THE LOOICAL ATOMISM

OF F. P. RAMSEY

J. D. MACKENZIE

Date submitted

Thesis submitted for the Degree of Master of Arts in the School of Philosophy University of New South Wales (i)

SYNOPSIS

The first Chapter sets Ramsey in histor:iealperspective as a Logical

Atomist. Chapter Two is concerned with the impasse in which Russell

found himself ,d.th general propositions, Wittgenstein's putative solution

in terms of his Doctrine of Showing, and Ramsey's "Wittgensteinian"

solution, which is not satisfactory. An attempt is then ma.de to describe

a Ramseian solution on the basis of what he says about the Axiom of Infi-

nity, and to criticize this solution. In Chapter Three Ramsay's

objections to the Pl4 definition of identity are considered, and consequences

of his rejection of that definition for the Theory of Classes and the

Axiom of Choice are drawn. In Chapter Four, Ramsey•s modifications to

Russell's Theory of Types are discussed. His division of the Paradoxes

into two groups is defended, but his redefinition of 'predicative' is

rejected. Chapter Five deals with Ra.msey's analysis of propositional

attitudes and negative propositions, and Chapter Six considers the

dispute between Russell and Ramsey over the nature and status of

universals. In Chapter Seven, the conclusions are summarized, and

Ramsay's contribution to Logical Atom.ism are assessed. His main fail­

ing is found to be his lack of understanding of impossibility,

especially with regard to the concept of infinity. (ii)

PREFACE

The thesis is divided into chapters, which are in turn divided into sections. Internal references are given in the form 14.17', i.e. t~e seventeenth section of chapter four. To avoid footnotes, references are in general given in parentheses in the body of the test. If an article has been reprinted in a book, page references will be to the pagination of the book rather than of the journal in which the article originally appeared. The titles of many works have been abbreviated to facilitate reference. In general, these abbreviations are the initials of important wards in the title (e.g. 'PIA' for "The Philosophy of Logical Atom.aim").

Sections 2.11, 2.12, 2.13 state my conclusions about Wittgenstein's Doctrine of Showing and related matters. I have argued for these in my ''Wittgenstein's Tractatus Logico-Philosophicus and the Theory of Types", a B.A. Thesis submitted to Monash University in 1967, and do not pretend to prove them here. They are stated to indicate what I mean when I refer to Wittgenstein's Doctrine of Showing, and compare Ramsey•s position with it. I have bad to do this because there is great disagreement about the Doctrine of Showing and the reader therefore would not know to which of the various interpretations I was referring without this statement.

A full list of abbreviations follows. (111)

ABBREVIATIONS

Aris.Sec. Supp. The Proceedings of the Aristotelian SocietyJ Su~plementary Volume. Bull.Am.Math.Sec. The Bulletin of the American Mathematical Society. CWT Black, M. Companion to Wittgenstein's "Tractatus ". Cambridge; The University Press, 1964. DL Kneale, w. and Kneale, M. The Development of Logic. OXford; Clarendon Press, 1962. EAC Rubin, H. and Rubin, J.E. Equivalents of the Axiom

of Choice. Amsterdam; North-Holland, 1963. ECA Flew, A. (ed.) Essays in Conceptual Analysis.

London; Macmillan, 1956, 1966. EWT Copi, I.M. and Beard, R.W. (eds.) Essays on Wittgenstein's "Tractatus". London; Routledge and Kegan Paul 1966.

F and P Ramsey, F.P. "Facts and Propositions" (1927).repr.

in FM.

FL Prior, A.N. Formal Logic. OXford; Clarendon

Press, 1963.

FM Ramsey, F.P. The Foundations of Mathematics and other Logical Essays. London; Routledge and Kegan

Paul, 1931, 1965. FST Fraenkel, A.A. and Bar-Hillel, Y. Foundations of Set Theory. Amsterdam; North-Holland, 1958. (iv)

I and F Brouwer, L.E.J. "Intuitionism and Formalism" (trans. Dresden) Bull. Am. Math. Soc. xx 2, New York, 1914.

IMP Russell, B. Introduction to Mathematical Philosoph.y London; Allen and Unwin, 1919, 1967.

IWT Anscombe, G.E.M. An Introduction to Wittgenstein's "Tractatus ". London; Hutchinson, 1963.

JSL The Journal of Symbolic Logic.

LA Russell, B. "Logical Atomism" ( 1924), repr. 1n I.K.

LK Russell, B. Logic and Knowledge. London; Allen and Unwin, 1964. Chwistek, L. The Limits of Science. (trans H.C. Brodie) London; Routledge and Kegan Paul,1949.

I.SM Tarski, A. Logic, Semantics, Metamathematics. Oxford; Clarendon Press, 1956.

ML Quine, W.V.O. Mathematical Logic. New York; Harper Torchbooks, 1962. MLT Russell, B. ''Mathematical Logic as based on the

Theory of Types" (1908), repr. in I.K.

MPD Russell, B. My Philosophical Development. London;

Allen and Unwin, 1959. Nbks Wittgenstein, L. Notebooks 1914-1916. Oxford; Blackwell, 1961.

NL Wittgenstein, L. "Notes on Logic" (September

1913). repr. in Nbks. (v)

NM Wittgenstein, L. ''Notes dictated to G.E. Moore in NorW¥" (April 1914). repr. in Nbks. PA Urmson, J.O. Philosophical Anal.ysis. OXford; Clarendon Press, 196o. PI Wittgenstein, L. Philosophical Investigations. Oxford; Blackwell, 1963. PL Russell, B. A critical Exposition of the Philosophy of Leibniz. wndon; Allen and Unwin, 1937, 1964.

PIA Russell, B. "The Philosophy of Logical Atomism" ( 1918), repr. in LK.

PM Whitehead, A.N. and Russell, B. Principia Mathematica. Cambridge; The University Press, 1925-7, 1950. PMpbk Whitehead, A.N. and Russell, B. Principia Mathematica to *56. Cambridge; University Paperback, 1962, 1964.

PW Pitcher, G. The Philosophy of Wittgenstein. New Jersey; Prentice Hall, 1964.

RUP Russell, B. "On the Relations of Universals and

Particulars" ( 1911 ) • repr. in LK. TLP Wittgenstein, L. Tractatus wgico-Philosophicus. wndon; Routledge and Kegan Paul, 1961, 1963. TPW Frege, G. Tran_5-lations from the Philosophical Writings.

OXford; Blackwell, 1966. (vi)

WIA Griffin, J. Wittgenstein's Logical Atomism. Oxford; The University Press, 1964.

WT Stenius, E. Wittgenstein's "Tractatus". Oxford;

Blackwell, 1964. (vii)

CERTIFICATE

None of the work in this thesis has been submitted for a higher degree to any other University. However, Sections 2.11, 2.12 and 2.13, summarize some conclusions from a B.A. Thesis submitted to Monash

University, so that these conclusions can be used in the present work.

The reason this was necessary is explained in the Preface (p.ii). (viii)

ACKNOWLEDGEMENTS

I wish to thank both my supervisors, Professor

C.L. Hamblin and Mr. R.S. Walters, for their encouragement and helpful suggestions, and Mrs H. Langley for her quick and efficient typing. (ix)

"A quantity is infinite if it is such that we can always take a part outside what has already been taken".

Aristotle Physics III vi. (x)

CONTENTS

1. INTRODUCTION 1

2. GENERALITY, SHOWING AND THE AXIOM OF INFINITY 2

3. IDENTITY, CLASSES AND THE AXIOM OF CHOICE 29 4. TYPES 45 5. INTENSIONAL FUNCTIONS AND NEGATIVE FACTS 63 6. UNIVERSALS 76

7. RAMSEY'& CONTRIBUTION TO LOOICAL ATOMISM 102 1 •

CHAPTER I. INTRODUCTION

1.01 Frank Plumpton Ramsey was born on 22 February 1903 and died on 19 January 1930. In this short time, he made considerable contributions to Mathematics, Logic and Economics, and also to that Philosophy based on Modern Logic which may be conveniently labelled "Logical Atomism". This philosophy grew from two main sources; first :from the mathematical logic discovered by Frege and independently by Russell, which culminated in Whitehead and Russell's mammoth IM; and secondly, from the conversations between Russell and his brilliant pupil Ludwig Wittgenstein at Cambridge 1912-,. Russell and Wittgenstein were separated during the First World War, and published the two basic statements of Logical Atomismindependently - Russell in "The Philosophy of Logical Atomism" {a series of lectures delivered in Gordon Square London early in 1918, first published in the Monist of that year, and since reprinted in (IK) (PIA) and Wittgenstein in Logisch-Philosophische Abhandlung (first published in Annalen der Naturphilosophie 1921, first English translation as Tractatus Logico-Philosophicus in 1922, new translation 1961 (TLP) ).

Logical Atom.ism is no longer a fashionable philosophical position; and moreover many modern critics of the TLP would disagree with Ramsey's interpretation of that work. Thus it may be asked whether a study of Ramsey's version of Logical Atomism is worth while. I feel that, apart from its intrinsic interest, Ramsey's work is of historial importance, due to the fact that many later philosophers who acknowledged the TLP interpreted that obscure work along Ramseian lines. But more important philosophically is Ramsey's contribution to Logical Atomism. 2.

1.02 The condition of Logical Atomism when Ramsey began his work (1923) left something to be desired. Wittgenstein's TLP, with the first translation of which Ramsey had helped, was obscure in the extreme, and its author, believing himself to have found, on all essential points, the final solution of the problems of philosophy, had vanished :from its world. Russell, in PIA, had confessed himself unable to deal with certain problems, for example the nature of belief and other propositional attitudes (PLA. p.227), the analysis of general propositions (p.237) and negative propositions (p.215). Further, the great PM was even then not above criticism, particularly with rigard to the additional axioms required by the authors, the Multiplicative Axiom (equivalent to Zermelo's Axiom of Choice and still not a dead issue), and the Axiom of Infinity, neither of which could be proved or disproved (see PIA p.240).

1.03 Most importantly, few people were happy with the Ramified Theory of Types with its attendant Axiom of Reducibility, presented in PM as the solution to the Reflexive Antinomies. Even the authors themselves were not satisfied: "But although it seems improbable that the axiom [of red­ ucibility] should turn out to be false, it is by no means improbable that it should be found to be deducible :from some other more fundamental and more evident axiom. It is possible that the use of the vicious-circle principle, as embodied in the above hierarchy of types, is more drastic than it need be, and that by a less drastic use the necessity for the axiom might be avoided." (PM Intrcxiuction Chapter II svii p.59-6o) The first of the Refiexive Antinomies was discovered by Russell and communicated to Frege, who had just sent the second volume of the Grundgesetze der Arithmetik to the printers in 1902. Russell asked about the class of all classes which are not members of themselves, W. 3.

Is this class W a member of itself or not? Clearly it must be one or the other, and just as clearly it cannot be either:

X E W = (x e: x) df ~

WE W - ~(WE W)

Very soon it became clear that this is not an isolated problem of class theory; apparently similar cases arose everywhere, including the Bible, and seemed to reduce both Mathematics and Logic to confusion.

To eliminate these Antinomies Russell introduced the Ramified Theory of Types, which excised them satisfactorily, but unfortunately also excised important parts of mathematics, including almost the whole of mathematical analysis, which had to be saved by the Axiom of Reducibility. But the Axiom of Reducibility, like the other two Axioms, could neither be proved nor disproved.

1.o4 Ramsey attempted to repair these internal difficulties with the Atomist position. It is my purpose to outline his suggested rep:lirs and evaluate their effectiveness both in strengthening the Logical Atomist system in so far as this was possible, and in giving knowledge of permanent value to philosophy as a whole. 4.

CHA.Pl'ER II.

GENERALITY, SHOWING, and the AXIOM OF INFINITY.

2.01 One of Russell's main philosophical tools was Occam's razor. He preferred not to assert the existence of anything unless he had to (refusing to asRert is not, of course, the same as denying). However he felt that, as he did not know what the right analysis of general f'acts was (PIA, p.237) such f'acts could not be left out of an inventory of the worl.4. Because of the analysis of 1 ( 3:x)fx' as 1 ~ (x) ~ fx ', general facts include existential facts. His argument may be put this way. If you wish to prove that 'All men are mortal' is true, and try to do it without general :propositions, by enumeration you will get a long list beginning 'Adam is a man and is mortal, ••••'• But the whole of this long list does not impzy the conclusion 'All men are mortal' without the additional premiss 'All men are included in the list•, which is itself a general proposition. Thus, if we know general propositions to be true, we cannot have deduced them from strings of :re,rticular propositions. (v. PIA p.235). Furthermore, "It is perfectly clear, I think, that when you have em.me rated all the atomic facts in the world, it is a further fact about the. world that those are all the atomic facts there are about the world, and that is just as much an objective fact about the world as any of them are. It is clear, I think, that you must admit general facts as distinct fran and over and above particular facts."

(PIA, p.236). 5. 2.02 According to Moore, Wittgenstein

"said there was a temptation, to which he had yielded in the

Tractatus, to say that (x).fx is identical with the

logical product 'fa.:fb.fc ••• •, and ( :3: x) .fx identical

with the logical sum 1 fa V :fb V f C • • • t j • • • • 11 (G. E. Moore, ''Wittgenstein's Lectures in

1930-33", repr. in Moore, Philosophical

Papers, p.297)

That is, when he wrote the TLP, Wittgenstein would have denied Russell's argument that to the list of :i;articular propositions we must add an extra general proposition saying that the list is complete; for Wittgenstein, that the list is complete is shown. Ramsey endorses this. (FM, p.8) believing that •(x)fx' asserts no more than the logical product of all propositions of the form •:6:•, i.e. •ta.:fb.fc ••• •, and there is no need to add •and "a,b,c,. ••" is a complete list of the possible arguments of the function tx•. We know that no object other than those in the list could be an argument of the function fx, just as we know the converse

11 2.0123 If I know an object I also know all its possible occurrences in states of affairs. (Every one of these possibilities must be part of the nature of the object.) A new possibility cannot be discovered later."

(Vittgenstein, TLP.). 6.

2.03 Wittgenstein and Ramsey, by analysing general propositions into truth-

functions of particular propositions, have no reason to postulate general facts to correspond to irreducible general propositions. If adequate, this would be an improvement on Russell's position, on the

Occamist ground that it is simpler, It is not, however, certain that it is adequate.

The problem is that it is not clear how it is shown that the list is complete. Ramsey does not rely explicitly on the TLP doctrine

of showing, although he acknowledges a debt to Wittgenstein:

''Mr. Wittgenstein has perceived that, if we accept this

account of truth-functions as expressing agreement and

disagreement with truth-possibilities, there is no reason

why the arguments to a truth-function should not be infinite

in number (footnote: Thus the logical sum of a set of

propositions is the proposition that one at least of the set

is true, and it is immaterial whether the set is finite of

infinite. On the other hand, an infinite algebraic sum

is not really a sum at all, but a limit, and so cannot be

treated as a sum except subject to certain restrictions). 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 way, as we can by 7. using propositional functions" (FM, p.7-8)

There is something very strange about truth-functions with infinite numbers of arguments. Ramsey apparently thinks that because other functions seem to be able to have infinite numbers, e.g. the propositional function'••• is a real number' and the algebraic 2 2 function 'x -y = (x+y).(x-y)' are both true for an infinite number of arguments, truth-functions should also be able to have infinite numbers of arguments. Why shouldn't they1 This question is misconceived.

We can assert any one of an infinite number of values of 2 2 '••• is a real number' or of 'x -y = (x+y).(x-y)' and get a true proposition, but on each such occasion we have only~ argument. Whether we can assert ill values of these functions, however, involves the very question at issue, that of generality.

Russell himself had emphasized the distinction between 'all' and 'any' - e.g. •(x)x = x•, 'all things are identical with themselves', in which the variable 'x, is apparent (bound), in contrast to

'x = x', 'anything is identical with itself', where the variable is real (free). (v. "Mathematical Logic as based on the Theory of Types",

(Land K), p.64f., and PM, Introduction, Chapter I (p.14-22) ). Either can be asserted: if we assert '(x)x=x', we assert a general proposition; if we assert 1X = X 1 1 we assert a propositional function.

Russell also expresses this (even more oddly) by saying that in the first case we assert a definite proposition whilst in the second we assert an ambiguous proposition. 8.

2.07 In the introduction to the second edition of PM in 1927, however,

Russell, before acknowledging his debt to Ramsey, claimed that

"... there is no need of the distinction between real and

apparent variables, nor of the primitive idea "assertion

of a propositional function" (pxiii). thus changing his position on this matter. The formal differences are minimal; the first edition had included as a primitive proposition that ''What is true in~ case, however the case may be selected, is true in all cases" (*9, p.132), or in technical language, "A real variable may be turned into an apparent variable" (*9.13, p.132)

This primitive proposition is used only for inference (i.e. as a rule), not for implication (i.e. as a thesis of the system). In the second edition, the only change is that this rule should be applied automatically, eliminating real variables altogether. (v. qq. Appendix A, added in the second edition (1927, post-Ramsey) and replacing *9; PM pbk.p.385).

2.09 Von Wright has given an interesting argument against the analysis of general statements as conjunctions, and the conflation of

'all' and 'any' in his 1949 paper, "Form and Content in wgic", reprinted with appendices in his wgical Studies (1957), p.10, on the basis that there are propositions which are false in~ finite world, but which are true in all finite worlds; his example is 'Every prime is smaller than some prime'. (The reason it is true in all finite worlds is that the world of all finite worlds is not itself finite). And, as he notes

(loc. cit.), if •tautology' (or 'truth-function' in general) is defined 9o

in terms of truth-tables, since that method is inapplicable in

infinite cases, the concepts of tautology and truth-function are as

yet undefined in these cases. Wittgenstein and Ramsey, we can see,

have a task ahead of them in reducing general propositions to truth­

functions.

2.09. In fact, on the question of how a general proposition has

meaning, Ramsey seems at his weakest. We can only strengthen his

position by drawing consequences from what he says on other matters,

which in turn can most easily understood in relation to the other

Atomists.

