F. P. RAMSEY
CRITICAL REASSESSMENTS
Continuum Studies in British Philosophy: Duncan Richter, Wittgenstein at his Word Wilfrid E. Rumble, Doing Austin Justice Maria J. Frapolli (ed.), F. P. Ramsey: Critical Reassessments William R. Eaton, Boyle on Fire Colin Tyler, Radical Philosophy Stephen Lalor, The Egregious Matthew Tindal James E. Crimmins, Jeremy Bentham's Final Years
F. P. RAMSEY
CRITICAL REASSESSMENTS
Edited by Maria J. Frapolli
continuum
L O N D O N • N E WY O R K
Continuum The Tower Building, 11 York Road, London SE1 7NX 15 East 26th Street, New York, NY 10010 © Maria J. Frapolli and contributors 2005 All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage or retrieval system, without prior permission in writing from the publishers. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN:0-8264-7600-7 Library of Congress Cataloging-in-Publication Data F. P. Ramsey: critical reassessments / edited by Maria J. Frapolli. p. cm.-(Continuum studies in British philosophy) Includes bibliographical references and indexes. ISBN 0-8264-7600-7 1. Ramsey, Frank Plumpton, 1903-1930.1. Frapolli, Maria Jose. II. Series. B1649. R254F67 2005 192-dc22 2004059841
Typeset by Aarontype Limited, Easton, Bristol Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wilts
Contents
Acknowledgementss Contributors
vii viii
Prefacee
ix
Introduction Maria J. Frdpolli
1
1
Mind, Intentionality, and Language. The Impact of Russell's Pragmatism on Ramsey Juan Jose Acero
7
2
Ramsey and Wittgenstein: Mutual Influences Hans-Johann Glock
41
3
The Ramsey Sentence and Theoretical Content Jose A. Diez Calzada
70
4
The Contributions of Ramsey to Economics Jodo Ricardo Faria
104
5
Ramsey's Theory of Truth and the Origin of the Pro-sentential account Maria J. Frdpolli
113
6
Ramsey's Big Idea Colin Howson
139
1
Ramsey's Removal of Russell's 'Axiom of Reducibility' in the Light of Hilbert's Critique of Russell's Logicism Ulrich Majer
161
8
Ramsey and Pragmatism: The Influence of Peirce Christopher Hookway
182
vi 9
Contents Ramsey and the Prospects for Reliabilism Daniel Quesada
194
10
Ontology from Language? Ramsey on Universals Francisco Rodriguez-Consuegra
220
11
Ramsey and the Notion of Arbitrary Function Gabriel Sandu
237
Bibliography of Ramsey's Works
257
Index
260
Acknowledgements
It would not be possible to list everybody who has had a beneficial influence on the development and final stages of this book - there have been many who have helped, either directly or indirectly. Nevertheless, I would like to mention some of them: I warmly thank my colleagues Juan A. Nicolas and Juan J. Acero for their continuous support and friendship, and for having believed in this project from the beginning. I am also grateful to John R. Shook for his advice and help, and to the authors who have generously and enthusiastically contributed their works. The contact with them has been an unexpected pleasure added to the expected pleasure of working on Ramsey. I am also thankful to Philip de Bary, the Thoemmes editor for philosophy, who has handled the project with professional interest and affection, something that I highly appreciate. Neftali Villanueva and Francesc Gamos, PhD students of the Department of Philosophy, University of Granada, have been always ready to assist with their various technical and philosophical skills. To all of them I am deeply indebted, and I hope that the time and effort they have devoted to thinking about Ramsey for this volume has been at least as fruitful for them as it has been for me. Maria J. Frapolli
Contributors
Juan J. Acero: Departamento de Filosofia, Universidad de Granada, Spain Jose Antonio Diez Calzada: Universitat Rovira i Virgili, Tarragona, Spain Joao Ricardo Faria: University of Texas at Dallas, USA Maria J. Frapolli: Departamento de Filosofia, Universidad de Granada, Spain Hans-Johann Glock: University of Reading, UK Christopher Hookway: University of Sheffield, UK Colin Howson: London School of Economics, UK Ulrich Majer: Institut fur Wissenschaftsgeschichte, Universitat Gottingen, Germany Daniel Quesada: Universitat Autonoma de Barcelona, Spain Francisco Rodriguez Consuegra: Universidad de Valencia, Spain Gabriel Sandu: University of Helsinki, Finland; Director of Research, CNRS, JHPST, Paris.
Preface
F. P. Ramsey: Critical Reassessments is designed as a Festsschrift to mark the centenary of Frank Plumpton Ramsey's birth. Our way of honouring Ramsey has been to think with him and, wherever possible, to go beyond that, putting his ideas to work and seeing how far they can reach. In this regard, our main interest is not historical, although we have approached his work with textual and philosophical accuracy, and have tried not to put in Ramsey's mouth what is primarily in our minds. We share the opinion common among Ramsey's scholars that his work was ahead of his time. Many of Ramsey's ideas in philosophy and mathematics reappeared in mainstream twentieth-century thought, often independently of his writings, years and even decades after they were first discovered, created or entertained by the genius of Cambridge. Ramsey was too modern for his time, and his ideas were too revolutionary, difficult and deep to be understood by his contemporaries, even though he lived in a privileged city that was host to an extraordinary assembly of the greatest thinkers of the day. In this book, we have approached topics that we considered would be of interest to logicians, philosophers of mind, philosophers of science, philosophers of language, mathematicians, probability theorists and historians. We have touched on logic and foundations of mathematics (Howson, Majer and Sandu), philosophy of mind (Acero), philosophy of language and ontology (Frapolli and Rodriguez-Consuegra), pragmatism (Hookway), epistemology (Quesada), economics (Faria), philosophy of science (Diez) and the mutual influences between Ramsey and Wittgenstein (Glock). We are naturally well aware that the book does not exhaust the richness of Ramsey's thought, but if we contribute in some measure to spreading the ideas of this British philosopher and thereby bring them into contact with and fertilize contemporary thought, we will feel that the purpose of the book has been accomplished. The world of Ramsey scholarship is widening slowly but surely. Over the last decades we have been fortunate to see such excellent work as Sahling's (1990) pioneering monograph, Moore and Braithwaite's illuminating comments in the preface and introduction to (1978), Mellor's profound insights in his introductions to (1978) and (1990), and his edition of Prospects for Pragmatism (1990) and, more recently, Maria Carla Gavalotti's edition (1991),
x
Preface
and Dokic and Engel's monograph (2002). But Ramsey deserves much more attention, not only to do justice to the depth and brilliance of his work, but also so that we can learn from him the myriad lessons that he still has to teach. We only hope that the present volume will take a further step along the road of this exciting venture. Maria J. Frapolli
Introduction Maria J. Frdpolli
Ramsey was born on 22 February 1903 in Cambridge and died on 19 January 1930. During his short life he wrote some of the most profound pages on philosophy, economics, and mathematics of the twentieth century. He lived in an extraordinarily stimulating milieu, surrounded by figures such as Russell, Whitehead, Keynes, Moore, C. K. Ogden and I. A. Richards, W. E.Johnson and Wittgenstein. From them and others Ramsey picked up the master threads with which he was to weave his thought, although the resulting fabric was entirely due to his own genius. During the first quarter of the twentieth century the philosophical world was not prepared for a talent like that of Ramsey. As has been pointed out by several authors, Braithwaite (1931), Mellor (1990, p.xvi) and Sahling (1990) among them, other factors contributed to the relative silence with which Ramsey's work was received. One factor was that he did not publish his complete production, another was his humility that kept him from showing himself as the genius he was, and a third was the disproportionate influence of Wittgenstein on the philosophical world during the first three quarters of the last century. Joao Faria (see his contribution below) points to the technical innovations in Ramsey's approach to economics as one reason for its delay in being drawn into the mainstream of the discipline. Mathematics was his professional calling and in fact he became a Fellow at King's College, where he taught this subject from 1926 until his death. Although his only production in this discipline was the nine pages of §1 of 'On a problem of formal logic' (1928), his mathematical expertise was of great value in his incursions into economics and philosophy of mathematics, as well as in probability theory and general philosophy. In mathematics, he proved two theorems, both of them known as 'Ramsey's theorem', which lay down the basis for partition calculus. Its development and related discussions are referred to as 'Ramsey's theory'. Ramsey intended to give a solution to the Entscheidungsproblem, which years later Alonzo Church proved unsolvable. In the way of doing it, he proved these theorems that, although not needed for the general decision problem, have been shown to be extremely useful and fruitful. In his monograph on Ramsey's philosophy, Sahling illustrates the definite version of Ramsey's theorem with the
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F.P.Ramsey: Critical Reassessments
following example: 'In any collection of six people either three know each other or three of them do not know each other' (Sahling, 1990, p. 183). Ramsey's theorem follows from the Axiom of Choice, and in fact, Ramsey used the Axiom in his proof. L. Mirsky, in his section on mathematics in the Introduction to Foundations (1978), considers Ramsey's contribution to the subject as one of'the first magnitude to, and probably lasting significance for, mathematical research' (p. 10). In economics, he wrote two masterpieces, 'A Contribution to the Theory of Taxation' (1927) and'A Mathematical Theory of Saving' (1928), from which two branches of economic theory originated, the theory of optimal taxation and the theory of optimal saving, respectively. As has happened in the case of other contributions by Ramsey, the sheer novelty and depth of his insights meant that they made virtually no impression on the thought of his time. For Ramsey's view on economics, see Farias's chapter below. Nevertheless, although he was a mathematician 'by trade as well as by training', as Mellor says in the introduction to Ramsey's Philosophical Papers (1990, p. xiv), the core of his production and the topic that unified his apparently diverse writings is his account of belief. For this reason, Ramsey's philosophy of mind is crucial to understanding his thought. His philosophy of mind relies heavily on Russell's pragmatist ideas in The Analysis of Mind (1921). Ramsey is basically a functionalist as far as belief is concerned. Beliefs are dispositional properties that lead one to action. Ramsey answers a criticism by Russell, by saying that not all our beliefs have a real influence on our behaviour, and that it is not that we actually behave in a certain way, because we have such and such a belief, but that a belief is an idea that leads to action in appropriate circumstances. In this way, Ramsey improves the most naive behaviourist account using counterfactual considerations. For an analysis of Ramsey's philosophy of mind, see Acero below. His views on probability theory appeared in 'Truth and Probability' (1926), a paper rightly considered his most impressive piece of work. An analysis of (1926) and of Ramsey's theory of probability is found in Howson's chapter below. In (1926) Ramsey criticized Keynes's logical theory of probability proposed as a way of measuring the relative probability between two given propositions, and also the monolithical interpretation of probability as frequency. These criticisms do not have the same weight. While Ramsey clearly considers that the foundations of Keynes's position is erroneous, i.e. that we do not possess clear intuitions about the relative probability of every two propositions, his doubts about the frequency interpretation of probability are much weaker. Although some uses of probability support its interpretations as frequencies, Ramsey calls attention to another use, namely the use that interprets probability as a partial belief, thus understanding a theory of probability as a logic of partial belief. He shows in (1926) that a logic of partial
Introduction
3
belief understood in this way satisfies the mathematical restrictions for being a theory of probability. In this sense, the formal claim is justified. Its philosophical consequences are, nevertheless, much more far-reaching. Logic of partial belief such as Ramsey's has an application to the philosophical analysis of beliefs and offers a way of measuring them. The way in which Ramsey explains how to measure beliefs, using betting as an analogy, connects belief and action in a manner that exhibits his pragmatist perspective. His way of measuring beliefs is asking how far a subject is willing to go on the basis of this or that belief and offering an algorithm to determine their degree of belief. In this algorithm he takes into account not only beliefs (subjective probabilities) but also desires (subjective utilities) and respects the principle of mathematical expectation. The theory of probability, that is, the theory of partial belief, is, according to Ramsey, a branch of logic. This claim dovetails perfectly well with his logicist perspective on logic as well as with his pragmatist perspective on philosophy. In (1926) Ramsey defines logic as the science of rational thought (1990, p. 87) and under this umbrella he not only lists formal logic, the 'logic of consistency' as he calls it, but also the logic of truth. The logic of truth is a normative part of logic that tells us how we should think. It is a human logic that might occasionally be incompatible with formal logic. This human logic is supported by his pragmatism that, as we will see later, leads him to consider the world from a human perspective. This he reveals in 'Epilogue', an illuminating piece written in 1925 as a talk to be given at one of the Apostle meetings. And this human perspective supports his hospitable view on logic. As regards formal logic, Ramsey was an advanced disciple of Russell and Wittgenstein. Two of the few papers that he prepared for publication, 'The Foundations of Mathematics' (1925) and 'Facts and Propositions' (1927), explicitly show his logicist sympathies. In (1925) Ramsey offers his view on the foundations of mathematics, understanding this discipline as a part of logic. He focuses on Principia Mathematica, showing its drawbacks together with his proposals to solve them. Basically, Ramsey rejects Russell's characterization of mathematics as a set of unrestricted general propositions, and his ramified theory of types that requires the dubious Reducibility Axiom. Ramsey's alternative is to consider mathematical propositions as tautologies, following the path opened up by Wittgenstein, and to distinguish between semantical and logical paradoxes, making only the latter the concern of mathematics and leaving semantical paradoxes to epistemology. Once this move has been accepted, a single theory of types suffices to get round the relevant paradoxes. For Ramsey's view on the foundations of mathematics and the logicist programme, see Sandu's chapter below. Ramsey's position on logical constants developed from an orthodox Tractarian view into a pragmatist account. In (1927) there is a rapid treatment of quantifiers and logical
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F. P. Ramsey: Critical Reassessments
constants that closely follows the teachings of the Tractatus. Logical constants do not represent, he considers against Frege's view, and quantifiers are truth functions, i.e. reducible to series of conjunctions and disjunctions. Less than two years later, in 'General Propositions and Causality' (1929), Ramsey defends an account of quantifiers that coheres better with his general pragmatism. Quantified sentences do not express propositions, according to his new view. They express inferences we are prepared to do and not first order propositions. There is no longer a truth functional account of quantifiers but rather what we might dub as an inferential account close to the one later offered by Gilbert Ryle in (1949). As to epistemology, Ramsey only wrote one short paper called 'Knowledge' (1929), in which some authors, Sahling (1990) for instance, have found a reliabilist account, close to the one recently put forward in the philosophical arena by Alvin Goldman. In fact, at the beginning of this one-page work, he declares that he has always believed that a belief about which we are certain counts as knowledge when it is not only true but reached through a 'reliable process' (1990, 1929, p. 110) and he devotes the following paragraphs to analysing how we should understand the reliability of processes that produce knowledge. For Ramsey's view on epistemology and the fate of reliabilism, see Quesada's chapter below. Ramsey's ideas and influence have shown now and then in the brightest minds of the twentieth century. But above all, two topics outshine the rest. One of them is Ramsey's treatment of the theoretical terms of scientific theories as quantified variables and the other is his alleged proposal of a redundancy theory of truth. In 'Theories' (1929) Ramsey proved that, given a theory, it is always possible to offer an empirically equivalent one that does not contain theoretical terms. The procedure for obtaining the new theory is simply to substitute the theoretical terms in the former theory by existentially quantified variables. Ramsey's procedure had great influence on the positivist paradigm in philosophy of science. For an assessment and comments on Ramsey's philosophy of science, see Diez's chapter below. His view on truth appears in his unpublished paper 'The Nature of Truth' and very marginally in his published 'Facts and Propositions' (1927). In both, Ramsey puts forward an original view on the truth operator, considered as a means that natural languages possess to construe complex prepositional variables. This idea was recovered in the 1960s and is now known as the prosentential theory of truth. For Ramsey's account of truth, see Frapolli's chapter below. The general terrain in which Ramsey's thought resides ranges from positivism and scientific philosophy, on the one hand, to pragmatism, on the other. We might say that positivism is the form but pragmatism is the inspiration. From a philosophical point of view, two thinkers influenced Ramsey above the rest: Russell and Wittgenstein. Ramsey adopted Russell's views on logic
Introduction
5
and the foundation of mathematics but modified and improved them, following the path trodden by Wittgenstein most of the way. From Russell, he also took part of his philosophy of mind and his analysis of belief and the influence of William James. From Wittgenstein Ramsey took in his interpretation of logic, basically the idea that mathematics and logic are collections of tautologies, and in (1927) the analysis of logical constants as truth functions. His pragmatism, though, being at the heart of Ramsey's thinking, forced him to an alternative interpretation of the role played by quantifiers and by quantified sentences in the system of human beliefs. Part of his pragmatism, taken from William James through Russell, passed from him to Wittgenstein and was responsible for the revolutionary change in Wittgenstein's conception of philosophy. Wittgenstein himself acknowledges that Ramsey influenced him crucially in the evolution of his thought. Although Ramsey mentions Wittgenstein in most of his philosophical papers and considers himself a follower of the Austrian philosopher, Ramsey's influence on Wittgenstein was in fact far greater than Wittgenstein's impact on Ramsey. Having Russell and Wittgenstein as models, it is no wonder that the explicit ascription of Ramsey's thought was scientific philosophy. In 'Epilogue' (1925), where Ramsey analyses what science and religion have finally left to philosophy, he declares that the traditional philosophical questions have now become either technical, and so requiring specific training, or ridiculous. And he explicitly endorses the Tractarian view that philosophy is not a subject that is genuinely separate from science. He undertakes the explanation of what the realm of philosophy is in 'Philosophy' (1929), where he does not offer a very different view from that proposed in 'Epilogue'. In (1929) the task of philosophy is reduced to that of providing definitions or ways to construe definitions, although not only nominal definitions are allowed. In a certain sense one perceives here the idea that philosophy is a task of elucidation, but there is also an open-minded treatment of what a philosophical elucidation is, which is closer to pragmatism than to strict logical positivism. Ramsey's thought inhabits this tense terrain between positivism and pragmatism and, we might guess, he would have abandoned positivism altogether had he lived a little longer. In his introduction to Ramsey's Philosophical Papers (1990), Mellor reminds readers 'how much Ramsey repays close and repeated reading' (p. xii). This is very true. And the more one is aware of how philosophy, foundations of mathematics and probability theory have developed during the last hundred years, the more one is able to appreciate Ramsey's immense talent. Our hope now is to contribute to an ever-increasing acknowledgement of the scope and value of his thought and to honour him as a thinker in the best way, by taking his ideas as far as we possibly can, by thinking about, talking about and spreading them.
6
F. P. Ramsey: Critical Reassessments References
Braithwaite, R. B. (ed.) (1931). The Foundations of Mathematics and Other Logical Essays. London: Routledge and Kegan Paul. Dokic,J. and Engel, P. (2003). Frank Ramsey. Truth and Success. London and New York: Routledge. Gavalotti, M. C. (1991). The Philosophy ofF. P. Ramsey. Theoria. Special Issue ,LVII, 1. (ed.) (1980). Prospects for Pragmatism. Essays in the Memory ofF. P. Ramsey. Cambridge: Cambridge University Press. Mellor, D. H. (ed) (1978). Foundations: Essays in Philosophy, Logic, Mathematics and Economics. London: Routledge and Kegan Paul. (ed.) (1990). Philosophical Papers. Cambridge: CUP. Ryle, G. (1949). The Concept of Mind. New York: Harper Collins. Sahling, N.-E. (1990). The Philosophy of F. P. Ramsey. Cambridge: CUP.
1 Mind, intentionality and language. The impact of Russell's pragmatism on Ramsey Juan Jose Acero
1. Richard Braithwaite once commented that the philosophical thought in Cambridge in 1919 and for the next few years was dominated by Russell, that his books and articles \vere eagerly awaited and even discussed and criticized in E. G. Moore and W. E.Johnson's university lectures (Hacker 1996, p. 68). Not only in the general orientation of his thought but in tackling specific problems Ramsey stood long on Russell's shoulders. The philosophy of mind is a realm in which Ramsey's debt to Russell is particularly worth underlining. What are beliefs, what distinguishes one kind of mental state from others, i.e. believing from disbelieving, what is the relation of belief to action, in virtue of what do beliefs have intentional or representational properties, are all questions which help give this debt a neat profile. Concerning these questions Ramsey benefited from a number of proposals that Russell made both in 1910 in his essay'On the Nature of Truth and Falsehood' (Russell 1996) and, a decade later, in his influential book The Analysis of Mind (Russell 1921). If this work hypothesis is seriously taken, that is, if it were true that Russell's answers to those questions carried a significant weight on Ramsey's thoughts, it would be mandatory to admit that there is more than a personal sign of recognition in Ramsey's comment, made at the end of his 'Facts and Propositions', that his pragmatism in logic 'is derived from Mr Russell' (Ramsey 1990, p. 51). To determine the scope of such a debt is what the present chapter deals with. A brief overview of its content will help making its contents more perspicuous. The first steps towards Ramsey's conception of intentionality are taken in §2, where two arguments that Russell adduced against Brentano and Meinong's view of mental states are introduced and explained. Ramsey's acceptance of both their premises and conclusions explain why his views about belief inherit Russell's radical empiricism and naturalism. From the first of those, the Argument from the Non-existence of Mental Acts [NEAM], the non-existence of a transcendental subject, i.e. that the psychological subject is a construct, follows. The second one, the Argument from the Particular Nature of Content [PNC], concludes that human psychology is continuous with non-human, animal psychology. §3 goes more deeply into Ramsey's
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F. P. Ramsey: Critical Reassessments
debt to Russell by considering the latter's theory of belief's subjective dimension, i.e. what Russell referred to as the belief's content. A new argument, the Argument from Intentional Properties Pragmatism [IPP], is added to NEAM and PNC to picture a richer image of Russell and Ramsey's pragmatism in the philosophy of mind. IPP requires that the intentional properties of beliefs and other mental states derive from the causal curricula of the symbols that make up their contents. When NEAM, PNG and IPP are combined, a view of the mind arises that gives language a central role in the constitution of intentionality. The objective dimension of belief is dealt with in §§4, 6 and 7. §4 focuses on Ramsey's careful analysis of Russell's reasons for getting rid of propositions understood as objective complex entities and the difficulties this goal created for Russell. Those difficulties inevitably turn up when one, as Russell did in The Analysis of Mind, abides by the principle that the content of a mental state determines its objective component, the Principle of the Objective Component Determination [OCD]. Ramsey grasped the insight under Russell's view that as far as its objective dimension is concerned a belief is true if, and only if, its psychological subject is multiply related to a number of objects, properties and relationships. However, and unlike Russell, Ramsey realized that turning this insight into an articulated theory of belief's objective dimension required limiting OCD's scope. Before tackling this question, a new argument is put forward in §5 to complete picturing Ramsey's debt to Russell and to make the remaining work easier. It is the Argument from the Intrinsic Feature of Mental States [IFMS], according to which, besides its content, a mental state's subjective dimension also includes an intrinsic feature, a sensation or feeling perceived by introspection. Its truth-conditions constitute its objective dimension. As already said, Russell did not arrive at a general theory of the objective dimension. Ramsey, however, made a significant contribution to this end by combining Russell's pragmatism with Wittgenstein's theory of logical constants as non-denoting expressions. This was not a minor move, as explained in §6. It led to a severe restriction of the scope of OCD, thus opening a path that Russell had not thought of. Not only did it go against the core of Russell's views on mental content, but it brought about a deep modification of his pragmatism as well. §7 finally deals with how the views of Wittgenstein on what the truth-conditions of a proposition are were used by Ramsey to complete his theory of mental intentionality and how his pragmatism freed him from a number of difficulties Wittgenstein had to face. As a consequence of all this, it is reasonable to claim that Ramsey's debt does not mean a blind commitment to Russell's views. Whenever Ramsey sides with Russell, as openly happens in 'Facts and Propositions', this attitude is the consequence of carefully pondered arguments. Moreover, even during the stage in which his closeness to Russell was more evident disagreement between them is not exceptional. 'Truth and
Mind, Intentionality and Language
9
Probability' marked the beginning of a quick process in which Ramsey shifted away from ideas which he had previously embraced. The posthumously published book On Truth (Ramsey 1991) meant an even more radical departure from Russell's pragmatism. However, this matter lies outside the scope of the present chapter. Finally, Ramsey's views about a number of topics which clearly announce a resolute evolution towards an own-brand pragmatism are briefly summarized in §8. 2. What is, then, Russell's analysis of belief which Ramsey took on? (As synonyms of 'belief Ramsey also uses the terms 'judgement' and 'assertion'.) A number of reasons led Russell away from Brentano and - his disciple — Meinong's view of the mind. In his Psychologic vom empirischen Standpunkl (Brentano 1995), Brentano's main insight (quoted in extenso in Russell 1921, pp. 14ff.) had been that mental states are acts of a subject by means of which this subject becomes conscious of an object in a certain way. The object might be an unusual, sui generis, one. For these authors a mental state of a subject consists of three components: an act, a content and an object. The act is the characteristic mind modality under which the subject is conscious of its content. In (1): 1 Jones thinks that Caesar was murdered a mental state of Jones's is stated. Unlike that in (2), the act in (1) is the same, i.e. of the same kind, as in (3), namely the judgement (the thought or the belief) that Caesar was murdered: 2 3
Ernest denies that Caesar was murdered I think that Caesar was murdered
However, the acts stated in (1) and (3) are ascribed to different thinkers. On the other hand, (2) states a denial, not a judgement, whose content is the same as the judgements in (1) and (3). Thus, two mental states may be different, though they share the thinker and the kind of act carried out, if their contents are different. Echoing Meinong, Russell characterized content as that which 'exists when the thought exists, and is what distinguishes it, as an occurrence, from other thoughts' (Russell 1921, p. 16). This is why contents are catalogued as events happening in the thinker's mind. Neither Brentano nor Russell were careful enough in characterizing mental contents as events, because talking about events is systematically ambiguous. There is a sense in which Jones's thought in (1) as well as my thought in (3) are always distinct events, and there is another sense in which they are always the same. The threat of contradiction is dispelled out as soon as the token-type distinction is put an edge on this set of data. Acts in (1) and (3) are different tokens of the
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P.P. Ramsey: Critical Reassessments
same type of event, i.e. thinking that Caesar was murdered. Seen to this light, (1) and (3), but not (2), describe two instances of the same type of mental property, even though these three sentences might be used to state three different single mental events. Finally, the third component of a mental state is what Brentano calls its object. Unlike the act's content, its object does not have to exist in the thinker's mind when the act is done. The object can be a fact of the past, as in the above examples, but it does not have to. It may not even be something mental, but a physical or abstract entity (for example, the identity relation) or an imagined situation or state of affairs (as when someone thinks that there is a golden mountain) or a contradiction (as when someone thinks that there are round squares). Be that as it may, every mental act has an object, is a form of being conscious of an object. (See Russell 1921, pp. 16f.) Against this view of thought Russell deployed a battery of arguments. His logic of belief is like a jigsaw puzzle whose pieces have diverse origins. Some of them are of his own invention and fall into place as soon as we keep track of the evolution of Russell's philosophy; some others mean a debt of his to James' pragmatism and radical empiricism; a few others result from his calculated acceptance of the behaviourist conception of the mind. He was convinced that Brentano and Meinong's view was unable to resist a detailed analytic examination and unable to account for 'a host of facts in psycho-analysis and animal psychology' (Russell 1921, p. 15). This reference to animal psychology anticipates Russell's claim that 'I believe that the behaviourists somewhat overstate their case, yet there is an important element of truth in their contention, since the things which we discover by introspection do not seem to differ in any fundamental way from the things which we discover by external observation' (Russell 1921, p. 29). In Russell's programme of offering an explanation of mental states alternative to Brentano's a noteworthy role therefore corresponded to the naturalistic forces that were of a piece with both William James's conception of the mind and behaviourism. The final image that all these pieces come to picture openly contradicts the one promoted by Brentano (and Meinong). The first of Russell's arguments against Brentano's view of the mind is as follows:
The Argument from the Non-existence of Mental Acts [NEAM] No mental state has an act as one of its constituents. For a mental state to exist it is sufficient that there be one content, and there is mental content if (and only if) there happens to be the corresponding mental event. This argument goes against Brentano in holding that there is no act in any mental state. 'The first criticism I have to make is that the act seems unnecessary and fictitious' (Russell 1921, p. 17). Russell's reason to say so unfolds in
Mind, Intentionality and Language
11
two steps. On the one hand, an act demands an agent or subject, i.e. the thinker; on the other hand, the agent is undetectable. Our common forms of expression lead us astray to wrongly infer that there has to be a mental substance from the very fact that (l)-(3) have each a grammatical subject. This confusion feeds Brentano's tripartite analysis of mental states into a mental subject, a mental content and an object. However, Russell joined James in rejecting that behind any mental state there has to be an acting subject. Both of them emphatically held that existence can only be accorded to what can be observed, and then added that the acting subject is nowhere found. The conclusion follows that Jones, I and the thinker in general are useful fictions, logical constructions: Meinong's act is the ghost of the subject, or what once was the full-blooded soul. [...]! think that the person is not an ingredient in the single thought: he is rather constituted by relations of the thoughts to each other and to the body. [...] All that I am concerned with for the moment is that the grammatical forms 'I think', 'you think', and 'Mr. Jones thinks' are misleading if regarded as indicating an analysis of a single thought. It would be better to say 'it thinks in me', like 'it rains here'; or better still, 'there is a thought in me'. This is simply on the ground that what Meinong calls the act in thinking is not empirically discoverable, or logically deducible from what we can observe. (Russell 1921, p. 18) The criticism is, then, that, once the analysis is carried to its ultimate components, no agent remains. It does not follow from this that we have no right to say for example that Jones thinks that Caesar was murdered. Of course we have it. It just means that the Jones, I and any thinking subject are logical constructions, posits useful to make sense of the fleeting experience. This way out from the clash between radical empiricism and an uncritical reading of our mother-language grammar forces Russell to tell what the materials are from which logical constructions are assembled, that is, which those mental events are that give both thoughts and subjects their identity. The answer to this question constitutes Russell's second argument against Brentano.
The Argument from the Particular Nature of Content [PNC] There is no break in continuity between explaining human behaviour and explaining other animal species' behaviour. Therefore, a mental state's content has a particular nature, not a universal one, i.e. it consists of images, words or compounds of them, not of a counterpart of human understanding's conceptual abilities, but a product of perception and imagination.
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P.P. Ramsey: Critical Reassessments
Concerning the nature of mental content, Russell's choice consistently fits his naturalistic approach to intentionality. According to this naturalism, understanding human psychology and behaviour must be, in so far as possible, continuous with what animal psychology postulates to make sense of animal behaviour. Thus, if it lacks any justification to attribute a non-human a psychic state with an universal content, it also lacks justification to attribute a human a psychic state endowed with the same kind of content. Russell illustrates this point through the case of a horse that, on smelling a bear, behaves as if it knew the universal property being-a-bear. This sort of explanation must be discarded. It is simply incredible, Russell thinks, that horses possess the ability to get acquainted with universals. It follows, in virtue of the continuity between animal and human psychology, that when a human being either thinks or says to herself'I smell a bear', the thought's content, what is in her mind, cannot be universal or have universal constituents. Images and words, which constitute the content of our thoughts when either they come into the mind or are said to ourselves, are not universal. Mental content consists of tokens of words and images, as well as combinations of words and images, and being in a mental state is a tokening of them. Therefore, PNC translates into the theory of content Russell's attraction towards naturalism. The two arguments so far taken into account, NEAM and PNG, set forth neat differences between the analyses of intentionality put forward by Brentano, on the one side, and by Russell, on the other side. Ramsey was with the latter unambiguously: he accepted both their premises and their conclusions. He definitely took on these arguments and stood close to the terminology with which Russell had articulated them. In 'Facts and Propositions' he followed Russell in distinguishing with these very words the subjective dimension or dimensions of mental states from their objective dimension. While Russell chose the terms 'content' and 'objective' for those two dimensions, Ramsey did not use them. However, beyond words, they shared the fundamental issue. The content of a mental state of a thinker on a certain occasion consists of'[the] words or images in my mind' (Ramsey 1990, p. 334) on that occasion. When only words fix it, instead of'content', Russell used the word 'proposition' (Russell 1921, pp. 240f.). (As for this, Ramsey followed Wittgenstein and chose 'proposition' instead of Russell's 'objective'. See below, §7.) It has been remarked that this analysis of belief's subjective dimension is moulded to fit into PNG's demands, and sanctions the continuity between non-human and human psychology, between image-content and word-content. Because of it, that words play such an outstanding role in shaping human mentality should come as no surprise. Behind PNG lies Russell's conviction that '[t]he more familiar we are with words, the more our ((thinking)} goes on in words instead of images' (Russell 1921, p. 206). The more characteristically human mental content is the richer the deployment of linguistic resources.
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Ramsey's naturalism makes it plausible to think that he shared such a conviction. However, the only argument he gave to conclude that the content of human thoughts are made up with words comes up in another context, namely that of providing an account of the fact that human thoughts characteristically enter into logical relationships. In other words, that unlike non-human beliefs, i.e. a chicken's belief that a certain sort of caterpillar is poisonous, human beliefs, 'which are expressed in words, or possibly images or other symbols consciously asserted or denied, ... are the most proper subject for logical criticism' (Ramsey 1990, p. 40). A belief is subjected to logical criticism when, and only when, it has logical properties and keeps logical relationships to further beliefs. Now, Ramsey argues, in order for a belief to be subjected to logical analysis, it has to be articulated, i.e. it has to be compound. (Wittgenstein would say that a belief has to be a fact. Cf. Wittgenstein 1922/1961, §2.141.) Beliefs have content,-and mental contents are made up of elements of some sort. Language provides the answer to the question, what sort of things enter into their constitution. The logical properties of human beliefs are consequences of the linguistic or symbolic nature of their constituents. Lacking language abilities, non-human animal beliefs lack the complexity that would make it possible for them to have logical properties. Thus, Ramsey arrives at a point where PNC, a crucial argument against Brentano and Meinong's view of mind, smoothly follows. All this makes it natural to conclude that James' assault on an immaterial substance, the human soul, which is the subject of conscious acts, and Russell's naturalism, which demands the continuity of human thinking to animal psychology, constitute two fundamental commitments of Ramsey's philosophy of mind. 3. In addition to having a mental or subjective dimension, beliefs and in general mental states have an objective dimension as well. It is in virtue of playing intentional or representational roles that beliefs possess what Brentano and Meinong called the object and Russell the objective reference (Russell 1921, pp. 232f.) — prepositional reference when mental contents consist of words exclusively. By having objective reference beliefs relate to a more or less inclusive part of the world, and by being connected to the world, beliefs have truthconditions and mental states in general have satisfaction-conditions. This is not a contingent feature of mental states, but an essential one. Jones's belief stated in (1) has in its content the words 'Caesar', 'was', and 'murdered'. When combined in the proper way, these words represent that Caesar was murdered, the fact or existent state of affairs that constitutes the belief's objective factor. The logical details of such a dimension will be taken up below (in §§6 and 7). What has to be considered at once is how both the mental and objective factors relate to each other. As for this, Russell held that mental content somewhat determines objective reference. Put in a different way, that a belief or judgement's objective reference 'is, in general, in some way derivative
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from' the meanings of words or images of the mental content (Russell 1921, p. 235). Therefore, it is metaphysically impossible for a belief's content to be made up of constituents c\, ,.., cm whereas its reference objective does not depend at all upon any ofci,...,cm. (The reason why the condition is stated this way will be apparent in §6.) An explanation of the mechanism which yields objective references out of the intentional properties of words or images is, together with NEAM and PNC, a mandatory step that has to be taken to put forward an alternative to Brentano and Meinong's view of mind. Such an explanation not only hangs on accounting for the meaning of words and images, but it also requires that this task should be carried out in the spirit of Russell's continuity between non-human and human mentalities. The key to all this lies in an aspect of Russell's philosophy in which his influence on Ramsey reaches the highest point: his commitment to pragmatism, namely to the principle that 'the meaning of a sentence is to be defined by reference to the actions to which asserting it would lead, or, more vaguely still, by its possible causes and effects' (Ramsey 1990, p. 51). The stuff that makes this declaration of principles comprehensible is contained in Chapter 3 ('Words and Meanings') of Analysis of Mind, whose target is to account for the meaningfulness of words, images and their combinations. Russell's analysis of that relationship walks a good way within behaviourist courses, though a significant part of it explores new territories. His determination to understand human psychology in accordance with principles deemed to be valid for non-human psychology now confronts what at first sight would seem to the most formidable obstacle in the way of such a project, namely to account for human thought. Supposedly, the theory of meaning that Russell sets out there allows him to accomplish at least part of such a goal with the help of two premises: first, that, as far as psychology is concerned, thoughts, i.e. beliefs, and judgements (in general, mental states), are images or words (or combinations of both of them); and second, that thinking involves associating images and words with their meanings. From this point of view, there is no separate mental reality that underlies language. On the contrary, the way language affects thought, giving rise to it, is very important in this approach. 'Almost all higher intellectual activity is a matter of words, to the nearly total exclusion of everything else.' It is easily understood that the most complex kinds of beliefs and judgements 'tend to consist only of words' (Russell 1921, p. 238). The thought that planets revolve around the sun may have associated images of spherical bodies turning around a star, but the only images that may occur in its content'are, as a rule, images of words' (Russell 1921, p. 238). This example is not without a moral. It illustrates the general phenomenon that we humans think in and through words. Following a classical distinction of Sellars - and denying some hasty comment on the significance of Russell's views (cf. Carruthers 1996, p. 2) - language is not only a means of communication,
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but above all the medium of thought. Language is the primary form of thought and so mental intentionality is not primitive intentionality, but intentionality (Sellarsl969). The device that makes language, though not only language, a crucial medium for thought is that of conditioned response. Something, an image or a word, becomes a symbol b of something else x when b acquires the function (or the power) of replacing x, and this happens when b takes the place of x in the system of causes and effects in which x is engaged. For this to be the case a link between x and b has to arise that confers b x's causal history, i.e. both its causal antecedents and powers. In non-linguistic forms of thought, i.e. in image-thinking, x brings about an image i which, in virtue of its similarity to x, turns i into a symbol of x thereby gaining the role of meaning or representing x. Image i replaces its meaning or prototype because of its similarity to it. This means that the causal sensitivity and power of the image i becomes similar to those of x. 'If we find, in a given case, that our vague image, say, of a nondescript dog, has those associative effects which all dogs have, but not those belonging to any special dog or kind of dog, we may say that our image means "dog" in general. If it has all the associations appropriate to spaniels but no others, we shall say it means "spaniel" ' (Russell 1921, p. 209). When a symbol s is not an image but a word w the explanation of how w becomes meaningful is more involved, and depends upon a general law — Russell calls it the general law of telescoped processes — according to which if A causes B and B causes C, it will happen in time that A causes C without the intermediate operation of B. ('A', 'B' and 'C' are variables that range over events.) The learning process of a word w by its user S consists in the setting-up of a general causal law connecting the tokening of w, maybe because S says w to himself or herself, and on perceiving x. The sort of cases that Russell was mostly concerned with marks out what he called the private use of language, and it is exclusively this sort of use that truly gives thought the chance to reach its highest level. This happens thanks to a kidnapping-and-doubling effect, as it were: an image i of x, which is an effect of 5"s having perceived x in the past, is replaced by a word w and given the causal curriculum of x (or something close to it). For this to be possible, a causal link between tokenings of images and tokenings of words must have taken root. Once this has happened, w gains the capacity to replace i in the economy of mental causal transitions. This is why the more familiar we are with our language the higher the extent to which our thought goes in it. Well, that to which Ramsey alludes and in which his pragmatism finds inspiration is nothing but a consequence of the explanation of thought's intentionality just put forward. The explanation belongs to Russell's alternative to Brentano concerning the intentional properties of mental states. Therefore, to NEAM and PNC the following argument has to be added.
