appropriate to a satisfactory relation of implication: but
what the resulting systems describe are actually modes of
statement composition -- revised conditionals of a non-truth-

2
functional sort... ."3

 

32Quine, 02- cit., Mathematical Lo ic, p. 32.

310

Many other aspects of Quine's theory of logic, while
interesting, fall outside the limits of this work. We shall
therefore leave an account of them for another occasion. For
our present purposes, the recognition that his system of
logic is conceived by him to be language about language is

enough for us to draw some conclusions.
u. CONCLUSION.

Throughout this work, our main purpose has been to
exhibit the wide diversity of theories of logic adopted by
those who have made the most significant contributions to the
subject. This exhibition of diversity is of some interest in
itself simply because this historical phenomenon has far too
longpassed unnoticed. Logic has grown to maturity, while
almost no interest has been shown in the theory of logic,
as a separate philosophical discipline.

A promising direction for further work in the theory
of logic might include an attempt to establish the theory
more firmly as a field in philosophy with basic problems of
its own. One step toward accomplishing this would be to
raise such questions as whether or not there is some unifying
principle among the various theories which we have discussed.
If it could be established that there is an underlying unity
among these theories, the philosophy of logic would thereby
be provided with a unifying principle of the sort which it

seems to need.

311

We shall therefore conclude this work by suggesting
one possible direction which future work in this field might
take. Our conclusion will constitute a suggestion as to how
a unifying principle might be recognized amongst the diverse
theories already developed.

One way of characterizing all of the theories
discussed in this work is that each one of them recognizes
form, in some way or other, as the subject-matter of logic.
Thus we could hold that Boole thought of logic as being about
form as it appears in thought. Frege and Church could be
held to believe that logic is about form, especially as it
exists in the extra-linguistic world. Finally, Russell and
Quine could be held to maintain that logic is about linguistic
form. Taken separately, the characterization of these theories
in this way presents no real problem. The problem comes in
the attempt to discover whether or not each of the theories
is about various aspects of the same thing, i.e., of form.
Our suggestion will be that the advantages of being able to
hold that all theories of logic are fundamentally about the
same thing are great enough to warrant further effort in the
direction of establishing such a unifying basis.

Let us consider how such an underlying unity might be
characterized as holding between the theories of Quine and
Church. While Quine does not wish to allow that logic is
about objects of every sort, nonetheless, it is clear that
he considers logic to be about what are commonly recognized

as the formal properties of language. When he maintains, for

312

instance, that logic is about recurrences of sentences, or is
about the essentially occurring features of language, he is
maintaining that logic deals with form in language. On the
other hand, Church's positing of the existence of sense of
sentences can be described as an attempt to account for the
existence of form in the world. When he says that logic is
about propositions (i.e., senses), he, too, is maintaining that
logic deals with form. The question remains whether there is
any way in which the concept of form might be so conceived

as unambiguously to take both Quine's and Church's concept of
form into account. Unfortunately, while reCOgnition of form
appears to be a necessary ingredient in all theories of logic,
unanimity on the interpretation placed on the concept appears
to befhr from a necessary ingredient.

Future work in the theory of logic might profitably
concentrate on questions concerning the nature of form. The
manner in which we have seen form conceived in this work
follows traditional philosophic lines; it is a hand-me-down
from philosophic concerns which fall outside the theory of
logic. Thus the theories account for form by way of
conceptualism, or through formalism, or realism. The
suggestion which is being made would result in the setting
aside of traditional phllOSOphical prejudices in an attempt
to find a conception of form which would center specifically
on the needs of the theory of logic.

There are many difficulties which face any attempt to

develop a general theory Of form which would be adequate for

313

the needs of the theory of logic. Not only is the term 'form'
so vague as to require sharpening, but also, the tradiational
ways of conceiving of form seem less and less appropriate.
Modern science is showing us that conceiving of the world as
being broken up into discrete chunks of matter is largely
inaccurate, and that a process view of the fundamental
workings of the world works much better. Adopting such a
view presents difficulty for those who would begin thinking

33

of form by opposing it to matter. Some other ground for
conceiving of form may have to be found to replace such
traditional conceptions. For instance, instead of opposing
form to matter, we might begin by making the fundamental
Opposition between form and content. While such a shift would
circumvent problems arising from modern physics, we would
still be faced with the problem of explicating the concept
of content. Nonetheless, such a direction is well worth
investigating.