To recapitulate, Russell realized that 1fa.fb.fc' does not

imply '(x)fx' unless •a•, 1b' and •c• are all the possible argu­

ments of the function f'x, that if this is so it must be said to be

so, and that this can only be done in a further general proposition.

(By "possible arguments of a function", the Logical Atomists mean

"arguments for which that function gives either a true proposition or

a false proposition, but not nonsense"}. In the TLP, Wittgenstein

claimed to have solved the problem without requiring irreducibly general

propositions by use of his Doctrine of Showing. Ramsey, though he

agreed with Wittgenstein's answer, did not explicitly base it on the

Doctrine of Showing, but used less convincing arguments.

I shall first examine Wittgenstein's solution (which is very com­

plicated), then show what is wrong with Ramsey's explicit arguments for 10.

the same conclusion, and finally, by examining Ramsey's views on the

Axiom of Infinity, attempt to reconstruct and evaluate the strongest

possible argument for the reducibility of general facts that can be based

on Ramsey's discussion.

2. 10. We begin with Wittgenstein's Doctrine of Showing. This Doctrine is one of the most important pirts of the TLP:

''What ~ be shown cannot be said"

(4.1212)

"The whole sense of the book might be summed up in the

following words: what can be said at all can be said

clearly, and what we cannot talk about we must pass over

in silence"

(Author's Preface para 2 P• 3)

and one of the most commented upon:

Anscombe IWT p 162f

Black CWT p 190-2

Griffin WIA p 23f

Pitcher PW p 153-4 Stenius WT p 222

Urmson PA p 90f

Sellars ''Naming and Saying" in EWT p 249f

are the most important treatments of the subject of this part of the TLP.

A detailed examination of Wittgenstein's Doctrine of Showing is

outside the scope of the present work, and I shall merely state my 11 • conclusions about it. (I have dealt with it more fully in'~ittgenstein's

Tractatus Logico-Philosophicus and the Theory of Types", a B.A. thesis submitted to Monash University, 1967).

2.11. The reason what is shown must be unsayable has been given by

Anscombe:

"If it were sayable, then failure to accord with it would have

to be expressible too, and thus would be a possibility"

(Anscombe, IWT p.166)

Her point is that the Necessary must be inexpressible, for otherwise its negation, the Impossible, would also be expressible, and that which is expressible or picturable is for Wittgenstein a possible state of affairs, and for the Impossible to be possible is absurd.

Following Griffin (WLA. p.23), I agree that the most central of the thingsthat are shown is logical form. As the TLP itself tells us:

"Propositions can represent the whole of reality, but they

cannot represent what they must have in common with~Ji-ty

in order to represent it - logical form.

"In order to be able to represent logical form, we should

have to be able to station ourselves with propositions

somewhere outside logic, that is to say outside the world."

''Wittgenstein, TLP 4.12)

However, Griffin's analysis does not go far enough. We must continue the ar~nt thus: language am the world mirror each other 12.

(4.121); like the phonograph record, the musical idea, the written notes, and the sound waves (4.014), they are isomorphic. The logical form of objects (particulars) determines what can be (either is, or is not) the case. Similarly, the ways in which the signs for particulars can be arranged in propositions determines what can be (truly or falsely) said to be the case. What is impossible is impossible because of the logical form of particulars; and the signs cannot be arranged thus because the result would not be a proposition, but nonsense. What is necessary is the negation of what is impossible; and the negation of nonsense is still nonsense. Thus we cannot say what must be so; it is shown by the ways in which we can (sensibly) arrange our signs, or in more modern jargon, by the rules for well-formed formulae (wffs). This itself is shown, for in trying to state such rules we should not be governed by the rules, we should have to station ourselves with propo­ sitions outside logic, and hence should utter only nonsense. (The impossibility of stating the rules for wffs is what is fundamentally wrong with the Doctrine of Showing, vide infra 2.15). As Wittgenstein himself put it,

"The rules of logical syntax must go without saying, once

we know how each individual sign signifies."

(Wittgenstein, TLP 3.334)

2.12. We can now, incidentally, explain the difference between •structure'

('Struktur') and form ('Form') which puzzled Ramsey in his review of the

TLP (reprinted FM) P• 271-2. •

"Form is the possibility of structure 11 (TLP 2.033) 13.

The structure of a proposition 'aRb' is that of a proposition expressing a dyadic relation. The form is that 'aRb' is a wff, or, that the structureO of atomic facts is such that a proposition

like this can be true or false ('structure• is typically ambiguous; there are words 1 structre1 1 , 1 structure2', etc., one for each type;

'structureo•, the structure of atomic facts (Sachverhaltea) is 'form•

( V • 4 • 1 22a ) • 'Form• is not typically ambiguous, but confined to the lowest type.).

2.13. We can now explain such matters as why Wittgenstein dismiss

Russell's Theory of Types so shortly (TLP 3.331 and f). What the Theory tries to say, that there are symbols of different Types, is shown.

Furthermore, the Antinomies are in Wittgenstein's view exp.;ressions like

'F(F(fx))', and thus are exluded as malformed formulae without needing any Theory. We can also see why, though there were three inkblots on the :paper, Wittgenstein would not agree that there were at least three things in the world (v. Russell, MPD p.116; cf. TLP 4.1272) for

' a x a y a z I is not a wff, and '... is in the world I is not a legitimate predicate (it is typically ambiguous; and as it cannot be

used to make a false proposition, it cannot be used to make a true one).

2. 14. Even from what has been said, it is clear that Wittgenstein's

position on general propositions, based on the Doctrine of Showing, is

embedded in a sophisticated and profound theory of logical form. We may state this position's explanation of general propositions thus:

According to the conventional usage of logical symbols, in which small letters from the beginning of the alphabet (•a•, 1b', •c•, 'd', with or 14.

without subscripts) are names of individuals (or, of objects of the lowest type under consideration in that context), small letters from the end of the alphabet (•x•, 'y', 'z', With or Without subscripts)

are variables ranging over these, and capital letters such as

'F', 'G', 'H' are first order monadic predicates. By alloWing subscripts, we allow the possibility that there are an infinite number of individuals. However, when I assert • (x)Fx', I assert the truth of all propositions of the form 'F2'. If there are a finite number of these, say three, this will be shown by the fact that there are only three names in the system which~ be arguments in 'Fi'. There are 'a •, 'b ' and 'c ', and '(x)Fx' asserts 'Fa.Fb.Fc'. Unlike Russell, we do not have to say •a, b, and c are all there •·- 'd I 'd I are'• This is shown by the non-existence of the names I 1 1 No other sign ('G', 'R', •~•, or 'F' itself) could be an argument of the function •~•, for the result would not be a wff, a proposition; it would be nonsense. (Cf. the common idiom for 'He knows all about so-and-so', as 'He knows so-and-so

from A to Z'; it would be senseless to ask whether anything comes

a:fter Z). Conversely, if an infinite number of signs

'b I 'a' I 'b' I 'c' I 'd' I •a 1 ' I •a2 ' I ... , 1 I • • • I ••• have been introduced, this too would have to be shown. ''What the axiom of infinity is intended to say would express itself in language through the existence of infinitely many names with different meanings. "

(Wittgenstein, TLP 5.535c) 15.

To give an example Wittgenstein himself used in explaining his

TLP views later on,

" ••• he said that the class 'primary colour I is 'defined by grammar•, not by a proposition; that there is no such

proposition as 'red is a primary colour', and that such a proposition as 'In this square there is one of the primary

colours I realzy is identical with the logical sum 'In this square there is either red or green or blue of yellow'"

(Moore, Wittgenstein's Lectures in 1930-33"

repr. in Moore, Papers p.297)

It is evident if you consider the problem in this way that there are infinite classes which are defined by grammar (i.e. logical form) in this sense; e.g. the class of natural numbers. (There are an infinite number of symbols each of which we would call the name of a natural number.) "If objects are given, then at the same time we are given all objects"

(Wittgenstein, TLP 5.524a)

2.15. Wittgenstin1s explanation of general propositions, as presented above, is a courageous and profound attempt to avoid Russell's fate of having to postulate general facts. It is not acceptable, because the inexpressibility of logical form, the foundation on which the whole structure rests, is a consequence even worse than general facts. The

Doctrine of Showing resigns Philosophy to despair of understanding 16. the Universe - all philosophical questions must be confessed to be mystical. This is the course courageously taken by Wittgenstein in the TLP. It is, nevertheless, an admission of defeat, of the failure of philosophy, and not a result with which any philosopher can remain content. (It was, of course, rejected by Wittgenstein in his later works.). It is also incorrect. We.£!!! establish rules for wffs. Such rules are not of course true or false; but only a Logical Atomist would believe that !:!!.Y expression, if neither true nor false, must be nonsense.

2.16. Wittgenstein's solution of the problem of generality, unsatisfact­ ory as the grounds for it ultimately are, is better than Ramsey's explicit solution. Although accepting Wittgenstein as his guide,

Ramsey apparently does not understand either his guide or the depth of the problem, and seems unaware of the force of Russell's argument that we must complete the 11st, and that this completion must involve another general proposition.

2.17. To quote his discussion of the topic at length:

"A proposition which expresses agreement and disagreement

with the truth-possibilities of p, q, ••• (which need not be

atomic) is called a truth-function of the arguments p, q, •••

Or, more accurately, p is said to be the same truth-function

of P, q, ••• as R is of r, s, ••• if P expresses agreement with the truth-possibilities of p, q, ••• corresponding by the substitution of p for r, q for

s, ••• to the truth-possibilities or r, s, ••• with which R expresses agreement. Thus 'p and q' is the same truth•:f'unction of p, q as 'r and s' is of r, s, in each case the only possibility allowed being that both the arguments are true. Mr. Wittgenstein has perceived that, if we accept this account of truth-functions as expressing agreement and disagreement with truth-possibilities, there is no reason why the arguments to a truth-:f'unction should not be infinite in number.

(Thus the logical sum of a set of propositions is the proposition that one at least of the set is true, and it is immaterial whether the set is finite or infinite. On the other hand, an infinite algebraic sum is not really a sum at all, but a limit, and so cannot be treated as a sum except subject to certain restrictions.) As no previous writer has considered truth-functions as capible of more than a finite number of arguments, this is a most important innovation. Of course if the argu- ments 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 way, as we can by using propositional

functions. ''A propositional function is an expression of the form 'fx', which is such that it expresses a proposition when any symbol (of a certain

appropriate logical type depending on f) is substituted for ' X"r •

Thus ':t is a man' is a propositional function. We can use propositional functions to collect together the range of propositions

which are all the values of the function for all possible values of x.

Thus 'x" is a man' collects together all the propositions'a is a man',

'bis a man•, etc. Having now by means of a propositional function 19. defined a set of propositions, we can, by using an appropriate notation,

assert the logical sum or product of this set. Thus, by writing

1 (x) .fx I we assert the logical product of all propositions of the form

'fx'; by writing '(IDc).fx' we assert their logical sum. Thus

'(x).x is a man' would mean 'Everything is a man'; ''1:x).x is a man',

'There is something which is a man'. In the first case we allow only

the possibility that all the propositions of the form 'x is a man' are true; in the second we exclude only the possibility that all the

propositions of the form •x is a man' are false.

"Thus general propositions containing 'all' and 'some' are found to

be truth-functions, for which the arguments are not enumerated but given

in another way. "

(Ramsey, FM p.7-8. Ramsey gives essentially the same account,

attributing it to Wittgenstein, in ''Matrematical Logic"

(repr. in EMetc., P• 73-4), and in "Facts and Propositions" (in FMetc.,

P• 152-5) •

2.18. All this is of course true; general propositions do express

agreement or disagreement with truth-possibilities (i.e. with how things

do (or do not) stand in fact} or in other words, general propositions

state that all propositions of a certain form are true (and mutatis

mutandis for existential propositions). The question is how they do

this, particularly in the case where enumeration is impossible (because

infinite). Russell admitted that he could not answer this, except by

supposing that there are irreducible general facts which are meant by 19.

general propositions. Wittgenstein answered it by appealing to his

Doctrine of Showing. Given the truth of this Doctrine, Wittgenstein's

answer is satisfactory; the problem lies in his premiss; the Doctrine

of Showing is itself unacceptable, because it is incorrect.

Ramsey, however, apparently answered the question by saying

'TheYjust do'. This is not a sufficient answer from a philosopher.

(For some further rema.kks on Ramsey's explicit solution, vide infra 4.11).

Ramsey was led into this situation because of his tendency to j discount limitations as merely due to lack of human ability. (We

shall see this tendency causing him trouble on other points too.) On

the matter in hand, it leads him to discount the impossibility of

infinite enumeration:

"••• owing to our inability to write propositions of

infinite length, which is logically a mere accident.

(♦).♦ a cannot, like p.q, be elementarily expressed, but

must be expressed as the logical product of a set ••• II

(Ramsey, FM p.41)

This inability is not a mere accident logically. An infinite number

which could be completely enumerated would no longer be infinite; or as

Wittgenstein h!mself put it after he had rejected Logical Atomism:

"The fact that we cannot write down all the digits of it

is not a human shortcoming as mathematicians sometimes

think"

(Wittgenstein, PI s203 p 83) 20.

In the light of this, there seems to have been scmething lacking in Ramsey's understanding of infinity; which perhaps helps to explain his later (1929) conversion to finitism.

The only argument that Ramsey gives explicitly for his belief that the list is complete is as follows:

"The second objection that will be :made is more serious; it will be said that this view of general propositions makes what things there are

in the world not, as it really is, a contingent fact, but sOJRething presupposed by logic or at best a proposition of logic. Thus it will

be urged that even if I could have a list of everything in the world •a•, 'b', ••• 'z', 'For all x, fx' would still not be equivalent to

'fa, fb ••• fz', but rather to 'fa, fb ••• fz and a, b ••• z are everything'. To this Mr. Wittgenstein would reply that •a, b ••• z

are everything' is nonsense, and could not be written at all in his

improved symbolism for identity. A proper discussion of this answer

would involve the whole of his philosophy, and is, therefore, out of the question here; all that I propose to do is to retort with a tu

quoque! The objection would evidently have no force if 'a, b • • • z are everything' were, as with suitable definitions I think it can be made to be, a tautology; for then it could be left out without altering the meaning. The objectors will therefore claim that it_is not a tautology, or in their terminology not a necessary propositions; and this they will presumably hold with regard to any proposition of the sort, i.e. they will say that to assert of a set of things that they are or are not everything cannot be either necessarily true or necessarily

false. But they will, I conceive, admit that numerical identity and

difference are necessary relations, that 'There is an x such that

fx' necessarily follows from 'fa', and that whatever follows necesea-

rily frClll a necessary truth is itself necessary. If so, their

position cannot be maintained; for suppose a, b, c are in fact not

everything, but that there is another thing d. Then that d is

not identical with a, b, or c is a necessary fact; therefore it is

necessary that there is an x such that x is not identical with

a, b, or c, or that a, b, c are not the only things in the world.

This is therefore, even on the objector's view, a necessary and not

a contingent truth. "

(Ramsey "Facts and Propositions"(FM)pp.154-5)

This argument is invalid. If a, b, and c are in fact not everything, then it necessarily follows that there is a fourth thing. But since our premiss was contingent ( "in fact"), the conclusion itself is contingent, not necessary••• This is the fallacy to which G.E. Moore was objecting in his contribution to the same symposimn:

"He tries, instead to retort to it with a tu guogue. In this

retort, however, he makes a step, of which I, at least, should

deny the validity. He supposes that if the objector admits

(as I should admit) that numerical difference is a necessary

relation, he is bound also to admit that, suppose a, b, c

are not everything, but there is also another thing d, then

that d is not identical with a, b or c is a necessary fact. But I should hold that, though numerical difference is 22.

a necessary relation, yet, in the case supposed, that d is

other than a is not a necessary fact. For numerical

difference is a necessary relation only in the sense that if

a and d both exist, then a must be other than d.

But to say that 'a is other than d' is a necessary

fact would entail besides that •a exists' is necessary,

and that 'd exists• is necessary, which I should deny."

(G.E. Moore, "Facts and Propositions"

Aris. Soc. Supp. 7, p.206)

2.19. Ramsey•s position on generality is however, strengthened if

we consider in conjunction with it his treatment of the Axiom of

Infinity, to which we therefore now turn. In PM Whitehead and

Russell wished to show that mathematics followed from propositions

of logic (which must be accepted to be true) by logically impeccable

steps, and thus that mathematics was true. They did not wish to show

that mathematics is true only if there happens to be an infinite number

of things in the world. But since for some areas of mathematics this

was all that they could prove, they included as one of their axioms

the proposition that such is the case. (This was of putting their

position is perhaps unfair to Whitehead and Russell, but it helps to

emphasize the difficulty that some critics, including Ramsey and indeed

Russell himself, felt in accepting the Axiom of Infinity, and thus why

they wished to dispense with it if possible. Russell relatea how he

tried to prove the Axiom of Infinity (by taking classes and classes of classes•••••• together), IMP p.135). Ramsey thought that

" ••• mathematics ••• might reasonably be supposed to require

an Axiom of Infinity"

(FM p.56)

"on my system••• the Axiom of Infinity asserts merely that

there are an i infinite number of individuals"

(FM p.59)

Ramsey is being careless here. His mentor Wittgenstein did not say that the Axiom of Infinity asserted this, but that:

''What the axiom of infinity is intended to say would express

itself in language through the existence of infinitely many

names with different meanings"

(Wittgenstein, TLP 5.535c)

For Wittgenstein, the Axiom of Infinity does not assert anything, what it is intended to assert would be shown by the structure of language. With this view Ramsey agrees, (FM p.60, quoted below (2.20)). Raasey argues:

"Let us start with 'there is an individual', or writing it

as simply as possible in logical notation.

'( 3:x).x-.x'

"Now what is this proposition? It is the logical sum of the

tautologies x • x for all values of x, and is there-

fore a tautology. But suppose there were no individuals, and

therefore no values of x, then the above formula is abso- 24. lute nonsense. So, if it means anything, it must be a tautology.