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P.P. Ramsey: Critical Reassessments
The Argument from Intentional Properties Pragmatism [IPP] Thoughts are made up of symbols, i.e. words and images, whose meaning consists of their causal links. The intentional properties — the meaning - of thoughts (beliefs and mental states in general) are what they are by virtue of their causal antecedents and powers. Brentano and Meinong had it that the distinctive feature of a mental state is its making reference to a suigeneris object. It could be thought that, as a consequence of abiding by NEAM and PNG, which give images and words the truly key role in casting light on intentionality, Brentano's view has been definitely got over. This is not true. In the doctrine that holds that the meaning of a symbol is an idea in the mind of its user Brentano and Meinong have still a chance to give a new boost to their view of intentionality. Ideas in the mind would connect words (and images) to the objects or situations they represent and in so doing would provide the thinker with the sort of object whose peculiarity gives the mental its own character. IPP stands in the way of such a variant of Brentano's theory. The meaning, the intentional properties of images and words, constitutively depend on their causal history, i.e. their causal antecedents and their causal efficiency. In a word, their meaning is their use: 'The relation of a word to its meaning is of the nature of a causal law governing our use of the word and our actions when we hear it used' (Russell 1921, p. 198). Therefore, whoever understands a word is in the habit of using it properly. This means that, as a speaker of the word, she understands it because she acts in the right way and, as a hearer, she has got the ability of [being] affected by it in the way intended' (Russell 1921, p. 198). It is plain that this explanation makes no room for an idea's making itself present to a subject's consciousness. If while crossing a street with an absent-minded friend I say: 'Look out, there is a motor coming!', and she glances round and jumps aside, her behaviour is accounted for by imputing to her the capability of understanding my words. This explanation does not accord ideas any role to play in human understanding. To understand is to have got the right habits: '[tjhere need be no {(ideas)}, but only a stiffening of the muscles, followed quickly by action' (Russell 1921, p. 199). IPP makes it unnecessary to posit any idea or mental entity, of which a definition can be framed and communicated. The very idea of a lexicographical definition tempts us to take the wrong way of assuming that meaning is not the result of polishing the countless irregularities of its use. Far from being true, this idea is refuted by the fact that 'there is always a greater or lesser degree of vagueness' (Russell 1921, pp. 197f). That is why Russell compares the meaning of a word with a target, which can be reached with a gradually .diminishing degree of precision from its bull's eye. Rather than failing to score if the target is missed, a speaker is not deprived of the skill to use a word appropriately for not being able to define its meaning.
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This way of accounting for the intentional properties of mental states, and a fortiori of the faculty of understanding, answers behaviourism's naturalistic demands. A word or an expression has been understood if the right habits have been learned, i.e. if it is used in the right situations and it brings about the right reactions as the context requires. Now, having got the correct habits, adds Russell, 'may be taken to belong to the nerves and brain5 (Russell 1921, p. 199). It follows that the laws which govern understanding, and particularly language understanding, can be reduced to causal laws of nervous system physiology. Let us put this last point aside for a moment to highlight the word 'habit' as the key term in the previous explanation of language understanding - it is the identity stamp of the behaviourist's view of mind as well. Learning language, says Russell, is like learning to play cricket, 'a matter of habits, acquired in oneself and rightly presumed in others' (Russell 1921, p. 197). Given that thought is an essentially linguistic, symbol-involving activity, it follows from what has just been said that thinking too is essentially a matter of acquiring certain habits. Ramsey shared this point of view when he declared that the human mind works 'according to general rules or habits' (Ramsey 1990, p. 90). The package of arguments so far taken into account, NEAM, PNC and IPP, is implicit in Ramsey's recognition that his pragmatism derived from Russell's. Were it true that he accepted those three arguments, and it has been argued that he did, the nature of his debt to Russell would have a neat outline. (On the other hand, were such a hypothesis to be abandoned, we would be in the dark about what the point of Ramsey's pragmatism is.) On the interpretation here sketched, Ramsey's pragmatism finds inspiration in Russell's and the latter's commitment to behaviourist naturalism. (See below §6.) However, in order to get a more faithful picture of the whole situation, it has to be added that this commitment was not plain. His naturalism led Russell to maintain, firstly, that the content of a mental state consists of words or images and, secondly, that the intentional properties of mental contents are constituted by causal relations. In this very sense, meaning is use. Nevertheless, in the detailed analysis Russell worked out in Analysis of Mind., he goes beyond the limits of behaviourism in two crucial aspects. Russell distinguished three uses of words: the demonstrative, the narrative and the imaginative use. Only the first of these, the use words acquire when we learn to utter them in the right situations, to react properly after having heard them and to associate them with individuals, so that the causal efficiency these have is transferred to their corresponding symbols, can be accounted for within behaviorist strictures (see Russell 1921, pp. 199f.). The sphere of thought, as pointed out by Watson in his classic Behaviorism., 'is the field of language habits - habits which when exercised implicitly behind the closed doors of the lips we call thinking' (Watson 1970, p. 215). Russell shares with Watson a concept of thought as a faculty of
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P.P. Ramsey: Critical Reassessments
inner speech's being incipiently pronounced. However, Russell widens this concept by including in it the faculty of imagining inner speech's being pronounced. This difference, though a significant one, is minor when what matters is the nature of understanding and the relation between words and meanings, i.e. the demonstrative use of language. The narrative and imaginative uses, on the other hand, do not square with the more restrictive, i.e. the Watsonian, view of thinking. This is due to the very special role images play in both uses of language. In the narrative one, words are designed to capture the images going through the user's mind — whether the speaker is being conscious of them or he or she is able to recall them — and recreate them in the hearer's mind. In the imaginative use of language a previous connection of words with images is exploited to create new images. That the bounds of behaviourism are infringed is proved by the fact that causal links are set up not between words and actions but between words and images. As a consequence of adding this parameter, an image of thought emerges as a system of two-way causal relationships, linking words to images and backwards, that mediates between the world that human beings inhabit and their action on it. In such an intermediate system the pragmatist principle, according to which '[t]o understand the function that words perform in what is called ({thinking}), we must understand both the causes and the effects of their occurrence' (Russell 1921, p. 203) is still valid. The relation of non-linguistic stimuli to the thinker's actions is much more indirect. 4. While mental content is in the thinker's mind, its objective component is not. By virtue of having got an objective the world makes a difference for any mental state, thus making possible for beliefs to be true or false, for desires to be satisfied or unsatisfied, and so on. Concerning beliefs, Russell said that their being true or false 'does not depend upon anything intrinsic to the belief, but upon the nature of its relation to its objective' (Russell 1921, p. 232). What is the nature of the objective component which, together with the way the world is, determines beliefs' truth values? In Analysis of Mind Russell showed a less resolute attitude than in many other previous publications to answer that question. In fact, he never goes so far as to explain what the objective is. However, it is worth taking a short time over a remark he makes about it, because it helps understanding the way Ramsey dealt with this problem. The remark has to do with the relation between the subjective and the objective dimensions of belief, i.e. between the words or images that make up my believing that Caesar was murdered and the murder of Caesar — something outside the mind, a worldly entity of some sort. Because those two factors are somewhat related to each other, beliefs are true or false. Now, the idea is that all that is needed for beliefs to have a truth-value is for them to have the right sort of content. Once a number of words or images maintain the appropriate relations, the objective
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has been fixed, and now it is up to the world, so to say, to assign a truth-value. It is thus that Russell accepts the following principle.
The Principle of the Objective Component Determination [OCD] The content of a mental state determines its objective component. Since the content of a mental state is constituted by either words or images, OGD sets forth that either words or images fix its objective. Beliefs stated in (1) —(3) are true since they have the fact that Caesar was murdered as their objectives. However, since the objective component of (4)
Caesar died in bed
is an objective falsehood, i.e. that Caesar died in bed, the belief that Caesar died in bed and the judgement expressed by uttering (4) must be false. Russell was perfectly aware of this. Moreover, he did not either overlook that Brentano's view of intentionality lies in the background of OCD. Thus, in 'On the Nature of Truth and Falsehood' he had written: If every judgement, whether true or false, consists in a certain relation, called {(judging)} or {(believing)}, to a single object, which is what we judge or believe, then the distinction of true and false as applied to judgements is derivative from the distinction of true and false as applied to the objects of judgements. Assuming that there are such objects, let us, following Meinong, give them the name {{Objectives}}. Then every judgement has an Objective, and true judgements have true Objectives, while false judgements have false Objectives. (Russell 1992, p. 118) Russell noticed, therefore, that Brentano and Meinong's view of the objective dimension is exposed to two decisive objections: that in a complete inventory of the world objective falsehoods, i.e. 'not depending upon the existence ofjudgements' (Russell 1992, p. 119), should be included side by side with objective truths; and that by postulating objective falsehoods one runs the risk of blurring the difference between truth and falsehood if'we abandon the view that, in some way, the truth or falsehood of a judgement depends upon the presence or absence of a {{corresponding}} entity of some sort' (Russell 1992, p. 119). As for Ramsey, he was acquainted with these criticisms. Thus, when discussing the doctrine that beliefs have an objective component that determines their truth or falsehood, he writes: This was at one time the view of Mr Russell, and in his essay 'On the Nature of Truth and Falsehood' he explains the reasons which led him to abandon
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it. These were, in brief, the incredibility of the existence of such objects as {{that Caesar died in his bed)}, which could be described as objective falsehoods, and the mysterious nature of the difference, on this theory, between truth and falsehood. He therefore concluded, in my opinion rightly, that a judgement has no single object, but is a multiple relation of the mind or mental factors to many objects, those, namely, which we should ordinarily call constituents of the proposition judged. (Ramsey 1990, p. 34) In these lines Ramsey alludes to the following. The belief ascribed in (1) would, in accordance with Brentano and Meinong's analysis, express a relation, either the act of judging or the mental state of believing, between Jones and the fact that Caesar was murdered. That is, (IBM)
Believing (Jones, {Caesar, Having-been-murdered))
Russell, who disagreed with that analysis, recommended something close to the following: (!R)
Believing (Jones, Caesar, Having-been-murdered).
In (1R) the act of believing relates Jones to Caesar and to the property of having been murdered. Because the entities involved are the familiar particulars and universal, no sui generis object is brought up. In this respect, (!BM) differs very much from (!R). (IBM) analyses Jones's belief into a thinker and {Caesar, Having-been-murdered}, an objective fact or state of affairs. As such, (IBM) does not generate any inconvenience at all. The problem comes as a in deciding whether to take (IBM) as a model for further cases. (5) is a case in point: (5)
Jones believes that Caesar died in bed
awill resort to a complex entity {Caesar,aesar, If we stand by (IBM)Janalysis the Having-died-in-bed}, that would be included among the things put on record in the world inventory. In other words, the analysis, in welcoming objective falsehoods, would be exposed to Russell's objections Ramsey was familiar with. Moreover, he realized that what lies at the bottom of the problem is Russell's allegiance to OCD. Is there any way of putting aside these objections without having to give up OCD? In his Analysis of Mind Russell tried to stick to OCD while retaining the idea that true beliefs have a fact as their objective component - a trace of the Brentanian suggestion that mental acts have sui generis entities as their objectives. Russell now brought two novelties to bear. On the one hand, he
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split up what Brentano and Meinong called the mental act into the act proper and a pointing direction. A true belief points towards a fact proper whereas a false belief points away from the fact (Russell 1921, p. 272). The second novelty lies in how the act's pointing direction and the mental content fit into each other. The objective component of a true belief, i.e. its prepositional content, is both determined by the meaning of its content, i.e. the proposition that constitutes its content, and its pointing towards a fact. The objective component of a false belief is determined by the meaning of the prepositional content and by its pointing away from the right fact. Thus, the difference between (1) and (5) is respectively captured in (IEM+R) and (SEM+R): (1 EM+R)
Believing-true(Jones, 'Caesar was murdered')
(SEM+R)
Believing-false( Jones, 'Caesar died in bed')
Before the analysis is finished, it has to be added that the truth of (IEM+R) implies, and is implied by, the truth of (!EM) while the truth of (SEM+R) does not imply the truth of (SEM): (SEM)
Believing (Jones, (Caesar, Having-died-in-bed})
Since no objective falsehood is now part of analysis, (SEM) is not an allowed possibility. Russell added to all this the remark that his analysis has the 'practical inconvenience' of not permitting to tell what the right analysis of a mental state is until it is known whether it is true or false, if it is a belief, satisfied or unsatisfied, if it is a desire, and so on. Ramsey rejected this way of complying with OCD. His assault on Russell's manoeuvre is interesting in itself. Russell begins by accepting Russell's hypothesis that unlike beliefs and judgements, which point either to a fact or away from a fact, depending on whether they are true or false, perception is a kind of mental state that only points towards its objective component, i.e. the situation or state of affairs perceived. In other words, perceptive states, states of the form Jones sees that/? should only be understood as (SEM+R)
Seeing-true (Jones, >')
or, what amounts to the same, as (6EM+R)
Seeing (Jones, {/>))
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P.P. Ramsey: Critical Reassessments
The infallibility of perception would thus seems to open the way to capturing the meaning of (6) and, even more important, to motivating the need for two kinds of pointing directions. As against this, Ramsey suggests considering a case in which someone judges that Jones sees that a knife is to the left of a book, (7): (7)
Jones sees that the knife is to the left of the book.
In fact, Jones does not see that the knife is to the right of the book, so the one who judges what Jones sees makes a mistake. The false judgement states the following: (^EM+R)
Seeing-true (Jones, 'the knife, To-the-left-of, the book')
that is, Seeing( Jones, (the knife, To-the-left-of, the book}) As a matter of principle, (7EM+R.) cannot be the right analysis of any perceptive state on the conditions laid down by Russell, because the objective falsehood (the knife, To-the-left-of, the book) is required by the analysis. The infallibility of perception really demands that (7) be analysed as (7 EM+R ): C^EM+R)
Seeing-false( Jones, 'the knife, To-the-left-of, the book')
On the one hand, whether Jones's perceptive belief is true or not, the meaning of (7) does not change. On the other hand, (7EM+R) and (7 EM+R ) express obviously different perceptive beliefs, that is, (7 EM+R ) is not a paraphrase of (7). Therefore, we are pushed to choose a way out of this dilemma. The first option is Russell's, i.e. to reject that (7) means only one thing, so that what (7) expresses depends upon whether the knife is to the left or to the right of the book. The second option is Ramsey's, that is, the meaning of (7) 'cannot therefore be that there is a dual relation between the person and something (a fact) of which "that the knife is to the left of the book" is the name, because there is no such thing' (Ramsey 1990, p. 35f.). Russell's attempt in The Analysis of Mind to square Brentano and Meinong's view of mental states with OGD is finally unsuccessful. This is not the only attempt to stick to the idea that the objective component of a mental state has to be some sort of entity in the world. Though it concerns a rather narrow range of cases, another possibility is enhanced by construing (9) (9)
Jones is aware that Caesar died
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as setting up a relation between Jones and an entity that could be described by means of the description 'the death of Caesar': (9')
Being-aware-of( Jones, the death of Caesar).
However, Ramsey has got an argument to block the way to this analysis. Let us suppose that The death of Caesar = the murder of Caesar i.e. that the event that was Caesar's death is just the event that was his murder. From the truth of (9) and (10) the truth of (11) follows: Jones is aware that Caesar was murdered (11')
Being-aware-of( Jones, the murder of Caesar)
Now Ramsey argues that (9) (and (9')) can be true without ( l l ) ' s (and (11')) being true as well, because anyone might be aware of Caesar's death while ignoring that he was murdered. Therefore, this construal of (9) is wrong. The question is, then, how to safeguard the idea that the death of Caesar was an event without moving back towards an unacceptable theory of intentionality. To put it briefly, Ramsey's solution lies in pointing out — thus opening a route Davidson has explored much later (in Davidson 1980, pp. 105ff.) - that a mental state like the one expressed in (9) and (11) quantifies over events. The connection between the event which was the death of Caesar and the fact that Caesar died is, in my opinion, this: 'That Caesar died' is really an existential proposition, asserting the existence of an event of a certain sort [...] The event which is of this sort is called the death of Caesar. (Ramsey 1990, p. 37) These words strongly suggest that, unlike (9') and (11'),.the right analysis should be close to the following ones: (QD)
3
(1 ID)
Bg(Murder-of(Caesar, e} A Being-aware-of( Jones, e}}.
(Ramsey's anticipation of Davidson has been briefly pointed out in Dokic and Engel2001,p.21.) A consequence of all these criticisms, both of Russell's against Brentano and Meinong and of Ramsey's detailed examination of Russell's variations in The
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P.P. Ramsey: Critical Reassessments
Analysis of Mind, Ramsey concludes that sentences that state what someone sees, judges, believes and so on should be construed in such a way that their that-clauses do not behave as singular terms, i.e. do not function as name phrases. Thus, the that-clause '(that) the knife is to the left of the book' names nothing, not even what is currently called a proposition, i.e. the meaning of a declarative sentence. Because that-clauses are not names, Brentano and Meinong's view of intentionality, as well as Russell's attempt in The Analysis of Mind, are subject to fatal objections. In a certain sense, however, all the difficulties have one and the same diagnosis, one that Russell had hit upon more than twenty years before. In the sort of cases taken up so far that-clauses are seen in the same light as that in which 'the king of France' usually has been seen: (8)
The king of France is wise
i.e. as contributing an object or entity, namely the king of France, to its truthconditions. However, Russell's theory of descriptions has it that 'the king of France' does not behave as a name phrase. On the contrary, it is an incomplete symbol, an apparent syntactic constituent that does not contribute to the proposition expressed by (8) with a meaning of its own. Therefore, Ramsey concludes, if in analysing (l)-(3), (5) and (7) the possibility of pairing mental acts to either propositions or further kinds of entities is ruled out, thus giving up the idea that propositional contents determine facts or states of affairs as objective components of mental states, then Russell was right in holding that the objective component of a belief is a multiple relationship between the thinker and a number of objects, properties and relations - the analysis Russell favoured in 'On the Nature of Truth and Falsehood'. In Ramsey's words,' [w]e are driven, therefore, to Mr. Russell's conclusion that a judgement has not one object but many, to which the mental factor is multiply related' (Ramsey 1990, p. 38). Thus, Ramsey recommends following the line of (!R). In itself this is not of much help, because Russell hardly added anything useful to such a shy remark. In a way, it is like saying that 'the king of France' in (8) is an incomplete symbol and leaving things just there. Of course, Russell did the hard work and added the details that make his theory of incomplete symbols so very worthy. In the same spirit, Ramsey took up the challenge of filling up the gaps that Russell has left in the so-called multiple-relation theory of the mental state's objective component. 5. Before undertaking such a task one final component of mental states still remains to be studied, one that plays a particularly important role. In effect, if a belief just consisted of a complex of words or images related to an objective component, then it would be impossible to distinguish Jones's assertion that Caesar was murdered from his denial that Caesar was murdered. The two contents are one and the same, maybe the inner sentence 'Caesar was murdered',
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and the same is true of their objective components. For the same reason, believing that an escaped tiger is coming along the street and supposing it are different mental states, as are imagining being invited to become king of Georgia and believing having been (Russell 1921, pp. 245, 247). These obvious differences are not accounted for with the help of the distinction between content and objective reference. To do justice to them, Russell and Ramsey added one more conceptual tool. In 'Facts and Propositions' such a third factor of mental states is a specific sensation: The mental factors of such a belief I take to be words, spoken aloud or to oneself or merely imagined, connected together and accompanied by a feeling or feelings of belief or disbelief, related to them in a way I do not propose to discuss. (Ramsey 1990, p. 40). These words should be read in conjunction with the following footnote: I speak throughout as if the differences between belief, disbelief, and mere consideration lay in the presence or absence of'feelings'; but any other word may be substituted for 'feeling' which the reader prefers, e.g. 'specific quality' or 'act of assertion' and 'act of denial' (Ramsey 1990, p. 40, n. 2). Thus, (1) differs from (3) not only with regard to who the thinker of the mental state is — a logical construction in the end, but in their respective contents. They share the words (or images) and the way they are combined, but the contents thus made up constitute only one aspect of the mental or subjective component. Whoever thinks that Caesar was murdered feels in a certain way; whoever denies that Caesar was murdered feels in another way. And if a person first thinks that so-and-so and then, days or years later, denies it, her thoughts differ because on one occasion she feels assertively whereas in the other she feels disapprovingly. Contents, and the propositions expressed, are the same, but the respective feelings towards them are different. Ramsey adopted Russell's solution to the above difficulty. For Russell, bare assent, memory and expectation, i.e. the three kinds of belief, have each as a characteristic mark 'a certain feeling or complex of sensations, attached to the content believed' (Russell 1921, p. 250). The attachment of a feeling to content, however, poses a problem. It cannot consist of a mere juxtaposition of sensations and words or images, because in such a case a casual co-existence of one content and a sensation or feeling of a certain kind, e.g. assent, would give rise to an inner judgement or belief. Russell arrives at the conclusion that such co-existence is not sufficient for belief. If the proposition 'Caesar was murdered' came to a thinker's mind while having, say, a memory-feeling, that would amount to remembering that Caesar was murdered (Russell
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1921, p. 251). Though Russell is perfectly aware that this is false, he limits himself to expressing the need for an analysis that relates feelings and contents in a firmer way, a relation 'of the sort expressed by saying that the content is what is believed' (Russell 1921, p. 250). The analysis of the mental dimension of belief that Ramsey put forward in 'Facts and Propositions' does not differ, therefore, from the one Russell proposes in The Analysis of Mind, at the very end of Chapter 12. It is important not to lose sight of the fact that this theory of belief just follows two further theories of belief that Russell puts aside. The first of these has it that, as far as its mental dimension is concerned, a belief consists of one content and a disposition to behave in such-and-such way, which depends on its content. For example, in being invited to suppose that an escaped tiger is coming along the street, ordinary mortals' behaviour will significantly differ from a situation in which they are informed that an escaped tiger is coming along the street. It is easy to stay calm on one occasion but not on the other. 'According to this view, images or words are {(believed)) when they cause bodily movements' (Russell 1921, p. 245). Belief, then, is a matter of causal efficiency, so that distinct kinds of mental states correspond to distinct forms of such efficiency. The weak point of this approach lies for Russell in its bluntness in making sense of beliefs that do not give rise to actions. Our beliefs, says Russell, 'often exist actively (not as a mere disposition) without producing any voluntary movement whatever' (Russell 1921, p. 246). Beliefs that do not result in action are not taken to be beliefs because it is thought that for a mental state to be of such a type it has to possess an intrinsic feature. This is a crucial ingredient of Russell's analysis of belief.
The Argument from the Intrinsic Feature of Mental States [IFMS] Two types of mental act - to judge, to consider, to suppose, to reject and so on — stand out from each other by some intrinsic feature. Mental acts of different types are distinguished by their intrinsic features. Since the power of a mental state to bring about voluntary actions is not enough to single it out from the others, it follows from IFMS that bringing about voluntary action is an extrinsic property of beliefs, one unable to account for their nature. It is worth noting that IFSM is an argument that Russell shares with Brentano, for whom what makes the belief in a proposition different from its merely taking it into consideration must be something additional to the proposition itself (Russell 1921, p. 247). In looking for the intrinsic feature that characterizes a mental state Russell dwells upon the second theory of belief above alluded to. On this theory, which Russell associates to Spinoza and James,
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beliefs are attitudes we by default adopt towards things and situations we experience. It is not necessary to resort to a specific sensation or feeling of assent to explain what beliefs are, because any occasion in which such a feeling would be absent is so to speak a marked one, i.e. a case for which a separate explanation has to be provided. Thus, the person who doubts, denies or considers a proposition without assenting to it 'will be in a state, restraining the natural tendency to act upon the proposition which he would display if nothing interfered' (Russell 1921, p. 249). However, belief is the existence of images or words when no counteracting forces operate. Against this theory, Russell has one objection to adduce. Though, by way of an example, a dog can have images (visual and olfactory) of his absent master or of a hunting rabbit he chased, and this makes plausible to infer that an image may have the force of a belief of its own, neither a memory-image nor a mathematical proposition in themselves are enough to make up a belief. These belong to a family of cases in which a feeling or a sensation, or complex of sensations, should be attached to content. The feeling or sensation, therefore, is the intrinsic feature that singles out the type of mental state the thinker is in. Thus, when the explanation of belief supported by Spinoza and James is adequately modified, it leads to Russell's preferred option. It is also the theory Ramsey warily resorts to in 'Facts and Propositions' (see again Ramsey 1990, p. 40, note 2) and which allows him to arrive at his analysis of belief. According to it, a thinker S believes that/? if attached to words or images (or combinations of both of them) in his or her mind (or brain) there is a specific sensation or feeling of assent/, and those words or images express the proposition that p. A concise formula Ramsey uses repeatedly is this: S believes that q, if S feels/ towards the sentence '/>', and 'p' means q (cf. Ramsey 1990, pp. 43, 46). 6. It was pointed out above (in §3) that Russell held the Argument from the Intentional Properties Pragmatism (IPP), among a number of arguments, against Brentano and Meinong. The main premise of that argument has it that the causal trajectory of a mental state determines its intentional properties. Ramsey also accepted this premise and thus sided with Russell's pragmatism. While this conclusion seems unproblematic, there are some notable differences between Russell's and Ramsey's pragmatism that have to be brought up, differences that are alluded to in the following well-known lines: In conclusion, I must emphasise my indebtedness to Mr Wittgenstein, from whom my view of logic is derived. Everything that I have said is due to him, except the parts which have a pragmatist tendency, which seems to me to be needed in order to fill up a gap in his system. (Ramsey 1990, p. 51) These words make plain that Ramsey's analysis of belief mixes ideas from Russell as well as ideas from Wittgenstein, i.e. Wittgenstein's Tractatus, and,
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because of it, the sort of pragmatism that the former encouraged in him fits in well with what Ramsey here calls 'my view of logic'. The problem then lies both in determining what inadequacies Ramsey detects in Russell's theory of belief and in how he overcomes those inadequacies by turning to a few ideas of Wittgenstein's. (It would also be worth studying why Ramsey thinks that Wittgenstein's tractarian system is deeply inadequate, but this is not a topic to be taken up in what follows.) What then is Ramsey's pragmatism like? It is easy to confirm Ramsey's support of NEAM, PNG and IFSM. In his analysis of contents with a relational logical form these three ingredients of Ramsey's theory are easily seen. To hold that a thinker S believes that aRb is equivalent to hold, first, that there are names V, 'R' and '£' in 6"s inner language such that V means a, 'R' means R and '&' means b, and 'a', 'R' and '&' connect among them in the appropriate way within the stream of sensations, images and words out of which 5"s mental identity is logically constructed; and second, that such a complex of words and images is accompanied by a sensation or feeling of assent. The content of a relational belief is formed by words — in the notation of Principia Mathematica, according to the convention which Ramsey follows in the second part of 'Facts and Propositions' - that combine in the right order in obedience to grammatical rules. To such a complex of particulars a sensation is attached to form the belief's mental or subjective dimension. As for the objective dimension of relational beliefs, Ramsey implicitly appeals to OCD as well. Names V, 'J?' and 'b', their meanings and the constituency principles of >S"s inner language determine whether it is the belief that aRb or the belief that bRa, that is, to determine what the truthconditions the sentence 'aRb' are. In addition to NEAM, PNC and IFSM, IPP provides the pragmatist explanation of a word's meaning what it means. Thus, to believe that aRb is to attach a sensation of assent to an inner language sentence whose names, V, 'R' and 'b' have causal properties that 'are connected with a, R and b in such a way that the only things which can have them must be composed of names of a, R and b' (Ramsey 1990, p. 45). Taken together, both the subjective and the objective dimension, Ramsey is ready to solve two problems Russell was unable to tackle satisfactorily. The first, what exactly are beliefs' objective components? The second, what is the relation between the specific sensation on the subjective dimension and its content? As for the first problem, Skorupski has defended that, concerning relational beliefs, Ramsey stuck to Wittgenstein's picture theory. The text that supposedly backs such an interpretation is the following one: By means of names alone the thinker can form what we may call atomic sentences, which from our formal standpoint offer no very serious problem. If a, R, and b are things which are simple in relation to his language, i.e. of the types of instances of which he has names, he will believe that aRb by
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having names for a, R, and b connected in his mind and accompanied by a feeling of belief. This statement is, however, too simple, since the names must be united in a way appropriate to aRb rather than to bRa; this can be explained by saying that the name of R is not the word 'R', but the relation we make between 'a' and 'b' by writing 'aRb'. The sense in which this relation unites 'a' and 'b' then determines whether it is a belief that aRb or that bRa. (Ramsey 1990, p. 41) This construal raises two objections. On the one hand, Ramsey repeatedly opposed Wittgenstein's picture theory, not only in his critical notice of the Tractatus (see Ramsey 1923, pp. 471f.), but in his leaning towards Russell's pragmatism as soon as he deals with matters of meaning and intentionality. (See §7 below.) On the other hand, it should be pointed out that this text does not support the idea that Ramsey endorsed the picture theory. In fact, the text seems to prove that Ramsey is being consistent with his commitment to pragmatism. A thinker S believes that aRb, if £ connects the names 'a', 'R' and 'b' in the appropriate grammatical order and these names are causally linked to their meanings, a, R and b, respectively. In other words, 'the causal properties [of believing that aRb] are connected with a, R, and b in such a way that the only things which can have them must be composed of names a, R, and b' (Ramsey 1990, p. 45). This is no more than a necessary condition, though the one which really matters. In order to have got a necessary and sufficient condition to S's competence in using V, 'R' and 'b', S's competence in abiding by the grammatical rules of her inner language must be added. Depending upon how those rules are followed, S will believe that aRb or bRa, that is to say, S's belief will have as its objective component the proposition that aRb or the proposition that bRa. There is no need for a fact which either 'aRb' or 'bRa' be a picture of. Ramsey avoids the need for pairing the belief content to a tailormade entity, because in order for S to believe that aRb, it suffices that S knows the use of'a', 'R' and 'b' as well as the syntax of her inner language to provide the sentence 'aRb' with the right truth-conditions, and knowing the use is a matter of causal sensitivity and power. Thus, a pragmatist approach to intentionality, unlike a picture theory of thought, does not have to face a foreseeable objection to explaining the objective component of any belief by pairing it to a complex entity, i.e. a fact or a state of affairs. It is true that the text above sounds familiar to any reader of the Tractatus (particularly as regards §3.1432). However, one thing is the deep impact of Wittgenstein's Tractatus on Ramsey's thought; another one is that differences between Ramsey and Wittgenstein are not significant. As for the second problem, it is useful not to forget what NEAM and IFMS state, i.e. that the mental act's thinker is nothing but a logical construction carried out with images and words and that some intrinsic feature is needed
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to explain why mental states sharing the same content can be different in the end. Now the question is, how does the specific sensation of assent relate to the belief's content? Ramsey solves this problem in two steps. He begins by giving the sensation or feeling a role in the belief's causal efficiency. 'To say that feeling belief towards a sentence expresses such an attitude is to say that it has certain causal properties which vary with the attitude' (Ramsey 1990, p. 46). Then, as has been explained in the previous paragraph, Ramsey makes belief's causal properties dependent upon the meaning of either the words or images occurring in its content. Therefore, not only is there an easy explanation why the belief that aRb has a causal trajectory different from the belief that bRa, but — what matters now — that the belief that aRb and the supposition that aRb have causal histories of their own as well. As Russell had guessed but not proved, this is due to the fact that content and sensation are not simply juxtaposed. The mental or subjective dimension of a mental state consists of feeling or complex of feeling/towards a sentence '/)' of the thinker «S"s inner language. Alternatively, it consists of 5"s feeling f towards the proposition that q, if'/?' means q. The feeling/and the sentence lp' are put together to hold a certain relation, namely to feel/towards the sentence '/>'. Ramsey overcomes Russell's difficulty, since a mere juxtaposition off and '/)' does not amount to feeling /towards '/)'. Having solved these two problems, the programme of explaining relational — in general, atomic — beliefs is finished. It rests to extend the analysis given to atomic beliefs to non-atomic ones, i.e. to beliefs in whose contents logical constants ('not', 'and', 'or', and so on) may occur. Ramsey thus goes into a realm Russell did not explore. The challenge is not without interest, because this is the place where Ramsey moves away from Russell. Their disagreement concerns OCD, the Principle of the Objective Component Determination, a cornerstone in their philosophies of mind and logic. As has been explained, Russell held that the objective reference of a mental state is determined by the meaning of its words or images: 'Given the meanings of separate words, and the rules of syntax, the meaning of a proposition is determinate' (Russell 1921, p. 241). Ramsey thought that the validity of this principle has to be restricted when logical constants are in play. To understand his reasons, it is illuminating to follow Ramsey's analysis of negative beliefs or judgements closely. The question is, what are the subjective and the objective components of mental states of this form: (BelieveNeg)
S believes that not-/),
whose content is the proposition 'not-/)'? The truly delicate step in Ramsey's analysis is his giving up of the inference that 'not' is, like 'a','/?' or '£', another name of S's inner language, from the premise that has it that 'not' occurs in the
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belief's content in (BelieveNeg). The logical constant 'not' thereby makes a semantic contribution of an entirely different sort. It follows that when only names ofPrincipia Mathematical language occur in a belief's content, OCD, a cornerstone in Russell's analysis of mental content, is in force. For that sort of case it holds true that the meaning of the (mental) proposition is determined by its name constituents and by the language syntax. Ramsey did not disagree with Russell with regard to atomic contents. Commenting on the belief that aRb, he wrote the following: [W]e can say that the causal properties [of believing that aRb] are connected with a, R, and b in such a way that the only things which can have them must be composed of names of a, R, and b. (This is the doctrine that the meaning of a sentence must result from the meaning of the words in it.) (Ramsey 1990, p. 45) However, as far as non-atomic contents are concerned, Ramsey rejects OCD. This was a natural step to take after having accepted, as Ramsey had, what in his Tractatus Wittgenstein had described as 'my fundamental idea', namely that '{{logical constants)) are not representatives' (Wittgenstein 1922/1961, §4.0312). According to such an idea, 'not', 'or' and so on are not names of material functions (§5.44). As a consequence of it, there are no specifically logical objects (§5.4), i.e. objects referred to by the logical constants. And since there are no logical objects, it is wrong to conceive of logical operations like negation, conjunction and the rest as on the same footing as relations like being to the left or being to the right. Wittgenstein thought that unlike the latter, which are relations in their own right, the former are not relations at all (§5.42). The conclusion naturally follows that any negative sentence, 'not-/?', is not about negation; in other words, that in 'not-/?', nothing is said in '/?' about a presumed object referred to by 'not'. Truth-functions are not material functions. For example, an affirmation can be produced by double negation: in such a case does it follow that in some sense negation is contained in affirmation? Does c ~~p' negate '~/>, or does it affirm/) — or both? The proposition '~~p' is not about negation, as if negation were an object: on the other hand, the possibility of negation is already written into affirmation. And if there were an object called '~' it would follow that '~~/>' said something different from what '/?' said, just because the one proposition would then be about ~ and the other would not (§5.44). In 'Facts and Propositions', the last two paragraphs are nearly repeated by Ramsey in the service of his own project:
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It seems to me, therefore, that 'not' cannot be a name (for if it were, 'not-/?' would have to be about the object not and so different in meaning from '/?'), but must function in a radically different fashion. (Ramsey 1990, p. 43) Assuming that 'not' works in a radically different way, what is the analysis of (BelieveNeg)? This is the answer: S's believing thatp attaches a feeling of assent to content '/?' and as a result of such an attachment the belief will have certain causal properties. Because 'not' is not a name, it will not contribute to the causal properties of S's believing that not-/? in the same way as a true name occurring in '/>' contributes to. On the other hand, since there is a world of difference between believing that p and believing that not-/?, the word 'not' has to make a causal difference, whatever it may be. Ramsey solves the puzzles by giving 'not' a role in determining the kind of mental act and consequently making 'not' a symbol of the feeling or sensation involved. Thus, in thinking or uttering 'not-/?' a thinker S does not judge that not-p (nor assents to 'not-/?'). He or she expresses his or her incredulity about being the case that/?. It is the kind of mental act that has to be fixed when a word like 'not' occurs in its content. In fact, Ramsey sustains, the supposed belief that not-/? is not the belief in a negative fact, but not believing, i.e. disbelieving, that/?. Not believing that/? is another kind of mental state, singled out by a characteristic feeling, a sensation of incredulity. Far from being a name, 'not' indicates a type of mental state different from belief. The difference between uttering assertively '/?' and uttering assertively 'not-/?', betweenjudging and denying, 'consists in a difference of feeling and not in the absence or presence of a word like "not"' (Ramsey 1990, p. 43). That is, (BelieveNeg) and (Incred) symbolize 'equivalent occurrences': (Incred)
S does not believe that/?.
