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THE LOOICAL ATOMISM J. D. MACKENZIE Date Submitted Thesis 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.
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