"Next let us take 'There are at least two individuals' or

t ( :ii X, y), X f YI • This is the logical sum of the propositions x r Y, which are tautologies if x and y have different values, contradictions if they have the same value. Hence it is the logical sum of a set of tautologies and contradictions; and therefore a tautology if any one of the set is a tautology, but otherwise a contradiction. That is, it is a tautology if x and y can take different values (i.e. if there are two imividuals), but otherwise a contradiction.

"A little reflection will make it clear that this will hold not merely of 2, but of any other number, finite or infinite.

That is, 'There are at least n individuals' is always either a tautology or a contradiction, never a genuine proposition. We cannot, therefore, say anything about the number of individuals, since, when we attempt to do so, we never succeed 1n constructing a genuine proposition, but only a formula which is either tautological or self-contradictory.

The number of individuals can, in Wittgenstein's phrase, only be shown, and it will be shown by whether the above formulae are tautological or contradictory."

(Ramsey, FM p.59-6o) 25.

(It is historically interesting that this seems to be exactly the same argument that gave Wittgenstein his insight into the Axiom of Infinity, before the Doctrine of Showing had been fully elaborated; v.Nbks 9/10 114

(p.10), 13/10/114 (p.11) and TLP 5.535). From this statement it becomes evident that Ramsey too held a doctrine of showing, though a rather simpler one than Wittgenstein's. According to Ramsey, the

Axiom of Infinity is a tautology, which, like all tautologies, is shown to be true. (On p. 61, he admits that there is a possibility that it is a self-contradiction, a possibility which if realized would render mathematical analysis self-contradictory. However I do not think that Ramsey needs to admit this possibility, as I hope to show

in connection with his Theory of classes, vide infra, 3.07). This

Ramseian doctrine of showing also shows why Ramsey said (in "Facts and Propositions" (FM) p.154} that the statement that the list is complete is a tautology (and if it is a tautology, it adds nothing to the meaning of the general statement}, and that a tautology is a tautology is shown by the truth-possibilities of the world. If p

is an atomic fact, i.e. if 1p 1 is a true atomic proposition, then p and ~ p are both truth-possibilities. This explains clearly why

"Contradiction is the outer limit of propositions; tautology

is the unsubstantial point at their centre"

(Wittgenstein, TLP 5.143c}

a view which Ramsey and Wittgenstein share. In a tautology all truth-possibilities are agreed with; in a contradiction, none are. 26.

A tautology is equally balanced between the possibilities; either p or ~ p might be the case, it is neutral between them. With a tautology, we are neutral between lli such pairs of possibilities, in the centre. A contradiction, on the other hand, rules out both p and ~ p; it excludes all possibilitjes, it is the limit. "Tautology and contradiction are the limiting cases - indeed

the disintegration - of the combination of signs"

(Wittgenstein, TLP 4.466d)

"It ['p or not-p' ] should be regarded not as a

significant sentence, but as a sort of degenerate case

(footnote - In the mathematical sense in which two lines or

two points form a degenerate conic) and is called by

Mr Wittgenstein a tautology"

(Ramsey, F and P (FM etc.) p.151)

Wittgenstein analysed truth-possibilities further (a truth­ possibility for him is what can be said to be the case by a proposition, and what can be said is shown by the way names can be arranged in propositions, i.e. in wffs) and thus arrived at his Doctrine of

Showing. Ramsey did not analyse the notion of truth-possibility as far, and thus his doc~rine of showing is simpler (it is implied by, but does not imply, Wittgenstein's) and does not give him an automatic solution to the Reflexive Antinomies as Wittgenstein's Doctrine had

(vide supra, 2.13), and so he had to work out a new one. This was a good thing, because it resulted in some of Ramsey's best work (vide infra, chapter 4). 27.

2.21. Incidentally, it is important for Ramsey's theory of truth­ possibilities that the Decision Problem be solvable in the general case, as such a solution would enable us to exhibit the truth­ possibilities and "show" that tautologies are tautologies {vide

Wittgenstein TLP 6.122). This is why

"Ramsey's profound disagreement with Hilbert •s doctrine

of mathematics as a game with meaningless marks did not

prevent him from giving a good deal of attention to the

formalist's chief problem - that of finding a general

procedure for determing the consistency of a logical formula

( the Entscheidungsproblem) ••• "

(R. B. Braithwaite, in his

''Editor's Introduction" to Ramsey's

FM, p.xii)

2.22. We have not yet finished with the Axiom of Infinity, but we have got far enough to outline our reconstruction of Ramsey's answer to the question 'how do general propositions, i.e. propositions of the form "(x)fx", have meaning?'. Ramsey's answer is that a proposition of that form expresses agreement with all truth-possibilities of that form. Since we can express agreement with all truth-possibi­ lities whatever, with a tautology, we must be able to express agreement with a subset of these, viz. all truth-possibilities of a certain form; and this will hold even if that subset is infinite. 28.

2.23. This solution of the problem of generality, though an interest-

ing suggestion, is no more acceptable than Wittgenstein's O'Wll. With

a tautology, we do not so much express agreement with all possibilities,

but rather fail to express disagreement with any. For as Wittgenstein

saw

11 In a tautology the conditions of agreement with the

world - the representational relations - cancel one another

out, so that it does not stand in any representational

relation to reality."

(TLP 4.462b) "\ < A white shee·t; of paper is not a picture of everything, but a picture of nothing; and the same goesfor a black sheet of paper (cf. Anscombe's , j metaphor of the map, IWT p 75-6). A tautology does not express

agreement, but rather fails to express disagreement, with all truth-

possibilities. But a general proposition certainly expresses agreement

with all truth-possibilities of certain form. Ramsey's solution is

therefore wrong.

(vide infra, 4.17).

2.24. In showing what Ramsey•s theory of generality was, how it

differs fram Wittgenstein's, and why it is not satisfactory, we have

been led to consider many other matters. This is to be expected when

studying the work of a metaphysician who builds his structure with

logic; everything is interrelated, making exposition difficult. We

shall now look at the system from a different perspective. This will give, among other things, more illumination on Ramsey's treatment of the

Axiom of Tnf'ini tv: hnt it~ msi in l"nnl"P,..n m 71 h"" +h.,,. Avi,....,., ,-,-f' rh,-,.i ,..,,. 29

CHAPI'ER III IDENTITY, CLASSES and the AXIOM of CHOIC~

3.01. Following Wittgenstein, Ramsey rejected the definition of

Identity in PM (*13.01):

11x=y. _-df 0. • (.t.\.., ; .....t.l x. => ...... t.1 y II

He also rejects (p.30) PM's claim (Introduction Ch II s vi, pbk edn

P• 57; *12 (p 167); and *13.101 (p.169) ) that this definition

depends on the Axiom of Reducibility, using the same method he uses

for removing the Axiom of Reducibility itself (We shall consider this method later, in chapter 4). The objection to the PM definition of identity shared by

Wittgenstein and Ramsey is perhaps most concisely stated in the TLP:

"5.5302 Russell's definition of'=' is inadequate, because

according to it we cannot say that two objects have all their

properties in common (Even if this proposition is never correct,

it still has sense)

115.5303 Roughly speaking, to say of two things that they are

identical is nonsense, and to say of one thing that it is

identical with itself is to say nothing at all. 11

(TLP)

3.02. Wittgenstein derived this from his Doctrine of Showing.

''What .£!ill be shown cannot be said

"I believe that it would be possible wholly to exclude the

sign of identity :fran our notation and always to indicate identity merely by identity of the signs (in certain circ- 30.

umstances) ••• ,

(Nbks 29/11/ 114 (p.34e))

"The symbol of identity expresses the internal relation

between a function and its argument; i.e.

ta == ( a: x). ~. x=a"

( "Notes to Moore", in Nbks p.116)

(This would perhaps be more perspicuous if' rewritten 'The symbol of identity expresses the internal relation between a function am its argwmnt; i.e. in "( a: x).~.x=a", the identity sign expresses the internal relation between the function ♦ and the argument a•)

"It [ 1 (x) x=x •] doesn •t describe reality at all, and

deals solezy with symbols; and it says that they must

symbolize, but not what they symbolize"

( "Notes to Moore", Nbks p. 116)

"It is impossible, however, to assert by means of'

propositions that such properties am relations exist:

rather, they make themselves manifest in the propositions

that represent the relevant states of affairs and are

concerned ,r.i. th the relevant objects."

(TLP 4.122d)

From 4.122a,b, it is clear that the internal relations and internal properties of a fact are its logical form; _and logical form is that which is shown, and which cannot be said. (vide supra, 2.11 - 2. 14).

Further; 31.

"It -would be just as nonsensical to assert that a proposition

had a formal property [or relation] as to deny J.v•.J... • "

(TLP 4.124b)

In other words, for Wittgenstein, internal relations are shown, are made manifest, but are inexpressible.

In a logically perfect language, 'a=b' would be nonsense;

'a=a. I would say nothing. The same applies to •afa• and 1afb 1 respectively. (cf. 5.5303) • Neither the identity sign '=' nor the diversity sign 'r' can be used in any language to~ anything. They appear when we use two signs with one and the same meaning (which

is an imperfee1;ion, a redundancy, in the language).

"5.534 And now we see that in a correct logical notation

pseudo-propositions like 'a=a', 'a=b', 'a=b.b=c.::a.=c',

'(x).(x=x)', 1 (3:x).x=a', etc. cannot even be written down."

(TLP)

Wittgenstein's views on identity may be summarized thus. The

logical form o~ an object, or of a fact, is shown. What can be shown

cannot be said. What is shown is logical form, the comition for the

possibility of the structure of facts. The logical structure of facts

is mirrored in the logical structure of propositions (both true and false).

When we try to state logical form, we have a sequence of signs which is

neither true nor false. It is nonsense, it is not well-formed. The

internal properties am the internal relations of a fact make up the

logical structure of that fact. The sign for identity 1s used when

stating the internal relations of facts, and for resolving converse- 32. ambiguous signs. But internal relations are shown; in a correct conceptual notation there will be no ambiguity or converse-ambiguity.

Therefore, the sign for identity is superfluous. So we should observe conventions which avoid the use of this sign. Wittgenstein gives examples of how this might be done in 5.531.5321. Wittgenstein, and Ramsey, realized the convenience of '=' for "purely sumbolic" purposes (Russell), i.e. those merely to do with signs (Wittgenstein -

4.2412) and retained it in practice.

Ramsey's simpler theory of showing, in terms of truth-possibi­ lities (vide supra, 2.20), could be used to justify the elimination of the identity sign, as in Wittgenstein; for all propositions of the form 'x=x • must be true (agree with all truth-possibilities) while

(unless the language we are using is unnecessarily misleading by having two signs for one object) all propositions of the form 'x=y' must be false (agree with no truth-possibilities). However, Ramsey gives three other arguments to support his criticism of the PM definition of identity. (They all occur on p.31-2)

3.03. The first is that the PM definition makes it self-contradictory for two things to have all their elementary (and by his argument on p.30, 'all' without qualification) properties in common.

"Yet this is really perfectly possible, even if in fact,

it never happens."

(FM p.31) 33.

This is a rejection of the Principle of the Identity of

Indiscernibles, which was first explicitly formulated by Leibniz

(e.g. Monadology ix). Indeed Ramsey is rejecting more than the

Leibnizian Principle, because for Leibniz the discernibility had to be monadic (because of Leibniz's whole-hearted acceptance of

Aristolelean-type subject-predicate logic, and his consequent denial

of the reality of relations). By contrast, the Logical Atomists were firmly committed to belief in relations or polyadic predicates.

This was because, first, they rejected Traditional Logic and embraced

the new Mathematical Logic, which allowed relations, and secondly they were reacting against Bradley and his followers, who deduced the non-existence of relations from Traditional Logic just as Leibniz had

(One suspects that Russell's term 'monadic relation' for 'quality', though avowedly "to avoid circumlocution" (PLA p.198) perhaps owed

something to a desire to shock the Bradleians). Ramsey was denying the Identity of polyadic Indiscernibles, viz.:

If the thing we call 'A' has a certain set of qualities, and a thing we call 'B' has exactly the same set of

qualities, and moreover if there is any relation which A

has to anything in the universe, then B has that relation

to that thing, and has no other relations not shared by A, then the names 'A' and 'B' denote the same object.

If we accept, as both Ramsey and Wittgenstein did, the thesis of

extensionality, one of the properties of A which B must share is 34. the property of being called I A'; and if B is called 'A' and vice versa, it seems difficult to see how 'A' and 'B' can fail to denote the same object.. Indeed, even a God who knows by immediate intuition could not distinguish them, for such knowledge would also give a distinguishing property - 'known by God to be different from A'.

Of course this argument is merely ad haminen; but we are here concerned with Ramsey's Logical Atom.ism, not with the ultimate truth on all matters on which he had an opinion. An opinion may be true or false, but an inconsistency in a system is a fault in that system.

Let us consider Ramsey's explicit arguments against the Identity of

Indiscernibles.

3.04. Although the argument in the following passage

11 ••• there is nothing self-contradictory in a having any

self-consistent set of elementary properties, nor in b

having this set, nor therefore, obviously, in both a and

b having them, nor therefore in a and b having all their

elementary properties on common. Hence, since this is

logically possible, it is essential to have a symbolism

which allows us to consider this possibility and does not

exclude it by definition. 11

(FM p.31) contains a true premiss, viz. that anything may have any self­ consistent set of elementary properties, we need not accept its conclusion. Black argues: 35.

"But Ramsey here seems to commit the fallacy of composition,

or something like it. Given any elementary property, it

is possible that a and b might both have that property,

but it does not follows that a and b might have all

elementary properties in common. The inference from

(P) ◊ (Pa ■ Pb) to ◊(P)(Pa ■ Pb) needs a special

defence. (cf. the argument from 'For any given ticket in

a sweepstake, it is possible that it might win first prize'

to 'It is possible that all the tickets in the sweepstake

might win first prize 1 ) • "

(Black, CWT p.292)

Ramsey does not give any defence of his inference from 'For all, it is possible' to 'It is possible that, for all', which is in general not valid. Nor do I think that he could give such a defence in this case. Just as in the converse case, for all individuals it is possible that they should be ♦, it does not follow that it is possible that all individuals should be •.

"There is no sort of point in a predicated which could not

conceivably be false." (Russell PIA p.241)

Such a predicate would be senseless, in Wittgensein's terms (and hence in '(x) ♦ x v ~ , x 1 , the predicate function '<>x v ~ • x • lacks sense). Against Ramsey, we argue: if a and b have all their elementary properties in common, they must have all their properties in common, and properties include polyadic relations. This would mean that any proposition true of a would be true of b, and vice versa, 36. and mutatis mutandis for false propositions. In this case the names

•a• and 'b' meet the most rigorous requirements for synonymity.

We could not fall into error by identifying them. To fail to identify them, to think that there might be two things when we could not (logi­ cally could not, ex hyp} utter any false propositions by assuming there to be only one, would be to ignore Occam's razor. Two synonymous logically proper names denote the same thing.

3.05. Ramsey's two other arguments (also on p.31-2) do not help him either. The first is that the objection that one cannot know of any two particular indiscernibles holds no force, because one can still consider the possibility. He gives an ingenious argument by analogy:

"since there are more people on the earth than hairs on

any one person's head, I know that there must be at least

two people with the same number of hairs, but I do not

know which two people they are. 11

(FM p.31)

This argument is also invalid. It is again Ramsey's tendency to discount all impossibilities as due to merely human limitations. We do not know which two people have the same number of hairs because of the lack of appropriate statistics. This indicates a deplorable lack of industry by our social scientists, but nothing more. The situation presents no more difficulty than the fact that we know that the number of English husbands is the same as that of English wives, without knowing how many there of either. The identity we are talking about is logical indiscernibility; the necessary identity of indiscerneds has 37.

little plausibility. The Principle of the Identity of Indiscernibles states that two things must be identical if they cannot (logically

canno~ be distinguished; not that they must be identical if we frail humans happens to have not (yet) distinguished them. Once again,

Ramsey•s conflation of what cannot be done without self-contradiction and what cannot be done because we are mortal imperfect beings leads him into trouble.

3.06. Nor does Ramsey's third argument (p.31-2) give us a good

reason for rejecting the PM definition of identity. This is that the

identity sign can be replaced by conventions, a discovery due to

Wittgenstein (TLP 5.53.534) This proves nothing; it is like the

discovery that the negation sign can be replaced by conventions, by

Ramsey himself two years later ( "Facts and Propositions 11 (1927)

(repr. FM) p.146-7; contrast TLP 5.02b). Wittgenstein's discovery merely shows that we could manage (though with great practical

difficulty) to do without an identity sign. But this was obvious

from the beginning, for if it were not so, PM could never have defined

identity in the first place.

The denial of the Identity of Indiscernibles has some curious consequences, which Ramsey does not explicitly mention, for the theory

of classes. It renders every class, even one given by enumeration,

potentially infinite. This is because even if we give all the

properties of a class-member, and all its relations to other things,

there may still be things indiscernible from it which we have included 38.

in the class without realizing it. It is possible that this is the

explanation for Ramsey's curious theory of classes, although I think

there are more important reasons, as will became plain (3.oB-3.14).

But we should note in passing that given the non-identity of

indiscernibles, and the consequent potential ini'inity of any class, it

would be impossible to disprove the truth of the Axiom of Infinity

(Ramsey himself apparently did not realize this - v.p. 59-61). For

even the non-existence of an infinite number of names would not show

the Axiom of Infinity to be self-contradictoriJ: we could always

rope in a few indiscernibles, and if we rope in an infinite number of

them, the Axiom is tautologous. The only trouble with this is that

there seems to be no need to introduce indiscernibles to our system;

a move which would give Occam reason to wish his razor had been an axe.

Our difficulty in coping with logically indiscernible entities has

nothing to do with human limitations.