What sort of equivalence is this? Ramsey's commitment to pragmatism now reappears. What is truly fundamental about the word 'not' is the equivalence of (BelieveNeg) and (Incred), an equivalence which one does not do justice to by treating 'not' as a name. For Ramsey, the crux of the question lies in the equivalence of believing that not-/? and disbelieving that /?, i.e. in their 'having in common many of their causes and many of their effects' (Ramsey 1990, p. 44). The content 'not-/?' together with the feeling of assent attached to it and the content'/?' plus the feeling of incredulity attached to it are one and the same attitude: Feeling belief towards the words 'not-/?' and feeling disbelief towards the words'/?' have then in common certain causal properties. I propose to express this fact by saying that the occurrences express the same attitudes, the attitude of disbelieving/? or believing not-/?. On the other hand, feeling belief
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towards 'p' has different causal properties, and so expresses a different attitude, the attitude of believing p. It is evident that the importance of beliefs and disbeliefs lies not in their intrinsic nature, but in their causal properties, i.e. their causes and more especially their effects. (Ramsey 1990, p. 44) It follows from this that, although 'not' is a syntactical constituent of the content 'not-/;', its presence in the language is explained not by its contribution to generating new contents, but by its helping to express disbelief. This means that two ingredients of Russell's analysis of belief, IPP and OGD, cannot be simultaneously satisfied! Because OCD does not separate logical constants from the rest of words in language, (BelieveNeg) and (Incred) should be different kinds of mental state. However, if IPP is given the role that Russell and Ramsey thought of, believing that not-/; and disbelieving that/) are one and the same attitude. This forces seeing the place of logical constants in language in another light. Neither the analysis of believing that not-p requires positing negative facts nor the analysis of believing that (p V q) demands positing disjunctive facts. It is simply false that by attaching a feeling of assent to a sentence of the thinker's inner language we obtain a belief of his or hers. This assumption, which Ramsey considers as firmly established in many philosophers' minds as the one which holds that every proposition is formed by a subject and a predicate, is definitively wrong. (BelieveNeg) represents a kind of mental state different from believing, namely it represents disbelieving. Because what matters in beliefs is their causal — extrinsic — properties, it is not true that for every kind of content that a logician might catalogue, i.e. atomic, conjunctions, disjunctions and so on, there has to be a corresponding kind of belief. Negations and disjunctions are not assertions that something is not so-and-so and that something is so-and-so or something else is such-andsuch. Negations and disjunctions are not among positive assertions (Ramsey 1990, p. 43). There can be no positive assertion either of negations or of disjunctions. A tokening of either a negative sentence or a disjunctive one in the thinker's inner mind does not indicate a negative belief or a disjunctive belief, respectively, of his or hers. They are indicative of another kind of mental state. Brian Loar has criticized this manoeuvre of classifying beliefs in terms of their causal properties, instead of doing it by taking into account their logical forms. His reason is that Ramsey's move 'generates a gigantic incommodity' (Loar 1980, p. 52) for logical analysis. It is true that Ramsey gives up setting up a tight correspondence between kinds of logical forms (or kinds of equivalences of logical forms) and kinds of causal properties. It is also true that it seems to prevent the logician from doing her work, because Ramsey's giving up OCD forces the logician to look for logically interesting patterns among causal transitions. And why should there be any? Thus, the possibility of mapping causal properties on formal properties, and in the end on intentional
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properties, the possibility that the computational theory of mind has enthroned, defines an enterprise which now seems extremely difficult to carry out. This reproach is justified, only if the logician adheres to the assumptions Ramsey rejected. However, both IPP and OGD are not assumptions without any substance. On the contrary, their place in Brentano and Meinong's view of intentionality is a central one and they give rise to the difficulties which weaken it. If by simultaneously accepting IPP and restraining the scope of OCD those difficulties can be avoided, then Loar's criticism is no more than another symptom of how much ill at ease the one who believes that meaning is a contribution to truth-condition feels when he or she has to face the idea that meaning is use. Loar thus joins those who take the meaning of an expression to be its contribution to the truth-conditions of any sentence the expression might be a constituent of, and logical form is a neat representation of such a contribution. For him, logical analysis demands that logical constants be also understood as meaning their own contributions to the truth-conditions of those sentences they occur in. Thus, they are no exception to OCD. As regards Ramsey, he refuses to abide by that view of logical semantics, the reason being that in the end semantics, and the analysis of intentionality in general, can only make sense under a pragmatist point of view. 7. There is a further reply to Loar that even non-logicians might prefer to the one just given. According to it, far from there being something in what Ramsey says that required giving up logical analysis, he would be recommending understanding it in another guise. Beliefs of the form of (BelieveNeg) are particularly interesting because they help reveal a significant difference between Russell's and Ramsey's pragmatism concerning belief's content. It follows from OCD that if an expression e occurs in a content sentence 'p', e makes its own contribution to the meaning of'/»' and, therefore, to the belief's truth-conditions. By constraining the scope of OCD, e could have some other role to perform than meaning an individual, a property or a relation. Since Ramsey accepts all this, he has got the duty to explain how his pragmatism is not in the way of providing a general account of belief's subjective and objective components. By putting a new twist on a few doctrines that Wittgenstein had expounded in his Tractatus, Ramsey thought he had managed to achieve that end. The task of extending the pragmatist analysis of relational belief's objective component to non-atomic beliefs cannot be eluded. Conjunctive beliefs, i.e. beliefs of the form (ConjBelieve)
S believes that (p A q)
have got a quite simple analysis: to believe that (p A q) is to believe that p and to believe that q (Ramsey 1923, p. 472). Thus, a belief whose content is the
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sentence '(p A q}' is in fact two beliefs, one having lp' and the other '
S believes that ( p V q ] ,
which Ramsey takes to be 'relatively indefinite', pose a serious problem for the pragmatist, a problem which Ramsey solves by resorting to the tractarian notion of truth-possibility. What Ramsey borrows from Wittgenstein's Tractatus is the idea that a proposition is the expression of agreement and disagreement with truth-possibilities of its elementary propositions (Wittgenstein 1922/1961, §4.4). This means that the way the truth-value of a proposition q depends on the truth-values of its elementary propositions pi, p2 • • • , Pm constitutively defines what proposition q is. Wittgenstein says that a proposition is the expression of agreement with certain truth-possibilities, that is, those truth-possibilities under which the proposition is true. The remaining truth-possibilities are those under which the proposition is false. However, a proposition's expressing agreement with certain truth-possibilities and disagreement with the remaining ones is nothing but saying that it is true in certain conditions and false otherwise. A truth-possibility of elementary propositions pi, p2, ..., pm of a given proposition q is a condition under which q has a truth-value no matter what truth-value pj, p2, . . . , and pm may have. And each truth-possibility has been fixed when, and only when, p j , p2, ..., pm have one and only one truth-value. Summed up in few words: Truth-possibilities of elementary propositions are the conditions of the truth and falsity of propositions (§4.4). The expression of agreement and disagreement with the truthpossibilities of elementary propositions expresses the truth-conditions of a proposition. A proposition is the expression of its truth-conditions (§4.431). Thus, following the notation Wittgenstein introduces in his Tractatus, '(FTTT)(p,q)' is the proposition that could be put in words by uttering: 'Not simultaneously p and q', one that in logical notation can be put in a number of familiar ways: '~(p A V ~q}' and so on. It is the proposition that expresses disagreement ('F') with the first truth-possibility and agreement (' T"') with the remaining three. And lp' and (q' are the elementary propositions of which '(FTTT)(p,q)' consists. Wittgenstein's notation does
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not show the list of truth-possibilities — these are supposed to have been ordered in accordance with some pattern - but the main idea is clear enough: a proposition can be identified with a mapping from truth-possibilities of its elementary propositions to truth-values. To illustrate the analysis and assuming such an order, the first truth-possibility, i.e. '/>' 's being true and ' is for her to feel a quality/(of assent) towards a set of truth-possibilities and to feel f (of incredulity) towards the remaining truth-possibilities (Ramsey 1990, p. 46). From a pragmatist point of view this idea has a neat interpretation, namely S believes that p is nothing but S's behaving 'in disregard of the possibilities rejected' (Ramsey 1990, p. 46). Of course, there are some differences worth considering between Ramsey's use of the notion of truth-possibility and Wittgenstein's. In the Tractatus's system truth-possibilities result from a more basic difference, namely that between possibilities of existence and possibilities of non-existence. 'Truthpossibilities of elementary propositions mean possibilities of existence and non-existence of state of affairs' (Wittgenstein 1922/1961, §4.3). And possibilities of existence and of non-existence are what they are depending upon what objects there are in the world and what their (logical) forms are (Wittgenstein 1922/1961, §§2.01ff.). Thus, the concepts of possibility of existence and possibility of non-existence belong to the ontological background of Wittgenstein's picture theory of thought. Ramsey was reluctant to accept this theory. In his critical notice of the Tractatus he pointed out that the picture theory only allows to replace '/>' says p' by 'p' expresses agreement with these truthpossibilities and disagreement with these others'. Because of it, it constitutes an explanation of what meanings are there, but it says nothing about what sentences mean what (Ramsey 1923, p. 471). In Ramsey's opinion, Wittgenstein would not have showed in a satisfactory way how the analysis should have gone on, that is, how the notion of possibility of existence could be explained away. If in making sense of the notion of truth-possibility the idea of a coordination of names that make up atomic sentences with objects that make up atomic facts is stressed, the final analysis will have to face serious objections.
Mind, Intentionality and Language
37
One which Ramsey insists on in his critical notice of the Tractatus is that 'we will be unable to say what someone, e.g. a Chinese, asserts unless we know the language he or she speaks'. This is an unusually strong condition because, Ramsey replies, we very well manage to know many things people think even though we cannot understand their language, i.e. the logical notation they use. This limitation is due to the fact that Wittgenstein's theory 'will not give an analysis of "A asserts p" but only of "A asserts p using such and such logical notation"' (Ramsey 1923, p.472). Years later, in 'Facts and Propositions' Ramsey puts forward a second weakness, a classical one, which reiterates the objection that the explanation of meaning, i.e. sense, provided in the Tractatus is relative to a language or a linguistic framework. This objection points out that 'what was to [our thinker] an atomic sentence might after translation into a more refined language appear as nothing of the sort' (Ramsey 1990, p. 48). However, the objection goes on, to make the meaning of sentences depend on the linguistic framework is like limiting the movements of the players in a chess game by subjecting them to magnetic forces that prevent certain game positions from arising. Ramsey's construal of the tractarian theory of proposition just reviewed is free from these difficulties. As before, the key to this is his pragmatist orientation. Beliefs, as well as mental states in general, have causal properties which depend upon the causal properties of those words or images that make up their contents. Let's add to this that a belief's causal properties determine what truth-possibilities the thinker takes to be still open and what further truthpossibilities are closed to him or her, and we will get what we need: To say that feeling towards a sentence expresses such an attitude is to say that it has certain causal properties which vary with the attitude, i.e. with which possibilities are knocked out and which, so to speak, are still left in. Very roughly, the thinker will act in disregard of the possibilities rejected, but how to explain this accurately I do not know. (Ramsey 1990, p. 46) Thus, Ramsey's analysis meets a pragmatist construal of what is for a sentence, i.e. a mental state's content, to have this or that meaning with the best of Wittgenstein's insights on the nature of logical truth and formal inference. A logical truth is a tautology; what amounts to the same, it 'excludes no [truth-] possibility and so expresses no attitude of belief at all' (Ramsey 1990, p. 47). And an inference from the proposition that/? to the proposition that q is formally valid, if'/)' 's truth-possibilities are also '
38
F. P. Ramsey: Critical Reassessments
construal of logic were inadequate, that would not be due to its being pragmatist, but to having inherited whatever weaknesses its Wittgensteinian model had.) And it requires an independent argument to hold that a theory about one of these two families of questions has to dictate how to tackle the other (see Stalnaker 1999, pp. 225ff.). In giving up OCD and construing in a new way the tractarian notion of proposition — the one put forward above Ramsey did justice to these subtleties. 'Atomic content', 'conjunctive content' and so on are one with a theory of belief's subjective dimension and wholly alien to a theory of its objective dimension. John Skorupski has summarized the effects of Ramsey's manoeuvre in the following formula: Ramsey 'replace[d] the single attitude of belief towards compound thoughts by a multiplicity of attitudes towards atomic thoughts' (Skorupski 1980, p. 81). This penetrating remark is very close to the truth. It would perhaps be even closer to say that Ramsey replaced the single attitude of belief towards simple and compound thoughts by a mix of two attitudes, of assent and incredulity, towards atomic thoughts. These are not all the advantages of Ramsey's solution. He was impressed by Wittgenstein's being able to successfully argue that lp', '~~p' and '~~~~j&' mean one and the same proposition, that 'q' and 'q A (p V ~/>)' mean one and the same proposition (Ramsey 1990, p. 47), and that' (q ~^p] A (~q ~^p]' and '~(^v~/>)' mean the same proposition as 'p' (Ramsey 1990, p. 471). Wittgenstein's view of propositions allowed him not to count differences among contents as differences in the propositions expressed, i.e. differences in their truth-conditions. It follows that to state what a belief's objective component is there is no need to take into account the language in which the thinker thinks. The language in which S 's beliefs are coded does not need to be acquainted with in order to know what S believes. As recently argued in a similar vein (in Stalnaker 1999, pp. 169ff.), by exploring what possible states of the world S's actions leave open and what others she seems to consider to be closed, it would be possible to settle what her beliefs are. 8. So far the impact of Russell's pragmatism on Ramsey's philosophy of mind and philosophy of logic has been examined. On this analysis, besides having followed Russell's general orientation on the philosophy of intentionality and accepted a number of his specific doctrines, Ramsey developed hints that Russell had left unarticulated. Russell's view of beliefs as relations of a thinker to multiple objects is surely the main case in point. This does not mean, however, that Ramsey followed the trail of Russell everywhere. On the contrary, the question whether the subjective dimension of belief determines the objective one is not only a crucial subject, but one about which Russell's and Ramsey's pragmatism significantly diverged. After 'Facts and Propositions', differences kept on growing. In 'Truth and Probability' three of them became truly significant. The first was Ramsey's rejection of IFMS. From
Mind, Intentionality and Language
39
accepting — in an implicit way - that every mental act contains a sensation or feeling that is perceived introspectively, Ramsey went on to hold a theory of belief that Russell had explicitly abandoned in his Analysis of Mind, the one which has it that beliefs are essentially characterized by certain extrinsic properties of their own, namely their causal power to bring about voluntary action. Ramsey retrieves this theory after subjecting it to a deep conceptual modification - the second finding alluded. By distinguishing acts of thought from mental dispositions - judgements are acts of thought; beliefs are mental dispositions - Ramsey handles partial belief, i.e. attitudes of the form '£ believesto-the-degree m that p' as dispositions to rely, up to the point m, on the truth of '/>'. Ramsey's third finding in 'Truth and Probability' that matters here is his insight that partial beliefs have intentional properties correlatively with others' attitudes. Moreover, these other attitudes, namely desires, also possess intentional properties in tandem with beliefs. By setting up such a mutual relation between beliefs and desires, two kinds of mental dispositions, Ramsey leans towards a functionalist view of the mind (as has been pointed out in Mellor 1990), an input/output functionalism (Rey 1997, pp. 191ff.) and particularly psycho-functionalism. In his posthumous work On Truth (Ramsey 1991), his pragmatism becomes less behaviourist and gains tones close to those of the brain-mind identity theory. In spite of it, in its third chapter (entitled 'Judgements') Ramsey put forward a new kind of mental content to account for intentional properties of psychic (and organic) states that relate to actions in the way characteristic of conditioned reflexes. The objective component of those states are not propositions, i.e. sets of truth-possibilities, but propositional functions, that is, functions from individuals to propositions. This important novelty provides a hint of how far Ramsey walked out of Russell's shadow and how quickly he achieved new results. All this, however, is a subject for another occasion.
References Brentano, F. 1995. Psychology from an Empirical Point of View. London: Routledge. Carruthers, P. 1996. Language, Thought and Consciousness. Cambridge: Cambridge University Press. Davidson, D. 1980. Essays on Actions and Events. Oxford: Clarendon Press. Dokic, J. and Engel, P. 2001. Frank Ramsey. Truth and Success. London: Routledge, Taylor & Francis Group. Fodor J. 2000. The Mind Doesn't Work that Way. Cambridge, MA: The MIT Press. Hacker, P. M. S. 1996. Wittgenstein's Place in Twentieth-Century Analytic Philosophy. Oxford: Blackwell. Loar, B. 1980. 'Ramsey's Theory of Belief and Truth'. In Mellor 1980.
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Mellor, D. H. (ed.). 1980. Prospects for Pragmatism. Cambridge: Cambridge University Press. Mellor, D. H. 1990. 'Introduction' to Ramsey (1990). Ramsey, F. P. 1923. 'Critical Notice of Tractatus Logico-Philosophicus by Ludwig Wittgenstein'. Mind, 32, 465-78. 1990. Philosophical Papers, ed. D. H. Mellor. Cambridge: Cambridge University Press. 1991. On Truth, ed. N. Rescher and U. Maier. Dordrecht: Kluwer Academic Publishers. Rey, G. 1997. Contemporary Philosophy of Mind. Cambridge, MA: The MIT Press. Russell, B. 1992. 'On The Nature of Truth and Falsehood'. In The Collected Papers of Bertrand Russell, vol. 6: Logical and Philosophical Papers, 1909-1913, ed. J. G. Slater, with the assistance of B. Frohmann. London: Routledge. 1921. The Analysis of Mind. London: George Allen & Unwin. Sellars, W. 1969. 'Language as Thought and as Communication'. Philosophy and PhenomenologicalResearch, XXIX, 506-27. Reprinted in Geirsson, H. and Losonsky, M. (eds): Readings in Language and Mind. Cambridge, MA: The MIT Press, 1996. SkorupskiJ. 1980. 'Ramsey on Belief. In Mellor 1980. Stalnaker, R. 1999. Context and Content. Oxford: Oxford University Press, 1999. Watson, J. 1970. Behaviorism. New York: W. W. Norton & Company. Wittgenstein, L. 1922/1961. Tractatus Logico-Philosophicus. London: Routledge & Kegan Paul.
2 Ramsey and Wittgenstein: mutual influences1 Hans-Johann Glock
Some contemporary admirers of Ramsey have tried to beat his drums by casting aspersions on the originality and value of Wittgenstein's work. This tendency is motivated in part by genuine differences in their philosophical outlook, style and temperament. As regards this issue, I am somewhat split. In my estimate, Wittgenstein has made more important contributions to philosophy; and on the issues on which the two clearly diverged I prefer Wittgenstein's anti-naturalistic approach to the naturalistic pragmatism adopted by Ramsey. At the same time I have a moderate preference for Ramsey's style of philosophizing over Wittgenstein's. Even though the latter has a unique intellectual charisma and aesthetic appeal, it is not a suitable model for contemporary philosophizing. I share neither Ramsey's effortless command of formal techniques nor his confidence in their importance. Nevertheless, I prefer his more conventional — often charming and entertaining - way of writing to Wittgenstein's aphoristic and oracular style. The tendency to belittle Wittgenstein's achievements is also motivated by sore feelings about the fact that Ramsey's work was eclipsed by Wittgenstein's from the 1930s onwards. But while one must deplore the relative neglect of Ramsey's work until the 1970s, two wrongs do not make a right. Even less do they make for sound exegesis and fruitful philosophy. In this chapter I shall therefore avoid the question which exerts such a curious fascination on some of Ramsey's admirers, namely whether he was 'at least' the 'intellectual' or philosophical equal of figures like Moore, Keynes, Russell and Wittgenstein (Armendt 2001: 139; see also Mellor 1990: xi; Sahlin 1990: ix). Ramsey was a prodigy standing on the shoulders of giants. Both the stature of the giants and the premature death of the prodigy ought to dampen any hankering for personal rankings. In trying to assess the philosophical relationship between Ramsey and Wittgenstein I shall, however, have to take a stance on two questions. First, who influenced whom and in what respect? Second, on matters of disagreement, who was closer to the truth? For reasons of space, I shall concentrate on the first question, but the second will come to the fore occasionally. For the same reason I shall also leave aside some interesting issues connected with my topic, such as Wittgenstein's interactions with other Cambridge philosophers
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between the wars (Hacker 1996: ch. 4), his more general relation to pragmatism (Schulte 1999), and the development of his views on infinity (Wrigley 1995; Kienzler 1997: ch. 4; Marion 1998). As regards my first question, there are two polarized views. One extreme is epitomized by Sahlin. He holds not only that 'Wittgenstein was far more influenced by Ramsey than Ramsey was by Wittgenstein' (1990: 227), but also that 'Ramsey is not just another key, but the key to unlocking the interpretation of Wittgenstein' (1995: 158). The other extreme is Kienzler's. In his painstaking study of the emergence of Wittgenstein's later philosophy in the years 1930-32 he reaches the conclusion: Ramsey's constructive contribution to this whole development is very minor, however, with the result that the search for similarities between his philosophical attempts and the philosophy of the later Wittgenstein is entirely misleading. Ramsey's positive suggestions were almost always criticized by Wittgenstein, and nowhere have they led to an endorsement in Wittgenstein's notes; none of the basic ideas of the later Wittgenstein can convincingly be traced to a stimulus by Ramsey. (1997: 75-6) My position is closer to Kienzler's than to Sahlin's. I also think, however, that there is much to be gained by scrutinizing the way in which Wittgenstein and Ramsey influenced each other by way of inspiration and opposition alike. After a discussion of the nature of their philosophical interactions, I shall turn to the specific topics of Logic and Mathematics, Probability and Induction, Universals, Truth, Meaning and Content, and the Nature of Philosophy.
Philosophical relations Ramsey first came across Wittgenstein's ideas when he took a major part in Ogden's 1922 translation of Logisch-Philosophische Abhandlung. Ramsey was immediately fascinated by the book, and in 1923 he visited Wittgenstein in Puchberg, where the latter was working as an elementary school teacher. In a letter to his mother, Ramsey expresses his admiration for Wittgenstein: T used to think Moore a great man but beside W!'. He also recounts their lineby-line discussions of the Tractatus (cited LO 77-9). His strenuous efforts to understand the book bore fruit. The discussions led to some changes in the second edition of the Tractatus (see LO 85—6 and Probability and Induction below). More importantly, Ramsey's 'Critical Notice', written shortly before his first conversation with Wittgenstein, is perceptive in its interpretations and acute in its criticisms. In particular, Ramsey put his finger on a problem that
Ramsey and Wittgenstein
43
was to lead eventually to the unravelling of the Tractatus system. Ascriptions of different colours to a point in the visual field are inconsistent. (1)
a is red
is not just empirically but logically incompatible with (2)
a is green (blue, yellow, etc.)
This is an apparent counter-example to the Tractatus claim that all necessity is logical, that logical relations are a consequence of the inner complexity of molecular propositions. Wittgenstein tries to deal with this difficulty by showing that 1 and 2 can be analysed as logical products which 'contradict' each other, e.g. 1 &sp. q. r 2 as s. t. ~ r. To this end, he invokes physics and claims that 1 and 2 imply logically incompatible propositions about the velocity of particles (6.3751; NB 16.8./11.9.16). A more straightforward version suggests that they respectively entail something like 'a reflects mainly light of 620nm' and 'a reflects mainly light of 520nm'. Either way, the conjunction of 1 and 2 is a contradiction. As Ramsey pointed out, however, this procedure only pushes the problem one step back (FM 279—80). The resulting propositions once more exclude each other; they ascribe one out of a range of incompatible specifications, a determinate of a determinable. Eventually, colour-exclusion led Wittgenstein to realize that such statements cannot be analysed to yield logically independent elementary propositions, as demanded by the Tractatus (see PR ch. VIII; Glock 1996: 81—4). As a result he abandoned the requirement that elementary propositions are logically independent, holding instead that they form prepositional systems of mutual exclusion and implication. This move not only abandons logical atomism in favour of a kind of prepositional holism, it also brings down the logical system of the Tractatus. For it means that there are logical relations which are determined not by the truth-functional combination of elementary propositions, and ultimately that the logico-semantic rules of language are much more complicated than those catered for in the predicate calculus. Ramsey's observation only knocks over the first domino in a long row. Nevertheless, it is ironical that Ramsey scholars like Sahlin have ignored this very real and important impetus in favour of vague and unsubstantiated speculations about the alleged indebtedness of Wittgenstein's rule-following considerations to Ramsey's decision theory, or of Wittgenstein's holism to the idea of a Ramsey sentence (1995: 150—4). Wittgenstein's initial influence on Ramsey is both clear and momentous. Ramsey's guiding ambition was to reformulate the logicist foundations of mathematics on the basis of Wittgenstein's novel account of logic and of the
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F. P. Ramsey: Critical Reassessments
nature of the proposition (PP164). On his initial visit to Wittgenstein, Ramsey planned not only to understand the Tractatus, but also to 'pump him for ideas for its further development, which I shall attempt' (LO 78). Although the Tractatus rejects the logicist programme, Wittgenstein initially seems to have encouraged Ramsey at least to the extent of agreeing that something should take the place of Russell and Whitehead's Principia Mathematica (Monk 1990:216-17). But in the course of subsequent meetings and correspondence he rejected not only some details of Ramsey's reconstruction, but also the very attempt to provide mathematics with foundations. It is this realization, I submit, that led Ramsey to complain about his visit to Puchberg in May 1924 in a letter to his mother: '... he is no good for my work' (LO 85). And in an unpublished note he wrote about Wittgenstein: 'The immediate effect of his work is almost wholly destructive. But it can I think be made to give a constructive solution' (FR 002-29-01; quoted Kienzler 1997:259). Together with Keynes, Ramsey managed to lure Wittgenstein out of philosophical retirement and back to Cambridge. In 1929 Wittgenstein returned to England. He was officially being supervised by Ramsey, and the Tractatus was accepted as a PhD thesis in June 1929. During the same month, Ramsey wrote to Moore in support of Wittgenstein getting a grant from Trinity College: 'In my opinion, Mr. Wittgenstein is a philosophic genius of a different order from anyone else I know. ... From his work more than that of any other man I hope for a solution of the difficulties that perplex me both in philosophy generally and in the foundations of Mathematics in particular.' Moore also mentions the regular discussions between the two, which have become the stuff of legends. Wittgenstein told Moore that Ramsey once said to him: 'I don't like your method of arguing' (PO 48). Wittgenstein's reactions to these discussions are documented at greater length in his unpublished Nachlass. One can distinguish three phases. Initially, Wittgenstein describes them positively: I have very enjoyable discussions with Ramsey about logic etc. They have something of sport about them and are, I believe, conducted in a good spirit. There is something erotic and at the same time chivalrous about them. They also teach me a certain courage in thinking. There is almost nothing more pleasant for me than to have someone take my thoughts out of my mouth and spread them out in the open. Of course this is all mixed with vanity, but it is not just vanity. It is only reluctantly (\uri\gern) that I roam the fields of science (Wissenschafteri) on my own. (MS 105:4—5; 15.2.29; as Kienzler notes (1997: 58 and nl 10), sense requires insertion of the un.} Gradually, however, the mood changes. Some time before October 1929 Wittgenstein complains about Ramsey's objections being 'shallow' on the grounds
Ramsey and Wittgenstein
45
that they pass by the root of the problem and are unproductive (MS 107: 81). Shortly after Ramsey's death in January 1930 he is even more disparaging: Ramsey's mind (Geist] was very repugnant to me ... about some things I could communicate quite well with Ramsey but in the long run it did not really work well. Ramsey's inability for genuine enthusiasm or genuine reverence - which is the same - finally disgusted me more & more. On the other hand I had a certain fear of Ramsey. He was a very quick and skilful critic if he was presented with ideas. However, his criticism did not move things forward but delayed and sobered. The short period, as Schopenhauer calls it, between the two long ones in which a truth appears first paradoxical & then trivial to people had shrunk to a point in Ramsey's case. And thus one first tried long and unsuccessfully to explain something to him, until he suddenly shrugged his shoulders about it & said that it went without saying. He had an ugly mind (Geist), but not an ugly soul (Seele). (DB 20-1, 27.4.30) Finally, with the benefit of hindsight Wittgenstein's attitude improved. As Kienzler has pointed out, his later remarks about Ramsey are both less specific and more positive (1997: 57). In the Preface to the Philosophical Investigations, written in 1945, he acknowledges the contribution of Ramsey's 'always forceful and assured' criticism in making him realize the 'grave mistakes' of the Tractatus. Still, it is Sraffa whom he credits with having provided the 'stimulus for the most consequential ideas of this book'. Unlike Ramsey, Sraffa also made it onto a list of influences that Wittgenstein drew up in the early 1930s (CV 19). Although the testimony concerning these discussions comes from diverse quarters (see also Rhees 1984:50), it has to be taken with a grain of salt. Nevertheless, it seems that in the main Wittgenstein tried to explain his ideas, old and new, to Ramsey. What is also clear is that Wittgenstein respected and to a degree feared Ramsey as a critic, and that the discussions probably contributed more to the overthrow of the Tractatus vision than to the new approach of Philosophical Investigations. But now it is time to attend to some specific issues debated between Ramsey and Wittgenstein.
Logic and mathematics 'Foundations of Mathematics' is an explicit attempt to defend logicism by remedying certain shortcomings in Principia Mathematica with the aid of the Tractatus (PP 164, see 168-76, 196). There are three points at which Ramsey follows Wittgenstein, although in each case there is also a note of discord.
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F. P. Ramsey: Critical Reassessments
First, his account of logic is derived from the Tractatus (61, 86), and he adopts Wittgenstein's view that the propositions of logic are tautologies. This allows him to improve Russell's insufficiently demanding definition of mathematical propositions as purely general ones by requiring them to be tautologies as well (172-7), a position which is similar to the one that the Vienna Circle derived from Wittgenstein. The Tractatus itself, by contrast, distinguished mathematical propositions as equations from the tautologies of logic. Second, Ramsey followed Wittgenstein in criticizing Principia's reliance on axioms which are not purely logical and thereby undermine the logicist project of grounding mathematics in logic alone. For censure on this count, the Tractatus singles out the axiom of infinity, according to which the number of objects in the universe is not finite, and the axiom of reducibility, according to which for any prepositional function, whatever its order, there is an extensionally equivalent prepositional function, a function of an order just one higher than its arguments (CC 56-7; TLP 5.535, 5.55, 6.1232-3). As regards the axiom of reducibility, Ramsey followed suit. Furthermore, he avoids the need for it by redefining the notion of a prepositional function in a way partly inspired by Wittgenstein (195—6). As regards the axiom of infinity, he argued that, appearances to the contrary notwithstanding, it is actually a tautology. But this argument itself relies on Wittgenstein's treatment of generality (219-24). This treatment constitutes a third point of inspiration. The TLP treated general propositions as the logical products and logical sums of atomic propositions. But it does not straightforwardly identify ' (x}fx' with 'fa .fb .fc...' and l (3x)fx' with 'faVfbVfc fc ...'. l(x)fx' and e(3\fx' express truth-functions, i.e. operations on atomic propositions. But they differ from propositional connectives in the way in which the base of the operation is specified, namely not through listing these propositions, as in l(p . q) V r', but through a propositional variable — Russell's propositional function — 'fx}. Such a variable is a 'logical prototype'; it collects all the propositions of a certain form, since its values are all those propositions we get by substituting a name for the variable, i.e.fa,fb, fc, etc. (3.315-7,5.501,5.522). This approach accommodates the fact that one can understand a general proposition without ever having heard of any of its specific instances. Many Americans believe that all communists are evil, without being able to name a single communist. 'The world can be completely described by means of fully generalized propositions' (5.526; NB 17./19.10.14; PG 203-4). Nevertheless a connection with a logical product remains: some statement of the form 'fa.fb .fc ...' must be equivalent to the universal proposition. As Ramsey pointed out, this is why I know that e.g. '~fa' is incompatible with l(x}fx', whether or not I have heard of a (50).
47
Ramsey and Wittgenstein >
There remains a second problem. One can analyse' (x\fx into a specific conjunction 'fai .fa
There are at least three individuals with some property.
In Russellian notation this comes out as (!')
(Ex)(Ey) (3 X)(3y)(3z)(tt}(®x&y.3>z.x^y.x£z.y^z)
But in a world in which there are only two individuals and one property, Wittgenstein's account of generality seems to turn 1' into a contradiction. If we substitute '/ for '$', V for lx' and 'b' for y and 'z' (in our model-world no other individual constant is available), we get (1*)
Fa.Fb.Fb.a^b.a^b.b^b
a contradiction because of the last conjunct. By the same token, 'There is at least one individual' and 'There are at least two individuals' come out as tautologies.
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F. P. Ramsey: Critical Reassessments
In 1925 Ramsey credited the idea that the number of individuals in the universe is not 'a mere question of fact' to 'the profound analysis of Wittgenstein' (222). By 1927 he had realized that Wittgenstein solves the problem in a different fashion (50). The Tractatus classifies propositions about the numbers of objects in the universe as nonsense (4.1272, 5.535). It can do so because its ideal notation foregoes the use of the identity-sign (see Clock 1996: 164-9). Accordingly, it analyses 1 as (1")
(3*) (3?) (Hj) (3$) (**.$?.$<;).
Substitution into that proposition will not yield a contradiction. Nevertheless our commitment to objects in fully general propositions cannot outstrip the number of objects. In our fictional world of just two individuals the prepositional function <&x will not have values other than^a andyj, that is, there will be no other propositions for truth-functions to operate on, which means that propositions employing more variables than x andjy have no application and are hence nonsensical (see 5.5262). A cut-off clause like 'a and b are everything' suffers the same fate, since its formulation requires the use of the identity-sign. The number of objects in the universe, which both Russell's axiom of infinity and Ramsey's cut-off clauses try to state, can only be shown by the number of names employed by an ideal notation (5.535, NB9.10.14). In 'Facts and Propositions' Ramsey merely alludes to Wittgenstein's claim that a cut-off clause cannot be formulated in Wittgenstein's 'improved symbolism for identity'. He foregoes a discussion of this alternative answer (50). By 1929 his response is less diplomatic. He notes that if'(x\fx' quantifies over an infinite domain, we cannot treat it as a conjunction because it would require 'a theory of conjunctions which we cannot express for lack of symbolic power' (146). Then follows: [But what we can't say we can't say, and we can't whistle it either.] The jibe alludes not just to Wittgenstein's well-known expertise at whistling (Hacker 2001: 10 In). It is also a highly cryptic attack on the Tractatus attempt to avoid the aforementioned problems concerning quantifications over infinite domains by appeal to the saying/showing distinction. Wittgenstein for his part rejected Ramsey's solution on account of its reliance on the notion of identity. For Wittgenstein numerical identity, far from being a 'necessary relation', is not a genuine relation at all: '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' (5.5303). In an ideal notation, identity and difference is expressed not by a separate sign, but by ensuring a one-to-one correlation between logically proper names and objects.