While great problems face anyone who would make
precise the concept of form, still some modest beginnings can
be made on the way to solving them. For instance, as was
mentioned earlier, we might begin by attempting to establish

whether Quine and Church (and all the other theoreticians)

are really talking about various aspects of the same thing.

 

33I.e., specifically in the way in which Aristotle
does.

311.

If it could be held that form is homogeneous, i.e., that
is, is the same in kind in all of its occurrences, and that
Quine and Church are really talking about two aspects of the
same thing, this would work a great simplification for the theory
of logic. It could then be held that questions as to whether
logic is about the laws of thought, or about language, etc.,
are beside the point; that no specific category of things
(minds, languages, etc.) is £22 locus of form, and hence that
none of the alternatives presented in the theories considered
in this work provide the last word on the matter.

In order to regard the subject-matter of logic this
broadly, we might begin by maintaining that logic is about
any state of affairs. That is, we might begin by maintaining
that form is the "statehood" in any state of affairs, whether
mental or wnon-mental, linguistic of non-linguistic. Of course
such an initial conception is just the bare bones of a theory
with no meat at all, but given current philosophic theories
Which deal with propositions conceived of as states of affairs,
such a beginning looks promising. Whatever concept of form
is employed, it must be capable of providing the generality
which was suggested earlier.

By way of adding meat to such bones, we might attempt
to decide whether such topics as "deontic logic" or ”the
logic of change," etc., which have been much discussed lately,
are really parts of logic at all, by attempting to decide
whether they deal with the states in states of affairs, or

whether they are just instances of standard parts of logic

315

being put to interesting uses. What is called for is a deter-
mination of how the concept of a state is to be conceived.

Another interesting possibility which suggests itself
in accordance with the conception of logic as being homogenous
in the manner mentioned earlier concerns the place of language
in logic.

Besides the recognition of form, the only other constant
thread which runs through all of the theories we have considered
is that all of them provide a place for language in logic. This
position has been the source of a consideralbe amount of dif-
ficulty. 0n the one hand, if logic is held to be about all
form, it must lfl.i sense be about language, insofar as there is
a distinct formal element to language. On the other hand, since
language has been used as the medium for logic, in the sense
that it serves the 10gician as a useful model for form in
general, there has been a tendency among logicians to set aside
questions of theory for the more convenient ploy of thinking of
logic gp’i£_i£.gppp simply (i.e., only) about $£2_p!p model.

The situation would be much the same if the physicist were to
declare that physics is simply about the formulae in which its
laws are expressed; in both cases the procedure would be effec-
tive in allowing the scientist to escape pertinent philosophical
problems. In neither case, however (if the view which we have
been considering is correct), would the scientist be providing

a complete picture of the subject-matter of the science.

Taking logic to be about form in all of its occurrences

helps put the emphasis on languages in its proper perSpective.

316

Simply because form in language serves as a model for form in
general provides no reason for restricting the theory of logic
to a consideration of language alone. Wittgenstein's theory
of logic foreshadows such a view of the place of language by
clearly pointing out the model-like nature of language.

Concentration on the unity which underlies such diverse
theories as Quine's and Church's may indeed be the most
fruitful direction which subsequent work in the theory of
logic could take. It would not only allow for all of the
systematizing features already mentioned, but would also set
the theory of logic in its proper perspecitive among other
philosophic endeavours. We have seen how frequently questions
of theory have been relegated to places of relative
insignificance in the thought of many logicians; too often we
have had to pry a theory from its hiding place. Taking the
conception of form as the subject-matter of logic allows for
the introduction of metaphysical inquiry into the theory of
logic, as well as the more traditional concerns such as the
philosophy of language, and contemporary epistemology.
Rounding out the theory of logic in such a way may serve to
avoid having the theory of logic relegated to a place of
relative insignificance.

Our conclusion has not been a proper conclusion, for
we have concluded nothing, but have instead raised more
problems. If our discussion of the various theories can
help in their eventual solution, our effort will have been

well rewarded.

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