3.08. To return to Ra.msey•s theory of classes (and of relations in

extension, which as he says, is strictly analogous). Ra.msey's theory

may be stated tuite simply. He opposes the view that:

"•. ■ a class can be given to us either by enumeration of

its members, in which case it must be finite, or by giving a

propositional function which defines it. So that we cannot

be in any way concerned with infinite classes or aggregates,

if such there be, which are not defined by propositional

functions (footnote: for short I shall call such classes

'indefinable classes•). But this argument contains a common 39.

mistake, for it supposes that, because we cannot consider a

thing individually, we can have no concern with it at all.

Thus, although an infinite indefinable class cannot be

mentioned by itself, it is nevertheless involved in any

statement beginning 'All classes' or 'there is a class such

that', and if indefinable classes are excluded the meaning

of all such statements will be fume.mentally altered."

(FM p.22)

On this question Ramsey is rather unfair to the authors of PM.

An indefinable class is only included in such statements if our defi­ nition of ~lass I allows it. The definition of class in PM

(*20 (pbk p.187f)) includes only definable classes, as Ramsey agrees

(FM p.23). Thus when we see a statement in PM beginning 'All classes', this means, in Ramseian language, 'All definable classes'.

3.09. Ramsey's mistake here is elementary: he takes a defined term, gives a new, different definition of it, and then complains that state­ ments interpreted by the old definition are not equivalent to the same statements interpreted by the new definition. Of course they are not equivalent; one would be surprised of they were. The question to consider is rather which definition is better.

3° 10. Ramsey's argument that his definition is better than that of PM

is found on p.23 of FM. He actually states it for the case of relations in extension. 40.

"Consider the proposition 'x( ell x} sm x( '!.' x} '; (i.e. the

class defined by ell x has the same cardinal as that

defined by '!.' x}; that is defined to mean that there is a one-one relation in extension whose domain is x{CII x}

and whose converse domain is x(! x}. Now if by relation in extension we mean definable relation in extension, this

means that two classes have the same cardinal only when

there is a real relation on function f(x,y} correlating

them term by term. Whereas clearly what was meant by Cantor,

who first gave this definition, was merely that the two

classes were such that they could be correlated, and not that

there must be a propositional function which actually correlated

them. Thus the classes of male and female angels may be

infinite and equal in number, so that it would be possible to

pair off completely the male and the female, without there

being any real relation such as marriage correlating them."

(p.23)

3.11. The crux of this argument is the meaning of 'real'. Ramsey may mean 'real' as opposed to •artificial'. We feel there is something artificial about the alleged common property shared by Homer, Albert

Einstein, Bucephalus, the square root of minus one, and Mrs. Smith's kitchen table; the common property being that I have just mentioned

their names. Similarly, there is something artificial about pairing

off the angels, male and female, in order of {say) their creation, the oldest male with the oldest female and so on (other possibilities may be: in the order of their favour in the sight of God, in the order

in which Satan tried to tempt them, or in al~habetical order). These

relations are more artificial than the relation of marriage. If this

is what Ramsey means, his argument merely establishes that we must admit

artificial functions. But that is no criticism of PM, where artificial

functions occur continually (the best example not in symbols is when

the authors suggest defining the class of great generals by the dis­

junction of the exact instants of their births (Introduction, Chapter

II, Svi pbk, p.56)). Nor does Ramsey's argument establish that we

must admit indefinable classes and relations, for all the artificial

functions we have to admit so far are definable.

On the other hand, Ramsey may be using 'real' in an extensionalist

way, in accordance with his professed policy. In this case 'real

relation' means 'any dyadic function'. Such a function may or may not

be known to us. But it is very certain that before two classes _£!!!

be correlated taere must exist a correlating function known to, or

definable by, whoever is correlating them. Two classes may, of course,

be such that they could be correlated, though nobody has ever defined

a function which correlates them. But to say that they could be

correlated is to say exactly that such a function is definable.

(What could be differs from what is, definable differs from defined).

Thus under this interpretation too Ramsey has not established indefinable relations. 42.

Ramsey here has again neglected the difference between human and

lobical limitations. Two classes may be correlatable even though no

person can define a correlating :f'unction; but to say that they can

be correlated is to say that there is definable a :f'unction which

correlates them. These two sorts of 'can' are different, but Ramsey

apparently does not realize this, for he continually confuses them.

(vide supra, 2.18)

3.12. Apart from criticizing Ramsey's argument that his definition is

better, we should if we can advance positive arguments of our own

against his position. There are two.

The first, as is fitting when defending Russell, is an Occamist

argument. Russell did not need to postulate an indefinite range of

indefinable classes (he could not do without any, see below on the

Axiom of Choice, 3.13) and since Ramsey agrees that they cannot be

mentioned by themselves, what reason is there for postulating them?

If we are worried about the propriety of saying 'All classes' when we

mean what Ramsey would call 'All definable classes', we may always

qualify 'class' by 'definable'. The avoidable prolixity of this course

is evident.

3.13. The second argument which tells against Ramsey's proposed new

definition of 'class' is that it makes the Multiplicative Axiom a

tautology (FM, p.24, p.59). The Multiplicative Axiom of PM (*38, vol. I p.536r; not•in pbk.) is equivalent to Zermelo's Axiom of Choice,

by which name it is better known. Ramsey thinks "••.mathematics is largely independent of the Multiplicative

Axiom.•• II (FM p.56) 43. a judgement with which few subsequent mathematicians would concur.

11 • • • fundamental and general theorems and methods in the

theory of sets, as well as in analysis, algebra and topology

are based on the Axiom of Choice in the sense that we do

not know a way of avoiding its use 11

(Fraenkel and Bar-Hillel, FST

p 6o, their emphasis)

Ramsey's theory of classes makes the Axiom of Choice tautological in the

following way. The Axiom of Choice allows us, under certain circumstances,

to form a class which we could not otherwise define. (vide e.g. Rubin

and Rubin, EAC, p.5). In fact, it admits certain indefinable classes.

It is indeed this fact which has made it so unpopular with mathematicians,

some of whom go through extraordinary contortions to avoid its use

wherever possible. Ram.sey's admission of all indefinable classes

naturally implies the tautologousness of the Axiom of Choice, whose

purpose is to admit some of them; but for that very reason it cannot

be regarded as an improvement. Just because there are some diseases,

or indefinable classes, which we must admit are incurable, we do not

accept that all diseases, and indefinable classes, are unavoidable

and close our hospitals.

3.14. Finally, on the subject of Ramsey's proposed definition of

'class' to include 'indefinable class', we may point out that on this

matter too he disregards the limitations of logic. The indefinable

classes he talks about not merely have not been defined, but can not 44. be defined (except perhaps by infinite enumeration; he specifically excludes this as impossible in this connection, (FM p.22), though he says elsewhere that its impossibility is a mere accident, logically speaking (FM, p.41)). Once again Ramsey's fatal tendency of assimilating all impossibility to human shortcomings leads him astray.

In this chapter I have examined Ramsey's rejection of the

Identity of polyadic Indiscernibles, and decided that it will not do, noted the consequences of his view for the theory of classes and the

Axiom of Infinity, described his theory of classes, and rejected the proposed new definition on which it is based. 45.

CHAPTER IV

TYPES

4.01. The most important criticism Ramsey makes of PM is concerned with Russell's Theory of Types. This Theory was introduced by Russell to solve certain paradoxes known as the Reflexive Antinomies, the first of which was discovered by Russell in 1901 (IMP, p.136) and communicated to Frege in the following year (v. Appendix to

Grundgesetze vol. II, repr. TPW, p.234f). This first paradox is that of the class of all classes which are not members of themselves; is this class a member of itself or not? Clearly neither possibility is satisfactory.

4.02. Other paradoxes were invented, or discovered in previous literature (including the Bible) which Russell judged to be similar.

He lists seven in "Mathematical Logic as Based on the Theory of Types"

( =MLT - first published in the .American Journal of Mathematics 1908, repr. Land K). The similarity Russell notices was that each of the paradoxes turns on a kind of self-reference (hence the name,

'Reflexive Antinomies 1 ). To prevent such vicious self-reference,

Russell laid down the Vicious Circle Principle:

"Whatever involves all of a collection must not be one of

the collection."

(MLT (LK) p.63) and on this erected the Ramified Theory of Types. To give a necessarily very brief and inadequate account of this Theory, we may 46. say that it distinguishes objects into types, in a hierarchy beginning individuals, classes of individuals, classes of classes of individuals, .. . ,. and :further, because of the inter-relation between classes and propositions, propositional functions, etc., distinguishes these too into types. A function of the lowest possible type compatible with its arguments is called a predicative function or matrix; and the only functions which may be quantified over are predicative functions (vide e.g. MLT, P•75f; and PM, Introduction, Chapter II; *12; and *20). A "type" is a level of this hierarchy; it is on the structure of this hierarchy that the metaphysics of Logical Atomism was constructed.

This theory eliminates the Antinomies, but unfortunately it also eliminates almost all mathematical analysis (it at .first appeared to eliminate also mathematical induction. Whitehead and Russell showed that this is not so in the second edition of PM, (Introduction to the second edition pbk p xliv; Appendix B *99 (not in pbk)p.650f)). Such consequences had been foreseen by Russell in the paper in which he first introduced the Theory of Types (MLT p.80-1). To save mathematics from these undesirable side-effects of his Theory he proposed the Axiom of Reducibility, which would restore the lost parts of mathematics without readmitting the paradoxes.

4.03. And to be sure, it does both these jobs; but most people feel first that it is not "self-evident" (with which PM agrees, p.59-60); and secondly that it is not "logical" in the sense in which mathematics is supposed to be being deduced from "logic"• This latter objection gained force with Wittgenstein's claim that the propositions of logic are whelly tautological (TLP 6.1), a claim accepted by Russell during the

War (IMP p.203-5. For Wittgenstein's own Theory of Types see above

2.10-2.13, and my "Wittgenstein's Tractatus Logico-Philosophicus and

the Theory of Types")•

4.o4. The Axiom of Reducibility is the claim that, for any function ti, of whatever type, there is a formally equivalent function of some

type fixed relative to the (known) type of the argument(s) of tx(PM p.58).

The Axiom is formally introduced in FM *12 in the two forms:

''*12.1 (a f): c:,x= • f!x Pp I-: X

"*12.11 f): c>(x,y) f:(x,y) Pp" 1-: ( a =x,y •

(PM *12, pbk p.167; vide quoque PM Introduction Chapter II, p.55-60)

That is to say, the authors assumed that for any function (actually of

only one or two variables, though as they remark (PM p.167) if

relations of more variables were indispensible for the purpose in hand,

further corresponding assumptions would have to be made for them) there

is a formally equivalent matrix or predicative function (written

'f!x',etc.) of those variables. 43.

4.05~ Now this is not a rule of logic in any sense; on the contrary, if it were true, it would seem rather to be a fortunate coincidence, as Wittgenstein remarked (TLP 6.1232). It is

"equivalent to the assumption that any combination or

disjunction of predicates (given intensionally. If given

extensionally (i.e. by enumeration) no assumption is

required; but in this case the number of predicates must

be finite) is equivalent to a single predicate"

(PM Introduction Chapter II Svi

p.59).

From this statement of the Axiom Ramsey has no difficulty in showing it to be neither a self-contradiction nor a tautology (FM p.57).

Even in the first edition of PM it was realized that the Axiom of Reducibility is not satisfactory:

" ••• it is by no means improbable that it [the Axiom of

Reducibility] should be found to be deducible from some

other more fundamental axiom. It is possible that the use

of the vicious circle principle, as embodied in the above

hierarchy of types, is more drastic than it need be, and

that by a less drastic use the necessity for the axiom

might be avoided. "

(PM Introduction Chapter II p.6o) 4.06. The situation after the publication of the first edition of PM

(1910-13) seemed to be a straight choice: either you accept the

Axiom of Reducibility, or resign yourself to the loss of analysis and induction. Neither horn of this dilemma is very inviting, and even after induction had been saved (PM (2nd edn.) *89) it was obvious that something had to be done about it.

4.07. Ramsey's first move was to prune the banyan-like Ramified

Theory of Types. Because of the difficulties in quantifying functions,

Russell's Theory had involved not only functions of types O, 1, 2, 3, •••

(the Simple Theory), but also functions of types (o,o), (0,1), (1,1), ••• (o,o,o,), (0,0,1,), ••• (The Rami~ied Theory.

See also Quine, ML p.163). Chwistek had previously (in 1921) described the simple Theory (in an article in Polish v. Chwistek, LS

P• 152) •

Ramsey distinguished two groups of antinomies (p.20; p.27).

The second group, since called the "semantic", are not part of mathematics at all, but problems of all language. The first group, now commonly known as the "syntactic", can be outlawed by the Simple

Theory of Types (i.e. which recognizes only types adequately represented by the ordinal numbers alone), and the additional complications of the

Ramified Theory are thus avoided.

If Ramsey were able to do this successfully, it would be a greater gain than it might at first seem, because the Theory of Types 50.

is very important in the metaph7sics of Logical Atomism, and whereas

some people find the Simple Theory intuitively acceptable, nobody

regards the Ramified Theory as anything but a metaphysical nightmare.

Also, it is possible to formulate the Simple Theory without mentioning

the 'meanings' of symbols (as Church did in ''Aformulation of the simple

theory of types" JSL Vol. V No.2 (19li-O) p.56f), thus reinforcing the

Logistic Thesis that mathematics is logic, and can be developed within

an analytic system.

4.08. Ramsey's remarks about the syntactical antinomies (Group A)

(p.24 ), that they can be removed by pointing out that a propositional

function cannot significantly take itself as argument, are of course

derived from the remarks about types in the TLP (v.3.332.3). But in

dividing the antinomies into two groups, and seeking a different

solution for the second (semantic) group, Ramsey was departing from the

TLP. Despite his disclaimer that his new solution

"••• is a natural consequence of the logical theories of

Mr. Wittgenstein" ( P• 33)

the distinction between syntactic and semantic antinomies cannot be

found in the TLP, whether under Ramsey's, or any other, interpretation.

It was Ramsey's own innovation. (Brouwer ( "Intuitionism and Formalism"

in Bull. Am.. Math. Soc. XX:2 (1913-14) p.90n) points out that Richard's

and Burali-Forti's paradoxes should not be classified together because

whereas the latter rests on the axiom of inclusion, the former does not,

but rather on the word 'defined'. However Brouwer neither elaborates 51 • nor justifies his distinction). Chwistek did not accept the distinction until much later, see LS p.319.

4.09. Ramsey's second move was to attack PM's notion of a

"predicative function" or ''matrix". To replace it, he advances his own concept, which he also calls a "predicative function", but to avoid confusion I shall call it a 'Ramseian function', reserving the title 'predicative' for use in the sense of PM.

4.10. The basis for Ramsey's new functions is what he calls the replacement of 'construction' by 'meaning' (p.37f). PM defined the range of functions as all those which could be constructed in a certain way, namely by substitution (v. PM Introduction Chapter I,

"The range of values and total variation", pbk p.15). This Ramsey calls the "subjective method 11 (p.37). As an improvement he offers his own "objective method" (he warns us that these words are merely

labels, and should not be pressed), as follows. A function • ~ gives us, for each individual, a proposition-kind (p.35). A

function of functions f( $ ~) becomes a proposition-kind when we substitute any function of individuals in its blank spaces. But

functions of functions are different from functions of individuals, Ramsey believes (p.36) because

"•••whereas functions are symbols, individuals are objects •• "

This is a perfect exan:9le of the common tendency among Russell's followers ~o confuse symbols (language) with objects (the world) on a scale which is truly heroic. What Ramsey means is that whereas we 52. cannot construct new logical atoms (whatever they may be like) by inventing new names, we.£!!! invent new functions, and we do this simply by talking about the symbols that express them.

4.11. Ramsey then describes his ("Objective") method of defining functions: we are to determine the symbols which can be substituted in

'f( f ~) •, not by construction, but by meaning. A function ''means" by way of the propositions which are its values. We wish to fix on a definite set of propositions as the values of 'f(f t)•; then and only then may we assert their conjunction and disjunction (i.e. for Ramsey, quantify over them).

We begin by defining an atomic function; this is the result of substituting a variable for a name in an atomic proposition. As for Wittgenstein (TLP 4.22) an atcxnic proposition is one that may be expressed by using names alone (this point of view generates difficulties even within the TLP - cf. 4.24). We know (or at least we hope) that all propositions are truth-functions of these atomic propositions (by the Thesis of Extensionality). We now define truth-functions of atcxnic functions. If y i is the truth-function

• x.v.p, we take this to mean that~ value of y x, e.g. ta, is the corresponding value of • x and the proposition p, i.e. the expression ~ a.v.p. Thus to say ''I x is the conjunction

• x.p• means that 'I a is the conjunction • a.p, and Y b is the conjunction ❖ b.p, and so on. Thus we are enabled to 53. include functions among arguments of any truth-functions. (This is at least more comprehensible than the PM explanation which allows truth­ functions of "ambiguous values" ( e.g. ♦ y) of propositional functions (e.g. $ i) (PM Introduction p.15, p.20) though both are open to similar minor criticisms on such things as the use of quotation-rnarkso In this section I have followed Ramsey's usage of both quotes and the circumflex on p.33, but simplifying the functions to monadic. )

A Bamseian function is now defined (p.39) as a function which is any truth-function of arguments that are all atomic functions or atomic propositions. Since Ramsey believes that all propositions are truth-functions of atomic propositions anyway, this can be simplified to read 'a function which is any truth-function of propositions and atomic functions, and of nothing else'. For this definition to allow quantification, we must follow the reduction of quantified expressions to truth-functions, which involves us in infinitely long formulae, as we have seen. (vide supra, 2.22).

4.13. In examining Ramsey's theory of types, there are three things

to be done. First, we must see if Ramsey's division of the antinomies

into two groups is valid. The next, to discover whether Ramseian

functions do indeed allow us to do analysis, without requiring the

Axiom of Reducibility. Thirdly we must look at the new functions for

acceptability; do they, e.g., allow the antinomies to be reintroduced, or do they involve us in some other difficulty. 54.