Ramsey and Wittgenstein
49
Identity played a major role in Ramsey's attempt to improve Russell's logicism. He followed Wittgenstein's criticism of the Principia (*13.01) definition of identity, namely that it implies the principle of the identity of indiscernables, i.e. that two objects cannot have all their qualities in common (193—4; see TLP 5.5302). Unfortunately, like Russell and Wittgenstein he ignored the question of whether these qualities are to include spatio-temporal location, which would make the principle plausible. At the same time Ramsey tried to retain identity in a way which accommodates TLP. He defined 'x =y> through a two-place function Q(x,y) which is a logical product of material equivalences:/^ =f\j -fixx =/2j • > etc. This yields a tautology when x andjy have the same value or meaning —f\a =f\a .faa =fza, etc. — but otherwise a contradiction, because there will be a function/^* and two objects a and b such that ~ (fka =fka). Wittgenstein protested that in the case of x andjy having different values, 'a — b' is not contradictory but nonsensical, and so is 'a ^ b\ since the negation of a nonsense is a nonsense. It involves a misuse of the logically proper names involved, since in an ideal notation each object must have its own distinctive name. But if'a = b' is nonsense, so is 'a ^ b', since the negation of a nonsense is a nonsense. To this, Ramsey responded in effect that his Qjfunction was not synonymous with '=', but only performed an equivalent role (CG 216-21; see WVG 189-92). The exchange nicely displays their contrasting yet complementary philosophical talents: Ramsey tries to counter the paralysing effects of Wittgenstein's dialectical acuity through ingenious technical expedients. As we shall see shortly, Ramsey is right to resist Wittgenstein's appeal to the saying/showing distinction. Shorn of this distinction, Wittgenstein's treatment of identity turns into a meta-linguistic or de dicto account of identity statements, as prefigured by the early Frege. Even on a de re account, however, Ramsey's solution to the problem of infinite domains cannot stand. While it is necessary that an+\ differs from a\ to an ifh t exists, it is not necessary and not a tautology that there is an object that so differs from a\ to an, and hence that there is a particular number of objects (see Kripke 1980). It emerges that neither Wittgenstein's nor Ramsey's solution to the problem of infinite domains can stand. Both came round to this view and abandoned the Tractatus account of generality. Some of their reasons diverge. For Wittgenstein, the collapse of logical atomism implied that there is no well-defined and enumerable totality of conjuncts of the form '/x' into which '(x)fx> can be analysed, and that the quantifiers are not 'topic neutral' (Clock 1996: 149— 50). Ramsey for his part was worried that we could not know or be certain of an infinite conjunction or disjunction, and hence could not guide our actions by it (146-7). There is, however, one central criticism both share. The TLP analysis only works in cases in which the domain of quantification is a closed class, as in
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'In this picture I see all the primary colours', not in the case of what Ramsey calls 'variable hypotheticals' such as 'All men are mortal', where we would require a cut-off clause. On current evidence it cannot be ascertained whether this objection originated in Wittgenstein or in Ramsey. Wittgenstein's reflections on this issue postdate the drafting of Ramsey's 'General Propositions and Causality' in 1929, but are also more elaborate. What can be ascertained is the origin of the alternative which both Wittgenstein and Ramsey adopted around 1929. Since both remained reluctant to accept Russell's idea that general propositions state sui generis general facts, they were driven to the conclusion that they are not genuine propositions at all, but rather rules or laws for the construction of propositions. This view originated in Weyl, and seems to have reached Wittgenstein via Ramsey, who excerpted Weyl's discussion in 1925 (Sahlin 1990: 104-7; 1995: 153-4). But on a related and more general issue the influence runs the other way. Initially Ramsey defended logicism and classical two-valued logic against what he termed the 'Bolshevik menace' posed by Brouwer's and Weyl's intuitionism. By 1929 he seems to have become a fellow traveller. Shortly before his death, Ramsey underwent a conversion from logicism to the kind of finitism associated with intuitionism (Braithwaite 1931, xii; NPM 178—81 and 197—202). Accordingly, at that time Ramsey's and Wittgenstein's views on mathematics were not further apart than ever, as Sahlin speculates (1990: 179). For in 1930 Wittgenstein also rejected the extensional notion of the infinite, the idea that there can be an actual infinity of things. Furthermore, although the issue is controversial, this position is already anticipated in the Tractatus. The early Wittgenstein certainly rejected not just Russell's theory of types, but the axiomatic conception of logic on which logicism is predicated. Furthermore, its definition of number is constructivist rather than logicist (see Glock 1996: 231-6, 264-8), and Ramsey alluded to that definition in his move towards finitism. But even if the Tractatus did accept the actual infinite (as argued by Wrigley 1995), there are strong indications that Wittgenstein abandoned this position before Ramsey. In a note dating from February 1929 he remarks: I once said that there was no extensional infinity. Ramsey replied: 'Can't we imagine that a man lives forever, that is simply, never dies, and isn't that extensional infinity'. I can surely imagine that a wheel spins and never comes to rest. What a strange argument 'I can imagine ...' (PR 304, see also 155).
Probability and induction With respect to these topics, Ramsey worked in the context of a Cambridge tradition that comprised not just Wittgenstein but, before him, Keynes,
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Russell and Johnson. Like Keynes, TLP provides a logical account of probability as a relation between the structures of propositions which can be displayed through truth-tables. It elaborates Laplace's classical definition of probability as a ratio of the number of possibilities which are favourable to the occurrence of an event and the number of overall possibilities. Like Bolzano it drops from the definition the principle of indifference, the requirement that the possibilities be equally likely, although this requirement recurs in the idea that all propositions are analysable into logically independent and equally probable elementary propositions (see von Wright 1982:144). Ramsey probably had a hand in the correction of a mistake in the first edition of TLP (5.152; see von Wright 1982: 141; Sahlin 1995: 155 and n35). More importantly, Ramsey reinvigorated the subjective theory of probability. His two main complaints against Wittgenstein's theory are that it is of no practical use, and that it does not furnish a justification of induction (see 86 and Sahlin 1995: 156; 1990:48-51). Both observations are correct. The question is whether they constitute a serious failure on Wittgenstein's part. Throughout his career Wittgenstein rejected two paradigmatic defences of inductive reasoning against Hume's attack — the invocation of a Principle of the Uniformity of Nature and the suggestion that it is covertly probabilistic. TLP adopts a Humean scepticism about inductive reasoning (6.3-6.31, 6.363-6.36311). Induction is a procedure, namely of'accepting as true the simplest law that can be reconciled with our experiences'. It has only a psychological justification, 'there are no grounds for believing that the simplest eventuality will in fact be realized'. For the 'law of induction', according to which nature is uniform — will carry on the way it has in the past — is a proposition with a sense, and has no logical justification. However, everything outside logic, in the domain of empirical science is 'accidental'. In the same spirit, Wittgenstein's account of probability delivers the axioms of a standard a priori probability calculus. It does not even purport to be of practical use by explaining contingent statistical observations. Successive draws from an urn containing an equal number of black and white balls (after each of which the ball is returned) will show that the number of black balls approximates that of white balls. This does not confirm the a priori judgement that the probability of drawing a white ball is ^. Rather, it shows that relative to the 'hypothesized laws of nature', and to the initial conditions of the experiment, the two events are equipossible, i.e. that a condition for the application of Wittgenstein's probability calculus is satisfied (5.154). This discovery is empirical, since there might be an unknown physical link between the colour of objects and their propensity to be drawn. The probability calculus of TLP collapsed with logical atomism. But Wittgenstein did not come round to Ramsey's idea of vindicating induction through non-deductive reasoning (pace Sahlin 1995, 156-8). For one thing,
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he held on to the insight that there is a difference between a. priorijudgements of probability and empirical statistical judgements (PR ch. XXII; PG 213-35; WVC 93—100; PI §§482,484). It seems that probability judgements are confirmed by statistical observations about the relative frequency of alternative outcomes in a limited series of experiments. Attempts to construe inductive reasoning as a form of probabilistic reasoning in which observations of past regularities render a prediction probable trade on this illusion. Statistical observations, e.g. that in the past 20 per cent of smokers have died of lung cancer, may lead to an inductive extrapolation which assigns a certain probability to N.N.'s dying of lung cancer. One cannot, however, assign a probability to the induction itself. Further experience may confirm that the initial regularity continues, but this merely confirms the specific extrapolation, which is itself an inductive hypothesis. Consequently probability cannot vindicate induction. It either amounts to statistical extrapolations (smoker-case) which are themselves inductive, or to applications of a calculus (urn-case) which presuppose rather than explain natural regularities and their continuation. It is true that the later Wittgenstein, like Ramsey, stresses the central role of inductive reasoning in our practices. 'It is our acting which lies at the bottom of the language-game' of inductive reasoning (OC§§204, 273, 298, 613-19; PI §472—4). It is also true that he abandons the Humean scepticism about inductive reasoning. But he equally resists Russell's and Ramsey's pragmatist proposal that induction is justified through its success or usefulness. 'Thinking has been found to pay' itself exemplifies the pattern of reasoning it is supposed to vindicate (PI §§467-9; OC §§130-1). We cannot 'establish by induction that induction was reasonable', as Ramsey suggested in a 1923 paper (NPM 301). Nor is it clear how Ramsey's later appeal to reliable mechanisms (93) improves matters. For if we establish the reliability of induction by way of induction, then circularity once more looms. Current malpractice notwithstanding, one cannot improve circular reasoning by professions of pragmatism or naturalism. Unlike pragmatism, Wittgenstein denies that the success of inductive reasoning provides a rationally compelling response to inductive scepticism. Experience provides 'a hundred reasons' for our specific predictions (PI §475). Yet it does not provide compelling grounds for the general practice of taking relevant experience as grounds for prediction (OC §§130—1). Wittgenstein also insists that the demand for such grounds is itself misguided. But his reasons for doing so are much closer to Strawson's analytic approach to induction (1952: ch. 9) than to pragmatism. Inductive reasoning is not a method for predicting the future which may be more or less adequate, more or less reliable by inductive standards. Rather, it defines what it is to make rational predictions. We call a prediction 'reasonable' precisely if it is supported by previous experience. On a more specific level it is a grammatical proposition that making a transition from a specific kind of evidence to a
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certain conclusion is rational. 'A good ground is one that looks like this' (PI §483). If the sceptic replies that our patterns of reasoning themselves are inadequate, because these regularities have only been observed in the past, he ignores that there can be no such thing as now having evidence from the future (although we can have evidence/or future events). The sceptic's point cannot be that there are good reasons for empirical beliefs, only past experience is not one of them. Instead, he simply refuses to call information about the past evidence for the future. But this could at best be a recommendation for a terminological shift. The reasons we have for resisting this shift may ultimately be pragmatic, however. For the sceptic's novel terminology removes the vital distinction between conclusive, good and weak evidence.
Universals It is now time to look at the Tractatus account of propositions, on which Ramsey professed to erect his theories of mathematics and probability. According to Sahlin, Ramsey found Wittgenstein's theory in the Tractatus, 'that neither is there a copula, nor one specially connected constituent, but that ... the objects hang one in another like the links of a chain' (page 17) dogmatic, for it 'does not really explain any difference in the mode of functioning of subject and predicate' (page 17). But it was the insufficient work of Wittgenstein, Johnson and Russell that inspired Ramsey to work on this classical problem. (1995: 159; see 1990: 194) In his eagerness to display Wittgenstein as Ramsey's philosophical inferior at each and every turn, Sahlin has misinterpreted his hero. Ramsey does not describe Wittgenstein's account of the proposition as dogmatic. Rather, he claims that, like Johnson's appeal to a 'characterizing tie' between the constituents of the proposition, it leaves the distinction between subject and predicate 'a mere dogma'. This is not presented as a criticism of the Tractatus, however, since Ramsey does not ascribe to Wittgenstein the ambition which he does ascribe to Johnson and Russell, namely to vindicate such a distinction. On the contrary, 'Universals' constantly draws on Wittgenstein in its own denial that 'there is a fundamental antithesis between subject and predicate' and the corresponding denial that there is a 'fundamental division of objects into two classes, particulars and universals' (12, 8). For one thing, Ramsey invokes the Tractatus to block Russell's claim that the distinction between particulars and universals is intuitively compelling, since a particular like Socrates is an independent entity and a universal like
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wisdom a quality of something else (18-19). According to Russell, this appeal to what we would nowadays call an intuition fails. Neither 'Socrates' nor 'wise' are names of genuine objects by the standards of logical atomism. And according to Wittgenstein, with whom I agree, this will be the case with any other instance that may be suggested, since we are not acquainted with any genuine objects or atomic propositions, but merely infer them as presupposed by other propositions. Hence the distinction we feel is one between two sorts of incomplete symbols, or logical constructions, and we cannot infer without further investigation that there is any corresponding distinction between two sorts of names or objects. (19) In this further investigation, moreover, Ramsey invokes Wittgenstein's notion of an 'expression' (Ausdruck) and his ideas concerning prepositional variables (TLP 3.3Iff.). A proposition like 'aRb' does not naturally divide into 'a' and 'Rb'. So why should we so divide it. The answer is that if it were a matter of this proposition alone, there would be no point in dividing it in this way, but that the importance of expressions arises, as Wittgenstein pointed out, just in connection with generalization. It is not 'aRb' but '(x).xRb' which makes Rb prominent. (19) Accordingly, the distinction between particulars and universals is not an ontological one, but an artefact of quantification. Particulars are simply those things over which we quantify first. Wittgenstein's principled agnosticism about what objects and elementary propositions look like (NB 14.-17.6.15; TLP 5.55/7, 4.221; CC 19.8.19) also helps Ramsey in resisting another attempt at distinguishing particulars and universals. In the second edition of Principia Russell maintained that all atomic propositions are of the forms 'Rl (x)', 'R2 (x, y)', 'R3 (x, y, z)', etc. This would allow him to define the names of particulars as terms which can occur in propositions with any number of terms. However, according to 'Universals' this solution is spurious, since 'we know and can know nothing whatever about the forms of atomic propositions'. Of all philosophers, Wittgenstein alone has seen through this muddle and declared that about the forms of atomic propositions we can know nothing whatever. (29-30, butcf. 31) Finally, Ramsey derived the inspiration of his own theory from the Tractatus. He discusses 'Wittgenstein's claim that the objects hang in one another like the links of a chain' (25). The reference is to TLP 2.03: 'In an atomic fact (Sachverhalt] objects hang in one another like the links of a chain'. Far from criticizing Wittgenstein along with Russell and Johnson, Ramsey prefers 'Wittgenstein's
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view that in the atomic fact the objects are connected together without the help of any mediator', to the traditional view, condoned by Johnson, that there is a need for a tie such as the one signified by the copula (29). Frege accounted for the unity of propositions through a chemical analogy: concept-words (concepts) are 'unsaturated' - they contain a variable - and hence combine with 'saturated' argument-expressions (objects) to form a saturated proposition (1891). Russell maintained that among the components of facts are logical forms, which hold together the components of the complex. But he faced a problem in that aRb and bRa have the same logical form. His solution was that a and b are linked to R through further relations, which differ in these two cases (1984:80-8), a proposal which invites Bradley's regress-argument against the reality of relations. The TLP agrees with Frege and Russell that even logical propositions and the states of affairs they depict must be 'logically articulated' or composite. A proposition contains two or more constituents; nevertheless it is not a mere list of names, since what represents is not just the assembly of constituents (TLP 3.14ff., 4.024-32). A proposition is a. fact which constitutes a description of a state of affairs. That propositions are facts also provides Wittgenstein's explanation of'how the prepositional union comes about' (4.221). After a fashion, all components (names) of propositions are unsaturated, since 'only in the context of a proposition does a name have a meaning' (3.3). Names combine immediately, without the help of logical glue, just as the components ofSachverhalte fit into one another like links in a chain, without the need of mediating entities or relations. The elementary proposition consists of names. It is a connexion, a concatenation of names. It is obvious that the analysis of propositions must bring us to elementary propositions which consist of names in immediate combination. (4.22f.; my emphasis) The combination is immediate because the possibility of and the need for combining with other objects of a certain kind is already written into each object through its internal properties or logical form (2.012—2.031). This dispenses with the need for a copula, as in Johnson, or for an additional logical form, as in Russell before the war, or for a distinction of saturated and unsaturated expressions and entities, as in Frege and in Russell's 'Lectures on Logical Atomism' (to which Ramsey refers on p. 10). Ramsey faithfully echoes Wittgenstein's position: 'In a sense, it might be urged, all objects are incomplete; they cannot occur in facts except in conjunction with other objects, and they contain the forms of propositions of which they are constituents' (11). Whether he ascribes to Wittgenstein his own conclusion that there is no distinction between particular and universal is
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unclear. And whether such an ascription would be accurate is controversial. Nominalist interpreters have long denied that universals are among the objects of the Tractatus, and that elementary propositions contain names for properties and relations. They insist that far from abolishing the distinction, Wittgenstein makes it even more radical, in that universals are not objects, but mere forms or matrices (Anscombe 1959: ch. 7). In my view, this interpretation is mistaken (Clock 1996: 2 70-4). But I also believe that Wittgenstein held on to the idea that particulars and universals are objects of different kinds, which is why he used symbols of different kinds to signify them.
Truth In 'Facts and Propositions', Ramsey writes: But before we proceed further with the analysis ofjudgement, it is necessary to say something about truth and falsehood, in order to show that there is really no separate problem of truth but merely a linguistic muddle. Truth and falsity are ascribed primarily to propositions. The proposition to which they are ascribed may be either explicitly given or described. Suppose first that it is explicitly given; then it is evident that Tt is true that Caesar was murdered' means no more than that Caesar was murdered, and 'It is false that Caesar was murdered' means that Caesar was not murdered. They are phrases which we sometimes use for emphasis or for stylistic reasons, or to indicate the position occupied by the statement in our argument. ... In the second case in which the proposition is described and not given explicitly we have perhaps more of a problem, for we get statements from which we cannot in ordinary language eliminate the words 'true' and 'false'. Thus if I say 'He is always right', I mean that the propositions he asserts are always true, and there does not seem to be any way of expressing this without using the word 'true'. But suppose we put it thus 'For all/), if he asserts p,p is true', then we see that the propositional functionjb is true is simply the same asp, as e.g. its value 'Caesar was murdered is true' is the same as 'Caesar was murdered'(38-9). This passage commits Ramsey not just to the schema (2)
It is true that p iffp
which he rightly regarded as a truism to which any account of truth must pay heed (OT 14). It also commits him to the idea that 'it is true that p' means the same, or has the same content, as 'p': (3)
The proposition that p = the proposition that it is true that p
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Furthermore, he seems to accept that this identity provides a complete explanation of the notion of truth, since he insists that there is no separate problem 'as to the nature of truth and falsehood', only one 'as to the nature of judgement or assertion'. This suggests that he indeed held the kind of redundancy theory often attributed to him (see Kiinne 2003: 34-7). It has been objected that unlike a redundancy theorist, Ramsey does not simply take for granted the content of judgement (Dokic and Engel 2002: 20). But it is precisely the punch-line of a redundancy theory that it explains truth in a way that requires a further, independent explanation of meaning or content. What one cannot do, as Dummett has shown (1978: ch. 1), is to treat equivalences like 1 both as a complete explanation of truth and as a complete explanation of content. Ramsey refrains from the latter. As we shall see, he explains content through a causal theory, i.e. without reference to truth. What is true, and unfortunate, is that this theory commits him to identifying the conditions under which a belief is true with the conditions under which it is successful. The later Wittgenstein is also widely credited with a redundancy theory. According to Philosophical Investigations (§136). (4)
'/>' is true —p
Elsewhere Wittgenstein explicitly declares the identity to be one of meaning: For what does it mean, a sentence 'is true'? 'p' is true = p. (This is the answer). (RFM117) At first sight, there is nevertheless an important difference between this version of the redundancy and Ramsey's. 4 treats sentences as truth-bearers, and hence the truth-predicate as a device for disquotation - like Tarski and Quine. 3 treats propositions as truth-bearers, and the truth-predicate as a device for denominalization (of the that-clause). However, this difference is more apparent than real. In earlier writings Wittgenstein insisted that ' "/>" is true' can be understood only if one treats the sign 'p' as an interpreted sign, a symbol rather than a mere ink mark. By contrast with Tarskian theories Wittgenstein rightly denied that 'is true' applies to sentences. In the idiom of the Tractates, it applies to propositions (Satze] rather than prepositional signs (Satzzeicheri). Furthermore, like Ramsey, Wittgenstein had no qualms about quantifying over propositions, which is necessary to account for blind truth-ascriptions that would otherwise defy the redundancy theory: What he says is true = Things are as he says. (PG 123)
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The convergence makes it even more tempting to assume that Wittgenstein took over Ramsey's redundancy theory. But the temptation should be resisted. Wittgenstein had toyed with a redundancy theory as early as 1914: '/>' is true, says nothing else butjfr! (NB 6.10.14; see 113) And in the Tractatus he wrote . . . a proposition is true if things are as we say they are by using it ( . . . wahrist ein Satz, wenn es sich so verhdlt, wie wir es durch ihn sageri]. (4.062) This equates to the quote from Philosophical Grammar., except that propositions rather than people are presented as saying things. Whether the Tractatus propounds a correspondence theory, as traditionally assumed, or a deflationary account, as some recent commentators have maintained, is a complex question that I discuss elsewhere (Clock 2004). What is clear is that in his 'Critical Notice' Ramsey already interpreted it as expounding a deflationary account. Commenting on Wittgenstein's answer to the question 'What is it for a proposition token to have a certain sense?' he writes: First, it may be remarked that if we can answer our question we incidentally solve the problem of truth; or rather, it is already evident that there is no such problem. For if a thought or proposition token 'p' says p, then it is called true ifp, and false if~p. We can say that it is true if its sense agrees with reality, or if the possible state of affairs which it represents is the actual one, but these formulations only express the above definition in other words. (FM 275) These other words are exactly the ones used in the Tractatus. The sense of a proposition is the state of affairs it depicts. And The agreement or disagreement of its sense with reality constitutes its truth or falsity. (2.222) If an elementary sentence is true, the state of affairs obtains: if an elementary sentence is false, the state of affairs does not obtain. (4.25) Ramsey credits Aristotle with having anticipated the redundancy theory (OT 11), and he had access to Johnson's version of it (1921: 52—3). Nevertheless, his adoption of a redundancy theory was at least partly stimulated by Wittgenstein, rather than the other way around.
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Meaning and content This conjecture receives additional support from the culminating passages of 'Facts and Propositions': In conclusion, I must emphasize my indebtedness to Mr Wittgenstein, from whom my view of logic is derived. Everything that I have said is due to him, except the parts which have a pragmatist tendency, [fn: and the suggestion that the notion of an atomic proposition may be relative to a language] which seem to be needed in order to fill a gap in his system. But whatever may be thought of these additions of mine, and however this gap should be filled in, his conception of formal logic seems to me indubitably an enormous advance on that of any previous thinker. My pragmatism is derived from Mr Russell; and is, of course, very vague and undeveloped. The essence of pragmatism I take to be that the meaning of a sentence is to be defined by reference to the actions to which asserting it would lead, or, more vaguely still, by its possible causes and effects. Of this I feel certain, but of nothing more definite. (51) Ramsey was right to detect a gap in the Tractatus. The book states that the logically proper names that are the elements of elementary propositions go proxy for the objects that are the elements of possible states of affairs, and that the meaning of a name is the object for which it deputizes (3.203-3.22). But it ignores the question of how signs (names) are connected with what they mean (objects). Worse still, Wittgenstein explicitly defended this omission, insisting that the relation between language or thought and reality was merely a 'matter of psychology' (NB 130). Like Russell (1921) and Ogden and Richards (1923), Ramsey envisaged filling this very lacuna by a causal theory of meaning. On his return to philosophy Wittgenstein also recognized the need for remedial action. But his account was developed in direct contrast to the causal theories propounded by Russell, Ogden, Richards and Ramsey. Wittgenstein abandoned the idea that the meaning of a word is the object it stands for. Instead, he elaborated an idea, dimly anticipated in the Tractatus, that it is 'its use in the language' (§41, see TLP 3.326ff., 6.211). Causal and behaviourist theories also tend to explain the meaning of an expression by reference to its use. Accordingly, the idea that meaning is use links Wittgenstein's later philosophy not just to Russell, Ogden, Richards and Ramsey but also to American pragmatists like G. H. Mead. But whereas Wittgenstein's conception is a normativist one, linking meaning to rules of use (see Clock 2000), the pragmatist conception of use tends to be purely causal and behaviourist.
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The meaning of a word is equated either with the conditions which cause a speaker to utter it, or with the effects which such an utterance has on hearers. Wittgenstein grants that the initial acquisition of language is a kind of training which includes the conditioning of verbal reactions to external stimuli. But the causes and effects of uttering an expression determine neither whether the expression has a linguistic meaning, nor what meaning it has. As regards the causes of our utterances Wittgenstein, like Chomsky after him, points out that fully developed linguistic behaviour is not strictly conditioned. Whether an expression is used in a particular situation depends not just on its meaning, but on extrinsic factors. For example, the fact that few people would dare call Tony Blair a liar is due neither to the meaning of'liar' nor to Tony Blair, but to legal and political considerations. As regards the effects of utterances, Wittgenstein reasons as follows. Whether a sign is meaningful and what meaning it has does not depend on either its actual or even its intended effect, whether on a particular occasion or in general. 'This sign means X' does not mean 'When I utter this sign X will happen' or 'When I utter the sign I intend X to happen'. If I say 'Milk me sugar!' this may well have the result that my hearer stares at me and gapes. But it does not follow that this combination of words means 'Stare at me and gape!'. It doesn't even follow if this entertaining effect can be repeated. Indeed, it does not even follow if I utter these words with the intention of bringing about this reaction (PI §§493-8; PR 64; PG 68-9, 187-92). The causal analysis of the linguistic meaning of expressions that Ramsey declares to be the essence of his pragmatism is a definite failure, therefore. But Ramsey also suggests a 'pragmatist view' of the content of beliefs and utterances: 'any set of actions for whose utility/? is a necessary and sufficient condition might be called a belief that/?, and so would be true ifp, i.e. if they are useful'. In a footnote he explains 'It is useful to believe aRb would mean that it is useful to do things which are useful if, and only if, aRb; which is evidently equivalent to aRb' (40). Ramsey is right to hold (5)
11 is useful to do things which are useful iff aRb 4=^ aRb
But it is a stipulation to hold that (5')
It is useful to believe that aRb 4=> it is useful to do things which are useful iff aRb
On the ordinary understanding, even if acting on a belief would not be useful, having the belief might be useful, simply because it improves the believer's mood, and thereby helps her to do things which have nothing to do with things one might do on account of the belief.
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More seriously, neither a believing - Ramsey's 'mental factor in judgement' (34) - nor what is believed — Ramsey's 'objective factor' — can be a set with members that occur in space and time. Furthermore, the passage commits Ramsey to a strong form of behaviourism that both Wittgenstein and contemporary Ramseyians would reject. It identifies beliefs not just with dispositions to act, but with actions. In order to avoid these problems, Ramsey's account would have to take the following form: (6)
a believes that/? <=$• a is disposed to act in a way which is useful iffp
But what about beliefs that aren't linked even to dispositions to behave? And what about weakness of will? I may believe that excessive training is bad for me, and yet be disposed to act in a way that is useful iff excessive training has no harmful effects. Ramsey confines this account to non-linguistic belief, and neither of these phenomena gets a foothold there. However, he also writes without such qualifications: 'For a belief that A is B, means on this view, a belief which is useful if and only if A is B, i.e. if and only if it is true; and so conversely it will be true if and only if it is useful' (OT 91). This in effect identifies the content of a belief and the conditions under which it is true with the conditions under which it is useful. But not only can the conditions under which a belief is true differ from those under which holding the belief is useful, they often actually do. According to Ramsey (7)
The belief that p = the belief which is useful iffp
By this token, (8)
The belief that God exists = the belief that is useful iff God exists
The 'only if does not hold, however. The belief that God exists is useful even if God does not exist. People with religious beliefs tend to lead longer and happier lives, and to recover more quickly from trauma and disease. They also tend to have greater biological fitness, since their beliefs inspire them to produce more offspring. The success conditions of the theistic belief in God are amply satisfied, while the truth conditions are not. It might be objected that the belief in the existence of God does not contribute to the success of any particular action. But to the extent to which this holds in this case, it also holds for many other theoretical beliefs, which would therefore remain outside the scope of the pragmatist account. Furthermore, the point also holds for beliefs that do contribute to the success of actions, such as overestimating one's strength (see Russell 1910: ch. 5).
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Dokic and Engel (2002:46-8) protest that such a false belief can also have disastrous consequences, and they conclude that the problem can be solved by insisting that true beliefs, and only they, guarantee success whatever desire and action are involved. But this defence fails. For the true belief that I'm weaker than a competitor can and often does lead to failure. This is simply the reverse side of the point that false beliefs can be advantageous. And it is all that is needed to disprove what they call 'Ramsey's Principle', namely that 'true beliefs are those that lead to successful actions whatever the underlying motivating desires'. That principle also faces the objection of cognitive overload (Brandom 1994). It implies that any failure of my actions must be due to at least one of my beliefs being false. But there is a non-denumerable set of reasons for possible failure, ranging from things that contradict my beliefs to things I never thought of, like a sudden change in the gravitational constant. This would imply that I have a non-denumerable set of beliefs. Dokic and Engel bite this bullet, since they think that we hold an indefinite number of implicit beliefs (2002: ch. 5). Alas, this response lumbers agents with a host of beliefs that they are not even able to avow when asked. In fact, it lumbers them with a host of theoretical beliefs even if they lack the concepts that feature in those beliefs, like the concept of the gravitational constant. By the same token, animals entertain highly complex beliefs that most humans would not even be able to understand, simply because the corresponding facts are preconditions for the success of animal behaviour. Such an uncontrolled attribution of beliefs can be acceptable only to someone who is willing to tailor the concept of belief to his analysis, rather than the other way around. As we shall see in the next section, the readiness to even consider such a move is one respect in which Ramsey's conception of analysis differs from Wittgenstein's. Wittgenstein never addressed Ramsey's fledgling success semantics. But he convincingly criticized Ramsey's causal account of content (Russell 1923, ch. XII). Russell connects a thought and what satisfies it through a 'tertium quid': for example, my desire is fulfilled if I have a feeling of satisfaction. As Wittgenstein pointed out, this implies that 'if I wanted to eat an apple, and someone punched me in the stomach taking away my appetite, then it was this punch that I originally wanted'. To avoid this absurd consequence, Wittgenstein invokes a key element of his earlier 'picture conception' (PR 64; see TLP 4.014, 4.023, 4.03). The relation between a thought (belief/desire) and its content is not causal but logical or internal, i.e. constitutive of the relata. My thought that/? could not be made true by any fact other than that/?, irrespective of what feelings or actions it leads to. Equally, 'I should like an apple' does not mean T believe an apple will quell my feeling of non-satisfaction' (PI §440; PG 134).
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This criticism applies equally to Ramsey. The connection between a belief and the conditions under which actions motivated by it are successful is external, whereas the connection between a belief and the conditions under which it is true are internal, independent of any mediating link or extrinsic factors.
The nature of philosophy According to the Tractatus, the only propositions with a sense are the bipolar propositions of empirical science. Tautologies are a limiting case of such propositions, in that they have zero sense. But all other attempts to express necessary truths must result in nonsense. This holds not just for logico-metaphysical pronouncements about the essence of language and reality, but also for 'mystical' pronouncements about matters such as value and death. At best, these pronouncements are attempts to say, to express in propositions, what can only be shown, namely by empirical propositions properly analysed. The TLP consists almost entirely of such attempts. That is why the book ends: My propositions serve as elucidations in the following way: anyone who understands me eventually recognizes them as nonsensical, when he has used them — as steps — to climb up beyond them. (He must, so to speak, throw away the ladder after he has climbed up it.) He must transcend these propositions, and then he will see the world aright. Whereof one cannot speak, thereof one must remain silent. (6.54-7) Proper philosophy cannot be a doctrine, since there are no philosophical propositions. It is an activity, namely 'the logical clarification of thought' through logical analysis. Philosophy demonstrates 'what cannot be said', namely ineffable truths about the nature of language and reality, 'by presenting clearly what can be said'. Without propounding any propositions of its own, it clarifies the logical form of the meaningful propositions of science, but also demonstrates that the propositions of metaphysics violate the rules of logical syntax (4.112,4.115,6.53). What is surprising is not that Ramsey dissented from parts of this extraordinary picture, but how much of it he accepted. He shared the basic idea that philosophy aims at clarity and that it proceeds by way of logical analysis. Philosophy issues in a 'logical system of primitive terms and definitions' (1), a system which has the purpose of clarifying concepts. He also draws on the more specific idea that some words or 'symbols' cannot be defined nominally, but instead are to be clarified by 'elucidations' or explanations of their use (2-3). This is an allusion to TLP 3.263. 'Names' or 'simple signs' cannot be explained through definitions, since they are the unanalysable constituents of
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elementary propositions. At the same time, unlike proposition names need to be explained and given meaning (4.03). This is to be achieved through 'elucidations (Erlauterungeri}\ elementary propositions 'which already contain the primitive sign' and explain it by demonstrating its use. At the same time, Ramsey repudiated the notorious saying/showing distinction: Philosophy must be of some use and we must take it seriously; it must clear our thoughts and so our actions. Or else it is a disposition we have to check, and an inquiry to see that this is so; i.e. the chief proposition of philosophy is that philosophy is nonsense. And again we must then take seriously that it is nonsense, and not pretend, as Wittgenstein does, that it is important nonsense! (1) According to Ramsey, the Tractatus was indeed committed to the idea of metaphysical truths that cannot be said but can be shown, contrary to recent proponents of a 'New Wittgenstein'. As we have seen in Logic and Mathematics, above, Ramsey's famous 'what we can't say we can't say and we can't whistle it either' is explicitly directed against Wittgenstein's idea that the number of objects in the universe can be shown rather than said. Ramsey was right to balk at the saying/showing distinction. Even if logical and conceptual analysis is an activity that differs from both science and traditional philosophy, it cannot proceed on the basis of a once-and-for-all insight into the essence of language which is totally independent of the results of analysis, as the Tractatus assumed. Rather, it results in and in turn needs to be guided by propositions that clarify the use of philosophically relevant terms. Wittgenstein himself later abandoned the saying/showing distinction (see Hacker 2001: ch. 5), perhaps partly because of Ramsey. A second contrast is connected to Ramsey's animadversions against scholasticism, the essence of which is treating what is vague as if it were precise and trying to fit it into an exact logical category. A typical piece of scholasticism is Wittgenstein's view that all our everyday propositions are completely in order and that it is impossible to think illogically. (This last is like saying that it is impossible to break the rules of bridge because if you break them you are not playing bridge but, as Mrs G. says, not bridge.) (7) This complaint points back at the Tractatus and forward to the PI. In his Introduction, Russell had alleged that the Tractatus was intent on establishing 'the conditions that would have to be fulfilled by a logically perfect language'. In his 'Critical Notice' Ramsey conceded that some passages are concerned with a logically perfect language, but pointed out that others apply to ordinary languages (FM 270). The truth of the matter is that for the
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Trcctatus all languages are logically perfect, since they are all capable of 'expressing every sense'. Even natural languages do not suffer from the shortcomings deplored by Frege and Russell, such as referential failure, type confusions or indeterminacy. It is just that their grammatical surface conceals their underlying logical form, the fact that under the surface they are governed by the strict and precise rules of logical syntax. Accordingly, 'all propositions of oui everyday language, just as they stand, are in perfect logical order' (5.5563, see4.002, LO50). Ramsey's criticism points at what Wittgenstein himself later termed the 'ca.culus model of language', the idea that speaking a natural language is to operate a logical calculus the workings of which must be discovered by logical analysis. By the time of the PI Wittgenstein had entirely abandoned this picture. In this context he referred approvingly to a remark by Ramsey in conversation, according to which 'logic is a normative science' (PI §81). The idea that logic is normative was a commonplace at the beginning of the twentieth cer.tury, and Ramsey excerpted it from Johnson (1921: XX; see Kienzler 63— 76). It should also be noted that in this passage it has no immediate connection with the normativity of meaning that features in Wittgenstein's aforementioned attacks on causal theories. Rather, the point is that the clear, strict rubs of logic do not describe a depth-grammar that allegedly underlies the fluctuating and vague practices of ordinary language. It is misguided to argue that every .Anglophone is playing a game according to 'determinate, rig.d rules', and if he appears to violate these rules, then he is in reality playing a distinct yet equally determinate game, not English as it were (TS 213: 248). Such scholasticism is implicit in the Tractatus. It is explicit in a notorious argument by Davidson (1986): no two speakers use words in exactly the same manner, therefore every individual speaker must speak a distinct but precisely determined idiolect. Wittgenstein's alternative is to recognize that natural languages are loosely denned and historically evolving practices. At the same time he continues to deny that ordinary language is logically inferior to the formal languages of logic. Formal calculi neither reveal the 'depth-grammar' of knguage nor do they constitute a superior alternative. Their only legitimate philosophical role is as objects of comparison that bring into sharp relief philosophically relevant features of natural languages (PI §131; BB 28). On this issue Ramsey inclines more towards the Frege-Russell-Carnap tradition of ideal language philosophy. This tradition holds that natural languages in general and specific ordinary expressions suffer from various logical defects and should therefore be replaced by an ideal language or imoroved terms - at least for the purpose of philosophical and scientific incuiry. In this vein, Ramsey contradicts Moore and insists that philosophical definitions do not necessarily 'explain what we have hitherto meant by our propositions, but rather that they show how we intend to use them in the
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future'. For 'sometimes philosophy should clarify and distinguish notions previously vague and confused, and clearly this is meant to fix our future use only' (1). Wittgenstein, by contrast, denies both the premise that ordinary language is defective and the conclusion that philosophical clarity can only be achieved through linguistic reform. In this context he inveighed not just against Carnap but also against Ramsey. One of the temptations that we must resist while philosophizing is to think that we must make our concepts more exact than they are, according to the current state of our insight. This deviation leads to a kind of mathematical philosophy, which believes that it must solve mathematical problems to achieve philosophical clarity. (Ramsey). We only need a correct description of the status quo. (MS 115: 71; see PI §133, Glock 2002) Philosophical problems concern expressions which already have a nonphilosophical use (RPP I §550). This would be granted by Ideal Language philosophers. But they blame philosophical problems on the logical defects of ordinary language, and try to achieve clarity through the introduction of new concepts for which these problems cannot arise. But if the philosophical problems arise from our ordinary concepts, their resolution must clarify those concepts. As Strawson (1963) put it: artificial concepts can cast light on these difficulties only if their relation to our ordinary concepts is understood, which presupposes an accurate description of the latter. For this reason I am inclined to follow the descriptivist conception of philosophical analysis elaborated by Wittgenstein rather than the revisionist conception intimated by Ramsey. Let me end with an observation about philosophy in a wider sense. Ramsey's 'Epilogue' ends with the following striking words: My picture of the world is drawn in perspective, and not like a model to scale. The foreground is occupied by human beings and the stars are all as small as threepenny bits. I don't really believe in astronomy, except as a complicated description of part of the course of human and possibly animal sensation. I apply my perspective not merely to space but also to time. In time the world will cool and everything will die; but that is a long time offstill, and its present value at compound discount is almost nothing. Nor is the present less valuable because the future will be blank. Humanity, which fills the foreground of my picture, I find interesting and on the whole admirable. I find, just now at least, the world a pleasant and exciting place. You may find it depressing; I am sorry for you, and you despise me. But I have reason and you have none; you would only have a reason for despising me if your feeling corresponded to the fact in a way mine didn't. But neither can correspond to the fact. The fact is not in itself good or bad; it is just that
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it thrills me but depresses you. On the other hand, I pity you with reason, because it is pleasanter to be thrilled than to be depressed, and not merely pleasanter but better for all one's activities. (250) Jeffrey (1991) detects an attack on Wittgenstein here. Yet both the idea that fulfilment is to be found by living in the present and the idea that the facts as such have neither a positive nor a negative value derive from the mystical passages of the Tractatus (6.4—6.45). The anthropocentric perspective is in some respects foreshadowed by the solipsism of the Tractatus, according to which 'the world is my world' (5.62). But Ramsey's pragmatist anthropocentrism avoids the egocentrism and the idealism of that position. A similar anthropocentric perspective was later adopted by Wittgenstein, partly, perhaps, under the influence of Ramsey and Sraffa. Unlike Wittgenstein, however, Ramsey introduces a utilitarian note into this picture. Even though the facts are neither good nor bad, one has reason to find them pleasant and exciting, since to do so promises pleasure and success. Wittgenstein also recommends looking at the morally neutral world with 'a happy eye', i.e. accepting whatever happens with equanimity (6.43). The reason is not, however, that this promises success in action, but rather that stoic contemplation of the world sub specie aeternitatis is its own reward. Furthermore, in his war diaries Wittgenstein declared Christianity to be 'the only safe way to happiness', while at the same time recognizing that someone like Nietzsche could coherently resist this Pascalian siren song, albeit at the price of misery (GT 49-50). As far as genuinely non-factual questions are concerned, Ramsey's perspective is indeed rational. But this cannot be said for his uncritical acceptance of Wittgenstein's claim that ethics is non-factual (see also 248). Finally, as far as factual beliefs are concerned, considerations about pleasure and success are immaterial to their content, to their truth and to their rationality. Or so I have argued by appeal to Wittgenstein.