4.14. First, then, the distinction. Ramsey cites (p.21) Peano as having pointed out the linguistic (or epistemological or semantic) nature of the paradoxes. Brouwer, in a footnote to "Intuitionism and Formalism", had pointed out that Richard 1s paradox rests on the word 'defined', and is therefore unlike Burali-Forti's (I and F p.9()n).

But it was Ramsey who realized that a two fold exhaustive division of the antinomies can be made, and that it is useful to make it. Peano's observation does not apply to all the paradoxes; Brouwer•s distinction

can be generalized; and Ramsey it was who did these two things.

Ramsey 1s distinction has since become an accepted starting-point for discussion, particularly after the publication (in 1933) of Tarski's

paper "The Concept of Truth in Formalized languages" (repr. in

Tarski, I.SM (tr. J.H. Woodger); v.esp. p.161-2).

hl2.! It is evident that there is a distinction here. The syntactic ('A') paradoxes are concerned with the extensions of certain symbols;

the semantic ('B') involve words which, whether they are common like

'lie' or uncommon like 'heterological', are of the kind that led to

the Logical Atomists problem about Intensional Functions (to use this

merely as a label). The Liar (what I am now saying is false'),

the least integer that cannot be named ••• , the least ordinal and

Richard's decimals, which cannot be defined ••• , and the strange

predicate 1heterological' which means 'the meaning is not a predicate

of ••• •; all these contain one of the words that are so difficult

to reduce to extensional form. 55.

On the other hand, the syntactic paradoxes can be cleared of linguistic factors and made wholly extensional (v. Prior, FL p.290-1): if we define the predicate 'non-self'predicable ', which I shall write

'W' by the following definition

w Q = ~ •• df so that w 4> - ~ •• and by Q / w, we get

ilW ; ,v WW

('Non-self-predicability is non-self-predicable if and only if non-self- predicability is not non-self-predicable'). But the semantic paradoxes cannot be so cleared; in their case the paradox rests on linguistic factors.

The syntactic antinomies can be solved merely by banning the definition of sucn predicates (and so on) as 'non-self-predicable', i.e. by bannins functions that take themselves as possible arguments.

This was done by the Simple Theory of Ty-pes, which forbids such functions by making expressions containing them malforned {see also above, 2.13, and TLP 3.331 and f). Since class-symbolisr:i is translatable into propositional function symbolism ( which is ·what Russell means when he calls classes "incomplete symbols"(!), "fictitious entities", "logical constructions 11 , etc.) we have forbidden simultaneously the class of all classes that arc (or are not) members of themselves, for expressions of 56. the form 'x € a:' and 'x € f(•x)' may be written 'Ax' and

'♦ x ', am the offending 'x € x' is then of the forbidden form

'♦(♦ x)'.

Hence we distinguish

'~ypes of propositional :f'unctions by their arguments; thus

there are :functions of imividuals, functions of functions

of individuals ••• and so on. 11

(Ramsey, FM, p.24)

The syntactic antinomies rest on the presumption that it is significant to say that a function takes itself as argument. This presumption is denied by the Simple Theory of Types, which thus disposes of the syntactic antinomies. The semantic antinanies cannot be eliminated so easily, and in PM require the additional apparatus of the Ramified Theory, which distinguishes orders within types by the appearance of apparent

(bound) variables. If we are studying a group of entities which are known to have common characteristics ( they are paradoxes; they are viciously circular), the perception that they can be divided into two mutually exclusive and collectively exhaustive species is an important

further step. Ramsey's distinction is not only valid but important.

4.16. Our second task is to see whether Ramseian functions allow us

to do analysis without requiring an Axiom of Reducibility. Because

Ramseian functions include PM's elementary functions (FM p.39) and

infinite truth-functions of these (i.e. quantification), anything

that PM can do with its definition of 'predicative function• Ramsey 57. can do with his. Whether Ramsey requires an Axiom of Reducibility, however, is a vexed issue. He requires no explicit additional axiom.

But if we turn to PM Introduction (pbk p.58-9), we find that

"The axiom of reducibility is equivalent to the assumption

that 'any combination or disjunction of predicates (footnote:

Here the combination or disjunction is supposed to be given

intensionally. If given extensionally (i.e. by enumeration)

no assumption is required; but in this case the number of

predicates concerned must be finite) is equivalent to a single

predicate".

It is clear that Ramsey, by claiming that the combination or disjunction can be given extensionally even if infinite, has built something very like an axiom of reducibility into his sytem. Indeed, Prior has remarked that the reduction of universal quantificatio~ to conjunction

"perhaps amounts in itself to a kind of 'axiom of

reducibility'. "

(Prior, FL, p.287n)

Obviously too, if the bound variables in quantified expressions disappear into infinite (but enumerated) conjunctions and disjunctions, the distinctions of orders within types, made on the basis of appearances of bound variables, are no longer really necessary. Distinctions of orders appear in Ramsey•s system only because we frail humans use bound variables as abbreviations for infinitely long lists. They are involved in our symbolism alone. (Ramsey, FM p.47-9). 58.

Ultimately, whether we want to say that Ramsey has, or has not,

included an Axiom of Reducibility is a verbal dispute, which may be

ignored provided we realize that, on the one hand his assumptions imply

something very like Russell's Axian, and on the other hand he has

avoided the introduction of an obviously extra-logical axiom like

Russell's, for the claim that quantification can be reduced to truth­

functions is more like a "Law of Logic" than Russell's statement of the

Axiom of Reducibility.

4 .1 J. I would suggest that the most appropriate thing to say here is that Ramsey has saved analysis by giving a form of the Axiom of

Reducibility that is not open to the criticism that it is obviously

not a Law of Logic, i.e. that if the Axiom were true this would be a

fortunate accident (TLP 6.1232). By doing this he has strengthened

the logicist thesis that mathematics can be deduced from logic. Ramsey's

claim that quantification can be reduced to truth-functions is a better

candidate for a Law of Logic than Russell's statement of the Axiom of

Reducibility. By this I mean, first, that it is more in the form we

expect a Law of Logic to take, being essentially just the statement that

the universal quantifier can be regarded as repeated conjunction (the

existential quantifier being then defined in terms of this) which is a

claim about the nature of a basic logical operator; and secondly, it

is the sort of statement which, if true, is tautological (and if false,

self-contradictory), as opposed to "accidental". 59.

But Ramsey's claim is still open to very destructive criticism, as has in fact been given above (2.23). The quantified expression

(i) '(x)fx' is admittedly very like the list (11) 1 fa.fb.fc •••• fz'.

But they are only equivalent, i.e. inter-reducible, if we know that

(iii) a,b,c, ••• , z are all the things of which f-ness can be predicated. Wittgenstein tried to explain (inter alia) how we know

(111) by his Doctrine of Showing; Russell confessed that he could not explain it without introducing general propositions (like (i)), and so

-was forced to allow the existence of general facts. Ramsey seems to ignore the problem (vide supra, 2.18) but a Ra.mseian solution of how we know (iii) can be given in terms of his theory of truth-possibilities, and in fact has been given, in section 2.22.

If the Ramseian solution of the problem of generality (i.e. how we know (iii), viz. by truth-possibilities) is satisfactory, then the reduction of quantification to truth-functions is satisfactory, and as this reduction does in Ramsey's system what the Axiom of

Reducibility does in PM, Ramsey will have rescued analysis from the

Vicious Circle Principle, which saved mathematics from the Reflexive

Antinomies only at the expense of analysis. Let us consider again the footnote to PM p.59 quoted above (4.16) I paraphrase:

If the combination of predicates is not given intensionally

but extensionally (i.e. by enumeration) then no Axiom of

Reducibility is required; but in this case the number of predicates must be finite. 60.

Now if Ramsey thinks of the combination being given extensionally by enumeration in the infinite case, he is clearly wrong; for it is self-contradictory to speak of an infinite enumeration having been completed by any definition of 'infinite', whether mathematical or common-sensical. The reason we cannot do so is not human frailty but logic itself: to confuse these two, as we have seen in 2.18, 3.05,

3.14 was Ramsey's great weakness as a philosopher. On the other hand, a Ramseian might argue, the list can be given extensionally in some other way, according to the theory of truth-possibilities, by a general proposition which has meaning by expressing agreement with all truth­ possibilities of a certain form (which is possible because we can express agreement with all truth-possibilities whatsoever by a tautology, so we must be able to express agreement with a subset of these, 2.22). But has been pointed out (2.23) this solution is not satisfactory.

Tautologies do not express agreement with all truth-possibilities so much as fail to express disagreement with any. Whereas a general proposition (e.g. (1)) does express agreement with all truth-possibi­ lities of a certain form (e.g. (ii)), the question is how it does this, how it shows that (iii) is a complete list of possible arguments of the function (there is no question corresponding to this in the case of tautologies). Wittgenstein's Doctrine of Showing gives an

(unsatisfactory, v. 2.10 - 2.15) answer to this question - that whether something is a possible argtm.ent of fx or not is shown; an alternative would be in the statement of the rules for well-formed formulae (which since as rules they cannot be true or false, would not fit easily into a Logical Atomist system). And Ramsey's theory of truth-possibilities, which is essentially a sim'Jler version of the

Doctrine of Showing, does not give a satisfactory answer either,

because it ignores the difference between general propositions and

tautologies - tautologies say nothing. Thus Ramsey's explanation

of how quantified expressions are to be reduced to truth-functions

does not work. Consequently, the whole of his rescue of analysis

is based on false premisses, and will not do.

4. ,s. OUr third task is to examine the acceptability of Ramseian functions. They allow us to escape from the semantic antinomies,

and do not involve us in any further antinomies, as Ramsey capably

shows (FM p.42ff). However, the concept of a Ramseian predicative

function

"••• is essentially dependent on the notion of a truth­

function of an infinite number of arguments; if there

could only be a finite number of arguments our predicative

[i.e. Ramseian] functions would be simply the elementary [i.e. predicativeJ functions of Principia." (Ramsey, FM p.39)

and infinite truth-functions, as I have pointed out (2.23, 4.17) are

not an acceptable notion. A truth-function which cannot be written

out is not a truth-function as the term is generally understood. An

infinite truth-function cannot be written out, not for merely

inessential reasons like the expense of the paper, but for a logical reason. 'Infinite' means (both etymologically, in its present usage, 62. and in all definitions used in mathematics) that there will always be one more, that the list is unfinishable. To talk about the finish of an unfinishable process is absurd. It is Ra.msey's old confusion of what ~ cannot do and what would involve a self­ contradiction. Ramsey's "axiom of reducibility" is not true only by accident; it is false by logical necessity.

4 .19. Ramsey had great insight into the problems raised by the Reflexive Antinomies. This can be best seen in his discoveries

(4.07, 4.08) that there are two distinct groups of Antinomies, and that one of these groups can be eliminated merely by the Simple

Theory of Types. Despite this insight, Ramsey's solution to the problem of the remaining (semantic) Antinomies was not successful.

Again he was betrayed by his inadequate understanding of the difference between human impossibility and lpgical impossibility. CHAPI'ER V

INTENSIONAL FUNCTIONS AND fil:GATIVE FACTS

5.01. Belief was a particular problem for the Logical Atomists, who required that the truth or falsity of non-atomic propositions be determined solely by the truth or falsity of the atomic propositions like 'A believes that p' (and 'A wishes that p', etc.) do not seem to fit their requirement. For this reason,

Russell called such propositions "Intensional Functions" (i.eo not

(apparently) extensional). Russell had not been able to come to a final solution of this problem in PIA, concluding his remarks about it:

" .. o one has to be content on many points with pointing

out difficulties rather than laying down quite clear

solutions."

(PIA, po227)

He did make two points (vide PIA, p.226, last paragraph). The first is

"the impossibility of treating the proposition believed

as an independent entity", the other is

"the impossibility of putting the subordinate verb on a

level with its terms as an object term of the belief".

We may explain these by using his own example, 'othello believes that Desdemona loves Cassio 1 o With othello, Desdemona 64. and Cassio themselves, the logician (in contrast to the man of letters} has no trouble. They can be symbolized by 'o', 'd' and IC I respectively. The proposition 1Desdemona loves Cassio' asserts that between two objects there is a certain relation, and is written 'L(d,c)•.

Russell's second point is that the subordinate verb 'L' is not on a level with its terms, i.e. that the belief is not a tetradic relation between o, L, d and c, because of the possibility that the belief is false and that there is no relation L between d and c. Thus •e(o, L, d, c}' is not in Russell's opinion the correct analysis. His first point is that we should not think of the situation as a relation between othello am the proposition 'L(d, c}' either, because this would involve propo­ sitions, and in the case of false belief, false propositions, going about in the world. Such a notion Russell regards as ''monstrous"

(PIA, p.223).

5.02. Wittgenstein deals with intensional functions in TLP

5.54-1, 5.54-2, 5.54-21, 5.54-22. These are much commented upon: see especially Anscombe IWT, p.87f, Griffin WI.A, p.112, Black CWT, p.29Bf, Urmson PA, p.133, Copi "Tractatus 5.54-2 11 (in Copi and Beard,

EWT, p.163). The following account (i.e. section 5.02) is derived immediately from my thesis ''Wittgenstein's 1ra~~atus Logico-Philoso­ phicus and the Theory of Types", but represents a consensus of critical views. In 5.54-2, Wittgenstein says that "it is clear" that so-called intensional functions like 'A believes that p' are of the form • "p" says that p'. Lest this seem like the strange habit some people have of thinking that the assertion of a tautology proves their case ('After all, England is England'), Wittgenstein adds a word of explanation:

"And this does not involve a correlation of a fact with

an object, but rather the correlation of the facts by

means of the correlation of their objects."

(TLP from 5.54-2)

What he means by this enigmatic observation is that when A believes that a state of affairs obtains, the configuration of objects in the state of affairs is expressed by an isomorphic configuration of objects in A's mind or soul. What these latter objects may be Wittgenstein leaves wholly ambiguous. His account is compatible with any and every theory of the mind. The required configuration could be assembled from words, logical symbols, neurons, ideas, cogwheels, electrons or lumps of cheese, and the mind could be composed of any of these things. The only restriction

is that the mind has :i;arts, that it be composite; which is why he says of his theory of belief.

"5.54-21 This shows too that there is no such thing as

the soul - the subject, etc. - as it is conceived in the

superficial psychology of the present day. "Indeed a composite soul would no longer be a soul."

(TLP 5.54-21) 66.

Since a state of affairs ("Tatsache") is not a fact, which is the case, but rather a possible fact, which could be the case,

Wittgenstein's account deals with false beliefs, in the same way as he deals with false propositions (2.21). Since Desdemona does not love Cassio in reality, there is no fact that Desdemona loves

Cassio; but it is possible that these objects, Desdemona and

Cassio, could occur in this relation (of the first loving the second), and it is with what would be the case if this possibility were the case that the contents of othello's mind are isomorphic with.

A belief ('p') is true if it corresponds to reality (i.e. if p); it is false if it does not (i.e. if ~ p). Wittgenstein does not have false propositions going about in the world; a proposition-token is a fact about somebody's mind.

5.03. Ramsey was impressed with the TLP account when he read it (vide his review of the TLP, reprinted in FM, especially p.274-9).

He explicates it there essentially as I have done above (5.02), ending.

"Thus 1 "p" says that ~ aRb I would, supposing us to be dealing with the symbolism of Principia Mathematica,

be analysed as follows: call anything meaning a, 'a',

and so on, and call 'a' 'R' 1b', 'q'; then 'p' is

I ~ q I or t ~ ~ ~ q I or I - q V ~ q I or any of the other symbols constructed according to a definite rule.

(It may, of course, be doubted whether it is possible to

formulate this rule, as it seems to presuppose the whole of symbolic logic; but in any perfect notation it might be

possible; for example in Mr. Wittgenstein's notation with

T's and F's there would be no difficulty.) But it is

obvious that this is not enough; it will not give an

analysis of 'A asserts p', but only of 'A asserts p

using such-and-such a logical notation•. But we may well

know that a Chinaman has a certain opinion without having

an idea of the logical notation he uses. Also the evident-

ly significant statement that Germans use 'nicht' for not

becomes part of the definition of such words as 'believe',

'think' when used of Germans."

(FM p.278)

His difficulty, as he realizes, is with the TLP doctrine that what negates is not I~ I but rather

11 • •• the common rule that governs the construction of

'~ P', '~ ~ ~ P', '~P V~ P 1, '~P •~ P 1, etc. etc. (ad inf.). And this common feature mirrors negation."

(TLP 5.512c) Ramsey says

"I cannot understand how it mirrors denial"

(Review of the TLP, repr. FM p.279)

('denial' was the Ogden translation of 'Verneinung'; Pears and

MaGuinness translate it as 'negation'). 63.

5.04. Ramsey•s dilemma here is understandable. However it can be resolved within the TLP. What the Chinese, German and English beliefs h!ve in common is that they are all intertranslatable by definitions

(v. 3.343), that is, the German 'nicht' is equivalent to the English 'not', PM's ' ~' , and a particular Chinese ideogram.

"3 .344 What signifies in a symbol is what is common to all

symbols that the rules of logical syntax allow us to sub­

stitute for it."

(TLP 3.344 see also Black's discussion

of 5.512r, CWT p.278f).

Wittgenstein rejects Russell's notion of negative facts as facts to explain negative propositions (PIA, p.214):

''Must the sign of a negative proposition be constructed with

that of a positive proposition? Wl:l;y' should it not be

possible to express a negative proposition by means of a

negative fact? (E.g. suppose that •a• does not stand in

a certain relation to 'b'; then this might be used to say

that aRb was not the case.)"

(TLP 5.5151a)

'I'hat is to say, suppose that we wish to symbolize that aRb is not the case (a negative fact in that it is a fact which makes a negative proposition true) - why should we not express this by another negative fact, viz. that the name of a does not stand in a certain relation to to the name of b? An example makes it clear why Wittgenstein rejects this suggestion - we cannot express 'Desdemona does not love Cassio' by failing to write 'Desdemona loves Cassio'. The negative proposition must determine which proposition is being negated. Or as Wittgenstein continues:

"But really even in this case the negative proposition is

constructed by an indirect use of the positive.

"The positive proposition necessarily presupposes the

existence of the negative proposition and vice versa. 11

(TLP 5.515b,c)

The negative-ness lies in the proposition, not in the fact - a consequence of the Wittgensteinian discovery known to Russell, that

11 ••• there are two propositions corresponding to each fact. 11

(Russell PIA, p.187)

•~.0621 But it is important that the signs 'p 1 and

I...., p' can say the same thing. For it shows that nothing

in reality corresponds to the sign I ...., I •

"The occurrence of negation in a proposition is not enough to

characterize its sense (~~ p=p}.

"The propositions 'p' and '~ p' have opposite sense, but there corresponds to them one and the same reality."

Reality does not consist of positive facts corresponding to affirmative propositions, and negative facts corresponding to negative propositions. 70.

A negative fact is not, as Russell thought, a special breed of fact which gives meaning to negative propositions; Wittgenstein is only prepared to talk about negative facts in the sense of facts (simpliciter) which make negative propositions true.

The problem of negative propositions is rather like that of general propositions (vide supra, chapter 2). In PIA Russell was unable to analyse either of these kinds of propositions, and reluctantly has to postulate general facts and negative facts to correspond to them. Wittgenstein claimed to eliminate the need for postulating either. In the case of negative facts, he proceeds thus: a negative fact is what makes a negative proposition true (not: what makes a negative proposition meaningful), i.e. the non-existence of a possible state of affairs (2.06). From the existence or non-existence of one state of affairs it is impossible to infer the existence or non-existence of another (2.062), i.e. states of affairs are independent (2.061).

A proposition is a picture of a possible state of affairs. It is itself _a fact (2.141). One fact (the proposition) mirrors another fact. This is because both have the same logical form (2.2, 4.12).

Form is the possibility of structure (2.033). This is, in the case of the fact pictured, the possibility that objects should stand in that relation to each other (2.031, 2.032); in the case of the fact picturing, i.e. the proposition, it is the possibility that the signs for those objects should occur in that relation to each other. Form determines what can (truly or falsely, but not nonsensically) be said; i.e. the ways in which signs may be arranged in wffs. It is shown 71.

(4.12). Thus he concludes that negation applies to propositions, not to facts:

"4.o64 Every proposition must already have a sense: it

cannot be given a sense by affirmation. Indeed its sense

is just what is affirmed. And the same applies to negation,

etc.

"4 .0641 One could say that negation must be related to the

logical place determined by the negated proposition.

''The negating proposition determines a logical place

different from that of the ?Egated proposition.

"The negating proposition determines a logical place with the

help of the negated proposition. For it describes it as

lying outside the latter's logical place.

"The negated proposition can be negated again, and this in

itself shows that what is negated is already a proposition,

and not merely something that is preliminary to a proposition."

In short, a proposition has sense, there are two propositions for every fact. Facts are neither effirmative nor negative (although we may, by ellipsis, refer to the fact that makes a negative proposition true as a "negative fact", this is not a special kind of fact).

Wittgenstein's escape from the problem of negative facts is totally integrated into the TLP philosophy. It could not be understood without the concepts of propositions picturing facts (the Picture

Theory) and the inexpressibility of Logical form (the Doctrine of 72

Showing). As neither of these positions have been found to be acceptable by subsequent philosophers, Wittgenstein's analysis of negative propositions is as it stands not satisfactory. (Viewed internally, of course, it is a strength of the TLP system that it can account for negative facts; the objections apply to the TLP

system itself, or even to the whole Logical Atanist approach.)

5.06. Ramsey accepted the Picture Theory (as will becane plain) and was able to adapt these Wittgensteinian solutions of intensional functions and negative facts to his own theory of Truth-possibilities (which corresponds to Wittgenstein's Doctrine of Showing, see above,

2. 20 - 2. 22) He did this in "Facts and Propositions " ( 1927, repr. in FM). With regard to intensional functions, his exposition is

clear, in contrast to Wittgenstein's remarks, which are concise to

the point of obscurity. Two improvements Ramsey made may be noted.

First Ramsey states explicitly that he is concerned not with absolutely atanic propositions, but only with propositions which are atanic relative to a particular language (p.145). This obvious consequence of the Theory of Types was known to both Russell and Wittgenstein, but they do not make it as clear as they might (though

vide PM pbk P• 65). Secondly, unlike Wittgenstein, Ramsey camnits

himself to the mental entities in a belief being words. By doing this, he immediately raises the problem of belief (or other propositional attitudes) in animals, e.g. does a chicken which avoids eating a caterpillar believe that the caterpillar is poisonous? (F and P FM p.144). He suggests that this may be analysable in 73. terms of the chicken's behaviour, in a pragmatist way.

"Thus any se~f actions to whose utility p is a necessary and sufficient condition might be called a belief that p, and so would be true if p, i.e. i:t' they are usef'ul. " (p.144)

However, such problems were not central to the Atomist view o:t' the world (in contrast to some later philosophers, e.g. Professor

Ryle), and Ramsey dropped the subject to consider belief in terms of words.

Apart from these details, Ramsey•s analysis of belief in "Facts and Propositions" (1927) is that given by Wittgenstein in the TLP and which :first excited Ramsey in his review in 1923. In that account, he had been stopped :from completely following Wittgenstein by his inability to understand Wittgenstein's analysis of negative propositions. In the paper of 1927, however, Ramsey was able to go f'urther, by giving at least part of an analysis of negation.

He did this by means of an example which has become famous and is often used to explain the TLP view. He begins by saying that 'A is not not red', 'A is both red and not not red', etc. are dif:t'erent propositions pointing to the same :fact, and not new facts. Next, he points out that that there are such possibilities is an 74. accidental, not an essential, feature of our symbolism by his example:

''We might, for instance, express negation not by inserting a word 'not•, but by writing what we negate

upside down. Such a system is only inconvenient because we are not trained to perceive canplicated

symmetry about a horizontal axis, but if we adopted it we should be rid of the redundant 'not-not', for the

result of negating the sentence 1p' twice would be

simply the sentence 'p' itself."

(F and P FM p.146-7)

Fran this it is easy to see what Wittgenstein meant by negative

propositions indicating a possible state of affairs in a way dependent

on the corresponding affirmative proposition. 'p' and • ~ p 1 both picture a possible state of affairs, viz. that P• They have meaning by arranging names as objects would be arranged in p if

it were the case that p. The proposition 'p' says that P•

'lhe proposition '~ p 1 says that this state of affairs does not exist, that this configuration does not obtain. It has meaning by the state of affairs that exists of the affirmative proposition is true, not by some negative fact. Negative propositions do not point to negative facts, they point at the facts negatively. Ramsey has here reached the Wittgensteinian position on negative facts f'ran his own premisses, and his worries with the Wittgensteinian analysis 75. of intensional functions are over. On p.149-150, he describes belief directly in terms of truth-possibilities.

5.07. Ramsey however continues on from here, cooibining both negation and belief to try to define the equivalence between believ- ing 'not-p• and disbelieving 'p' in terms of causation (p.148).

This may be regarded either as an application of Wittgenstein's exceedingly abstract theory to something closer to real life (presum­ ing that the real life concept of •cause' can be given sane significance, perhaps as an incanplete symbol; for see TLP 5.1361), or as a move by Ramsey from Logical Atomism to a different kind of philosophy, not so strict and consistent, but not so prone to force the world into an a priori structure. We may see this philosophyasa deYelopnent of

Pragmatism, or the beginnings of "Ordinary Language" Philosophy. In view of Ramsey's later work (vide especially Paper F "Philosophy"

(1929) (printed in FM p.263)), it would be reasonable to conclude that Ramsey was already moving away from Logical Atomism in "Facts and Propositions". However, both in improving the exposition of Wittgenstein's solution of the two problems of negative facts and of intensional functions, and in showing that the Wittgensteinian solutions can be based on less than the full Wittgensteinian position, he was still contributing to the Atanist Philosophy. "Facts and Propositions" is unmistakeably the work of a Logical Atomist, but in these two paragraphs (p.148) we can glimpse the first tentative steps toward a less abstract and a less a priori kind of philosophy. 76.

CHAPI'ER VI. UNIVERSALS

6.01. The subject of the Logical Atomists • views on the problem of universals is most complex. In 1911, Russell said:

''We have thus a division of all entities into two classes: (1) particulars, which enter into complexes only as the

subjects of predicates or the terms of relations, and,

if they belong to the world of which we have experience,

exist in time, and cannot occupy more than one place in

the space to which they belong; (2) universals, which

can occur as predicates or relations in complexes, do

not exist in time, and have no relation to one place

which they may not simultaneously have to another. 11

(RUP in LK p.124)

That is, in 1911 Russell recognized what I shall call two categories - particulars and universals. After he had met Wittgenstein (who arrived at Cambridge in January 1912), and the philosophical ideas

"forced on him by thinking about mathematics II had begun to gell, the situation became much more complicated:

''As to universals, when I say of a particular that it exists, I certainly do not mean the same thing as if I were to say that it is not a universal. The statement

concerning any particular that it is not a universal is quite strictly nonsense - not false, but strictly and

exactly nonsense. You never can place a particular in the sort of place where a universal ought to be, and vice versa. If I say 'a is not b', or if I say

'a is b', that implies that a and b are of the same logical type • When I say of a universal

that it exists, I should be meaning it in a different sense from that in which one says that particulars exist. E.g., you might say 'Colours exist in the

spectrum between blue and yellow'. That would be a

perfectly respectable statement, the colours being

taken as universals. You mean simply that the propo- sitional function 'x is a colour between blue and

yellow' is one which is capable of truth. But the x which occurs there is not a particular, it is a universal."

(Russell, PIA, LK, p.258)

Both Russell's Theory of Types and the Simple theory of types preferred by Ramsey distinguish a (strictly) infinite nUJ1.ber of different types of classes, properties, and propositional functions (see above 4.02). A property F of a property f is of a different logical type from the property f of which it (F) is a property. This difference, on the lowest level, is that between particulars and properties of particulars, i.e. universals. Parti- 78. culars are of type o. Properties of particulars are of type 1. Properties of those properties are of the next type up again i.e. of type 2. Et cetera ad infinitum. Taking the passage from PLA. quoted above in the light of this, we may say that for Russell, there are an infinite number of categories, the difference between each corresponding to that between ptrticulars and universals. 'Existence' means something different for each Type. Consequently,

"The word 'there is' is a word having 'systematic

ambiguity! i.e., having a strictly infinite number of different meanings which it is important to distinguish." (PLA p.268)

6.02. In his article "Universals" (1925), Ramsey criticizes

Russell's view in ''RUP" (1911) which dates from before Russell had developed Logical Atomism, and in which he admitted only two categories. However, Ramsey's arguments apply also to Russell's later position (PIA). Ramsey does not disagree with Russell that there are objects of infinitely many different types. He argues that the difference between types is not that between universal and particular, as Russell construes it. Russell, following Frege ("Function and Concept" p.6 (TPW p.24), ''What is a J'unction" p.664 (TPW p.114), Grundgesetze der

Arithmetik S1 (TPW p.152)) held that there is something essentially incomplete about a universal which must be filled by a ptrticular, whereas the particular itself in contrast is complete and independent

(PLA. p.199f; LA p.338;PM pxix) (except for the Lowest two types, 79. this difference is relative). The form of the proposition is for Russell given by the object of the highest type occurring in that proposition - thus e.g. the dyadic relation 'R' can occur only in propositions of the form 'aRb ', and 'R I should be written with blanks (indicated by capped variables) ~1)-.~fx4.., •

Ramsey bases his argument on the TLP, specifically

"3.314 An expression has meaning only 1n a proposition.

All variables can be construed as propositional variables.

"(Even variable names) "

(Thus language is the totality of propositions, not of names, and consequently TLP 1. 1, "the world is the totality of facts, not of things.") Whether Ramsey's view is indeed that of the TLP Wittgenstein is a vexed issue, and one not relevant to the validity of Ramsey's arguments. Ramsey's view is in accord with TLP 4.22

''An elementary proposition consists of names. It is a nexus, a concatenation, of names." but is not so easily reconciled with 4.24b

"I write elementary propositions as functions of names, so

that they have the form 'fx ', ' • (x,y) •, etc. "

Whether the function-sympol is a name or not in the TLP (which is the problem of universals) has given rise to a major controversy ••• There are three main views. Firstly that of Edna Daitz-O'Shaugnessy ( "The So.

Picture T~ 1eory of Meaning" in EWT p.115f and ECA P•53f) and (with variations) Ellis Evans ( "Tractatus 3. 1432" in EWT p.133f), that

it is. Secondly that of Elizabeth Anscombe (IWT p.1o8-11O; and

''Mr. Copi on Objects, Properties and Relations in the Tractatus" in

EWT p.187f) and Irving Copi ("Objects, Properties and Relations in the Tractatus"), that it is not. And thirdly that of Stenius, accord­

ing to which the name of a function is itself a function - an incomplete

name (W's T, p.13O-3).

It is not necessary for us to decide who is right on this knotty point of TLP scholarship; our interest is not in what the TLP

says, but rather in what Ramsey believed it to say. Ramsey agreed with Mrs Daitz, that the function-symbol is a name, but his reasons are very different, and not obviously in confiict with the TLP (Mrs Daitz 1 criticism that all ordinary sentences have one word too many (EWT p.120, ECA p.59) conflicts with TLP 5.5563, that they are in perfect logical order. Her mistake is due to Bradley's regress of the relation of a relation to its relata). A close reading of Ramsey would strengthen those on this side of the debate on universals in the TLP.

G.o4. By TLP 3.314 and 4.22, Ramsey was convinced firstly that names have no meaning on their own ( which weakens the "independence" of particulars) and secondly that, at least in elementary propositions, function-symbols are names just as much as symbols for particu.la.rs are. He claims ("Universals" FM p. 116) that in principle a proposition may

be expressed with either term as subject ('Socrates is wise','Wisdom is 81. a characteristic of Socrates' both express the same proposition}, thus weakening the traditional grounds for distinguishing substances, the symbols for which can occur only as subjects, from universals, the symbols for which can also occur as predicates. That is to say, Ramsey regards '♦ a' as a :9roposition which is a concatenation of two names, ' .' which names an entity of type 1, and 'a I which names an entity of type o. Neither is the subject and neither the predicate of the proposition, for that distinction does not apply to propositions, but to English sentences such as 'a is cl> ' and ' ♦ is characteristic of a•. This is Ramsey's view, and it is a better argument for thinking that function-symbols~ names in the TLP than either Mrs Daitz or Mr. Evans give, because it is not in such obvious conflict with the TLP.

6.05. The traditional grounds for the substance-Universal distinction were very shaky anyway due to the discovery of propositions not of subject-predicate form. Ramsey tries to increase the impact of this discovery by an ingenious argument against such complex universals as 'being wise unless Plato is foolish' or 'being the teacher of Plato' (p.118-9). (The postulation of these canplex universals

is itself an attempt to reduce all propositions back to subject­

predicate form}. Let us take the second example: 'Socrates is the teacher of Plato' may be symbolized 1 sTp 1 • We may abbreviate this to '. s'. If ' ♦' is the name of a complex universal, then '~s• is a subject-predicate proposition attributing this property to Socrates; 82. and so is not identical with the relational proposition 'sTp' •

Since ' • x' was defined as meaning the same as !xTp ', this is absurd. But if on the other hand, complex universals cannot be named in this way, how can they be named, and if they cannot be named, what reason is there for postulating them?

Although this argument is ingenious, it is not as conclusive as it at first seems. The believer in ccnplex universals may insist on a very rigid and somewhat heterodox sentence-proposition distinct­ ion, by which the two sentences of different forms • • s' and 1sTp' both express the same proposition, which is of sane determinate form.

Ramsey also points (p.120) out that we can say 'a has all the properties b has• without assuming the existence of properties, only of propositions (every proposition true of a is also a true proposition when the name 1b • has been substituted for 'a' through­ out).

6.06. Ramsey then (p. 122) begins attacking Russell in earnest, by wondering how one sort of object can be specially incomplete (i.e. except as all objects are incomplete in that they must occur in facts, TLP 1.1). 'Wise' does involve the form of a proposition in this letter sense, but so does 'Socrates•, and thus this is no ground for Russell to base his distinction on. Ramsey imagines that Russell would have two replies to this; first, that in PM's symbolism, involves the form of a proposition, whereas 'a' does not, and that this feature of the notation is reflected in reality because the use- f'ulness of PM's symbolism results from the fact that it reflects reality (p.122).. Secondly, that everyone can feel the difference between universals and particulars.

6.07. This second objection may seem very weak as thus stated. But it has a powerful intuitive effect, which Ramsey brilliantly analyses

(p.123f) as follows. As we do not know of any definitely genuine objects, the feeling of the difference between universals and parti­ culars which we have is a feeling about logical constructions, i.e. what are symbolized by incomplete symbols. ('incomplete' here, of course, means 'has no meaning on its own, definable only by definitions­ in-use', not the mysterious kind of incompleteness that Russell and

Frege claim is possessed by universals, above 6.06).

An incomplete symbol occurs as an isolated unit immediately we introduced quantification. It is not 'aRb', but '{x).xRb' that makes 'Rb' prominent (p.123). '{x).xRb' expressed what is cOIDI1on to all propositions of·the form 'x:Rb'. Now 'Socrates' is common to all propositions on which it occurs. The same applies to

'wise'. But with 'wise', there is also a set of propositions to which the form 'x is wise' is what is common. Thus 'Socrates' picks out only one set of propositions, but 'wise' picks out both a corresponding set, and also a narrower set, in which 'wise• occurs in a special way. This is why we feel a difference between 'Socrates' and 'wise'. 'Socrates• is incomplete in that it does not express 84. a proposition of its own, and is to be completed; 'wise' is incomplete too, but is different in that it is to be completed not only in any way like 'Socrates', but also in a special class of ways, viz. the ones in which it is the predicate, which are of the form 'x is wise'.