Endnotes 1
For helpful discussions, I would like to thank Peter Hacker. I am also very grateful to the Arts and Humanities Research Council for a Research leave award during which this paper was written.
References Unless otherwise indicated, all references are to pages of Ramsey, F. P. (1990) Philosophical Papers, ed. D. H. Mellor (Cambridge: Cambridge University Press).
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Other works by Ramsey are referred to by the following abbreviations: FM OT NPM FR
The Foundations of Mathematics, ed. R.B. Braithwaite (London: Routledge and Kegan Paul (1931)). On Truth, ed. N. Rescher and U. Majer (Dordrecht: Kluwer 1991). Notes on Philosophy, Probability and Mathematics, ed. M. G. Galavotti. Naples :Bibliopolis 1991. Frank Ramsey Collection, Archives of Scientific Philosophy in the Twentieth Century, University of Pittsburgh.
I use the following abbreviations for Wittgenstein's work: NB: Notebooks 1914-16 (Oxford: Blackwell 1979); GT: Geheime Tagebiicher 1914-16 (Vienna: Turia and Kant); TLP: Tractatus Logico-Philosophicus (London: Routledge and Kegan Paul 1961); LO: Ludwig Wittgenstein: Letters to C. K. Ogden (Oxford: Blackwell 1973); PR: Philosophical Remarks (Oxford: Blackwell 1975); WVC: Ludwig Wittgenstein and the Vienna Circle (Oxford: Blackwell 1979); DB: Denkbewegungen (Frankfurt: Fischer 1999); PG: Philosophical Grammar (Oxford: Blackwell, 1974); BB: Blue and Brown Books (Oxford: Blackwell 1958); RFM: Remarks on the Foundations of Mathematics (Oxford: Blackwell 1978); PI: Philosophical Investigations (Oxford: Blackwell 1958; first edition 1953); RPP I Remarks on the Philosophy of Psychology Volume I (Oxford: Blackwell 1980); CV: Culture and Value (Oxford: Blackwell 1980); PO: Philosophical Occasions (Indianapolis: Hackett 1993); CC: Cambridge Correspondence (Oxford: Blackwell 1995); OC: On Certainty (Oxford: Blackwell 1969). References to Wittgenstein's Nachlass follow von Wright's catalogue. Where necessary, I have provided my own translations from the German. Anscombe, G. E. M. (1959) An Introduction to W.'s Tractatus (London: Hutchinson). Armendt, B. (2001) 'F. P. Ramsey', in Martinich/Sosa (eds) A Companion to Analytic Philosophy (Oxford: Blackwell). Braithwaite, R. (1931). 'Editor's Introduction', in Ramsey, The Foundations of Arithmetic, ix-xiv. Brandom, R. (1994) 'Unsuccessful Semantics', Analysis 54, 175-8. Davidson, D. (1986) 'A Nice Derangement of Epitaphs', in LePore, E. (ed.) (1986) Truth and Interpretation (Oxford: Blackwell). Dokic,J. and Engel, P. (2002) Frank Ramsey: Truth and Success (London: Routledge). Dummett, M. A. E. (1978) Truth and Other Enigmas (London: Duckworth). Frege,,G. (1891) 'FunktionundBegriff', English translation in Translations from the Philosophical Writings ofGottlob Frege (Oxford: Blackwell 1980) ,21-41. Glock, H. J. (1996) A Wittgenstein Dictionary (Oxford: Blackwell). (2000) 'Wie kam die Bedeutung zur Regel?', Deutsche ^eitschriftfur Philosophic 48, 429-47.
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— (2002) ' "Clarity" Is Not Enough!', in R. Haller and K. Puhl (eds), Wittgenstein and the Future of Philosophy (Vienna: obv & hpt), 81-98. — (2004) 'Truth in the Tractatus', Synthese, forthcoming. Hacker, P. M. S. (1996) Wittgenstein's Place in Twentieth Century Analytical Philosophy (Oxford: Blackwell). (2001) Wittgenstein: Connections and Controversies (Oxford: Oxford University Press). Hintikka, J. and K. Puhl (eds.) (1995) The British Tradition in 20th Century Philosophy (Vienna: Holder-Pilcher-Tempksy). Jeffrey, R. (1991) 'Thrilled by the Facts', Times Literary Supplement May 17,5—6. Johnson, W. E. (1921) Logic Vol. I (Cambridge: Cambridge University Press). Kienzler, W. (1997) Wittgensteins Wendezu seiner Spdtphilosophie (Frankfurt: Suhrkamp). Kripke, S. (1980) Naming and Necessity (Oxford: Blackwell). Kunne, W. (2003) Conceptions of Truth (Oxford: Oxford University Press). Marion, M. (1998) Wittgenstein, Finitism, and the Foundations of Mathematics (Oxford: Oxford University Press). Mellor, D. H. (1990) 'Introduction', in Ramsey Philosophical Papers, xi-xxvii. Monk, R. (1990) Wittgenstein: The Duty of Genius (London: Cape). Ogden, C. K. and Richards, I. A. (1923) The Meaning of Meaning (London: Kegan Paul). Rhees, R. (1984) Recollections of Wittgenstein (Oxford: Oxford University Press). Russell, B. (1910) Philosophical Essays (London: Longmans). (1921) Analysis of Mind (London: Allen and Unwin). (1927) Principia Mathematica (Cambridge: Cambridge University Press; 1st edn 1910). (1984) Theory of Knowledge: The 1913 Manuscript (London: Allen and Unwin). Sahlin, N.E. (1990) The Philosophy of F. P. Ramsey (Cambridge: Cambridge University Press). (1995) 'On the Philosophical Relations Between Ramsey and Wittgenstein', in Hintikka/Puhl (eds), 150-63. Schulte, J. (1999) 'Wittgenstein - auch ein Pragmatist?', in R. Raatzsch (ed.) Philosophen iiber Philosophic (Leipzig: Leipziger Universitatsverlag), 303—20. Strawson, P. F. (1952) Introduction to Logical Theory (London: Methuen). von Wright (1982) Wittgenstein (Oxford: Blackwell). Wrigley, M. (1995) 'W., Ramsey and the Infinite', in Hintikka/Puhl (eds), 164-9.
3
The Ramsey sentence and theoretical content1 Jose A. Diez Calzada
Introduction The aim of this chapter is to critically review the main ways in which the Ramsey sentence has been used to account for theoretical content. I present, first, the Ramsey sentence in the context Ramsey introduces it in 'Theories'. Second, I deal with the eliminability of theoretical terms the Ramsey sentence allegedly allows, distinguishing semantic, ontological and methodological eliminability. Third, I present, and make some criticisms to, the way Carnap and Lewis use the Ramsey sentence for analysing, respectively, theoretical analyticity and theoretical meaning. Fourth, I introduce the model-theoretic account that analyses theory's empirical claim as a so-called Ramsey-Sneed sentence. Finally, I sketch my own analysis of the content of theoretical concepts stressing the aspects related to the Ramsey sentence in connection with the criticisms of Lewis' definition of theoretical terms.
The Ramsey sentence It is well known that Ramsey introduces what has been generally called the Ramsey sentence (RS) in a 1929 paper entitled 'Theories', first published by Braithwaite in 1931 as a chapter of Ramsey's collected papers The Foundations of Mathematics and other Logical Essays. What is probably not so well known is that in this paper the discussion of RS occupies less than two pages. Here, as in many other topics, the brevity of Ramsey's writing contrasts with the extent of its influence. In this article Ramsey analyses the relation between the two levels or 'systems' that constitute a theory. A theory is characterized as a system/language where certain facts are (said to be) explained. In such a system we can distinguish two subsystems: the primary system, composed of terms and propositions for 'the facts to be explained' (p. 112), and the secondary system, 'the theoretical construction' (p. 114). Here is the example he presents as a model:2
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Variables n, n\, n^ ... range over integers, interpreted as time instants. These are the variables to which the following primary predicates or prepositional functions apply: A(n): I see blue atrc B(w): I see red at n C(n): Between n — 1 and n I feel my eyes open D(w): Between n—\ and n I feel my eyes shut E(ra): I move forward a step at n F(w): I move backward a step at n This suffices for the primary system. For the secondary system, new variables m, mi, m%,... are introduced. These variables range over (1, 2, 3}, interpreted as spatial places. A function/(m) is defined so that:/(I) = 2,/(2) = 3 and /(3) = 1. The secondary predicates are the following: a(n, m): At time n I am at place m {3(n, m}: At time n place m is blue 7(w): At time n my eyes are open So much for the vocabulary; now let us look at the primitive propositions. Ramsey uses the term axioms to refer to the primitive propositions whose nonlogical terms belong only to the secondary level, and he chooses the following: Ax 1 Ax 2 Ax 3 Ax4 4
Vw, m, m'(a(n, m} A ot(n, m') —> m — m'} Vrc3m a(n,m] Vn (3(n,l) Vra (/?(», 2) <->-./3(n+l,2))
Besides the axioms, the theory establishes other primitive propositions whose non-logical terms belong to both secondary and primary systems, what 4 Ramsey calls the dictionary. 3 In this example: Dil Di2 Di3 Di4 Di5 Di6
Vw(A(«) <-» 3m(a(n, m} A (3(n,m)} A 7(72)) Vn(B(w) <-»• 3m(a(«, m) A ^/3(n, m)) A 7(n)) V»(G(n) *-> -.7(n - 1) A 7(«)) Vn(D(n) < - > 7 ( n - 1 ) A-i7(n)) Vw(E(w) <-^ 3m(a(w — 1, m) A a(n,f(m}}}} Vn(F(n) <-> 3m(a(w - l,/(m)) A a(n, m}}}}
Finally, Ramsey uses the term laws to refer to the (general) propositions that follow from the axioms (AX) and the dictionary (DIG) whose non-logical terms belong only to the primary level, for instance, L 1 Vn((-nA(n) V -iB(n)) A (--G(n) V --D(n)) A (-•£(«) V -iG(«))
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and he calls consequences the particular propositions in the primary system that follow from AX and DIG when we add propositions involving particular values of n. What Ramsey wants to elucidate here, using this example as a model, is whether it is possible (and if possible, necessary) to 'reproduce the structure of the theory by means of explicit definitions [of secondary terms] within the primary system' (p. 120).5 In the example: is it possible to define a, /3 and 7 in terms of A, B, C, D, E and F so that AX and DIG follow from laws and consequences together with the definitions? The question, then, is clearly eliminative, or at least reductive. Ramsey explores a variety of possibilities, concluding in general that when such definitions are possible, they are either unacceptably arbitrary (when non-disjunctive) for there are many different (non-disjunctive) possible definitions, or unacceptably complex/disjunctive (when unique). But, at least in principle, the answer to the first question is affirmative. Now he examines the second question: 'We have seen that we can always reproduce the structure of our theory by means of explicit definitions. Our next question is "Is this necessary for the legitimate use of the theory?" ' (p. 129). Here the answer is negative, and the immediate reason is simply that 'rather than give all these definitions it would be simpler to leave the facts, laws and consequences in the language of primary system' (p. 130). But there is another, more profound, reason, at least for the case of arbitrary definitions: 'the arbitrariness of the definitions makes it impossible for them to be adequate to the theory as something in process of growth' (ibid., my emphasis). These definitions would make it impossible, or highly unlikely, that the theory could change by adding new truths and that, at the same time, its terms preserve their meanings. In our example, nothing is said yet about e.g. the colour of place 3. If in the process of growth we arrive at a new hypothesis about the colour of place 3 and we want to incorporate this into the theory, 'this would appear simply as an addition to the axioms, the other axioms and the dictionary being unaltered. But if the theory had been constructed by explicit definitions, this new axiom would not be true unless we changed the definitions... That is to say, if we proceed by explicit definition we cannot add to our theory without changing the definitions, and so the meaning of the whole' (ibid.}. This line of reasoning is somewhat puzzling. For it is clear that the fact that addition of axioms implies change in meaning can count against explicit definitions only if the addition of axioms does not have the same consequence in alternative proposals for the meaning of secondary vocabulary. Although Ramsey does not mention any such alternative explicitly, the immediate proposal (actually, the one that will become the standard proposal in the Received View) is to consider that secondary terms acquire their meaning through the axioms (and the dictionary/correspondence rules) functioning as
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implicit definitions.6 But this implicit definition theory of secondary meanings (when not qualified) also has the consequence that any change in the axiom system amounts to a change in secondary meanings. Therefore, had Ramsey considered this alternative, he would have had to reject it for exactly the same reason that he rejects the explicit definition theory, namely, that any theory change implies meaning change, making it impossible for a theory to preserve secondary meanings throughout its evolution. Maybe he had in mind a qualified version of the implicit definition theory that does not have this counterintuitive 7 consequence. Be this as it may, this is the context in which Ramsey introduces his famous RS. And he does it answering the following question: 'Taking it then that explicit definitions are not necessary, how are we to explain the functioning of our theory without them?' (my emphasis). Here is his answer: Clearly in such a theory judgment is involved, and the judgments in question could be given by the laws and consequences, the theory being simply a language in which they are clothed, and which we can use without working out the laws and consequences. The best way to write our theory seems to be this Ba, /3, 7: dictionary • axioms Here it is evident that a., (3, 7 are to be taken purely extensionally. Their extensions may be filled with intensions or not, but this is irrelevant to what can be deduced in the primary system, (p. 130, my emphasis) Since Ramsey says that the judgements involved in the theory 'could be given by the laws and consequences' and, as we saw, laws and consequences are pure primary judgements, it follows that, according to him, the judgements involved in a theory are purely primary, i.e. concerning only 'the facts to be explained' (in the Received View's version, 'observational'). The secondary system provides just a cloth for using these primary judgements 'without working out laws and consequences'. If this is so, then the best way to write the theory is8 TR:
3;M2,tf(AXADIC)^ 3
Although Ramsey does not say so explicitly, and far less offers a demonstration, this presupposes that TR and T = AX A DIG have the same primary system consequences, as they actually do.9 Both TR and T can then be used to express the relevant judgements without working out laws and consequences. But, why is TR better than T as a way of writing the (relevant judgements of the) theory? The answer lies in the last sentences of the quote: as far as only consequences in the primary system are concerned, the only thing that
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matters about constants a, (3 and 7 is their extension. As predicative constants, their extensions may go together with intensions, but this is totally irrelevant to what can be deduced in the primary system. That is, secondary intensions are irrelevant to the relevant primary judgements of the theory. And this is what TR shows better than T. Secondary intensions, or (non-extensional parts of) meanings, are irrelevant for what according to Ramsey a theory judges, and the only things lost from T to TR are precisely these irrelevant intensions/ (non-extensional parts of) meanings. Therefore T is somewhat misleading, for secondary intensions seem essential to its understanding. It is much better to use TR instead, for TR is exactly like T in the relevant respects but does not have T's misleading part. TR is, so to speak, the purely (secondary) extensional part of T. So, in defending that the best way of writing the theory is TR rather than T, there are two essential steps. The first is the fact that TR and T have the same primary consequences. This is a purely mathematical fact, without any epistemological insight; a fact that, as has been emphasized, applies with full generality to any division of the vocabulary of a theory into two different sets, no matter how arbitrary it is (for instance, let the primary terms be those with no more than two syllables, and the secondary terms those with three or more). The second is the thesis that the relevant judgements involved in a scientific theory are those in the primary system, when the primary/secondary system distinction is made in terms of facts or phenomena to be explained versus the theoretical explanatory apparatus. This is a highly substantive, and controversial, epistemological thesis about 'what the theory says'. Actually, it is a version of the instrumentalist thesis according to which only the judgements about things/phenomena to be explained are really 'involved' in the theory, all other alleged judgements (i.e. secondary 'judgements'), are instruments, or 'cloths' to use Ramsey's talk, for dealing with the formers. It is not clear, though, how Ramsey's primary 'facts to be explained' are to be understood. The example he gives is quite phenomenistic, although the predicates E and F are not clearly phenomenistic. Or maybe he had in mind something 'observable' along the lines of what would become standard in the Received View.! But, however he may have understood the nature of phenomena to be explained, his assertion that TR is the best way to express the theory rests upon the instrumentalist assumption that the 'relevant'judgements involved in the theory are all judgements confined to such phenomena (remember, though, that primary judgements include existential quantification on nonprimary properties, which as we'll see in the next section — mitigates the ontological scope of this instrumentalism). This is then the way Ramsey introduces RS in 'Theories', which has became one of his most famous contributions. Yet RS was practically ignored during almost two decades until his friend and editor, R. Braithwaite, mentions it
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as an answer to the question about the reality of theoretical concepts. From that point onwards, Ramsey's primary/secondary distinction will be rephrased almost always as the Received View's observational/theoretical distinction. We will now focus briefly on the main criticisms and applications of RS understood in terms of this last distinction.
RS and the eliminability of theoretical terms Semantics One of the main worries of the empiricist-positivist account of scientific knowledge concerned the 'legitimacy' of theoretical terms not directly connected to observable entities/phenomena. This concern was already present in several phenomenistic reductive programme during the first decades of the twentieth century, and was still present at the beginnings of the second half of the century, when the Received View began to be articulated. Once the reductivist/ eliminativist programme of defining theoretical terms by observational ones was abandoned, RS was welcomed as a way, not of solving but of eliminating the problem. The problem was a semantic one, a problem about the meaning of theoretical terms, i.e. of theoretical predicative constants like 'electron', 'magnetic field', 'temperature', 'mass' and the like. And RS seems to provide a way out, through the elimination of these terms in TR: if we accept that, as far as relevant judges are involved, we can substitute the full theory T, which includes theoretical terms, by the Ramsey sentence TR, which does not include theoretical terms but quantified second order variables, then we don't need to ask about the meaning of theoretical terms in order to understand the work the theory does. No theoretical terms, no problem about the meaning of theoretical terms; in Carnap's words: 'the puzzling questions [about the meaning of theoretical terms] are neatly side-stepped by the elimination of the very terms about which the questions are raised'.13 Is this so? It is so only if we accept the Ramseyian instrumentalist thesis according to which the only relevant judgements involved in a theory are primary/observational judgements. Well, one can say, even if we don't want to endorse Ramsey's instrumentalism, RS has a, let's say, conditional insight, interesting enough to dispense the semantic problem: as long as only observational judgements are involved, we don't have to face any semantic problem, thanks to RS and the disappearance of theoretical terms. Yet, this seems a Pyrrhic victory: at least as far as we confine ourselves to non-theoretical judgements, we don't have to face any problem about the meaning of theoretical terms. It seems to me, then, that even if we focus only on the semantic part of the problem of theoreticity, RS is of little help for
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by-passing the problem, for only if we have decided in advance not to deal with non-observational, i.e. theoretical, judgements/terms, can we side-step the problem of theoretical meanings. It is true, though, that the interest of RS remains in showing that we can, in principle, account for prediction and retrodiction, and more importantly, empirical systematization, without dealing with the problem of the meaning of theoretical terms}5 Ontology Be the semantic problem as it may, there is another problem involving theoreticity, namely the problem of theoretical entities. Even if we accept RS as a (dis) solution of the semantic problem, the question about theoretical entities still remains. And as Hempel pointed out, Ramsey eliminability does not apply to it: 'the Ramsey-sentence avoids reference to hypothetical entities only in letter rather than in spirit. For it still asserts the existence of certain entities of the kind postulated by T, without guaranteeing any more than does T that those entities are observable or at least fully characterizable in terms of observables.'16 It is true, though, as Hempel recognizes, that Ramsey never claimed that RS avoided reference to these entities, or that TR is better (i.e. weaker) than T in ontological respects. Yet, to some critics, it is not that TR is not ontologically weaker than T, but just the opposite, TR is ontologically stronger than T: 'the Ramsey method has expansive effects on ontological commitment, since it replaces non-committal theoreticalpredicate constants with existentially quantified predicate variables ranging over non-individuals.' According to this criticism, TR'S second order quantification has ontological commitments with the existence of non-individual entities in the quantification domain, whereas T has no such commitment. But, as Bonhert has replied, 'the alleged expansion is only in comparison with 1 a hoped-for nominalistic reformulation of scientific theories'.18 That is, only upon a nominalistic interpretation do T's theoretical predicates have no ontological commitment, but (according to Bonhert and many others) this interpretation is controversial, to say the least. Therefore, TR is not (nonnominalistically) ontologically stronger than T. Yet, one can say, even if Bonhert is right, there seems to be an ontological difference, in a sense. We are probably not obliged to accept that 'to be is only to be the value of a variable', but it seems that we are obliged to accept that 'to be is at least to be the value of a variable'. If so, then T commits us at least to the existence of individuals whereas TR commits us at least to the existence of individuals and properties (or the like). That is, T would leave open the nominalistic interpretation (controversial as it is), whereas TR would not. This seems to me a sense in which TR is ontologically stronger than T. Of course this difference vanishes if we accept a second order logic in which the existentialization
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rule is valid, for then T logically implies TR. When second order logic ontological commitments are accepted, the whole question of ontological comparison between T and TR is pointless. Epistemology
One last respect in which T and TR have been compared is neither semantic nor ontological but epistemological, namely, whether T and TR have the same epistemic status, whether they behave alike in epistemic respects like confirmation, inductive power and aposteriority. At first sight, there seems to be a surprising difference between T and TR in epistemic character, at least in some cases. For simplicity's sake, let's take a very simple theory with only one theoretical term 'R' (for instance, 'having high kinetic energy' or 'having chromosome XV) and one observational term 'O' (for instance, 'hot' or 'male'), and made up of only one sentence connecting them: Vx(Rx —>• Ox}. This seems to be a non-trivial empirical theory. Maybe it is too simple to be interesting, but this does not make it trivial, one would say. Yet, its Ramsey version 3tVx(tx —> Ox) is obviously trivial; in fact it is a (second order) logical truth. It seems, then, that at least in some cases there are non-trivial empirical theories T whose Ramsey version TR is empirically trivial since it is a logical truth. This problem is related with another one, first posed by ShefHer, concerning differences in confirmatory relationships.20 Take the following T, where 'M' is the only theoretical term and 'P' and 'R' are observational terms: Vx((Mx -> Px) A (Mx -» R*)) Suppose we observe Ra, then we have, given T, some inductive ground (even if weak) for Ma, from which we can, given T, deduce Pa. Therefore, T supports the inductive relationship between Ra and Pa. But it is clear that its Ramsey version TR 3A/x((ta -»• P*) A (tx -» Rx))
does not. If it did, then Ra would provide (the same) inductive grounds for -iPa too; that is, the same inductive grounds for both Pa and ->Pa, which contradicts our notion of inductive justification. The reason has to do with the above-mentioned fact, namely, that TR is analytic. If TR supported the inductive relation between Ra and Pa, then, since it is analytic, Ra would inductively support Pa. But since 3A/x((/x -» ^Px) A (tx -> R*))
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is also analytic, then Ra would inductively support ->Pa as well. This does not happen with T, for, because of its non-analyticity, the acceptance of T does not come with the acceptance of Vx((Mx -> ^Px) A (M* -> R*)) Therefore, according to Scheffler, T and TR do not preserve in general the same inductive relationships. The Ramsey sentence does not have then the same epistemic virtues than the original theory. But, as Bonhert points out, 21 this criticism rests upon the unjustified assumption that T supports an inductive relation between Ra and Pa. It is simply false that Vx(M.xx —> R#), together with Ra, and nothing else, inductively support Ma. No inductive logic accepts this argument as valid. It is a deductive fallacy and an inductive fallacy as well. It is true that TR does not inductively connect Ra with Pa, but T does not either. There is no asymmetry, then, between T and TR in inductive relationships. This answer to Scheffler's criticism is totally sound, but the other fact upon which this criticism rests remains to be accounted for, namely, that T is a nontrivial empirical theory whereas TR is logically true. Bonhert suggests a line of response elaborated further by Stegmiiller. The key is to notice, actually thanks to Ramsey, that when TR is logically true and hence has no (nonlogically true) observational consequences, then, given that as Ramsey showed us T's observational consequences are exactly the same as TR'S, T itself has no (non-logically true) observational consequences. In such a case, contrary to initial appearances, T must be (yet not logically true) empirically trivial, for there can be no empirical reason for believing it. We can then use RS as a criterion of empirical triviality: a theory T 'is empirically trivial if and only if either it is logically true or its Ramsey substitute is logically true'.23 Since (if we accept the second order existentialization rule) T logically implies TR, when T is logically true so is TR. Therefore we can say that, irrespective of first sight appearances, a theory T is empirically trivial if and only if TR is logically true. This would be, according to Bonhert, Stegmuller and others, one of the lessons of the Ramsey sentence.
RS and the meaning of theoretical terms We have seen that one of the earliest applications of RS was its use in the alleged 'side-stepping' of the problem of theoretical terms, or better, of the semantic part of this problem. Yet, we saw that properly understood it meant only that we can side-step the semantic problem as far as prediction and
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retrodiction are concerned. The semantic problem, though, may have other motivations independent of our predictive practices, like interests in theoretical analyticity or theoretical content. We are going to see now how RS was used by Garnap and Lewis to deal with questions of this kind. Carnap on analyticity of theoretical language
Garnap is one of the champions of the analytic/synthetic distinction. Even after Quine's criticisms he continues to think that such a distinction can be made in principle, irrespective of the problems of particular applications. In his Philosophical Foundations of Physics (1966) he defends a way in which such a distinction can be traced for the language of science. Garnap articulates here the observational/theoretical distinction for the vocabulary of science and offers different procedures for distinguishing the analytic and the synthetic in each case. His strategy for observational terms is similar to that in previous works. For a set of observational terms there are, in principle, a set of meaning postulates, now 'A-postulates' (analyticity postulates), specifying meaning relations between these terms. This can easily be shown for an artificial observational language, whereas for a natural language things are more complicated and 'empirical investigation of speaking habits' 4 is needed. With these A-postulates in hand, Carnap gives the natural definition of analyticity: a sentence is an analytic truth, or A-true, if it logically follows from the A-postulates. We are not interested now in the well-known problems of this approach, but in the different way Garnap faces the problem of analyticity for theoretical terms, and the use he makes of RS for solving it. Garnap confesses that he almost despaired of applying the analytic/synthetic distinction to theoretical discourse, for a reason very similar to the one that motivated Hempel's scepticism.25 Let AX be the set of T-postulates (theoretical postulates, Ramsey's axioms), i.e. of primitive sentences with only theoretical (and logical) terms, and CR the set of C-postulates (correspondence rules, Ramsey's dictionary), that is of primitive sentences with both theoretical and observational terms; a theory T is the conjunction T = AX A CR.26 Carnap begins by acknowledging that the reductive project failed, and that there is no complete empirical interpretation for theoretical terms. The question then is: 'How can comparable A-postulates be formulated to identify analytic statements in a theoretical language containing theoretical terms for which there is no complete interpretation?' (p, 267). The first option seems to be to look at AX or CR. Starting with AX: do T-postulates alone serve as A-postulates? No, they don't, for T-postulates don't provide any empirical interpretation, and theoretical terms of empirical theories must have some empirical interpretation. This interpretation is provided by C-postulates, without which the theory is a purely formal calculus
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where the problem 'of distinguishing the analytic from the synthetic does not even arise' (ibid.}. Do then C-postulates do the job? Are they the A-postulates? Not alone. Carnap is not explicit on this point, but it is easy to see why C-postulates alone won't do. If they did, the theoretical terms of two theories with the same G-postulates and different (non-equivalent) T-postulates would have the same meaning, which is clearly counterintuitive. So, maybe both T-postulates and G-postulates together will do? No, this is clearly too much, for AX A GR equals the full theory and the full theory cannot be analytic. So we have to take 'only a portion' of AX A CR, the portion 'responsible' for meanings. We have to 'split up' AX A CR into two parts, one fully, and maximally, analytic; and the rest, maximally synthetic, empirical. Here is where Hempel gave up and became sceptical about theoretical analyticity. Carnap, though, thinks that using RS he can face the problem. Carnap searches a portion of T = AX A CR that is purely and maximally analytic and a portion that keeps all the empirical/observational content of T. But, he claims, this last portion is precisely what RS provides. The Ramsey sentence TR is a proper part of T, since it is logically implied by T but does not logically imply T, and (according to Carnap's understanding of the empirical) has exactly the same empirical content as T (hence all T's empirical content), since every (purely) observational consequence of T is a consequence of TR too. Once the synthetic portion has been identified, the rest seems easy: the analytical portion will be T minus TR, that is, the smallest content we have to add to TR to obtain T. Which is the weakest content that together with TR implies T? Answer: the material conditional 'TR —>• T'. Since TR itself is the weakest observable sentence observably equivalent to T (there is no observable sentence observably equivalent to T and weaker than TR), 27 TR —>• T is the maximal analytic sentence, that is, all analytic truths will be consequences of TR —> T. We have then identified all the T's analytic truths: a T-sentence is analytically true if and only if it is a logical consequence of TR —> T.28 This conditional is the candidate for expressing theoretical A-postulates, the full part of T which is analytically true, true in virtue of the . 9Q meanings of T's theoretical terms. The conditional TR —> T, then, although not logically true, is analytically true and provides the global A-postulate for T's theoretical terms. It may seem empirical, but it is not: [This] A-postulate seems to tell something about the world, but it actually does not. It does not tell us whether the theory is true. It does not say that this is the way the world is. It says only that if the world is that way, then the theoretical terms must be understood as satisfying the theory [...] the postulate says in effect that, if there are entities that satisfy the theory, then the theoretical terms are to be interpreted as denoting [such entities], (p. 271)
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To put it the other way around: the meanings of theoretical terms must be so that TR —> T happens to be analytic. That is, had we a semantic analysis of theoretical terms, the result T* of substituting in T the theoretical terms by their analysans should be a logical consequence of TR (since TR —> T* would be a logical truth). Note that the whole strategy rests upon the assumption that all T's empirical/synthetic content is confined in the set of T's (purely) observational consequences. This assumption is less controversial than the instrumentalist one, mentioned in the preceding section, according to which all T's relevant judgements are the T's (purely) observational consequences. But, even if less controversial, is it completely uncontroversial? What is uncontroversial is that T and TR have the same (pure) observational consequences, i.e. that they imply the same sentences that use only observational terms (although these sentences may also existentially quantify over 'other' entities). What about sentences with both observational and theoretical terms? For sure, some of these sentences (e.g. T itself) are consequences neither of TR nor of TR —> T. Since they are not consequences of TR -* T, they are not analytic. Therefore they are synthetic. But they are not consequences of TR either. It is clear that not every synthetic T-consequence is a TR-consequence. Does this mean that not all synthetic content is in TR? Carnap does not address this question but his answer is sure to have been: 'Not at all'. For let $ be one of these sentences. It is synthetic, but its (synthetic) content can be divided into two parts: $R, which is synthetic and keeps all the synthetic content of$, and is a consequence of TR, and 'the rest' I?R —•> i?3 which is analytic and a consequence of TR —> T. This seems to me to be close to question-begging: in order to defend that, even though there are synthetic T-consequences which are not TR-consequences, all T's synthetic content is present in TR, he must presuppose that all the synthetic content of a synthetic sentence $ which is a consequence of T but not of TR is present in its T?R. I don't want to say that Garnap is explicitly begging the question (he didn't even raise it), I only want to point out that he does not justify his assumption and that the immediate justification at hand for him would merely be to apply the very same assumption. Put it in another way: Garnap's assumption is not self-evident and seems in need of justification, and the fact that he feels no need to justify it, but just takes it as obvious, is the core of his Ramseyan solution of the problem of theoretical analyticity.30 Lewis on the definition of theoretical terms
Carnap's proposal for theoretical analyticity presents TR —> T as a single global A-postulate for T-theoretical terms, but he does not try to analyse theoretical terms, much less to define them. Nevertheless, this proposal serves as a constraint for such an analysis, for we have seen that if TR —>• T is analytic,
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then any analysis/definition of theoretical terms must be such that the result of substituting in T the theoretical terms by their analysans must be a logical consequence of TR. This is what is behind the Lewisian proposal of'explicit definitions of theoretical terms (although, contrary to what a definitional account might seem, Lewis' aim is not 'eliminativist' but 'vindicationalist' about theo31 retical terms). Lewis does not follow the traditional observational/theoretical distinction. He finds this distinction misleading for the same reasons given by Putnam (1962), and divides T's terms, the terms used by a theory T, into two groups: T-terms, terms introduced^ the theory T; and O-terms, terms 'already understood before the new theory T with its new T-terms was proposed'.32 O-terms are 'old', not 'observational'; that is, O-terms may, or may not, be observational, but their only characteristic trait is that they are understood (not necessarily analysed'] prior to the introduction of new terms by the theory. This distinction, therefore, is implicitly relativized to a given theory; a T-term of a given theory may be an O-term in another theory. Leaving this essential difference aside, Lewis explicitly says that his proposal is 'in the spirit of Ramsey's and Carnap's' (p. 78). A sentence is an O-sentence when it is free of T-terms, otherwise it is a T-sentence. Let T(^/,... ,^fm, O / , . . . ,Om) be a theory, or theory-postulate, with n new T-terms #/,... ,^fn and m old O-terms O/,... ,Om Let us assume that the O-terms have conventionally established standard interpretations well known to us. The T-terms, on the other hand, are unfamiliar. Our only clue to their meaning is the postulate of T that introduced them. We are accustomed to say that it implicitly defines them; but we would prefer explicit definitions (p. 80, my emphasis) How can we get these explicit definitions? Let T(J?I, ... ,tn, O/,... ,Om) be the T-realization formula, an open formula obtained replacing the n T-terms ^i, ...,^!n by n variables (of the same type) ti,...,tn; let 3£i,...,3^«T (t\,... ,tn, O / , . . . ,Om) be the Ramsey sentence for T and 3^,...,3^T (ti,...,tn,Oi, ...,Om) -> T(*/, ...,*„, O/,...,O m ) the Carnap sentence. Lewis assumes that a necessary condition for a sentence to be true is that its constant terms denote. If the realization formula has a unique realization then T(\l/;,... ,\f n , O / , . . . ,Om) is true and T-terms denote. If the realization formula has no realization then T-terms do not denote and T(\I//,... ,^fn, O/,... ,Om) is false, but the Carnap sentence is still true, for its antecedent, the Ramsey sentence, is false. What if the realization formula has more than one realization? According to Carnap in this case T(\E r /,... ,tyn, O / , . . . ,OOT) is true, for the Carnap sentence is (analytically) true and the antecedent Ramsey sentence is also true. Therefore, in such a case T-terms denote as well as in the unique
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realization case. To Lewis this seems counterintuitive. What, among the different realizations, do they denote? There is no non-arbitrary way of answering this question. 'Therefore I contend that we ought to say that the theoretical terms of multiply realized theories are denotationless' (p. 83). But this has as a consequence that T(^/,... ,W W , O/,... ,Om) is false, and so it is the Carnap sentence since the Ramsey sentence is still true. The conclusion, then, is that the Carnap sentence is not analytic, it is not the right A-postulate. We have to replace this meaning postulate by the three following ones:33 (i)
3 1 *!,... ,3 1 ^T(^ 1 ,... ,tn, O / , . . . ,0m) -> T(ty,... ,#„, O/,... ,0m)
(ii)
-i3tl,...,3tnT(tl,...,tn,Oi,...,Om}) -> (-^3xx = ^l A - - - A - i 3 * * = tfB)
(iii) ) (3*i,... ,3*BT(*i, • • • ,*n, O / , . . . ,0m) A -.31*!, ...,3ltnT x ( f l 9 . . . ,tn, Oi,..., 0 M )) -»• (-^Bxx = #/ A • • • A -.3** = *„)
That is, if T is uniquely realized then it is (true and) realized by the entities named by T-terms; and if T is not realized, or if it is multiply realized, then T-terms do not name anything. These are the specifications for the denotation of T-terms in all different situations. Since giving the denotation in all possible situations amounts to giving the sense or meaning, these three sentences together provide senses or meanings for T-terms, that is, they are the A-postulates for T's theoretical terms. Once Lewis arrives at this conclusion, the rest of the way towards explicit definitions of T-terms is easy (p. 87, V is the symbol for the definite descriptions operator): Given our conclusion that the T-terms \1//,... tyn denote the components of the unique realization of T when there is one, and should not denote anything otherwise, it is natural to define T-terms by means of definite descriptions as follows: ty = l*i 31 %, • • • ,3'Wi, . . . ,tn, O/, . . . ,0ra) *„ = Ltn^ti, . . . ,3ViT(*i, - - - ,tn, O/, . . . ,0m]
These then are the explicit definitions of T-terms. When T is not uniquely realized the definiens is an improper description and T-terms are denotationless, as Lewis wants; note, though, that multiple realization affects only the denotation or reference, it does not affect the sense/meaning of the theoretical terms, for even if the description is denotationless as a consequence of multiple realizability, its sense/meaning is perfectly determined.