6.o8. However, Ramsey argues (p.125), this difference would vanish if we could find a subset of propositions, about Socrates of the form 'Socrates is q', where 'q' is a quality, or simple property, for then 'Socrates' would pick out two sets of propositions, one a subset of the other, like 'wise'.

Ramsey admits (p.125) that we do not distinguish qualities fran other properties, and considers the possibility that this may be due to an underlying difference between particulars and universals, a view he atributes to W.E. Johnson. According to Johnson, 'Socrates is neither wise nor just• attributes a property (that of being neither wise nor just) to Socrates, and so implies '(::3: t ) • ♦ Socrates'.

But in contrast, 'Neither Socrates nor Plato is wise' does not attribute the property of wisdom. to a being called 'neither Socrates nor Plato', and does not imply ' ( ::3: x) .x is wise '. Ramsey answers this Johnsonian argument by pointing out that if we distinguish pro­ perties from qualities, 'neither wise nor just' would not refer to a quality. Against this there is the objection that we do not distinguish qualities and properties, with which Ramsey now (p.126) deals. 6.09. Ramsey claims that a distinction between qualities and other proper­ ties is arbitrary and artificial only in the case of logical construct­ ions (such as wisdom), and that there is a clear distinction in the case of genuine objects. If '• a' is a two-termed atomic propo­ sition, then '~, is the name of a quality; and this distinguishes qualities from other properties on the atomic level. If they represent logical atoms, both ' ♦ • and 'a' each pick out two sets of propositions (for if we can distinguish qualities from other properties, 'a' picks out two sets, as shown above, 6.07). So as far as the constituents of atanic facts are concerned, the contrast between particulars and universals does not apply.

Ramsey next (p.127) shows that incomplete symbols also pick out two sets of propositions. If 'Nx' means 'aRx', then 'N' gives us two sets of propositions, viz. (i) all propositions of the form

'aRx ', and ( 11) all propositions in which the symbol 'N' appears. (For Ramsey, who holds the extensionalist theory of language, these will all be truth-functions of 'aRx' and other propositions).

The narrower range (1) is the set of propositions in which the incanplete symbol has primary occurrence in PM's sense (v. PM p.68-9. P• 182), i.e. its scope is the whole proposition in which it occurs; the wider range (11) is the set of propositions in which the incomplete symbol has secondary occurrence, i.e. its scope is less than the whole proposition in which it occurs. 86.

From this Ramsey concludes (p.127) that any incomplete spmbol is really adjectival, in the sense of picking out two sets of propositions (as everyday language adjectives like 'wise' do), and those incomplete symbols which appear to be substantival (like 'Socrates') only appear so because we do not distinguish between their primary and secondary occurrences. He concludes (p.128), validly, that the fundamental distinction regarding incomplete symbols is not that between substantive and adjective (i.e. between particular and universal) which reflects nothing more important than the human situation, but is rather that between primary and secondary occurrences, which we human beings, for our own purposes, sometimes blur. The distinction between universals and particulars originated frcn (what the Logical Atomists regarded as) logical constructions, e.g. material objects and their properties. But logical constructions are not divided into universals and :particulars. Rather, the symbols for logical constructions, i.e. incomplete symbols, can occur in two ways. We ignore this for some of them but not for others, so that it seems as if some can occur in two ways and others in only one. It is from this illusory contrast that the distinction between universals and particulars arose. So by having shown the distinction to be ill-founded in that case, Ramsey has weakened the plausibility of its being a fundamental one among genuine objects.

Against Ramsey here, we may ask why it is so natural to ignore the difference between primary and secondary occurrence in one class of cases, and not in the other; and point out that the cases in which we ignore it, i.e. in which we think of the symbol as picking out only

one set of propositions, i.e. as substantival, are those of material

objects, which are the objects (or logical constructions) of the

lowest type we customarily deal with in everyday language. This

would only begin to re-establish Russell's position, that the differences

between types is that between particulars and universals (see above, 6.01). Ramsey•s achievement of giving an analysis of the felt

distinction between particulars and universals in purely formal terms

is one which could not be omitted from aey final account of universals,

be it Russellian or not.

6. 1O. ~t Ramsey has done so far he describes as follows : ''We saw above that the distinction between particular and

universal was derived from that between subject and predicate

which we found only to occur in atan.ic propositions. We _then

examined the three theories of atomic propoaitions or rather of atomic facts, Mr. Johnson's theory of a tie, Mr. Russell's that the copulation is performed by universals, of which

there must be one and only one in each atomic fact, and Mr. Wittgenstein's that the objects hang in one another like the links of a chain. We observed that of these theories only Mr. Russell's really assigned a different f'unction to subject and predicate and so gave meaning to the distinction between them, and we proceeded to discuss this theory. We found that to Mr. Johnson's criticisms Mr. Russell had two

possible answers; one being to argue that his theory alone 88.

took account of the difference we feel there to be between

Socrates and wisdom, the other that his notation was far more convenient than any other and must therefore correspond

more closely, to the facts. We then took the first of these arguments, and examined the difference between Socrates and

wisdom. This we found to consist in the fact that whereas Socrates determined only one range of propositions in which it occurred, wise determined two such ranges, the complete

range 'f wise', and the narrower range 'x is wise•.

We then examined the reason for this difference between the two incomplete symbols Socrates and wise, and decided that it

was of a subjective character and depended on human interests

and needs. "

(''Universals" FM p.128-9)

6.11.However, Ramsey aims at refuting rather than merely undermining plausibility, and returns (p. 129) to Russell's first reply (see above

6.06), that FM's symbolism reflects reality, a:a:l that the distinction between particulars and universals being essential in this symbolism, it is also essential in reality. PM symbolizes particulars and universals differently. Universals are symbolized not by single letters, but by propositional functions containing a variable 'x' (or more than one, 'x', 'y', .•• ). The indispensible advantage of this is that it enables us to symbolize compound properties such as

'either having R to a, or having S to b'. To define a predicate, say t • I 1 for this can not be done by saying

'◊ = df Ra .v. Sb', for

''we should not know whether the blanks were to be filled with the same or different arguments, and so whether • was to be a property or relation. Instead we must put

$ x. = .xRa.v. xSb; which explains not what is meant by

$ by itself but that followed by any symbol x it is short for xRa.v.xSb. And this is the reason which makes

inevitable the introduction of propositional functions.

It simply means that in such a case 1 $ 1 is not a name but an incomplete symbol and cannot be defined in isolation

or stand by itself. 11

( Ramsey ''Universals 11 , FM, p. 130)

6.12. This argument seems to be based on the necessity for symbolizing complex properties, the existence of which Ramsey has rejected in this same article (see above, 6.05, and ''Universals", FM, p.118-9). However by an analogous argument to Ramsey's about saying •a has all the properties that b has' without assuming the existence of properties (''Universals' p.120; discussed above 6.06), we may talk about complex universals, i.e. about such properties as 'either having R to a or having S to b ', without involving anything other than simple properties (qualities) propositions, truth-functions, and (though only if we reject Ramsey's reduction of quantification to truth-functions), generalization. Thus Ramsey can accept the 90.

necessity for having complex predicates without committing himself to

the ontological necessity of complex universals.

6.13. So far Ramsey has shown that the universal-particular distinction

is not fundamental to incanplete symbols, and that the apparrent

necessity for symbolizing universals by signs which involve the

form of a proposition is a real necessity in the case of complex predicates. The traditional distinction between universals and

particulars is not yet refuted. Ramsey•s attempt to do this occurs

in a passage which is, at least at first sight, devastating:

"But this conclusion about xRa. v. xSb will not apply to

all propositional :f'u.nctions. If ci, a is a two-termed

atomic proposition, • •• is a name of the term other than a, and can perfectly well stand by itself; so, it will be

asked, why do we -write '• x' instead of • •• in this case also? The reason for this lies in a :f'u.ndamental

characteristic of mathematical logic, its extensionality, by

which I mean its primary interest in classes and relations in

extension. Now if 1n any proposition whatever we change

any individual name into a variable, the resulting propositional

function defines a class; and the class may be the same for

two functions of quite different forms, in one of which '•• is an incomplete symbol, in the other a name.

al logic, being only interested in :f'u.nctions as a means to

classes, sees no need to distinguish these two sorts of functions, 91 • because the difference between them, though all-important to philosophy, will not correspond to any difference between the classes they define. So because some Is are incomplete and cannot stand alone, and all t • s are to be treated alike in order to avoid useless complication, the only solution is to allow none to stand alone.

"Such is the justification of Mr. Russell 1s practice; but it is also the refutation of his theory, which fails to appreciate the distinction between those functions which are names and those which are incomplete symbols, a distinction which, as remarked above, though immaterial for mathematics is essential for philosophy. I do not mean that Mr. Russell would now deny this distinction; on the contrary it is clear from the Second

Edition of Principia that he would accept it; but I think that his present theory of universals is the relic of his previous

failure to appreciate it.

"It will be remembered that we found two possible arguments for his theory of universals. One was from the efficiency of the functional notation; this clearly lapses because, as we have seen, the functional notation merely overlooks an essential distinction which happens not to interest the mathematician, and the fact that some functions cannot stand alone is no argument that all cannot. The other argument was from the difference we feel between Socrates and wise, which corresponds to a 92. difference in his logical system between individuals and functions.

Just as Socrates determines one range of propositions, but wise two, so a determines the one range ~ a, but • z the two ranges • x and f(• £). But what is this difference between individuals and functions due to? Again simply to the fact that certain things do not interest the mathematician. Anyone who was interested not only in classes of things, but also in their qualities, would want do distinguish from among the others those functions which were names; and if we called the objects of which they are names qualities, and denoted a variable quality by q, we should have not only the range

♦ a but also the narrower range qa, and the difference analogous to that between 'Socrates' and 'wisdom' would have disappeared. We should have complete symmetry between qualities and individuals; each could have names which could stand alone, each would determine two ranges of propositions, for a would determine the ranges qa and

~ a, where q and • are variables, and q would determine the ranges qx and fq, where x and f are variables.

"So were it not for the mathematician's biassed interest he would invent a symbolism which was completely symmetrical as regards indivi­ dual and quality; all we are talking about is two different types of objects, such that two objects, one of each type, could be sole constituents of an atomic fact. The two types being in every way sym- metrically related, nothing can be meant by calling one type the type of individuals and the other that of qualities, and these two words are devoid of connotation." ( "Universals" FM, P• 130-2) ~- 6.14. This final argument by Ramsey against the universal- particular distinction may be expressed thus. Russell's second argument (11), the felt difference between 'Socrates• and 'wise•, whereby 'Socrates• determines one set of propositions but wise determines two sets, is due merely to the peculiar (and logically accidental) interests of human beings, in that we do not distinguish between primary and secondary occurrence in the case of some incomplete symbols, whereas we do so distinguish in the case of others. This sloppiness of human ha~its does not imply a corresponding difference among logical constructions. Ramsey is right in saying that this human inability is not logically necessary, and so should not be an essential feature of our symbolism. (The difference between this inability and our inability to e.g. write propositions of infinite length (2.18, 3.11, 3.14, 4.19) is evident.) That it is an essential feature of PM's symbolism (i.e. that universals should be symbolized in a way that involves the form of a proposition: ' ♦ AlX I whereas particulars are not: 'x') is a blemish on the PM system.

Russell's argument (1), that PM's utility proves the close correspondence between its system and the structure of reality,

Ramsey rejects because in the instance umer discussion, PM ignores a distinction essential to philosophy but not essential to mathematics: for although it is necessary to have complex predicate-symbols which have the form of propositions, this is not true of qualities (i.e. properties which are not complex). PM being extensionalist, that is, interested only in classes, it ignores the distinction between quali- 94. ties and other properties, am symbolizes both by symbols with the form of propositions; but though this procedure simplifies things for mathematical purposes, it is misleading from a philosophical viewpoint, for it ignores the distinction between qualities and other properties. Complex properties,which are logical constructions, must be symbolized by propositional functions, as we found in the case of 'xRa .v. xSb'; but simple properties need not be so symbolized :

"If • a is a two-termed atomic proposition, '• ' is a name of the term other than a, and can perfectly

well stand by itself; so, it will be asked, why do we write I. x' instead of r • r in this case also?"

(Ramsey, "Universals" p.130)

6.15. The fallacy in Ramsey's argument here, though difficult to see at first, is obvious once it is pointed out. Ramsey, by profound analysis, bas shown that the question 'Are objects diYided into two kinds, universals and particulars?' is equivalent to the question

'Is it logically possible to divide properties into simple properties

( "qualities") and others?' For if properties can be so divided, the symbol for a particular will pick out two sets of propositions; which (by an equally profound analysis) Ramsey bas shown to be the logical basis for the naive distinction between universals (the symbols for which pick out two sets of propositions) and particulars

(the symbols for which pick out only one set of propositions) 95.

("Universals" p.123-5; and above, 6.07, 6.09)

However, this argument shows no more than that if there

are qualities which do not need to be symbolized by propositional

functions ( ' ♦ ~' ) , but which can be symbolized by names alone

('•'), then the distinction between universals and particulars is invalid. Nowhere does he show that atomic facts do include

qualities in this sense. In both places where he mentions this

consideration (the last p:1.ragraph on p.126; and the last paragraph

on p.130), he assumes that atomic facts include qualities, i.e. may

be symbolized by propositions like , ♦ a, where '♦ ' is a name (and can stand alone as much as any name can). And this, by his own

analysis, is the vecy point at issue. Ramsey's argument is invalid

because he begs the question.

From the fact that he argues against the reality of complex

universals, we may feel that he had an intuitive conviction that there

are only simple qualities, and that other properties are logical

constructions; but he did not nrove this, for as we say (6.05) the

advocate of complex universals is by no means at a loss to reply to Ramsey's arguments.

6.16. It is necessacy to be clear exactly what Ramsey is objecting

to in the Russellian doctrine of universals. He does not object to

the doctrine that there are an infinite number of categories (vide supra 6.01); he objects to Russell's notion that the difference between one category and the next one up is analogous to the tradition­ al distinction between particulars and universals. Ramsey contends

that the traditional distinction does not assist us in understanding the difference between different types. But although his analysis

of the traditional distinction is far more profound than that of Russell, his arguments establish merely the possibility that atomic

facts should consist of objects which are all universals so far as the traditional distinction is concerned (i.e. they pick out two sets of propositions). Because he begs the question of whether there is a distinction between qualities and properties in atcmic facts, he does not establish that his thesis is~ correct one.

6.17. And Russell himself was not left without reply. First, there is the suggestion which Russell made in the second edition of

PM (pbk p.xix) that if we discover that all atanic propositions are of

the forms 1R1(x)', 1R2(x,y)', •~(x,y,z)', •••, we can define particulars as objects which can occur in any form of atomic fact, and n-adic universals as objects which can occur only in facts that

have at least (n + 1) terms. Ramsey dismisses this in ''Universals" (p.133) on the grounds:

"••• that we know and can know nothing whatever about the the forms of atODlic propositions; we do not know whether some or all objects can occur in more than one form of

atomic proposition; and there is obviously no way of

deciding any such question." (FM p.133) 97.

The following year, in "Note on the Preceding Paper" (1926) (FM p.135f)

(which was in fact part of a symposium, see Aristotelian Society

Supplemantary Volume VI (July 1926) p.17-26), Ramsey admitted that

he was now not so sure that we know and can know nothing whatever about the form of atOI1.ic propositions (unfortunately he gives no grounds

either for his earlier belief or for his change of mind), and that if

the form of atomic propositions were in fact discovered, and if it turned out to be in accord with Russell's prediction, then the universal-particular distinction would be able to be made in the way

Russell suggests.

A second Russellian reply could be based on the passage from

PLA (p.258) quoted above (6.01), that we cannot say of a thing of type n that it either is, or is not, the same as a thing of any

other type, and that •existence' has different meanings for each

type. Ramsey's idea of an atomic proposition •• a' consisting of two objects of different types, is on this position nonsense.

Though we can assert ' ♦ a', we cannot assert that there are two objects involved. 'Existence' has different meanings for each type; things of different types 'are there' in different senses of the word. To try to add them up as Ramsey does would be to commit the same mistake as the Chinese Philosopher who thought that because he had a horse and a cow, he had three things - a horse, a cow, and a

herd (PIA LK P• 26o). The problem can be seen as soon as we try to

use the PM definition of '2' in Ramsey's statement, for we are 98.

immediately involved in denying the identity of ♦ and a - and if they are of different types, this raust be nonsense. By strictly adhering to (even the Siraple) Theory of Types, Russell can reply that Ramsey's position is nonsensical, and does not even prove the possibility (see above, 6.15) that the distinction between parti­ culars and universals cannot be applied to genuine objects. That is to say, to deny the distinction is inconsistent with Russell's system, and also with Ramsey's system in that he accepts the Simple Theory of

Types.

6. 13. It may seem that Ramsey has not touched the traditional distinction at all, if by the traditional distinction we mean something like Russell's claim (made at the end of RUP (IK p.124), and quoted above, 6.01) that particulars exist in time and cannot occupy more than one place, whereas universals do not exist in time, and have no special relation to any one place. However, Ramsey, or any follower of the

TLP, would reply that though space and time are certainly forms of objects (TLP 2.0251), they are subsidiary to logical form (every object has logical form, but only spatial objects have spatial form, cf.TLP

2. 182); and that completely general logic would :I.snore the particular spatial and temporal forms of objects and concentrate on their logical

form. From the point of view of logical form, the difference between

12rticulars and universals is the difference between how such objects

can occur in states of affairs, i.e. how the s~T.:bols for them can occur

in atomic pro:i,io::;i tions; and thus we are driven back on the second of 99. the two traditional distinctions between particulars and universals, viz. that the symbols for particulars can occur only as subjects.