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Note that, as is to be expected, when we substitute the T-terms by their definiens in an A-postulate what we obtain is a logical truth. Of course this is not the case for the Carnap sentence, a perfectly acceptable result now, for under this construction we have seen that the Carnap sentence is not analytic. But it is the case with Lewis' substitutes of the Garnap sentence, (i)-(iii) above: if we substitute there every \I/Z- by its definienssthen we obtain logical truths, the consequent logically follows from the antecedent.34 All this then is in line with Carnap's proposal, but for the multiple realizability (and the nature of O-terms). Actually, Carnap himself was very close to this kind of explicit definitions. In defending that RS allows theoretical terms eliminability in principle but not in practice, he says that the Ramsey equivalent to a very simple particular theoretical sentence like 'mass (a) =5' would be too complex to manage (I accommodate the notation of the example and suppose that 35 'mass'is our ^3): 3*i,... ,3tn(T(ti,... ,tn, O / , . . . , O m ) A * 3 (a) = 5) We have then in general that for a monary predicate, for example, Wz-, W;(^) is equivalent to 3t\,... ,3tn(T(ti,... ,tn, O/,... ,Om) A ti(x}}. From this one could (but for uniqueness) define ^z as 'the' ti so that 3t\ ... 3ti-\3ti+\ ... 3tn (T^,...,^,,...^. Lewis's proposal is on the origin of conceptual role theories, or functional theories, of meaning: the idea is that the meaning of a term that denotes an entity belonging to a system is determined by the role/function this entity has in the system; the meaning of a term is the meaning of a description describing such a role/function. This idea is the legacy of the old idea of implicit definitions and Lewis shows how can they be turned into explicit ones. It has been applied to general problems as well as in particular fields, mainly in philosophy of mind. Psychological functionalism states that the meaning of psychological terms is the role 'the' denoted property has in person's psychology. But there seem to be many cases (if not all) in which these roles are multiply realizable so that, according to Lewis, the terms are denotationless. One way to face this problem 6 is 'second order' functionalism, the idea that the properties named are themselves functional properties, properties whose essence is that role. These properties are 'second order properties', i.e. 'the' (second order) property of being 'a' (first order) property that fits the role. With these functional/second order properties there is no room for multiple realizability: the theory is either uniquely realized or not realized at all (functional/second order properties have, though, other problems).37 The proposal Lewis offers for the definition, hence the meaning, of T's theoretical terms has to face a 'diachronic' problem. Lewis' definitions provide the meaning of T-terms at a given time in T's history, say, at the beginning
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of its life, when T-terms are introduced. But theories, like many other beings, are enduring entities, they change over time while preserving their identity. On the other hand, Lewis' definitions take 'the whole theory' to build up the realization formula; he puts every theoretical axiom and correspondence rule in the definiens. Therefore, if meanings of T-terms are what Lewis's definitions provide and at every stage of T the meanings of its T-terms are given by all the axioms and correspondence rules at that moment, then T-terms change their meaning along T's history. Tet, we would like to say, users of a given theory at different stages understand each other, there must be a sense in which they mean the same by using T-terms. Lewis recognizes this difficulty and accounts for it by distinguishing two senses in which the meaning of T-terms changes. We could accept, in a sense, that there are such continuous changes of meanings; 'but these are very peculiar changes of meaning - so peculiar that this position seems to change the meaning of "change the meaning of". They occur continually, unnoticed, without impeding communication' (p. 94). Even if we want to keep this sense, we have to accept that there is also a sense in which the meaning of Tterms is preserved along T's life. Which are these meanings? Lewis's answer is 'the original ones': 'the T-terms keep the meanings they received at their first introduction. They should still be defined using the original version of T even after it has been superseded by revised versions' (ibid.)?8 Leaving aside problems of vagueness in determining 'which is the first version', there are other problems related to the nowadays recognized 'stratified' nature of scientific theory's postulates; that is, to the fact stressed by Kuhn and Lakatos, and after them by many others, that not every postulate is alike in its contribution to the content of theoretical concepts; and, as a consequence, that to take 'all the postulates', even if at the first stage, is too much for meaning-constitution. There seems to be no need for this Lewisian 'full content of the first version' theory of theoretical meanings. There is an alternative answer, one that does not take 'the full theory' (at a given stage) as constitutive of meaning but only 'part of the theory', a part that will be (almost) preserved throughout its history. The last section offers a sketch of this alternative.
RS and a model theoretic T-empirical claim Our last stop on this historical/conceptual survey of the Ramsey sentence has to do with its usage in certain semantic approaches, mainly structuralism, to specify the nature of the empirical claim of a theory; or better, with the way that theory's empirical claim, structuralistically reconstructed, may be seen as a model-theoretic version of RS. 39
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According to semantic approaches, a theory T is identified with a set of models better than with a set of statements/axioms. But as a propositional or belief object, the theory brings with it an empirical statement or empirical claim, which is the direct bearer of truth values and what allows the indirect attribution of truth values to theories (which, as sets of models, are not the kind of entity apt for truth/falsity). The intuitive idea of T-empirical claim behind all semantic approaches is roughly the same: the empirical claim associated with a theory says that certain empirical systems 'satisfy' the formal constraints that define theory's models, i.e. that such systems are 'embeddable' in the set of models. But the specific reconstruction that every semantic approach makes of the theory's empirical claim is different in relevant respects, and the way structuralism does it shows particularly well the connection between theory's empirical claim and theory's Ramsey sentence. This connection is important for it suggests the use of theory's empirical claim to deal with semantic issues in a way similar to which axiomatic approaches used the Ramsey sentence. This suggestion is elaborated in the last section. Like Lewis and Hempel, Sneed does not set the question in terms of the observational/theoretical distinction. This distinction seems to him misleading, for it confuses two different distinctions: observational versus nonobservational and non-theoretical versus theoretical, and the one relevant for the analysis of theory's empirical claim is the latter. Sneed's idea is similar to Lewis's and Hempel's: the theoretical/non-theoretical distinction is relative to theories, a term or concept is theoretical for a given theory T if it is 'posed' by the theory as part of its explanatory conceptual machinery, and it is not theoretical for T if it is 'previously available'. But Sneed and structuralists go further and provide a precise criterion for characterizing T-theoreticity (so answering Putnam's challenge): a term or concept used by T (i.e. appearing in T's laws) is T-theoretical if and only if it is not possible to determine/measure it without presupposing T's laws, that is, if and only if every determination method uses some law of T. Otherwise the term (used by T) is T-non theoretical, that is, when it is possible to determine it without presupposing T, when there exists at least one determination method that does not use any T-law. For instance, Classical Mechanics (CM) uses five (primitive) concepts: 'particle', 'space', 'time', 'mass' and 'force'. Among them, 'mass' and 'force' are CMtheoretical, for we cannot measure them without using some mechanical law. In contrast, for example, even if we sometimes measure space/distance using mechanical laws, we can also measure it some other times without using them (for instance by means of geometric optics); analogously with the others. Hence, 'particle', 'space' and 'time' are CM-non theoretical. This distinction indirectly applies to T's models, for T's models are made of the denotations of T's terms, both T-theoretical and T-non theoretical. Hence, if P, s, T, m and f are the corresponding denotations of 'particle', 'space', 'time', 'mass' and
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'force', models of CM are of the form (P, s, T, m,f), with a 'part' or sub-model (P, s, t) which is CM-non theoretical, the 'purely' cinematic part. T-models are defined by axioms. The first important models are those that express T's conceptual apparatus: T-characterizations, or 'improper' axioms, are the formulas that determine only the logical type of the components of the models (e.g. 's is a function from P x T into "K 3 ). The potential models of T are the structures that satisfy these characterizations, that is the structures that have the appropriate logical type for them to make sense when asking whether they satisfy the laws or proper axioms of T; let M/>(T) be the set of such structures. The partial (potential) models of T are the result of cutting off the T-theoretical components of the potential models; let Mp/>(T) be the set of such structures. For example, in CM potential and partial models are, respectively, of the form {P, s, T, m,f} and (P, s, T}. According to structuralism, and to other semantic approaches, the individuation of a theory includes the systems to which the theory aims to apply or account for (Kuhn's exemplars). We call these systems the intended applications of T; let I(T) be their set. Members of I(T) are singled out pragmatically, by means of scientists' intentional actions. And - this is essential for our present concerns - qua members of I (T) intended applications are described/identified T-non theoretically, i.e. I(T) is a specific subset of M.pp(T) singled out by scientists in their selection of phenomena to account for. In CM, for instance, members of I (CM) are specific cinematic systems, i.e. space-time trajectories of planets around the sun, of the moon around the earth, of a pendulum, of a cannonball, of a body at the end of an elastic string, of a person skiing on a hill, etc. Members of I(T) are the T-facts or T-data to be explained/predicted. Since they are described T-non theoretically, they can be identified/measured without presupposing the validity of T (hence, although T-facts may be 'theory laden' — if T-non theoretical concepts are theoretical in some other theories — they are not 'T-laden' they are not theory laden by the very same theory relative to which they are Tacts'). In order to account for members of I(T), T introduces 'new' conceptual machinery, its T-theoretical concepts, and postulates that the alleged new entities interact with the T-non theoretical ones in a variety of ways. In model-theoretic talk, this means that T imposes different types of'theoretical postulates/restrictions' on potential models for accounting for intended applications. Structuralism distinguishes three different kinds of such theoretical restrictions. The main ones are T-laws, those that involve T-concepts only (T-theoretical and T-non theoretical; e.g. Second Law, gravitation law, pendulum law, etc.). These laws define the set M(T)M/> C (T) of actual mo dels, i.e. potential models that satisfy T-laws. A second kind of theoretical restrictions are 'bridge principles', laws that involve concepts of other theories as well (e.g. relating pressure and volume with temperature). These bridge laws also define
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a subset of M/>(T), the set L(T) of links. The third type of restrictions are transmodelic restrictions or constraints (e.g. that the same particle appearing in different models has the same mass-value, i.e. that mass is conservative; another is that masses are additive). These constraints define the set (of sets) of models C(T). We have then that Mp(T) expresses the total conceptual apparatus of T, and Mpp(T) the part of this apparatus inherited from previous theories, hence, 'the difference' is T's new conceptual machinery. With the T-non theoretical apparatus the phenomena to be explained, the set of intended applications I(T), are described/identified. Using the new concepts introduced by T, i.e. T-theoretical ones, scientists try to deal with intended applications imposing different theoretical restrictions or postulates: M(T), L(T) and C(T) express, together, the total theoretical postulates/restrictions with which T tries to account for, i.e. to explain, data/facts in I(T). This modeltheoretic approach is in fact more complicated, but this simplified version will do for our first approach to theory's empirical claim. Theory's empirical claim states, simply, that theoretical restrictions do account for intended applications; that is, that facts, identified as .certain specific T-non theoretical models, do fit into theoretical models appropriately restricted by theory's postulates. We can see now the precise form that this empirical claim has and how it is closely related to the Ramsey sentence. For simplicity's sake we will leave aside the set C(T) of transmodelic constrains. The empirical claim may then be expressed as: EC1 The set I(T) is embeddable into M(T) nL(T) where 'embeddable' has the following precise meaning: a set of partial models (like I(T)) is embeddable into a set of potential models (like M(T) PlL(T)) if and only if for every model of the first set there is a model in the second set that has it as a T-non theoretical sub-model; that is, every model of the first set is the T-non theoretical part of some model in the second set: EC2
For everyy G I(T) there is x G M(T) fl L(T) such that x hasjy as its T-non theoretical part
If we denote by V[M(T) D L(T)]' the result of cutting off the T-theoretical components from the models in M(T) D L(T), then the empirical claim is EC3
I(T) C r[M(T) D L(T)]
Intuitively: the theoretical restrictions are so that there are potential models that satisfy them and have the facts to be explained as their T-non theoretical
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part. That is, if T-theoretical entities interact with T-non theoretical ones in the way theory's postulates specify, then the obtained effect at a T-non theore43 tical level coincides with the data independently identified. Now we can see why theory's empirical claim, so reconstructed, is very close to the Ramsey sentence. For another equivalent reading of this same empirical claim is: empirical applications can be extended with T-theoretical entities obtaining models that actually satisfy all theory's restrictions. This reading quickly brings us to the Ramsey-Sneed version of the empirical claim. Let potential models of a theory T be of the kind ( 0 i , . . . ,om,ti,... ,tn), where o\,... ,om are them ('old') T-non theoretical components and t\,... ,tn are the n ('new') T-theoretical ones. Then this reading of T's empirical claim is expressed in the following sentence (equivalent to EC1-EC3): ECR
For every (o\,... ,om) G I(T): 3^ ... 3ttt(o\,... ,om, t\,... ,tm) eM(T)HL(T)
Since M(T) fl L(T) is what now does the work previously done by AX A CR, namely, to express theory's postulates (although there is no correspondence at all between the single components), the existential part of this sentence is a model-theoretic version of the Ramsey sentence: 3*1 ... 3tn: (GI, . . . , om., ti, . . . , * „ ) satisfies £ ( o 1 } . . . ,om, t\,..., tn,...), £ being the axioms that define M(T) flL(T). But with one important difference: now the validity of the Ramsey sentence 3*i . . . 3* n £(0i,..., om, t\,..., tn,. ..) is not irrestrictively stated, but stated only for a particular system or range of systems: a specific (set of) o\ — • • • — om- system(s) satisfies 45 3*i ... tn£,(oi,..., om, t\,..., * „ , . . . ) . This is then the way in which the model-theoretic reconstruction of theory's empirical claim resembles the Ramsey sentence. It is easy to see that this empirical claim can be false, since members of I(T) are identified without using T's postulates at all. And actually scientists do discover many times that it is false, namely, when predictions/retrodictions fail. The failure of theory's empirical claim is the origin of theory change. When things don't fit, i.e. when anomalies appear, something must be done. And basically two things can be done: either an intended application is dropped or some laws are modified. If changes are not drastic, we continue to work within the same theory, a case of intratheoretical change (Kuhnian normal science}; if changes are drastic, we change the theory and replace it by a new one, a case of intertheoretical change (Kuhnian revolution). To account for these diachronic phenomena some complexities are needed, even at a synchronic level of analysis. If intratheoretical and intertheoretical changes differ, as they do, theories must have both
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'essential' and 'accidental' parts: the former being the parts whose substitution amounts to a substitution of the theory by a different one; the latter being the parts that can be changed without abandoning the theory, their change amounts only to the move from one stage of the theory to another stage of the very same theory. This implies that, in a theory at a given time, not all laws/ restrictions are at the same level. Some laws or principles are more important or 'central' than others (as Kuhn and Lakatos already emphasized); e.g., Newton's Second Law is very central, the general law for elastic movements is less central, and the specific law for the simple pendulum still less so. At a given time, a theory has the structure of an inverted tree-like net, with its central parts/laws at the top, from where different specialization-branches open down making room for specific laws for specific applications; a knot that specializes another above it imposes additional empirical constraints (additional laws) on the actual models of the higher theory knot.46
At a single moment, in the synchronic sense, a theory can then be identified with this kind of theory net. During its history, in a diachronic sense, a theory can be identified with a sequence of theory nets so that posterior nets come from 'non-essential' changes in anterior ones (the lower the position of a knot in the net, the more specific it is, the more likely it is that it will be a target of the changes; the higher the position, the more general it is, the less empirical strength it has, the less likely it is that it will be subject to change; the highest knot is the least open to empirical refutation, e.g. Newton's Second Law is irrefutable taken in isolation]. We call such entities theory evolutions, and the change of its essential part amounts to the end of a theory evolution and the beginning of another, that is, a scientific revolution:
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Intratheoretical change
Interth. change
Intrath. change
We are going to use some of these complexities to make plausible a non fullfirst-stage theory of theoretical meanings.
RS and a multifactorial theory of theoretical contents We saw that Lewis applies the Ramsey sentence to the determination of the meaning of theoretical terms. Actually, this reading of RS may be found behind all conceptual-role accounts: the meaning/content of theoretical terms/concepts is determined by their role in the theory. Every conceptualrole analysis would then be Ramseyan in spirit, at least under this Lewisian reading of RS. And so it is the one we favour as an alternative to Lewis's. The proposal we will sketch differs from Lewis's in four main respects:47 (a) the general metatheoretical background is not axiomatic but model-theoretic net-like, along the lines of the preceding section; (b) instead of determining theoretical meanings as the sense of theoretical terms identified with descriptions that describe the role that the alleged denoted entity plays in the theory, we determine them as the content of theoretical concepts identified by their possession condition-role fixed by the theory; (c) the two-dimensional approach that takes into account only theoretical axioms and correspondence rules is replaced/augmented by a five-dimensional account; (d) what determines theoretical meanings is not the full theory in its first stage, but something weaker (almost) preserved throughout theory's history. Everybody agrees that the content of theoretical concepts of empirical theories is determined by both its 'formal' and its 'empirical' connections. But, although almost everybody also agrees in understanding formal
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connections as connections between the concepts by means of'laws', there has been less agreement about how to understand the second, empirical component. According to the orthodox Received View, the empirical content enters through a set of correspondence rules, sentences/propositions that relate the term/ concept C with some observational terms/concepts. According to the operationalist variant, the connection is made up of operational definitions or, in the non-reductivist version, operational rules that state the way C is related to a set of operational procedures of measurement or (for non metrical concepts) determination. According to Kuhn (and others), empirical content enters when scientists apply laws to specific systems (Kuhn's exemplars, which do not need to be described within an observational vocabulary). According to us, these different proposals are not in conflict but complement each other. There is no single source where empirical content comes from, and I take these traditional accounts as a basis for a multifactorial analysis of 'the' empirical component, and hence of theoretical contents. The content of a theoretical concept G of a theory T has the following components determined by the following relations: (1)
(2)
(3)
Lawful-formal component: determined by relations between C and (i) other theoretical concepts of T, through the T-laws, and (ii) other theoretical concepts of other theories T', T", ... , through bridge-laws that link T with T, T", ... This component intends to capture the formal (i.e. non empirical) part of the content. Because of the difference between nomic and conceptual necessities,48 not every T-law/bridgelaw can be constitutive of theoretical content. The main problem here is to specify which laws are constitutive, and to what extent. Applicative component: determined by relations between C and concrete empirical application/systems of T (Kuhnian exemplars, structuralist intended applications, e.g. the moon moving around the earth, a child playing on a swing, . . . ) . These empirical systems are described with T-non-theoretical concepts (which, remember, need not be observational). Again, not every empirical application is essential to the content, only a specific subclass of them, the paradigmatic ones, are. This part is the legacy of Kuhn's understanding of the empirical component. Observational component: determined by relations between C and some family of observational neutral pre-scientific concepts (or, perhaps, non-conceptual contents). This relation need not be direct, and generally it is not. It does not relate C directly with observational concepts, but indirectly through empirical applications. So the indirect relation is a compound of the direct relation between G and the T-non-theoretical concepts that conceptualize the intended applications, and a direct relation linking intended applications with observational 'scenes' ('red
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(5)
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spot between two white ones on the sky', 'increasing shape on the lens' surface', and the like). This component is in line with the Received View's correspondence rules, but links observational concepts not with T-theoretical concepts but with T-non-theoretical ones. Operational component (if any): determined by relations between C and some fundamental operational procedures of measurement/determination. This component applies only to T-theoretical concepts for which such operational procedures exist. Not every theoretical concept has it (e.g. quark does not), but for those that do, these determination procedures do matter for their content. If a theory has at least one T-theoretical concept C with operational content, its other concepts acquire part of this operational content via their relation to C. If a very abstract theory T has no concept with (fundamental) operational procedures, its concepts may acquire operational content via constitutive intertheoretical links (if any) between T and other theories with concepts with operational content. This operational component, contrary to classical operationalism, is clearly non reductive. Folk-ancestry component (if any): determined by relations between G and some pre-scientific folk explanatory practices. The motivation for this component is a view of scientific knowledge in continuity with pre-scientific, folk knowledge. Some parts of pre-scientific knowledge, though not fully structured, are explanatory, and some explanatory concepts of scientific theories are refinements of their folk ancestors. When this is the case, such a relation matters for the identity of the scientific concept. Not every scientific concept has, or is strongly related to, folk ancestors (e.g. quark doesn't have them, and entropy probably doesn't either), but some do (e.g. mass] ,49 As before, when a theory has some concept with this component, then its other concepts indirectly acquire it via their relation with the former; and again, even if no concept of a theory T directly contains this component, T-concepts may partially acquire it via constitutive intertheoretical relations (if any) between T and other theories with concepts with ancestral content. I think this component is implicitly present in some of the Received View's correspondence rules, but I won't argue for this here; I will give an independent, nonhistorical motivation later.
I cannot argue now for the plausibility of each of these five components. The first three, though not presented exactly in this way, have already been convincingly defended. The only thing to emphasize now is that the second and the third, even if in some cases they can extensionally coincide, do not collapse because of the conceptual distinction between T-non theoreticity and observability. That is, even if sometimes intended applications may be
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observational, they need not be and, importantly, on many occasions they actually are not. As we have pointed out, the last two components may be absent in some (even many) concepts, but when they are present, they are constitutive. In the case of operational procedures, like the fundamental measurement of mass using pan balances, the intuition for their constitutive character is the following. We have seen that, for instance, the classical concept of mass is CM-theoretical, which means that it cannot be measured without presupposing CM-laws, i.e. that there are successful GM-intended applications. The existence of fundamental measurement, however, seems to amount to a different conclusion, for in measuring masses with pan balances we have not made explicit use of any law. But this is an illusion. The illusion is not that we have measured something without using GM-laws. This is true. The illusion is that we have measured (classical) mass, i.e. the property that CM talks about. If we do not suppose that the pan balance satisfies momentum CM-laws, i.e. if we do not suppose that it is an intended application of (one branch) of CM, we cannot say that we are measuring the very same mass that CM talks about, that we are determining the concept CM-mass. Hence, as far as it is clear (as it is) that with such procedures we are measuring the very same mass that CM talks about, it is then constitutive of CM-mass that there are certain relations between CM-intended applications and operational fundamental measurement procedures. Regarding the last, ancestral component, the idea is that, taking again as an example the concept 'mass' in CM, when such an ancestor does exist, a scientist who does not relate the sophisticated mechanical explanation with the folk one hardly fully possesses our concept of mass, the very concept we do possess. But, one might object, could not a Martian become a CM physicist without our folk explanations? I don't think that in such a case he would use our concepts, hence CM-concepts neither. The issue is not whether he can refer to the same property, or to make the same right predictions, i.e. to be equally successful with intended applications. This is perfectly conceivable without 'folk connections' if he shares the other four components. The issue is whether sharing the previous four components, but lacking the connection to our folk explanations, he does all that with our very same concepts, i.e. with literally the same conceptual tools we use in doing so. I think he does not. So theoretical contents (of empirical theories) are made up of these five components. In terms of concepts, this means that theoretical concepts are individuated by these five components. To individuate a specific theoretical concept consists in identifying its particular formal, applicative, observational, operational (if any) and ancestral (if any) components. Although it is not essential for this proposal, and some other ways would probably do, we favour a possession condition version of conceptual-role theories for concept
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individuation. According to it, to individuate a concept is to identify its possession conditions, each possession condition being specified in terms of circumstances, prepositional attitudes and contents:51 (PC 1)
C = the concept X to possess which a thinker must: in circumstances DI have the attitude AI towards content K!(...X...), ..., in circumstances DK have the attitude An towards content Kn(...X...).
Actually, there is no need that G appears in every content Kz, it suffices if it appears at least once. And it is also possible, and common, that contents have other concepts as constituents as well, among them, other theoretical concepts C', G",.... We can then simplify PCI for a moment showing these facts (variables range over concepts): (PC2)
C = the X 0 ( . . . X , . . . , C', C",... .)
In this version, it is clear that this is the concept-possession condition version of Lewisian definitions for theoretical terms. As an identification scheme, it only works in case the definite description does not fail to refer, that is, in case the following uniqueness sentence obtains: 3 1 X0(...X,...,C',C",....) On the other side, C is not well individualized, as a theoretical concept, until the other theoretical concepts C', C",... that C's possession condition involves are identified. The corresponding identification schema for such concepts will also individuate something only if the corresponding uniqueness sentences are true. If possession conditions of these other concepts involve in turn the concept C, as is common in the case of theoretical concepts of the same theory, we have a case of a holistic family of concepts with the same . In such a case, putting all the uniqueness conditions together we obtain the following Lewisian Ramsey sentence for T-theoretical concepts for a given theory T (0 will include other concepts as well): 31X31X'31X" ...
(,... ,X', X",...),
which allows for Lewisian definitions/individuations. Or, better, for a holistic individuation: (HPC)
C / , . . . , C n are (respectively) the concepts X / , . . . , X K such that ^(...,X/,...,XK)
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Now, what remains to be done is to find out the specific (j) for a given theory T, the specific possession conditions for T-theoretical concepts of a given theory. And, if our five-dimensional account is sound, there should be possession conditions for every component (the last two may be lacking in some theories). Of course this is a task that needs specific meta-empirical research, but something about the logical form of every such possession condition can (and must) be said in advance. We cannot go into the details here, but the general form of these clauses is roughly the following:53 Formal:
Every possible intended application is embeddable under these ... laws/models Applicative: Confronted with a specific paradigmatic application, the scientist believes that it is embeddable under these ... laws Observational: Confronted with a specific application, the scientist expects that a certain specific observational scene obtains Operational: Confronted with a fundamental measurement system, the scientist takes it as being a specific successful intended application Ancestral: Confronted with an ancestor folk-explanation of a phenomenon, the scientist believes that the corresponding intended application is embedded under the corresponding set of models I have left the two first clauses open to connote that we need to say something about the laws that count as constitutive in each case. This is an important problem, for the plausibility of the account depends on providing a plausible answer to this question. And it has been objected that something along these lines is not plausible, for either we put something specific enough to be individuative, but then this is probably too specific to count as constitutive (i.e. to be part of the possession conditions), or we put something which may count as constitutive, but then it is probably not specific enough to be individuative. It is here that the hierarchized net-like structure of theories comes into play, for it makes plausible an answer consisting in 'cutting' the hierarchized net of laws 'somewhere in between' the almost empty laws at the top and the highly specific ones at the bottom. That is, what is constitutive may be the laws at the very top plus some of the more specific (but not too specific) ones below. The hierarchized net-like structure makes plausible the existence of a set of lawful constraints unspecific enough to be constitutive and specific enough to be individuative. Of course the concrete place/level where we cut depends on the specific concepts at stake. For instance, it is probably sound to consider Newton's Second Law as constitutive of the concept mass (and force), but it is highly implausible to consider the Gravitational Law in
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all its details as constitutive. And there probably may be more than the Second Law that is constitutive. For instance, that masses exert forces each other might also be constitutive. Or even that attract each other. Or even that this attraction decreases with distance. All this is purely speculative and must be justified by meta-empirical research, but illustrates the way the hierarchized net-like structure of theories allows for a plausible articulation of the first, formal component. And analogously with the second, applicative one. Suppose that the Earth-Moon system is a paradigmatic application (this, again, should be meta-empiricallyjustified), it is totally implausible to say that what is constitutive is that this system is embeddable under (among others) the Gravitational Law in all its details. What is constitutive is a weaker content, maybe only that this system is a case of decreasing attractive forces (or whatever the meta-empirical research discovers). This, again, illustrates a way in which the embedding content may be both specific enough to be individuative (i.e. to do the individuative work that must be done by this component) and unspecific enough to be constitutive of the concept. To conclude, it is obvious that this proposal is not of the Lewisian 'full-firststage' kind. As far as laws-roles are concerned (first and second components), what is constitutive of meaning/concepts is not the full theory, either in its first stage or in any other; what is constitutive is something weaker, to repeat: something in between the almost empty top and the extremely specific bottom of the theory net. And something (almost) preserved throughout theory's history, so that contents of T-theoretical concepts (almost) do not change. This is our Ramseyan non full-first-stage solution to the constancy of meanings in theory-evolutions.
Endnotes 1. This work has been partially funded by the research projects BF2002—04454C10-05 (Spanish Ministry of Science and Technology) and BFF2002-10164-E (European Science Foundation). I want to thank Manuel Garcia-Carpintero, Pablo Lorenzano, Ulises Moulines, David Pineda and Mauricio Suarez for their comments and criticisms. 2. Ramsey's example is probably motivated by the different phenomenalist proposals for 'constructing the external world' that emerged during the early 1920s; in another passage (p. 120), he explicitly refers to proposals of this kind made by Nicod (1924), Russell (1927) and Carnap (1928). Note, though, that the intended interpretation of E and F is not phenomenalist. 3. Ramsey's 'axioms plus dictionary' follows closely Campbell's (1920) 'hypothesis plus dictionary'. 4. These are what in the Received View will be called correspondence rules (Carnap, Nagel) or interpretative sentences (Hempel). Interestingly, in Ramsey's example these propositions have the form of definitions of primary ('observational', in
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P.P. Ramsey: Critical Reassessments Received View's approach) terms. Ramsey writes these formulas, not as universal biconditional sentences but as identity open formulas, e.g: (Dil)
5.
6.
7.
8.
9.
10. 11. 12. 13. 14.
15.
16. 17. 18. 19. 20. 21.
A(n) = 3m(a(n,«) A/3(n,m)) A7(n).
But Ramsey himself (cf. pp. 115 and 131) also refers to the dictionary in our form. Here is where he explicitly mentions the phenomenalist programmes: 'This question is important because Russell, Whitehead, Nicod and Carnap all seem to suppose that we can and must do this' (my emphasis). Actually, some passages of'Causal Properties' (1929b) seem to implicitly advocate a kind of implicit definition: 'No proposition of the secondary system can be understood apart from the whole theory to which it belongs [...] "there is such a quality as mass" is nonsense unless it means merely to affirm the consequences of a mechanical theory' (pp. 137—8). One of our main goals in the last section is to show how to qualify the implicit definition theory for it not to have this consequence. The only possibility is, of course, that not every axiom had the same weight. This can be articulated making use of some Kuhnian and structuralist ideas (cf. 'RS and a multifactorial theory of theoretical contents', below). In Ramsey's writing Greek characters are both secondary predicative constants and secondary predicative variables. To avoid this ambiguous notation, I use, as common, different symbols for constants and variables. As usual, if V is a constant and (t' a variable of the same type, '<&*' means the substitution of V by '£ in '$'. For a complete formal demonstration, cf. e.g. Razeboom 1962, sec. II, spec, pp. 291—3. Note, also, thatTR is 'the only' primary sentence primarily equivalent to T; every primary sentence i? primarily equivalent to T is logically equivalent to TR. Cf. e.g. Maxwell 1962, p. 17. Note, though, that the example he gives does not fit the observable/theoretical distinction, for the secondary vocabulary is clearly observable. Braithwaite 1953, p. 79. Carnap, 1966, p. 249; see also Stegmiiller 1970, Chapter VII, §5. It is common to say 'prediction and explanation', but I prefer 'retrodiction' instead of 'explanation' for it is far from clear that explanation involves only observational j udgements. And this is probably what Carnap wanted to say in the above quotation, for otherwise all his work on theoretical meaning/analyticity would be pointless (cf. '[the theory and RS] "say the same thing" in so far as observable consequences are concerned', 1966 p. 254). Hempel 1958, p. 81; see also Maxwell 1962 p. 19. Schemer 1968, p. 270. Bonhert 1968, p. 278. For more complex and realistic examples, see Stegmiiller 1970, chapter 7 §7. Cf. 1963, pp. 218-22 and 1968, pp. 273. Bonhert 1968, p. 280.
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22. Bonhert ibid., Stegmiiller 1970 chapter 7, §8. 23. Stegmiiller, op. cit. This criterion rests upon the assumption that all T's empirical content is contained in TR (hence, if TR has no empirical content then T has no empirical content either). For the not totally uncontroversial character of this assumption, see the next section. 24. Op. cit., p. 262. 25. Cf. Hempel 1958. 26. Carnap, and many others, use 'T' for my 'AX', 'C' for my 'CR' and 'TG' for my 'T', but I'll be using this notation cross-authors for expository reasons. 27. Gf. fn. 9 above. 28. This does not amount to a decision procedure, for the set of TR —> T-consequences need not be decidable, and generally won't be (remember that TR, hence TR —>• T, is not only first but second order). But the aim was not this but just to identify such (undecidable) set. 29. Remember that the A-postulates for T's observational terms are supposed to be previously given. 30. Of course there is an instrumentalist justification of this assumption, but then it is not different than the above-mentioned instrumentalist assumption. I leave open whether there can be a non-instrumentalist justification of the assumption. 31. 'My proposal could be called an elimination of theoretical terms, if you insist: for to define them is to show how to do without them. But it is better called a vindication of theoretical terms; for to define them is to show that there is no good reason to want to do without them. They are no less fully interpreted and no less well understood than the old terms we had beforehand' (p. 79). 32. Lewis 1970, p. 79. This distinction was partially advanced by Putnam: 'A theoretical term, properly so-called, is one which comes from a scientific theory (and the almost untouched problem, in thirty years of writing about 'theoretical terms' is what is really distinctive about such terms)' (1962, p. 219). Lewis' suggestion is similar to Hempel's (1966): a theoretical term of a theory is a term 'characteristic' of such theory, whereas the other terms used by the theory are 'antecedently available or pretheoretical terms' (chapter 8; cf. also 1970 and 1973). The distinction will be fully articulated, and so answering Putnam's challenge, by Sneed and the structuralist view with the 'T-theoretical/T-non theoretical' distinction for T's terms (a distinction relative to a given theory T; cf. next section). 33. p. 85.1 simplify here Lewis's notation. '3lx(p(xy abbreviates '3xtfj>((p(j>) <->jy = #)'. In addition, I have not mentioned his simplification of taking all T-terms to be singular terms; we can skip this simplification taking variables here as ranging over both individuals and properties and applying the descriptor also to both. 34. For, e.g., (i) we have it as simply an application of the general fact that 3lx<j>(x) logically implies , then if we label it 'a', and all we know about it isjust that itis the only
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36. Remember that the problem is only a referential one, for even if denotationless the role-description makes perfect sense. 37. For a second order functionalism, see for example Putnam 1975; for some problems cf. Kim 1998, chapter 5. 38. As Lewis points out, this 'will work only if we permit the T-terms to name components of the nearest near-realization of T, even if it is not a realization of T itself. For after T has been corrected, no matter how slightly, we will believe that the original version of T is unrealized. We will want the T-terms to name components of the unique realization (if any) of the corrected version of T. They can do so without change of meaning if a realization of the corrected version is also a nearrealization of the original version' (p. 95). 39. The idea is first presented, in a somewhat cumbersome way, in Sneed 1971, and slightly simplified in Stegmuller 1973; the most detailed and accurate exposition of structuralist ideas is Balzer, Moulines and Sneed 1987; for a brief intuitive overview see Moulines 2002. We present here, in a very simplified and informal way, the main ideas necessary to understand (a simplified version of) the so-called Ramsey-Sneed account of theory's empirical claim. 40. For the first precise formulation of the criterion, cf. Sneed 1971, pp. 3 Iff.; for a completely rigorous presentation, cf. Balzer, Moulines and Sneed 1987, II.3.3. This criterion is in accordance with Lewis's (and Hempel's) intuitive characterization according to which 'an old term [is one] we already understood before the new theory with its new terms was proposed' (1970, p. 79). As we have just done, we will be using 'T-term' to refer now to all the terms used by T (i.e. appearing in T-laws), both T-theoretical and T-non theoretical. 41. The T-non theoretical/T-theoretical distinction does not coincide, neither intensionally nor extensionally, with the observational/non-observational one. The former is local, relative to theories (a T-concept may be T-non-theoretical but T'-theoretical), the latter is global (a concept - perhaps at a given time - is observational or it is not, period). T-theoreticity has to do with T-dependence of determination methods, irrespective of whether these methods are, or are not, 'observational'. And they do not coincide extensionally either: there are observable T-theoretical entities (e.g. genes are Genetics-theoretical, enzymes are Biochemistry-theoretical) and non-observational T-non-theoretical terms (e.g. utility is T-non-theoretical for some economic theories, weight is Stoichiometrynon-theoretical). 42. The only intended meaning of this modal talk is that these are the sort of structures that can be models of T, i.e. with the appropriate logical type for them to make sense to ask whether they actually are models of T. 43. If we take transmodelic constraints C(T) into account, the form of EC becomes more complicated. Since C(T) is a set of sets of potential models, the total theoretical restriction must be expressed as 7(M(T) fl L(T)) fl C(T), and EC is now I(T) G r[7(M(T) n L(T)) n C(T)], applying r now to one higher set-theoretical level. 44. M(T) and L(T) are not defined using the observational/theoretical distinction. And, even in terms of the T-theoretical/T-non theoretical distinction,
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45.
46.
47. 48.
49.
50.
51.
52.
101
neither do they represent 'only T-theoreticaP postulates (i.e. postulates with only T-theoretical terms) versus 'T-theoretical and T-non theoretical' ones (i.e. postulates which use both T-theoretical and T-non theoretical terms). Informally: the traditional RS says something like '3t\ ... 3tn^(o\,... ,om, t\,..., tn,...) is true' (i.e. is true 'of the world'), whereas the new model-theoretic version says '3^ ... 3tn^(o\,..., om, t\,..., tn,...) is true of the specific (set) of system^) (oi,. ..,omy. E.g. the theory net of CM has at the top a theory element whose actual models are constrained only by Newton's Second Law (and perhaps the Action-Reaction principle). At the second level, theory elements impose additional constraints. One element imposes the constraint for all forces dependent on distance, i.e. it opens the 'distance dependent forces' branch. Another imposes the constraint for all forces dependent on velocity, opening the 'velocity dependent' branch. And so on. At the third level, the first branch specializes, opening a sub-branch with additional constraints for forces direct-dependent on distance and a sub-branch for forces inverse-dependent on distance. And so on. At the end points of these many sub-branches we find the most specific empirical constraints: gravitation law, simple pendulum law, etc. For a detailed, but still programmatic, presentation of this proposal, see Diez 2002. Here I offer only the main traits of the picture. Our notion of conceptual necessity is broader than what is standard in structuralism. According to standard structuralism, the only conceptual truths are the socalled characterizations, the 'improper axioms' that define potential models. According to our approach, some (parts of the) 'proper axioms' will be apriori as well, namely those constitutive of the concept possession. Typical folk explanations involving an ancestor of'mass' are 'the truck went off the road because it was too heavy and it took the turn too fast', or 'the ship will sink because the cargo is too heavy'. E.g. Carnap's 'if temp(x) is greater than temp(jy) then x is hotter thanjy', for the folk concept of 'hotter than' is not only observational but clearly explicative as well ('the pressure cooker exploded because we heated it too much'). Cf. e.g. Peacocke 1992, chapter 1 (which I favour because of its neutrality for the internalism/externalism issue). This is not to psychologize concept individuation, at least not in a problematic sense. To individuate concepts by possession conditions simply acknowledges that concepts are constituents of the contents to which prepositional attitudes are addressed, hence two concepts are the same if they play the same role in prepositional attitudes. For this to make sense it is essential to distinguish (full) possession from (mere) deferential use. For simplicity's sake we only take into account conceptual contents, but it might be necessary to take into account non-conceptual contents (if such things do exist) as objects of thinkers' attitudes. But note that now variables range over concepts/contents/meanings/intensions, not over properties/entities/extensions, as in Lewis. This has as a consequence that, if uniqueness fails, there is no concept/content/meaning/intension. It is then essential that, in order to be able to account for the content of denotationless
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concepts (e.g. phlogiston), the possession condition for a concept C can be uniquely realized even if C does not refer to anything; that is, individuative possession conditions cannot be so that uniqueness fails merely because the concept is denotationless. 53. Cf. Diez 2002, sections 4-8. These are conditions for theories believed true. In order to be possible, as it is, to fully possess concepts of theories believed false, the belief attitude must be a conditional one. Which is the precise antecedent of this conditional is one of the things that needs to be clarified. 54. 'There are of course often theoretical roles associated with theoretical concepts. [...] One cannot plausibly say that it is a condition for possession of a theoretical concept of an empirical science that anyone who possesses it must be willing to judge that something falling under the concept has a certain highly specific theoretical role. If on the other hand we make the role unspecific, it can hardly be individuative of the theoretical concept. Such [theoretical] roles as theoretical concepts have are not constitutive of those concepts' (Peacocke 1999, pp. 14-15, my emphasis). Note, though, that in our case we have other components for the individuation as well. But the challenge is still fairly addressed to the formal component (Lewis, for example, includes too much to be constitutive).
References Balzer, W., Moulines, C. U. and Sneed, J. D., 1987, An Architectonic for Science. TheStructuralistProgram, Dordrecht: Reidel. Bonhert, H. G., 1968, 'In Defense of Ramsey's Elimination Method', The Journal of Philosophy, LXV, 275-81. Braithwaite, R., 1953, Scientific Explanation, Cambridge: CUP. Campbell, N., 1920, Physics: The Elements, Cambridge: CUP. Carnap, R., 1928, DerlogischeAufbau der Welt, Berlin. 1966, Philosophical Foundations of Physics, New York: Basic Books. Diez, J. A., 2002, 'A Program for the Individuation of Scientific Concepts', Synthese, 130, 13-48. Hempel, C. G., 1958, 'The Theoretician's Dilemma: A Study in the Logic of Theory Construction', in Fiegl-Scriven-Maxwell (eds), Minnesota Studies in the Philosophy of Science II. 1970, 'On the "Standard Conception" of Scientific Theories', in Radner-Winokur (eds), Minnesota Studies in the Philosophy of Science IV, pp. 142-63. 1973, 'The Meaning of Theoretical Terms: A Critique of the Standard Empiricist ConstruaP, in Suppes, Henkin, Joja and Moisel (eds), Logic, Methodology and Philosophy of Science 4, pp. 367—78. Kim,J., 1998, Philosophy of Mind, Oxford: Westview. Lewis, D., 1970, 'How to Define Theoretical Terms', in D. Lewis, Philosophical Papers I, Oxford: Oxford University Press, 78-95. Maxwell, G., 1962, 'The Ontological Status of Theoretical Entities', in Feigl and Maxwell (eds), Minnesota Studies in the Philosophy of Science HI, pp. 3-27.
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Moulines, C. U., 2002, 'Structuralism as a Program for Modeling Theoretical Science', Synthese, 130,1-11. Nicod, J., 1924, La Geometric dans le Monde Sensible, Paris. Peacocke, C., 1992, A Study of Concepts, Cambridge: MIT Press. 1999, Being Known, Oxford: Clarendon. Putnam, H., 1962, 'What Theories Are Not', in Putnam, Mathematics, Matter and Method, Cambridge: CUP, 215-24. 1975a, 'The Nature of Mental States', in Putnam, Mind, Language and Reality, 429-40. 1975b, 'The Meaning of "Meaning" ', in Putnam, Mind, Language and Reality, New York: Cambridge University Press, 215-71. Ramsey, F., 1929, 'Theories', Philosophical Papers, ed. J. Mellor, Cambridge: CUP, 112-36. 1929b, 'Causal Properties', Philosophical Papers, ed. J. Mellor, Cambridge: CUP, 137-9. Razeboom, W., 1962, 'The Factual Content of Theoretical Concepts', in Feigl-Maxwell (eds), Minnesota Studies in the Philosophy of Science III, 273-57. Russell, B., 1927, Analysis of Matter, New York: Kegan Paul. Schemer, I., 1963, The Anatomy of Inquiry, New York. Scheffler, I., 1968, 'Reflections on the Ramsey Method', The Journal of Philosophy, LXV, 269-74. Sneed, J. D., 1971, The Logical Structure of Mathematical Physics, Dordrecht: Reidel, (revised 2nd ed. 1979). Stegmuller, W., 1970, TheorieundErfahrung, Heidelberg: Springer. 1973, Theorienstrukturen und Theoriendynamik, Heidelberg: Springer.
4 The contributions of Ramsey to economics1 Joao Ricardo Faria
Introduction Frank Plumpton Ramsey is a towering figure in economic theory. He laid the foundations of modern macroeconomics and public economics with only two papers published not long before his premature death. His papers were so advanced technically that they only made it into the mainstream economics three decades later. The enduring influence of his work, however, is not only due to its mathematical strength and innovations but also due to the intuitive insights his models give when applied to important economic problems. Both papers appeared in the Economic Journal, then edited by John Maynard Keynes with an incredible dose of discretion (Moogridge 1992). In the first paper, published in March 1927, entitled 'A Contribution to the Theory of Taxation' Ramsey aims at answering the following question suggested by Arthur Cecil Pigou: in an economy populated by identical individuals, where the State cannot raise revenues through income or profit taxes and must be financed solely by taxes upon commodities, how should these taxes be set in the least distortionary pattern? Should every commodity be taxed at the same rate? The model Ramsey built to answer this question is one of the cornerstones of the theory of optimal taxation and his answer became known as the Ramsey rule or the Ramsey taxes. In his second economic paper, published in December 1928, entitled 'A Mathematical Theory of Saving' which Keynes, in the obituary of Ramsey published in the Economic Journal of March 1930 (Keynes, 1972) described as 'one of the most remarkable contributions to mathematical economics ever made', Ramsey tackles the following problem proposed by Keynes: how much of its income should a nation save? Ramsey's answer became known as the Keynes-Ramsey rule and it lies at the core of modern intertemporal growth models. This chapter will focus on these major contributions of Ramsey to economics in detail in the next sections. However, it should be noted that Ramsey made also other small but important contributions. Ramsey's personal traits were such that they made him dearly loved by all that were lucky enough to
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have close contact with him. His brilliant and insightful mind allowed him to 2 influence the work of many economists. For instance, he helped Piero Sraffa to put forward his Ricardian ideas in a proper mathematical setup (see Kurz and Salvador! 1999). He was one of the three mathematicians acknowledged by Sraffa in the preface of his book The Production of Commodities by Means of Commodities (1961) - which is remarkable since there is no expression of gratitude to any of his fellow economists. There is also an interesting story about Ramsey concerning Roy Harrod's paper introducing the concept of the marginal revenue curve, submitted to the Economic Journal in 1928. Keynes asked Ramsey to review the paper. Ramsey raised some objections. Due to Harrod's poor health and college duties, Harrod had put the revision of the article away for eighteen months. 'I then took the matter up with Ramsey, who was an old friend, and he recanted. The article was re-submitted and appeared in June 1930', asserts Harrod. However, this came with a high cost, continues Harrod: 'Study of her preface [Joan Robinson's Economics of Imperfect Competition] indicates that if Keynes had not listened so readily to Ramsey's criticisms and the article had appeared in 1928, my claim to have "invented" this well known tool of economics would be without challenge' [Harrod 1972, p. 186]. The remainder of the chapter is organized as follows. The next section presents Ramsey's ideas on taxation. In the third section Ramsey's contributions in macroeconomics are analysed. The concluding remarks appear in the fourth section.
A contribution to the theory of taxation In his 1927 paper, Ramsey tackles the following problem suggested by Pigou: 'a given revenue is to be raised by proportionate taxes on some or all uses of income, the taxes on different uses being possibly at different rates; how should these rates be adjusted in order that the decrement of utility may be a minimum?' (1927,47). This problem can be summarized by the maximization below: n
max h,tl,..,tn
V(p\,... ,pn,I)
subject to
R = N t{X{ ^
where V is the indirect utility function, R is the revenue requirement of the State, t is the per unit tax, p is the after-tax price, X{ is the consumption level of good i and /is the income.
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Under simplifying conditions, Ramsey derives a formula, called Ramsey taxes, to answer Pigou's question: t f\ 1 - = k[ - + -
P
\p
£
where p is the compensated elasticity of demand, £ is the elasticity of supply, and A; is a proportionality factor that depends on the total amount of revenue the government is trying to raise. The Ramsey taxes are commodity taxes that minimize the deadweight loss. Assuming that the demand of one commodity does not depend on the price of another, the tax rates should be set so that the increase in excess burden (deadweight loss) per extra dollar raised is the same for each commodity (Stiglitz 2000). When the supply curves are infinitely elastic this implies that the optimal tax system should be such that the compensated demand for each good is reduced in the same proportion relative to the pre-tax position (Myles 1995). The main message of the Ramsey tax is that commodities with high elasticity of demand (or high elasticity of supply) have a higher marginal deadweight loss per marginal dollar of revenue raised, and thus should have lower marginal tax rates. That is, the less elastic the demand or supply of a commodity, the higher should be the tax rate. The major criticism to Ramsey taxes is that they can be very regressive, in the sense that the poor will bear a larger burden of the taxes than the rich. This happens because according to the Ramsey taxes, high tax rates should be imposed on commodities with low price elasticities, such as food and housing, and as it is generally the case, these commodities have low income elasticities (Beaton 1981). This result seems puzzling since Ramsey knew that one of the normative objectives of taxation is to redistribute income from the rich to the poor, or at least, to impose the burden of taxation on those who can afford it. In Ramsey's words: 'The effect of taxation is to transfer income in the first place from individuals to the State and then, in part, back again to rentiers and pensioners' (1927,47). So, what is precisely the contribution of Ramsey to the theory of taxation? His contribution lies in providing a framework and a method of analysis that can be generalized to more relevant settings. His model and the redistributive problems associated with it triggered an enormous body of research to include equity considerations. Actually, modern optimal taxation theory is founded on Ramsey's paper and on a couple of seminal papers by Diamond and Mirrlees (197la,b). Diamond and Mirrlees extended Ramsey's model by incorporating non-identical households. As a result, they derived a tax rule in which consumers regarded as socially important ought to be taken
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into account, as well as the ones whose tax payments change considerably as income changes. One of the most debatable topics in this literature concerns production efficiency. The relevant question is: should taxes be imposed on production processes? According to Diamond and Mirrlees (1971) it is preferable for the government to tax consumers rather than producers. However, it is hard for the government to find out whether a commodity is being consumed by a final consumer or by a business. Thus if a government is to impose a tax on consumers it must also impose a tax on business use. For Stiglitz and Dasgupta (1971) whenever the government is not able to identify and tax away all pure profits, it may be desirable to impose distortionary taxes on producers. In the same vein, if there are restrictions on taxes on final goods, then production efficiency is no longer necessarily desirable. Newberry (1986) shows that if there are goods that cannot be taxed, then input taxes should be used as partial substitutes for the missing final taxes. Finally, some minor details about the article may be of interest. The article is divided in four parts, plus an introduction and one appendix. In the first part Ramsey presents a general model that is then discussed with particular cases in the remaining parts. Among several equations, there are 14 numbered equations. There is also one figure. Ramsey just cites one economic reference, which is that of Marshall's Principles. The other reference is the mathematical textbook of de la Vallee Poussin, Corns d'Analyse.
A mathematical theory of saving As in his previous Economic Journal paper on the theory of taxation, Ramsey starts this article putting forward the research question: 'how much of its income should a nation save?', To answer this question he derives a simple rule which is presented in the next paragraph: 'The rate of saving multiplied by the marginal utility of money should always be equal to the amount by which the total net rate of enjoyment of utility falls short of the maximum possible rate of enjoyment' (Ramsey 1928, 543). Ramsey makes three assumptions: (i) there is no population growth; (ii) no technical progress; (iii) no discounting of utility. The last assumption is normative since for him discounting 'is ethically indefensible' (1928, 543). As there is no discounting, in order to avoid the problem of unbounded integral of utility, Ramsey assumes that there is a maximum obtainable rate of enjoyment that he calls Bliss or B. The economic problem is that 'the more we save the sooner we shall reach bliss, but the less enjoyment we shall have now, and we have to set the one against the other' (1928, 545). In other words, more savings today imply more consumption tomorrow. So we must
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contrast the cost of postponing our consumption today with the benefit of enjoying it tomorrow. Ramsey acknowledges that 'Mr. Keynes has shown me that the rule governing the amount to be saved can be determined at once from these considerations' (1928, 545). This is why the rule derived by Ramsey is called the Keynes-Ramsey rule. Formally the problem is to minimize the amount by which enjoyment (given by the difference between utility from consumption (C) and disutility from labour (Z)) falls short of bliss integrated throughout time: min
Jo
[B - (U(C) - V(L))]dt
subject to the budget constraint in which investment in capital equals saving:
£=w>-c where F(K, L] is the production function, Kis the stock of capital and dK/dt is investment. Ramsey solved this model using the calculus of variations. By inserting the budget constraint in the integral yields:
• r
[B - (U(C):V(LWK k = ^ 10 dK/dt Jo
m nn I
r {B -(V(C) - K(£))] dk Jo F(K,L)-C
Notice that the integral is of the form:
rx*"
Jo
\
dtJ
\dt
therefore using the Euler necessary condition for a minimum value of the functional gives the optimal path for savings over time: [F(K,L] - C]U'(Q =B- (U(C) - V(L)} where the left-hand side is saving multiplied by the marginal utility of consumption and the right-hand side is equal to the amount by which the total net rate of enjoyment of utility falls short of the maximum possible rate of enjoyment. This is precisely the Keynes-Ramsey rule. Ramsey draws as a conclusion that 'the rate of saving which the rule requires is greatly in excess of that which anyone would normally suggest' (1928, 548). In addition he says that population growth would increase saving even more. However, technical progress is likely to 'make income obtainable with less sacrifice than at the present' therefore it is 'a reason for
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saving less' (1928, 549). In addition, Ramsey considered some extensions, such as the choice of savings rate with constant factor prices, which provides a theory of life-cycle savings. It also considered the discounted utility case, and Ramsey shows that the discount rate should be constant in order to avoid the problem of successive generations having the same system of preferences. If individuals differ in their rate of discount, an 'equilibrium would be obtained by a division of society into two classes, the thrifty enjoying bliss and the improvident at the subsistence level' (1928, 559). Concerning the necessity of a high savings rate as argued by Ramsey, despite the agreement of Tinbergen (1956) - which, for instance, is one of the first papers to cite Ramsey's 1928 paper - Mirrlees (1967) and others found that Ramsey was misleading, since when population growth, technical progress and discounting were taken into account, the optimal rate of saving is different from that implied by the Keynes-Ramsey rule. According to Chakravarty (1987) Ramsey's paper received little attention for nearly three decades, partly 'because of the "Great Depression" where "excessive savings" in the sense of too high a propensity to save' appeared to be the problem (Chakravarty 1987, 207). As a consequence, if the main prediction of the model seems to be flawed, what are the main contributions of this paper for economics? Or, in other words, why did this paper make such an impact in economics? The Ramsey model was rediscovered by the literature on growth theory in the 1950s and 1960s (e.g., Goodwin 1961; Ghakravarty 1962). The motivation was to help planners arrive at optimal growth paths. Instrumental in its rediscovery was the development of new dynamic techniques such as optimal control methods (associated with the Soviet mathematician Pontryagin) and dynamic programming (associated with the American mathematician Bellman). The papers that 'updated' the Ramsey model and gave it its modern view and inception in the theoretical literature are Gass (1965) and Koopmans (1963; 1965). They introduced discounting and population growth in the Ramsey framework. These papers encompass the contributions of Solow (1956) to growth theory and Phelps (1961) concerning the golden rule of capital accumulation in which the optimal capital stock is found when maximizing consumption. They show that the optimal stock of capital is determined by the intertemporal maximization of discounted utility subjected to the budget constraint in which savings are allocated in capital investment. The solution of this problem leads to the modified golden rule, in which the interest rate (given by the marginal productivity of capital) equals the discount rate plus the rate of population growth. Newberry (1987) argues that the reason for the success of Ramsey's 1928 paper is that it poses a fruitful question and proposes a method of analysis, that of intertemporal welfare maximization using techniques of dynamic
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optimization. I would argue that its appeal lay in the fact that after the Rational Expectations revolution, lead by Lucas, Sargent and others in the 1970s, it became consensual that a robust macroeconomic model should be based on sound microeconomic foundations. That is, behavioural relationships should be derived form the intertemporal optimization of micro-economic agents. The Ramsey model, also known as the Ramsey-Cass-Koopmans model, became the 'prototype for studying the optimal intertemporal allocation of resources' (Blanchard and Fischer, 1989). The major characteristic of the Ramsey model is its malleability, because it allows us to deal with a number of problems in a very simple framework. For instance, the Ramsey model has been extended to study heterogeneous capital goods (Samuelson and Solow 1956); money and growth (Sidrauski 1967); labour supply (Brock 1974); to address investment and saving in an open economy (Abel and Blanchard 1983); habit formation (Ryder and Heal 1973); and many other important issues in macroeconomics and growth theory — an exhaustive list would run over a hundred citations. As minor, but important, details it is worth noting that the paper is divided into three sections. It departs from the general model and then tackles specific issues. It has ten numbered equations and three figures. It is remarkable that Ramsey does not cite any reference in the literature; he only mentions Keynes' opinions and examples.
Concluding remarks The enduring contributions of Ramsey to economics are amazing. It is hard to believe that a young man who published only two papers in this subject, one on taxation and the other on saving, could have such an impact. His papers share the same general properties. They depart from very interesting questions - suggested by the then most prestigious and influential British economists of the time, who happened to be from Cambridge as well: Pigou and Keynes — which are tackled through general models that give precise and clear answers. As a final step Ramsey explores some particularities of the models in order to check whether his general answers stand. Despite problems concerned with practical issues raised by the implications of his models, Ramsey's models attract the attention of the profession due to their simplicity, elegance, malleability and generality, which allowed economists to extend them to many other different problems. Ramsey's models became frameworks of analysis; they can be easily adapted and modified, to provide insightful and penetrating analysis with relatively low technical costs. Therefore they became invaluable tools themselves, which guarantee a place for Ramsey's name among the very best economists of all times.
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Endnotes 1.
Acknowledgements: I would like to thank, without implicating, M. Leon-Ledesma, for useful comments. Address for correspondence: School of Social Sciences, University of Texas at Dallas, P.O. Box 830688, M/S GR 31, Richardson, TX 750830688. Phone: +972-883-6402; Fax: +972-883-6297. E-mail: [email protected] 2. Concerning his personal traits, his wife says that he was a 'very easy-going, very natural person'. According to I. A. Richards, Ramsey was 'very modest, gentle and on the whole he refrained almost entirely from argumentative controversy'. 'Had a very loud laugh which was infectious', said Richard Braithwaite (these opinions are found in Mellor's Ramsey homepage). Actually, his laugh is something characteristic: 'He had a beautiful laugh, not loud or hearty, but sudden, genuine and convulsive; it sounded as if his huge frame was cracking under the strain of it', remarked Roy Harrod (1972). 3. Fonseca [http://cepa.newschool.edu/het/index.htm] considers Ramsey's paper 'Truth and Probability' (written in 1926, published 1931) as the third contribution of Ramsey to economics, since this was the first paper to lay out the theory of subjective probability and begin to axiomatize choice under uncertainty.
References: Abel, A. and O. L. Blanchard (1983) 'An intertemporal equilibrium model of saving and investment', Econometrica, 51, 675-92. Blanchard, O. L., and S. Fischer (1989) Lectures on Macroeconomics, Cambridge: MIT Press. Brock, W. (1974) 'Money and growth: the case of long run perfect foresight', International Economic Review, 15, 750—77. Cass, D. (1965) 'Optimal growth in an aggregative model of capital accumulation', Review of Economic Studies, 32, 233—40. Chakravarty, S. (1987) 'Optimal savings', in J. Eatwell, M. Milgate and P. Newman (eds) Capital Theory, TheNewPalgrave, London: Macmillan, 206-11. (1962) 'The existence of an optimum savings program', Econometrica, 30, 178—87. Deaton, A. S. (1981) 'Optimal taxes and the structure of preferences', Econometrica, 49, 1245-60. Diamond, P., and J. A. Mirrlees (1971a, b) 'Optimal taxation and public production, I: Production efficiency; and II: Tax rules', American Economic Review, 61, 8-27 and 261-78. Fonseca, G. The History of Economic Thought, http://cepa.newschool.edu/het/index.htm Goodwin, R. M. (1961) 'The optimal growth path for an underdeveloped economy', Economic Journal, 71, 756-74. Harrod, R. F. (1972) John MaynardKeynes, Harmondsworth: Penguin Books. Kurz, H. D. and N. Salvador! (1999) Sraffa and the Mathematicians: Frank Ramsey and Alister Watson, unpublished manuscript.
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Keynes, J. M. (1972) T. P. Ramsey', in Essays in Biography, The Collected Writings of John Maynard Keynes, Volume X, Cambridge: Macmillan [for the Royal Economic Society].
Koopmans, T. C. (1963) 'On the concept of optimal economic growth', Lowles Foundation Discussion Paper No. 163 [CFP 238], http:cowles.econ.yale.cdu/p/cp/p02a/ p0238.pdf (1965) 'On the concept of optimal economic growth', in The Econometric Approach to Development Planning, Amsterdam: North-Holland. Mellor, D. Ramsey's Homepage, http://www.dar.cam.ac.uk/dhml 1/RamseyLect.html Mirrlees, J. A. (1967) 'Optimum growth when technology is changing', Review of Economic Studies, 34, 95-124. Moogridge, D. E. (1992) Maynard Keynes, London: Routledge. Myles, G. D. (1995) Public Economics, Cambridge: Cambridge University Press. Newberry, D. M. (1986) 'On the desirability of input taxes', Economics Letters; 20, 26770. (1987) 'Ramsey model', in J. Eatwell, M. Milgate and P. Newman (eds) Capital Theory, The New Palgrave, London: Macmillan, 220-7. Phelps, E. S. (1961) 'The golden rule of accumulation: A fable for growth men', American Economic Review, 51, 638-42. Ramsey, F. P. (1927) 'A contribution to the theory of taxation', Economic Journal, 37, 47-61. (1928) 'A mathematical theory of saving', Economic Journal, 38, 543-59. Ryder, H. E. Jr, and G. M. Heal (1973) 'Optimum growth with intertemporally dependent preferences', Review of Economic Studies 40, 1-32. Samuelson, P. A., and R. M. Solow (1956) 'A complete capital model involving heterogeneous capital goods', Quarterly Journal of Economics 70, 537—62. Sidrauski, M. (1967) 'Rational choice and patterns of growth in a monetary economy', American Economic Review Papers and Proceedings 57, 534-44. Solow, R. M. (1956) 'A contribution to the theory of economic growth', Quarterly Journal of Economics 70, 65—94. Stiglitz, J. E. (2000) Economics of the Public Sector, New York: Norton. Stiglitz, J. E. and P. Dasgupta (1971) 'Differential taxation, public goods and economic efficiency', Review of Economic Studies 39, 151-74. Tinbergen, J. (1956) 'The optimal rate of savings', Economic Journal 66, 603-9.
5
Ramsey's theory of truth and the origin of the pro-sentential account Maria J. Frdpolli
Introduction The aim of this chapter is to discuss Ramsey's theory of truth. One of the (few) theses that everybody attributes to Ramsey's thought is the redundancy theory of truth, as it has been called. In the following pages I will maintain that Ramsey never supported such a position about truth, but rather that he proposed an analysis of this notion that is strikingly similar to the present prosentential account. In fact, the very word 'pro-sentence' appears in Ramsey's paper 'The Nature of Truth', written around 1927 and posthumously published by Ulrich Majer and Nicholas Rescher in 1991, for the very first time in the history of philosophy. Ramsey's main concern was to offer an analysis of belief and judgement. If his view about truth is to be properly understood, this general background must be borne in mind. Allegedly, Ramsey's theory of truth is to be found in his essay 'Facts and Propositions' (1927) where he asserts: 'it is necessary to say something about truth and falsehood, in order to show that there is really no separate problem of truth but merely a linguistic muddle' (1990, p. 38). Appearances notwithstanding, Ramsey does not argue for the vacuity of the truth operator. His aim is rather to explain the connections between truth and falsehood, on the one hand, and the notions of judgement, belief and assertion, on the other. Thus, what one encounters in (1927) is, basically, an analysis of the epistemic operators from a logico-semantical point of view. As far as truth is concerned, Ramsey's conclusion in this paper is that, once the philosophical treatment of judgement and belief is accomplished, the philosophical difficulties with which the notions of truth and falsity are fraught will dissolve. For the notions of judgement and belief pose serious difficulties to philosophy, while understanding the import of the terms 'true' and 'false' is a relatively easy task. Everybody knows what they mean by 'true' and 'false', although to explain it is far from simple. And a source of muddle comes from infecting the semantic operators of truth and falsehood with epistemic notions such as judgement and belief. Ramsey undertakes the project of developing a theory of truth in his paper 'The Nature of Truth' and here a more substantive account is offered.
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My purposes in this chapter will be the following. First, I will explain Ramsey's theory of truth and its place in Ramsey's thought. Second, I will discuss modern prosententialism and show how Ramsey's view not only fits the paradigm but is also the first clear and detailed formulation of it. And third, I will argue in favour of the accurateness of the prosentential account and thus of Ramsey's proposal as the correct view on truth from a logical point of view.
Ramsey's Theory of Truth Until the publication in 1991 of On Truth, the collection of previously unpublished material edited by Rescher and Majer, the classic place to search for Ramsey's position on the topic was his paper 'Facts and Propositions' (1927). His paper of 1926, 'Truth and Probability', in spite of what the title suggests, does not deal with the notion of truth. The first chapter of On Truth, entitled 'The Nature of Truth', is relatively self-contained, and is the only place in which Ramsey explicitly undertakes the task of providing an analysis of the semantic notion of truth. Both works, 'Facts and Propositions' and 'The Nature of Truth', were written at the same time, and the contrast between them is illuminating. In 'Facts and Propositions', the general context of a theory of truth is submitted; the context at issue is a philosophical treatment of the notions of belief and judgement. But the unfolded treatment of the notion of truth only appears in 'The Nature of Truth'. 'Facts and Propositions' was one of the few papers that were prepared for publication by Ramsey himself and thus we may suppose that it contains what he thought was his final view on the matter at that time. On the other hand, 'The Nature of Truth' is obviously an unfinished work. But for the purposes of this chapter, from what Ramsey explains there is more than enough to glean what would have been his mature view on truth. In this section, I will describe Ramsey's views closely following the two works mentioned. Their interpretation and evaluation will come later. Let us begin with 'Facts and Propositions'. As Ramsey explicitly states, the aim of the paper is to offer a logical analysis of what may be referred to by the terms 'belief, 'judgement' or 'assertion', that he takes as synonyms. Belief (judgement, assertion) is a relation between two poles, or two factors, as he calls them. There is a mental factor, my present mental state, or words or images in my mind, and an objective factor, facts or events in the world. And to say that I believe that Caesar was murdered is to maintain that a particular kind of relation holds between my mental state and the objective factor related to it. As regards what kind of relation belief is, Ramsey endorses Russell's position in 'The Nature of Truth and Falsehood' (1910, 1992: 115-24), where Russell rejects his previous view of belief as a binary relation
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between a subject and an object, a proposition in this case, in favour of a multiple relation between the subject and the ingredients of the proposition itself. Thus, my belief that Caesar was murdered is a relation between me, Caesar and murdering, and more complex beliefs might require more complex relations, although Ramsey allows the possibility that the multiple relation might be seen as a simpler one in which different ingredients might be somehow related to one other. Here the problem of the multiplicity of the belief relation arises. Ramsey does not enlarge upon it, but both Loar (1980) and Sahling (1990) defend that there are clues that point towards Ramsey's view as one in which the belief relation is a relation between a subject and the intentional abstraction of an ordered set, and thus the philosophical difficulty related to the adicity of the belief relation disappears. Ramsey acknowledges that his account of the objective factor of belief is sketchy and far from satisfactory, but leaves it there. In this setting, Ramsey poses the problem of truth. 'There is no separate problem of truth', he says (1991, p. 38), 'but merely a linguistic muddle'. The emphasis here is put on 'separate', for his view is that where essential philosophical questions are at issue is in the understanding of the notions of belief, judgement and assertion. Once this has been accomplished, the notion of truth will fall smoothly into place, free from epistemic contamination. Truth and falsehood paradigmatically play their part in the scenario defined by epistemic notions such as belief, but it is possible to offer a definition of them in which epistemic notions are not involved. What Ramsey upholds in this paper is that the logical analysis of truth is independent of the analysis of belief, and that most of the traditional difficulties in the analysis of truth are in fact difficulties in the analysis of other notions such as those already mentioned. To say that the (separate) problem of truth is no more than 'a linguistic muddle' does not commit one to embracing a redundancy theory of truth. Different authors with substantive views on truth have expressed more or less the same feeling. Austin, a champion of correspondence, maintained that 'the theory of truth is a series of truisms' (1950, p. 152) while at the same time vigorously rejecting that the truth predicate is logically superfluous. This is also Ramsey's case. In (1927), propositions are the truth-bearers. Ramsey considers that we ascribe truth and falsity primarily to propositions. In (1991) he sustains a slightly different view, mental states, such as beliefs, now being the class of thing to which 'true' and 'false' are applied. The possibility that the primary truth-bearers might be indicative sentences is dismissed as 'not a serious rival' (p. 7), because Ramsey considers it obvious that true or false is what people mean by these sentences and not the sentences themselves. But mental states are truth-bearers insofar as they possess what Ramsey calls a 'prepositional reference'. Thus his position in (1927) and in (1991) are arguably equivalent.