Plainly this argument has little power outside TLP-like systems, and if we accept its conclusion, that the traditional distinction is a grammatical one rather than a spatio-temporal one, we who do not accept logical atom.ism IIUSt advance independent arguments for our belief. We should not reject Ramsey merely for working within a system which we do not ourselves accept, for otherwise none of the great metaphysicians - Leibniz, Spinoza, Kant, Hegel, etc. - would be studied at all; rather we must first look at how the details of the system work and hang together, before we attack it as a whole (suspension of disbelief). The question of whether there are symbols which can occur only as subjects is clearly a valid question for a logician to consider; it is this question which is meant by "the universal-particular distinction" in this chapter. That there is another question, about the differing spatio-temporal natures of objects, which has often been called by the same name, is strictly irrelevant. It affects the value of Ramsey's remarks as little as his remarks answer it.

Ramsey's work on universals should not be discounted. His analysis of one form of the traditional distinction is profound and, subject to the limitations of working within a single a priori logical calculus, valid. The position 11&y be compared to that of Russell's Theory of Descriptions; Russell gave the conditions for a proposition containing a definite description being true in a logical calculus in 100. which everything is made explicit; but as Strawson has pointed out

( "On Referring" (in ECA) ), this does not automatically close the

question of what happens in corresponding cases in everyday language.

The insight gained in answering the question for the formal case may be very helpful in answering the CCIDDlon-usage case; but just because the formal case is formal, ideal, ignori.J1g air-resistance as it were,

it is an over-simplification for the real life case. Ramsey's

remarks on the problem of universals in formal languages - that the distinction between universal and particular is equivalent to the

postulation of qualities as distinct from other properties, and that

in many cases the distinction between primary and secondary occurrence

of an incomplete symbol is more appropriate - could be of use even to

philosophers more interested in conceptual confusion than in the development of formal systems. The problem of universals, after all,

arose first in ordinary language (albeit ordinary Greek language).

As I have indicated (above 6.15, 6.16), Ramsey does not establish his main point - that the distinction between universals and particulars

has no foundation and is incorrect within the Atomist system - beyond doubt. Russell could meet his objections successfully, though not easily, and I have shown how this can be done (6.17). I have not tried to answer the question of how right Ramsey was in attributing

his view to Wittgenstein, on the ground that it is more important

whether Ramsey was right or wrong in his opinions than whether he was

correct in attributing them to another. Further, the problem of 101.

Wittgenstein's attitude to universals in the TLP is so contentious and

involved that an adequate discussion of the matter would have to be

very much longer than its slight relevance to Ramsey's philosophy would

Justify (see above 6.03). I shall only say that Ramsey's theory of universals, considered as an interpretation of the TLP rather than as

a theory in its own right, is not an interpretation to be disregarded,

whatever our final conclusions on the subject may be, that his argu­

ments for his affirmative answer to the question 'In the elementary

proposition " ♦ a"• is " ♦ " a name'l' are better than those

most often advanced (e.g. by Daitz and Evans), and that the lack of

attention given to Ramsey•s position as an interpretation of the TLP

doctrine of universals by modern commentators is quite unjustified.

Although Ramsey does not establish the conclusion at which he

aims (viz. that the distinction between universals and particulars

is misconceived) in this paper, even within the framework of Logical

Atom.ism, "Universals" is a display of Ramsey philosophizing at his

best - lucid, comprehensive, and continually suggesting new approaches

and giving new insights to the problem. 102.

CHAPI'ER VII RAMSEY'S CONTRIBUTION TO LOOICAL ATOMISM

7.01. In the history of the philosophies which followed Logical Atomism, Ramsey's importance lies in that he typified the interpretat­

ion of Wittgenstein's TLP used as a starting-point with which to agree or disagree. I have indicated part of what this interpretation was, and some of the ways in which I believe it to be a misunderstanding

of the TLP (2.10-2.18, 2.20-2.22, 3.02-3.o4, 4.05, 5.03-5.06,

6.03-6.04). When writers before the second World War refer to

Wittgenstein's TLP they are, more often than not, thinking of a metaphy­ sic like Ramsey's rather than what I understand to be the system presented in the TLP itself; for example, the Doctrine of Showing (2.10-2.14) was often regarded as being no more than Ramsey's Theory

of Truth-Possibilities {2.20-2.22). I ,ave tried, however, to present Ramsey's metaphysic on its own, as a variant of the Logical Atomism of

Russell {PLA.) and Wittgenstein (TLP), rather than as a popularization of the TLP. Its influence as such I do not underestimate; but the importance of a metaphysical system, as indeed of a great scientific hypothesis, lies in its consistency, its elegance, its comprehensiveness and its closeness to truth, at least as much as in its influence on

other writers. Indeed, the latter is a consequent of the former. It

is the former qualities of Ramsey 1s system that I have discussed. I shall now summarize what I have found. 103.

7.02. Ontologically, Ramsey's theory of classes, including as it does ill indefinable classes (3.08), is less econom.ical than Russell's which includes only indefinable selection classes.

Furthermore, Ramsey's denial of the identity of polyadic indiscernibles makes every class potentially infinite (3.07), which is ontologically

very messy. On the other hand, the structure of Ramsey's universe

is simpler than that of Russell's. Ramsey, like Russell, distinguishes things into an infinite number of Types, but Ramsey does not then distinguish orders within Types (4.07). Ramsey also hoped to gain some simplicity by distinguishing qualities from other ( "complex") properties (6.08), which he regards as logical constructions of qualities (6.05). However, as he was not successful in showing how this distinction is to be made (6.15), and as Russell can convincingly

argue that because such a distinction would lead to a Type-mistake (6.17), i.e. that it is not even logically possible within a system employing a theory of types, Ramsey does not gain the simplicity of

removing the distinction between universals and particulars, or

reduce all properties to a few qualities.

7.03 •. Most of the internal difficulties of Logical Atomism, that is, the points at which those who felt sympathy with the assum.ptions of

Logical Atom.ism found difficulties with the system, were confessed by

Russell himself in PIA. Some of the more important were general

facts, propositional attitudes like belief, negative facts, and the

nature of the logical atoms themselves. To take these in order. 104.

Ra,msey claims to give an analysis of general propositions

(Chap. 2) which allows him to escape Russell's enforced postulation of general facts to correspond to them. As I have said, this analysis is not only unsatisfactory, but not as good an attempt as

Wittgenstein's earlier putative solution (2.18, 2.22-2.23, 4.17). It is in regard to general propositions that we see most clearly

Ramsey's confusion of logical impossibility with mere human limitation.

Russell had to postulate general facts because otherwise he could not explain how I cl> a. ♦ b ••• 1 implies 1 (x) • x' without knowing that the list 'a, b, ••• is complete, and because in the case where the list is infinitely long, it could not be written down. Ramsey's Theory of Truth-Possibilities (2.20-2.22), a simpler version of Wittgenstein's Doctrine of Showing (2.10-2.16), was an attempt to explain how we know that the list is complete (2.22). The general proposition expresses agreement with all propositions of that form. This needs no further analysis because, as we can express agreement with all propositions of 'Whatever form (by a tautology), so we can express agreement with a subset of these. My criticism of this was that tautologies do not express agreement, but rather fail to express disagreement; a tautology is not a map of everything, it is a blank sheet of paper (2.23). Ramsey answered the problem of the list being infinite by claiming that in principle an infinite list could be written out. This I flatly denied (2.18). It is impossible in principle to write out to its end an infinite list; for in any system in 'Which the word 'infinite' is used, it means 'there is 105. always one more', i.e. that there is no end, it is uni'inishable,

and to speak of the completion of an infinite list is to contradict

oneself.

7.04. On propositional attitudes, Ramsey was attracted by the TLP

solution, but could not accept it until he could also resolve the

problem of negative facts (5.03}. He was later able to give a

Wittgensteinian analysis of negative facts, in doing which he gave a

much clearer exposition of what this analysis was, with the help of

his negation-by-inversion notation (5.06). However, Ramsey thought

that the analysis of belief and negation should be continued by

trying to give a causal explanation (5.07}, in what is more the manner

of a Pragmatist or even of an "Ordinary Language" Philosopher than of

a Logical Atomist (neither Russell nor Wittgenstein accepted causation

in their Atomist periods). Ramsey was still a Logical Atomist when

the relevant article ("Facts and Propositions") was published(in

1927), but the first stirrings of a desire for a philosophy more rel­

ated to everyday life and less rigidly logical than strict Atomism

can be sensed on one page of it (p.148). Too much should not be made

of this, it is no more than a hint of dissatisfaction, but it is

interesting in the light of Ramsey's, and the other former Atomists', later views.

Ramsey's main improvement to the TLP theory of propositional attitudes and negative facts was largely expository - an achievement

not to be underestimated in view of the notorious obscurity of the TLP. 106.

By also demonstrating that the Wittgensteinian analysis of propositional attitudes could be accomplished, not by the Doctrine of Showing, but by Ramsey's weaker Theory of Truth-Possibilities, he pared some of the very little excess fat from the TLP. Here Ramsey's system is bette~ than that of Russell in PLA, in being more comprehensive.

Ramsey could deal with propositional attitudes (intensional functions) and negative facts, whereas Russell admitted that he could not. However the only reason for accepting the Wittgenstein-Ramsey solutions on these points could be a desire to be consistent with Iiogical Atomism, a reason which would have no force for any philosophers other than

Atomists. They strengthened their system by showing that it can account for more phenomena, but their solution of this problem has little intrinsic plausibility.

7.05. On the nature of atomic facts, Ramsey effects little improvement to Logical Atom.ism. He takes one particular interpretation of the

TLP, that within atomic facts there is no distinction between universals and particulars (6.o4), and gives very good arguments for it (6.05-6.14).

It is not an interpretation of the TLP with which all modern commen­ tators would agree (6.03). Arguments can be advanced (6.17) that what Ramsey says is not consistent with Logical Atom.ism (or in

Wittgenstein's case, the Doctrine of Showing (2.13)). Apart from. this, he does not tell us what sort of things atomic facts are supposed to be - whether they are perceivable ('Red here now') or not, for instance.

Ramsey believed that atomic propositions could not be reached by analysis, 107. but later changed his mind. Perhaps Ramsey's earlier position was due to the now well-known argument that given the Theory of Descriptions, which analyses names into quantified expression, together with the Thesis of Extensionality, which analyses quantifiers into truth-functions

of names, analysis must be infinite. Perhaps his change of mind was due to the realization that if analysis is infinite, there remains the

question of how we know what propositions 11.eant (If there are only

logical constructions all the way down, how does the analysis of such logical constructions help us understand what we are analysing, and what is the point of Logical Atomismt}. But certainly Ramsey does not answer our natural questions about atomic facts.

7.06. In his attempt to improve Atom.ism's philosophy of mathematics, Ramsey made his most significant advances. The most important difficulty left by the first edition of PM was the dilemma between accepting the Axiom of Reducibility and rejecting Mathematical Analysis (4.01-4.06)

Ramsey's solution was to divide the Reflezive Antinomies into two classes (4.07-4.08, 4.14-4.15) (a division since named that between "semantic" and "syntactic" antinomies), and to point out that the

syntactic need only the Simple Theory of Types, and no Axiom of Reducibility. The semantic on the otherhand are no :?lrt of ma.thematics. This part of Ramsey's work is correct (4.15). However, his solution

of the semantic antinomies, by invoking a new kind of functions which he called 'predicative' but which I referred to as 'Ramseian functions'

(4.09-4.12) rests on his solution to the problem of general propositions which was found to be unacceptable (2.17-2.23, 7.03). Ramsey reduces 108. general propositions (quantified expressions) to truth-functions; this is itself rather like the Axiom of Reducibility (4.16). Ramsey's conclusion here is better and clearer than Russell's, for what takes the place of Russell's AXiom of Reducibility in Ramsey•s system, namely the reduction of general propositions, is a much better candidate for a "Law of Logic" than the original Axiom; but the risk of increasing clarity is incurred - for RaJ1sey•s reduction of general propositions involves a self-contradiction in the notion of an infinite list (2.23,

4.17-4.19, 7.03). In his highest achievement, his work on Types,

Ramsey was betrayed by his inability to distinguish human and logical impossibility.

]_~

Axiom of Infinity is.tautologous (2.19-2.20, 3.07), and gave a new definition of 'class• which rendered the Axiom of Choice tautologous

(3.13). By rendering the three additional axioms of PM tautologous

(the Axiom of Reducibility is tautologous given Ramsey's analysis of general propositions (4.16, 7.06)) Ramsey could claim to have made PM wholly tautologous, and so wholly logical, in Wittgenstein's sense whereby tautologousness is the characteristic of propositions of logic.

I have, however, argued that Ramsey's claim is not justified because its premiss is false - his arguments against the PM definition 109. of identity are invalid (3.03-3.07), his theory of classes is not defensible (3.10-3.11), and his reduction of general propositions involves a self-contradiction (2.23, 4.17-4.19, 7.03). This is not to say that Ramsey's work on mathematics is worthless - it was a profound and well-executed attempt to deal with the inadequacies of PM. He discovered and displayed interconnections between concepts which nobody had every seen before. If Ram.sey's claims had been justified, the Logistic Thesis would have been proved, and the philo- sophy of mathematics would have been finished. But his work was not final; it raised as many questions as it solved. Perhaps this is the best that any philosopher can hope for.

7.09. Ramsey's work on both mathematical and metaphysical problems was flawed by one fatal mistake; his confusion of what cannot be done for logical reasons, in particular, the self-contradictor,ness" of completing an infinite process. If there has been a theme recurring throughout this thesis, it is this weakness on Ramsey's part

(see 2.19, 3.07, 3.11, 3.14, 4.17-4.19, 7.03, 7.06).. Although 'self-contradictory', like 'analytic', is system-relative,~ system which employes the word 'infinite• associates with this word the con- cept of there always being one more. To describe something as infinite is to say that it cannot (logically cannot) be completed, that if it were to be finished we would withdraw our description of it as infinite. To suggest that an infinite process can be finished is to fall into logical error. It is interesting in this respect to compare Ramsey's 110. reaction to the difficulties raised by PM with those of the

Intuitionists, such as Brouwer and Weyl. They refused to allow the possibility of something unless it was shown how it could be done

(by doing it), i.e. they restricted the concept of possibility. Ramsey, to save mathematics from the "Bolshevik menace" of Brouwer and Weyl, went too far in the opposite direction, and refused to allow the impossibility of anything, even if doing it would involve a self­ contradiction, like writing out an infinite number. Ramsey's mistake was due to a lack of understanding of the concept of infinity. The implications of this one mistake appear in many places 1n his system because of the close logical inter-relations between the different parts which were a common feature of the Logical Atomist, and other logic-based philosophies. Their greater internal consistency is achieved only at the expense of flexibility; a weakness in any pa.rt must affect many other parts. Ramsey•s analysis of general proposit­ ions, his theory of classes, his substitute for the Axiom of Reducibility and solution of the semantic antinomies, were all invalidated by his inadequate understanding of infinity ••

Ramsey was in an unequalled position to improve both the consistency and the plausibility of Logical Atom.ism, in that he, unlike the two other great Logical Atomists (Russell and Wittgenstein) had access to the mature thought of major figures in the school while he was a member of it. We, wise after the event, may be disappointed that he did not do a better job. His philosophy of matheI1Btics gave only one valid improvement on PM, the distinction between the syntactic 111 • and semantic paradoxes (Ch.3, Ch. 4, 7.06-7.07). His metaphysics are at best no simpler than Russell's, and in some ways worse (7.02).

On the traditional weaknesses of Logical Atom.ism, his attempt to resolve general propositions was unsuccessful (Ch.2,7.03), and his improvements on Wittgenstein on negative facts and propositional attitudes were mainly expositional (Ch. 5, 7.o4). Nor does he answer our most obvious questions about the nature of atomic facts (7.05), and although his investigation of this subject (Ch.6) leads to some very interesting conclusions about the difference between universals and particulars, Ramsey's ultimate answer to the problem of universals is inconsistent with the rest of his sytem. It may seem that Ramsey's achievement, as distinct from his influence, is negligible.

This is not just. Ramsey's arguments, though they do not prove what he hoped they had, are important. They are important even now that Logical Atom.ism is no longer a commonly held philosophical position. Their importance lies 1n the inter-relations between different concepts which they exhibit, and which Ramsey discovered.

To give just two examples: the inter-relations between logical impossibility, infinity, the Axiom of Reducibility, general propositions,

Wittgensteinian showing, the identity of polyadic indiscernibles, classes and the Axiom of Choice; or those between the felt difference between universals and particulars, complex properties, primacy and secondary occurrence of incomplete symbols and logical analysis. It is impossible, of course, to say what these inter-relations are in a few words, but I have tried to show and trace those which Ramsey discovered 112.

as clearly as I can in this thesis; for if philosophy is concerned with the structure of our Weltanschauung, it is inter-relations between the parts of this, our concepts, that are of philosophical importance. 113. BIBLIOORAPHY 1. Ramsey's Works. (If reprinted in FM, this is indicated)

Critical Notice of L. Wittgenstein's Tractatus Logico-Philosophicus

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Review of C.K. Ogden and I.A. Richards' The Meaning of Meaning.

In Mind N.S. Vol. 33 No.129 (January 1924) p.108.

Review of A.N. Whitehead and B. Russell's Principia Mathematica,

Volume 1, Second Edition. In~ N.S. Vol. 34 No.136

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"Universals" In~ N.S. Vol. 34 No.136 (October 1925) p.401.

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"Universals and the 'Methods of Analysis 1 "In Methods of Anal.ysis

the Aristotelian Society Supplementary Volume 6 (1926), a

symposium with H.W.B. Joseph (1) and R.B. Braithwaite (111), p.1. Ramsey•s paper begins on p.17. A very brief extract is reprinted in FM p.135.

''Mathematical Logic" In The Mathematical Gazette Vol. 13 No.184

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"On a Problem ef Formal Logic" Proceedings of the London Mathematical Society Ser. 2 Vol. 30 Part 4 p.338. FM p.82 114.

The Foundations of Mathematics and other Logical Essays (ed. R.B.

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