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If propositions and not sentences are the primary truth-bearers, then here we are seeing a straightforward escape from the semantic paradoxes, but I shall return to this topic later. In truth ascriptions truth is predicated of propositions. Propositions are referred to by expressions of two different types. In one, the referred proposition can be recovered from the referring phrase, as when quotation marks are used. In the other it cannot, as when the proposition is reached by a description. Sometimes, this distinction is marked by talking of expressions and designations of propositions. This topic will be developed in Propositions expressed and designated below. Without using this terminology, Ramsey takes on board both ways of pointing to propositions and only as regards the first he expresses the view that the truth predicate can be dispensed with. In fact, he says that 'It is true that Caesar was murdered' only means that Caesar was murdered, although stylistic reasons might advise the use of the former way. Nevertheless, this is not so when descriptions of propositions are at stake. When propositions are not explicitly given in the truth sentence, the truth predicate cannot be eliminated in natural languages (1927, pp. 38-9). Thus, there is no claim of redundancy here. Some uses of the truth predicate are dispensable, but others, those in which the truth predicate earns its living, are not. But in (1927) there is nothing else about the role of the truth predicate. The topic is dealt with in (1991). And here what is displayed is not a redundantist theory of truth but a prosentential account. The philosophical question that Ramsey addresses in this paper is not what is truth but what is the meaning of'is true'. The answer is obvious, because as many philosophers have defended before and after him, everybody knows what the predicate means. This is the answer of Aristotle, and also of Wittgenstein, Strawson, Austin, Prior, Tarski, Williams, Grover, etc. all through the twentieth century. What is the problem then? Tarski, for instance, expresses exactly the same idea in (1935, p. 152) and his famous diagnosis that truth cannot be defined in natural languages admits a charitable explanation if it is interpreted as a dim formulation of one of the tenets of the prosentential account. The problem, then, is not understanding what 'is true' means but to say what it means, because natural languages lack the appropriate expressive tools to explain the meaning of truth without using the predicate itself or a closely related one. That the truth predicate performs a role that cannot be explained without using it is what the prosentential theory of truth claims and is also a proof that the truth predicate is not redundant. But first let us go into Ramsey's definition. In (1991, p. 9), we read: 'We can say that a belief is true if it is a belief that/;, and/>.' And he explains that/? is a sentence variable that can represent any prepositional structure whatsoever. We might predicate truth of a disjunctive proposition, and then say: the belief that either p or q is a true belief because either/) or q\ or of a general proposition and say: the belief that
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every A is B is a true belief because every A is B, and so on. And p., as a sentential variable, already involves a verb, and thus there is no need to add 'is true' to the definition, which will render this view circular, as Ramsey explicitly acknowledges. Does this not show that the truth predicate is redundant? I am defending that the answer is negative. To understand Ramsey's position it is essential to take into account that there are ways of referring to propositions from which the proposition referred to is not recoverable. But this point will be argued for when we explain the prosentential account at length. For now suffice it to consider Ramsey's own words: As we claim to have defined truth we ought to be able to substitute our definition for the word 'true' wherever it occurs. But the difficulty we have mentioned renders this impossible in ordinary language which treats what should really be called pro-sentences as if they were pro-nouns. The only prosentences admitted by ordinary language are 'yes' and 'no', which are regarded as by themselves expressing a complete sentence, whereas 'that' and 'what' even when functioning as short for sentences always require to be supplied with a verb: this verb is often 'is true' and this peculiarity of language gives rise to artificial problems as to the nature of truth, which disappear at once when they are expressed in logical symbolism, in which we can render 'what he believed is true' by 'ifp was what he believed, p\ (1991, p. 10, Ramsey's italics) This is the first time that the expression 'pro-sentence' is used to mark the role of the truth predicate, although Ramsey's paper was not published until 1991. Thus, Ramsey's account of the meaning of the truth predicate is, in a nutshell, the following. The grammatical predicate 'is true' is a tool of natural languagues to build pro-sentences with the grammatical status of sentences. Words like 'yes' and 'no' act as prosentences from a logical point of view but have the grammatical category of adverbs. Words like 'it', 'that' and 'what' can assume the role of prosentences although they have the grammatical category of singular terms. It is necessary, then, to possess sentential pro-sentences, as it were, to perform certain logical operations that cannot be otherwise performed. As with the rest of pro-forms, pro-sentences cannot be eliminated altogether from natural languages without loss of expressive power, in the same sense that pro-nouns cannot be eliminated from natural languages without loss of expressive power. This does not mean that particular uses of pro-forms are not eliminable. Sometimes a pro-noun can be substituted by a noun, and a pro-sentence by a sentence. But the general category has a specific role to fulfil, that performed in artificial languages by variables of different categories. For this reason, it is easy to misunderstand the import of Ramsey's definition,
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'We can say that a belief is true if it is a belief that p, and/?', because we forget that it is not formulated in natural-language terms. As soon as we use prepositional variables, the truth predicate is dispensable. But now we have escaped from natural languages into the realm of artificial languages with prepositional variables. In these hybrid languages the truth predicate is no longer necessary but only because its job has been taken over by expressions added specifically to perform it. Ramsey thought of his position as a soundly formulated theory of truth as correspondence and rejected pragmatist and coherentist approaches to the problem as untenable. 'A belief that A is B is true if and only if A is B', he defined in (1991, p. 18), and according to him neither Pragmatism nor Coherentism can explain this basic intuition. Ramsey's early formulation of a prosentential view of truth will become clearer in what follows.
The prosentential account of truth A prosentential view of truth has been shown from time to time in the analytic tradition during the twentieth century. The origin of this view is Aristotle, with his: 'to say of what is that it is and of what is not that it is not is true' (Metaphysics 1011 b 27). Aristotle's sentence has been paradigmatically interpreted as an expression of a correspondence theory of truth. Although the intuitions under the correspondence theory are widely accepted, to embrace a developed and substantive correspondence approach is another story. The correspondence intuition allows a trivial and uncommitted implementation or a metaphysically burdened one. In the first case, it asserts that truth has to do with what is said by the users of language about the world, together with what the world is like, and it is a relatively harmless position. In the second case, it asserts that truth is a relational concept between two poles, the pole of language, and the pole of world. Here an account of the two poles is called for and in doing so room for philosophical disagreement emerges. Most proponents of theories of truth have vindicated Aristotle as a precursor. Ramsey, Williams and Grover, all defenders of prosentential accounts, are no exception. And it is significant that in 'The Nature of Truth' Ramsey struggles to distinguish his view from a correspondence theory, while presenting his own proposal on truth as an attempt to clarify Aristotle's dictum. What does a prosentential theory of truth aim to do? Basically, define the predicate 'is true' from a logical point of view, i.e. to offer an account of how the truth operator works. What we seek when we analyse the truth operator is to determine the logical form of truth ascriptions. A quick way into the topic is to ask what a speaker wants to say with a truth predicate or in which communicative situations a normal speaker (as opposed to a philosopher) puts the
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truth predicate to work. And the answer is illuminating for it shows that it is difficult to find real communicative exchanges in which the paradigmatic examples used in philosophical handbooks display their usefulness. That what the sentence 'snow is white' says is true if, and only if, snow is white (in a normal context of use) does not seem to be in need of explanation. Moreover, it hardly allows the truth operator to earn a respected position in language. If sentences like ' "snow is white" is true' were the only (or the favoured) contexts in which the truth operator appeared, then the redundantist conclusion would be hard to resist: that everything that can be done through a truth operator can be done without it. Fortunately, the truth operator has a task to accomplish and it cannot be dispensed with. The contents of sentences like 'What Victoria says is true', 'Everything the Pope says is true', 'The Theory of Relativity is true', etc. essentially involve the truth predicate. Let us call these sentences and the like 'blind truth ascriptions', because the speaker predicates truth of a content that the sentence does not display. In sentences like 'What Victoria says is true' the truth predicate might be contextually eliminated only if what Victoria says is known by the speaker and audience, otherwise the sentence is used to make an assertion in which the truth predicate cannot be suppressed. In the other cases, either because the content of what is said is indefinite or because it is strictly infinite, the truth operator becomes necessary. Prosentential theories of truth, contrary to other positions of the deflationist type, focus on the analysis of blind truth ascriptions. And what does it mean that what Victoria says is true? Any of a potentially infinite list of propositions. That if Victoria says that snow is white, snow is white, that if Victoria says that elephants can fly, elephants can fly, that if Victoria says that war is always unwelcome, war is always unwelcome, and in general that if Victoria says that p, then p. Notice that one thing is whether and when are we justified in saying that war is always unwelcome, and so asking when it is true that war is always unwelcome, and a very different thing is what a speaker means by her sentence 'what Victoria says is true' when Victoria has said that war is always unwelcome. In the first case we are looking for criteria and in the second we are determining the logical role. To properly understand the logical role of the truth operator one has to ask what the content of a sentence like 'What Victoria says is true' (uttered in an appropriate context) is. The content, or the proposition, expressed by a speaker who utters a sentence is what is said by her in the context in question. Suppose that at time t$ Victoria utters (1) War is always unwelcome, and at time t\ she utters (2) Mum does not like Mondays. After ( 1 ) 1 report
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(3) What Victoria says is true, and after (2) I add (4) What Victoria says is true. The basic intuition under the prosentential account is that, appearances notwithstanding, the contents of (3) and (4) diverge. And this occurs for several reasons, one of them being that the proposition that satisfies the description 'what Victoria says' is not the same in both cases. One might ask what, isolated from any context, a sentence like 'What Victoria says is true' says, i.e. what type of situation would make it true. The answer cannot be very precise, and not only because of the definite description. Compare (4) with (5)
The present President of the United States does not appreciate Japanese food.
In (5) there is also a definite description but if we needed it we might depict a situation in which a property were predicated of an individual. This cannot happen with (4). Although formally we might say that the content of (4) is that a proposition expressed by Victoria and identified in the context is true, this is compatible with infinitely many types of situation. The reason is that a sentence such as (4) is in fact a generalization from propositions of a certain structure and its role cannot be to describe a particular state of affairs. According to prosententialism, there are two basic roles of ascriptions of truth. Either they can be used for making a general assertion, as in 'The Theory of Relativity is true', in which in only one sentence we encode a piece of information that is strictly infinite, or else they can be used to inherit the content of a sentence distinct from itself. Sentences (3) and (4) are examples of this second use. The content of (3) in the situation depicted in the example is [War is always unwelcome] and that of (4) is [Mum does not like Mondays]. That the same linguistic expression expresses different contents depending on context is no longer something new. It is exactly what happens with indexicals. Pronouns and demonstratives keep their linguistic meaning constant while varying their content, their contribution to the proposition expressed, as the context varies. Because they are not ambiguous and linguistic meaning remains from one use to the next, they can be learned and taught to others. But any competent speaker knows that the reference of a demonstrative depends on the demonstratum on a particular occasion of use, and that a demonstration is needed in order to fix it. With pure indexicals there is no need for a demonstration, but the referent depends on prominent features of the situation: the speaker, the location, the time of utterance, etc. If we follow too closely our first thoughts about what a pronoun is we might say that a pronoun is an expression that marks the position that might be
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occupied by a noun (or of any other kind of singular term). A pronoun is usually interpreted as the counterpart in natural languages of bound nominal variables of first order calculi, and so as serving two basic functions, either generalization or anaphoric reference. Nevertheless, an English pronoun such as 'it', or a demonstrative such as 'this', performs these two functions even when related to categories other than singular terms. And also in these cases can be quantified over. Consider the following examples: (6) John loves Mary but she does not know it (7) Victoria said that the film was touching and Joan denies it (8) This is what I disapprove of in George, his lack of mercy where the two instances of'it' and the demonstrative 'this' do not stand for singular terms. Of course, if we expand (6) —(8) to avoid the anaphoric references, the expressions placed in their positions will have the status of singular terms from a grammatical point of view. But from a logical point of view, 'it' in (6) and (7) refers to a complete proposition, and 'this' in (8) points to an adjectival phrase. Grover, Gamp and Belnap introduced the term 'proform' (Grover 1992, p. 87) to cover the whole range of pronouns, proadjectives, proadverbs and prosentences. The fact that most preforms have the syntactic category of pronouns in natural languages is a historical accident with no philosophical significance, although it has had enormous philosophical consequences. Atomic proforms that are not pronouns are scarce, so natural languages have to supply them by means of complex proforms. The adverbs 'yes' and 'no' are the exception. But consider (9) (10)
Did you pick up your daughter from school? Yes Are you going to attend the seminar? No
In (9), and everywhere else, 'yes' has the import of a complete sentence, in this case T picked my daughter up from school,' and it is a truth bearer because its content is a proposition. And the same can be said about 'no'. In (10) 'no' has a negative proposition as its content: [I am not going to attend the seminar]. For this reason, although they occupy adverbial positions, they are logically prosentences. A prosentence is a prepositional variable, it is something capable of inheriting any prepositional content depending on context, it is a function from contexts to truth conditions. To give access to a wide enough repertoire of proforms is a useful feature of a language, and this also applies to prosentences. Proforms are sometimes dispensable, but not in every circumstance. When they are used to make general assertions or to encode a big or even infinite amount of information, language
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cannot do without them. But, as Ramsey himself acknowledges (1991, p. 10), the only single prosentences in natural languages are 'yes' and 'no'. Fortunately, language has mechanisms for building complex prosentences. Among them are 'It is a fact', 'It is true', 'Thus is as things are', 'You are right', and their negations. That both 'This is a fact' and 'This is true' are complex prosentences explains the appealing triviality of correspondence theories of truth. The main thesis of the prosentential account of truth is that the truth predicate 'is true' is a dummy predicate that allows complex prosentences to be constructed. Literally speaking, sentences like (4), 'What Victoria says is true', or just 'it is true', do not express a proposition unless used in an appropriate context to refer to a prominent proposition. They do not posses a content any more than 'it' or 'you' do in isolation. The description 'What Victoria says' has the grammatical status of a singular term. In the situation depicted in (1) it is satisfied by a proposition, to wit, [War is always unwelcome] that as such belongs to a different logical category. Neither a description such as the one mentioned nor any other single proform is, for grammatical reasons, a suitable entity to occupy a sentential position. Let us reconsider Victoria's utterance at time to, (10)
War is always unwelcome.
To endorse her opinion, it would not be enough to reply 'What Victoria says', or 'it' or 'this', but rather (3) or a contextually equivalent expression such as (11)
This is true.
From a logical point of view, this would not make any difference. If English allowed indexicals or descriptions to fulfil sentential roles, the predicate 'is true' would add nothing to the content of the pronoun or the denotation of the description. In this sense, the dummy truth predicate is a way of restoring grammaticality. This idea of truth as a way of restoring sentencehood has been defended by many authors. A contemporary one is Paul Horwich who expressed it saying that the truth predicate 'acts simply as a de-nominali^or' (Horwich 1998, p. 5). One of the consequences of an analysis of this kind is that truth is no longer considered as a first order property. Logicians define first order properties as properties of objects, although neither 'property' nor 'object' keeps its usual meaning in the mouths of logicians. The paradigm of the notion of first order predicate is a predicate that expresses observable qualities, a predicate that can be used in descriptions of the world at the lowest level. Nevertheless, not every first order predicate is of this kind, predicates of abstract entities are not observable qualities. But for the purposes of the present topic it is enough to
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bear in mind the paradigmatic cases. Once we have acknowledged that 'is true' is a dummy predicate that represents a formal operation, to wit, the operation of converting singular terms that designate propositions into whole sentences, the question of what kind of property truth is loses its sense. According to the prosentential view, 'is true' converts designations of propositions with the grammatical status of singular terms into expressions of propositions with the grammatical status of sentences. The contrast between designations versus expressions of propositions was introduced in the philosophical arena by the Kneales (1962, pp. 584—6) and has also been used by Christopher Williams (1995). Thus the task performed by the truth predicate is not expressing a property of things and so this alleged property cannot be found in the world. Fortunately, the appropriate interpretation of this apparently debatable thesis does not take us out of the realm of the philosophy of logic. That truth is not a property among others in the world does not imply any kind of relativism. Nor does it entail a subjectivist or idealist position that would make truth dependent on the subject. Truth is not a property of the world because it is not a property tout court. It is a second order logical operator like existence and identity. Identity and existence might seem not to be of the same category, but they are. In predicate calculus, existence is a second order function and identity a first order binary relation, but the most promising and puzzle-eliminatory interpretation of the identity operator interprets it as an n-order (n > 1) function on concepts, like quantifiers. I will come back to this interpretation in the Conclusion below. Propositions expressed and designated
Propositions can be pointed to by means of diverse kinds of terms. Paradigmatically, phrases with the role of pointing to something else have the grammatical category of singular terms. Let us call this kind of phrase 'designations'. Some designations somehow display the designated entity and we call them 'exhibitive' and others merely describe their target and we call them 'blind'. The paradigmatic way of building exhibitive designations in written language consists of the use of inverted commas around an expression. Thus, 'Gandalf is a designation of a name that shows the designated entity, in this case the name of the Grey Pilgrim. From a grammatical perspective, it is common to assume that the compound of an expression together with a pair of appropriately placed inverted commas has the category of a singular term. This claim is not completely true, because in cases of open quotation, the quoted material does not need to have this status (Recanati 2000, pp. 181—91). But in the context of the present discussion these troubles with open quotations are irrelevant. So, let us continue. Even in cases in which the demonstrated entity is a whole
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proposition, the compound of it and the above-mentioned device converts the whole into a singular term. Thus, while
(12)
The cat is on the mat
expresses a proposition (uttered in an appropriate situation),
(13)
'The cat is on the mat',
designates either a sentence or a proposition. And (14)
That the cat is on the mat,
designates a proposition. The way in which (13) and (14) designate the proposition at issue is exhibitive, for it is possible to recover the exhibited entity from the designation itself. But consider that Victoria utters (12), then (15)
What Victoria says,
designates in an appropriate context the proposition expressed by (12) or the sentence (12) itself, depending on the favoured sense of'say' here. (15) is a blind designation of a proposition while (13) and (14) are exhibitive. In all cases the content is a proposition, from a logical point of view. A prosentential account of truth defends that the difference between (13), (14) and (15), on the one hand, and (16), (17) and (18), on the other, (16) (17) (18)
'The cat is on the mat'is true It is true that the cat is on the mat What Victoria says is true,
is one of grammatical category. In some sense, the contents of (13)^(15), on the one hand, and (16)-(18), on the other, are the same, to wit, the proposition [The cat is on the mat]. This proposition is designated by (13), (14) and (15) and expressed by (12), (16), (17) and (18). What is the role of'is true'in (16), (17) and (18)? According to this account, it is to restore the sentential category of designations like (13), (14) and (15), among other things. In other words, the role of'is true' is to construe expressions of propositions out of designations of them, but the content of (12) and (16) is the same. This is the intuition under many deflationary accounts of truth. In particular, it is the main thesis of the so-called redundancy theory commonly attributed to Ramsey. But, in spite of this intuition, the truth predicate is not redundant, not even in the general account we are explaining
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here. And the reason is that the proposition expressed by a truth sentence is not always recoverable. A sentence like (18) can be used to express any proposition whatsoever providing that the proposition at issue has been said by Victoria. In the previous example the content of (18) was [the cat is on the mat]. But if Victoria says (2) (2)
Mum does not like Mondays,
and then somebody replies with (18), the content of this utterance of (18) would be [Victoria's mum does not like Mondays]. The moral to draw from the previous paragraphs is that truth ascriptions are apt to inherit propositional contents to which they contextually refer. The difference between exhibitive and blind truth ascriptions is that in the former but not in the latter the inherited proposition is displayed in the sentential heir. This is the sense in which truth adscriptions work like pro-nouns. They are not pro-nouns though, because the grammatical category acting as their antecedent must be a complete sentence. By analogy with the way in which pro-nouns work, some philosophers have called truth ascriptions prosentences. These prosentences express propositions that are designated by means of other devices, for instance, definite descriptions, demonstratives, quotation marks, etc. The logicalform of truth ascriptions: truth as a second order operator The main lines of prosententialism, (i) the idea that the truth predicate is a dummy predicate, i.e. a contentless predicate-like expression used to restore grammaticality when a sentence is required for grammatical reasons, and (ii) that truth adscriptions are natural-language counterparts of formal propositional variables, are shared by all philosophers that have explicitly defended this view. Nevertheless their proposals do not coincide in all their details. One source of divergence arises from the fact that a theory of truth, just like any other philosophically interesting proposal, cannot be built as an independent piece of knowledge. It relates to other substantial views in other realms of thought, in this case with a particular account of quantifiers, of the status of propositions and of abstract entities in general, and also to the correct interpretation of other logical constants like identity. The deepest prosentential proposal so far brought to the fore is the one developed by Christopher Williams in (1976) and (1992). To him is owed the thesis that the truth predicate works as a second order operator. It does not appear as such in any other proposal although it blends perfectly into the prosentential intuition and helps to explain the force of redundancy and correspondency feelings.
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In order to understand the logical role of the truth predicate completely a further explanation is required. If, as has been said, a truth ascription is a proform that inherits the content of a sentence prominent in the context, the content inherited by the prosentence must appear at least twice in the relevant situation. This is the core of the correspondentist intuition. When exhibitive sentences like (16) and (17) are considered as the paradigm of truth ascriptions, this fact is obscured. Therefore, in (16)
'The cat is on the mat' is true,
it would seem that just one proposition expressed once, that the cat is on the mat, is being taken into account and that, of it, somebody predicates truth, a monary first order predicate. This view is misleading. Things become clearer when the class of paradigmatic examples changes and instead of Tarski-style sentences one proposes examples like, (18)
What Victoria says is true.
These kinds of blind ascriptions are the paradigmatic cases considered by prosententialism. Using blind ascriptions instead of exhibitive ones is a mark, I would say, of a serious stance on the meaning of truth. Both Ramsey and Grover, as much as Prior, Strawson and Williams, focus their accounts on ascriptions of the blind case. And all of them analyse (18) as (19)
Victoria says that/; and/),
or alternatively, (20)
For all/>, Victoria says that/? andp.
This analysis appears, for instance, in Williams (1995, p. 152). Grover analyses cases like (18), explicitly endorsing Ramsey's interpretation (Grover 1992, pp.7 Iff). And both Williams and Grover support Ramsey's explanation of the logical reading of (19) and (20). Ramsey already foresaw an objection against his position that has been profusely addressed against the prosentential analysis. The objection goes like this: (19) and (20) cannot explain the meaning of the truth predicate because they are ill-formed. Conjunctions (and the rest of sentential connectives) are functions of truth and so their arguments must be truth bearers. In (19) and (20) a variable occupies the place of the second argument, but variables are grammatically singular terms. To restore grammatically in (19) and (20) we should add the predicate
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'is true' at the end, which will render this analysis circular. Thus, to be grammatical, (19) and (20) would need conversion into (19')
Victoria says that/? andj&is true,
and
(20')
For allp, Victoria says that/) and/? is true.
Ramsey answered this self-objection by reminding the reader that p is a sentential variable and that as such it already contains a verb. A similar question arises when we try to translate (19) into ordinary English. We would have (19")
Victoria says it [Mum does not like Mondays] and it,
which obviously is not grammatically well-formed. The reason, as has been explained before and this constitutes one of the main tenets of prosententialism, is that natural languages almost completely lack single prosentences and this role has to be performed by complexes of pro-nouns and prosentential formers like 'is true'. Ramsey expresses this objection and his answer as follows: 'We can then say that a belief is true if it is a belief that p, and p. This definition sounds odd because we do not at first realize that 'p' is a variable sentence and so should be regarded as containing a verb; 'and p' sounds nonsense because it seems to have no verb and we are apt to supply a verb such as 'is true' which would of course make nonsense of our definition by apparently reintroducing what was to be defined' (1991, pp. 9-10). A further objection against formalizations like (19) and (20) relies on the widespread view that bound variables commit our discourse to the existence of their values as objects. Quine famously championed this thesis but it had supporters before and after him. The thesis itself is not justified, at least there are no serious reasons to maintain that only name-like expressions can be generalized by means of quantifiers. However, this is not the place to fight this battle. All that is important is to know that Ramsey does not support the objectual interpretation of quantifiers and so the Quinean criticism does not affect his theory. Grover also rejects the objectual interpretation and in her case she favours the substitutional reading of formulae like (20) as would any other follower of prosententialism. Ramsey's account of quantifiers falls beyond the scope of this chapter, but it is enough to say that he rejects that general sentences express propositions, because they do not represent. Generalizations are better seen as maps, and quantifiers are merely higher level intralinguistic operations that make no claim about what the world is like.
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Prepositional variables in (19) and (20) have two kinds of roles: (i) they mark the place of a sentence, of any sentence, and thus allow generalization and (ii) the second occurrence anaphorically refers to the content of the first. In this they are like any other variable and fulfil some of the classic roles attributed to pronouns in natural languages (apart from direct reference). Christopher Williams has defended that in natural languages identity is an operator that forms n-l-adic predicables out from n-adic predicables, i.e. that serves to eliminate an 'argument' place. Identity is the same operation as reflexivity. Reflexive verbs are intransitive verbs, and so monadic predicates, constructed from transitive verbs that are typically diadic predicates. The transition between (21) and (22), (21) (22)
Fran shaves Fran Fran shaves himself,
is explained as the result of introducing in (22) an operator, the identity/ reflexivity operator, that converts 'shaves', a diadic first order predicate, into 'shaves himself, a complex monadic predicate. The same operation takes place in the transition between (23) and (24), (23) (24)
Fran is Maria's husband and Fran is Joan's father Fran is Maria's husband and Joan's father.
In (24) the predicable is the complex one 'being Maria's husband and Joan's father' that can be paraphrased as 'the same person is Maria's husband and Joan's father' predicated of Fran in which the identity operator becomes apparent. Identity is thus a second order operator in which arguments are predicables, an operator with the same status of quantifiers. Williams's version of the prosentential account makes of the truth predicate an instance of the identity operator in which the variables involved are prepositional variables. In (19)
Victoria says that/? and/),
the propositional variable appears twice. An instance of (19) might be (25)
Victoria says that war is always unwelcome and war is always unwelcome,
in which there is a diadic predicable, (26)
'Victoria says t h a t . . . and
',
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in which the two argument places have been saturated by the same proposition. The transition between (25) and (27), (27)
Victoria says that war is always unwelcome and this is true,
is the same as the one that occurred between (23) and (24). The only difference is that in the latter the argument-place that has been cancelled out was nominal, occupied this time by a name 'Fran', while in (27) the cancelled argument-place was propositional. According to Williams, the complex predicable presents in (27), and also in (18) and in any other truth ascription with the required modifications, is something like (28)
The same proposition/;, (p is said by Victoria and/)) . . . ,
predicated in this case of the proposition [war is always unwelcome]. That the truth predicate is a second order operator was not explicitly endorsed by Ramsey, but Williams's proposal, that also has Prior as a forerunner, is the natural development of Ramsey's intuition. Having truth as a second order operator involves acknowledging that it does not perform the role of representing the world, a logical feature that dovetails with the pragmatist view that language has a constellation of different uses. On this view the truth operator would acquire the status of quantifiers, and Ramsey's mature view on quantifiers is that they do not build sentences that express propositions. Generalized sentences are rules of action, not representations of the world. Ramsey's mature treatment of these logical operators (quantifiers and truth) is thus closer to the second Wittgenstein than the first, although he recognized Tractatus's influence that soon developed into a more flexible approach. Big points of prosententialism A symptom that a particular theory takes its object right usually is that difficulties that were previously considered insurmountable become tame or even vanish. This is what the Fregean account of quantifiers did with the paradoxes of existence, or the Russellian theory of descriptions did with the paradoxes derived from the use of non-denoting names, to mention just two examples. The notion of truth has its own paradox, the paradox of the Liar, that has attracted much attention, and even the paralysing diagnosis that the truth predicate cannot be defined in natural languages (Tarski 1935, p. 152). One of the big points of the prosentential account is that it shows why the Liar sentence is so puzzling and offers an elegant way out that emerges smoothly from the core of the theory.
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To Ramsey we owe the distinction between logical and semantic paradoxes (Ramsey 1925, 1990, p. 183). To the first ones he offers his modifications to the Russellian theory of types. To the second ones he endorses Peano's opinion to the extent that they are irrelevant to mathematics. While Peano pushes them into the realm of the linguistic, Ramsey places them in epistemology and relates them to 'faulty ideas concerning thought and language' (Ramsey op. cit., p. 184). But he never mentions the paradox of the liar when he talks about truth, either in (1927) or in (1991). On the other hand, an appropriate theory should explain the success of its rivals and being able to accommodate both commonsensical and theoretically sophisticated intuitions widely related to the topic it concerns. As regards the notion of truth there are several feelings that are difficult to overcome. One is that the truth predicate is redundant (at least in some uses), another is that truth is correspondence with facts. The prosentential theory of truth also has an answer for these intuitions. In what follows, we will see how the prosentential account explains the Liar sentence and how it accommodates the reduntantist and correspondentist challenges.
The Liar paradox The paradigmatic puzzle attached to the analysis of truth is the Liar paradox. The Liar sentence has the form (29)
This sentence is not true,
the sentence seems to predicate of itself its own falsehood and, in a bivalent setting, it will be true if, and only if, it is false and it will be false if, and only if, it is true. So the story goes. What has the prosententialist to say to this piece of philosophical common sense? First of all, it is worth remembering the Austinian dictum 'it takes two to make a truth' (1950, p. 154, n. 13). Of this, the believer in the seriousness of the paradox is well aware. The sentence says something of itself. There are two entities, the saying sentence and the sentence object of the first, but they are the same. Putting aside essential questions about how language works, such as if sentences say something as opposed to being used by somebody in context to say something, let us go to the crux of the matter. Some attempts to solve semantic paradoxes have put the blame on reflexivity for the troubles. In the particular case of the Liar sentence reflexivity has some responsibility but the prosentential account does not reject reflexivity as such. The prosententialist treatment has to do with the fact that ascriptions of truth are pro-forms and, as such, devoid of content.
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Let us analyse the issue in two steps. First there is the question of truth bearers. In Ramsey's account propositions are the primary bearers of truth, not sentences. Sentences are true or false only derivatively. So, as far as the Liar sentence is concerned, one has to ask what the sentence says, and in order to determine this a context of use needs to be provided. To ask whether, (29)
This sentence is not true,
out of any context, is true or not, is the same as asking whether (30)
I feel tired
is true or not, out of any context. Truth and falsehood are predicated of what is said by a sentence in a particular situation of use, they are predicated of a content. The content of (30) is something different from the sentence itself, it is another kind of entity, and it depends on who utters the sentence, and when, among other things. The personal pronoun needs a context to provide a content and the content in this case is the speaker. But this is an old and wellknown story. Prosententialism extends the standard treatment of pronouns to other kinds of expressions, and affirms that ascriptions of truth act as pro-sentences, that is, all-purpose sentential variables. If it is now obvious that (30), as it stands, is neither true nor false, then the same has to be said of (29). To determine the truth or falsehood of its content, it has to be previously identified. Pro-sentences inherit the content of the substantive sentences to which they refer. The grammatical subject of (29), 'This sentence', has to refer to a salient sentence in the context, and although it is a singular term from a grammatical point of view it has a proposition as its content. When this proposition is not available what we have is an empty sentence with no content and thus with no truth value. Exactly the same thing happens with (30) if, for instance, we see it written on a blackboard. A proform can inherit the contents of other expressions and for this very reason they do not possess any in their own right. They only indicate a grammatical category. A truth ascription indicates the grammatical category of sentences. (18)
What Victoria says is true,
is a complex sentential variable that endorses the content referred to by its subject, in this case, what Victoria says. If the content of Victoria's sentence is something like [Victoria's mum does not like Mondays], the content of (18) will be that Victoria's mum does not like Mondays and it will be true if, and
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only if, Victoria's mum does not like Mondays. If the content of Victoria's sentence is something like [War is always unwelcome], the content of (18) will be that war is always unwelcome, and it will be true if and only if war is always unwelcome. No point then in asking for the truth value of (18) tout court. And now for the second step. If truth ascriptions act as prepositional variables from a logical point of view, their standard translation to an artificial language will be a prepositional variable, single or complex, depending on the particular calculus. If it is the prepositional calculus, the standard translation will be a single prepositional variable, such asp, q, etc. and if it is the predicate calculus, it will depend on the inner structure of the proposition itself. It might be P(o), P(b] V P(a), Vx(P(x) -* Q»), etc. This being so, the standard translation of (29), (29)
This sentence is not true,
to, say, the propositional calculus, will be '—/>'. And to ask about its truthvalue will be as pointless as to ask about the truth value of—p. The prosentential theory of truth accepts the paradox of the Liar for what it is, a linguistic muddle, and shows why it is not a real problem for a theory of truth. Ramsey does not consider it worth treating at all although he offers clues as to what would have been his position, blaming our faulty understanding of linguistics and epistemology (Ramsey 1925, 1990, p. 184). The redundantist and correspondentist intuitions It is difficult to deny that there is some truth in the redundantist and correspondentist intuitions. And so, one simply cannot cast them off without a word. In fact, the prosentential account does not reject the background intuitions that support redundancy and correspondence as analyses of truth. On the contrary, it shows how they get things right (and why there is no need to go too far). As far as Ramsey is concerned, he saw himself as a defender of a mild version of the correspondence theory while the history of philosophy has blessed (or condemned) him as the father of redundantism. Let us begin with the redundantist view. It is a historical curiosity that the label 'The Redundancy Theory of Truth' had sailed throughout the twentieth century tied on to the name of Ramsey. Allegedly, redundantists defend that the truth predicate is eliminable from a language without loss of expressive power. If this were so, one might wonder how so useless a predicate has managed to survive in Indo-European languages (and in the rest, I suppose). The crux of the matter is to answer the following question: is the truth predicate eliminable? And, according to prosententialism, Ramsey included, the answer is: it is, as far as preforms can be. The question of whether or not
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the truth predicate can be eliminated is analogous to the question of whether or not pro-nouns can be eliminated. Pro-nouns, so we are taught by tradition, go in the place of nouns. So, it might seem that words of this kind are, in fact, laziness pro-forms, i.e., we sometimes use them for stylistic reasons and sometimes so as not to repeat a name, an adjective, a sentence, etc. that is already clear from the context. Now we know that this is not so. Anaphoric uses of pro-forms cannot be performed by other expressions of the same grammatical category (in the wide sense). Names, adjectives and ordinary sentences are not suitable for generalization or for anaphora. The truth predicate helps to build complex pro-sentences, as we have been maintaining here, and as complex expressions they are not mechanisms of direct reference. This fact might induce one to think that the pro-nouns analogy is not as close as the prosentential account would like. Nevertheless, what the prosententialist affirms is that pro-sentences are ones among proforms, and that the truth predicate indicates the concurrence of a complex pro-sentence. And to directly refer is not in general a mark of pro-forms. To refer is something that names do, and this job is taken on by pro-nouns. But it is not the task of adjectives, and so it is not taken on by pro-adjectives, nor of sentences, and so it is not taken on by pro-sentences. The truth predicate is eliminable from (31) and (32), (31) (32)
'Snow is white' is true, That Spain is a Kingdom is true
though not from (33) or (34), (33) (34)
Everything the Pope says is true The Big Bang theory is true.
Thus, it is eliminable from exhibitive truth ascriptions but not from blind ones. Christopher Williams (1995) has an illuminating explanation of why in Tarski-like sentences the truth predicate seems to be redundant. Quotation marks have in some contexts the effect of converting the expression that lies in between plus the marks themselves into a singular term from a grammatical point of view. The same function is fulfilled by the particle 'that' placed at the beginning of a sentence, rendering the whole into a single term. On the other hand, 'is true' is a sentence-builder. If what is required to guarantee grammaticality is a sentence-like expression, the dummy predicate 'is true' can be used to restore the category of being a sentence. This is what happens in (19'). Borrowing set theoretical terminology one might say that truth is the converse of quotation marks and 'that'. Putting both operations together neutralizes them. But this does not make any of them dispensable. In Williams's words:
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If you regard 'is true' and 'that' as operators, the one is seen to be the converse of the other. They are related, just as 'the double of is related to 'the half of. It is easy to see what happens if you apply them in succession to a string of words. We are not surprised if we think of a number, say seven, attach to it the phrase 'the double of, and to the result 'the double of seven', attach the phrase 'the half of, only to find that what we have at the end of it all, 'the half of the double of seven', is nothing other than what we first thought of, namely, seven. Nor should we be surprised if, when we use the word 'that' to convert 'Snow is white' into its own designation, 'that snow is white', and then append the words 'is true', we end up with something that is worth no more than the sentence we started with. (1995: p. 148) What about Ramsey's historical position? It leaves no room for doubt. As we have already seen in 'Ramsey's Theory of Truth' above, Ramsey acknowledges that truth expressions are sometimes used for emphasis or stylistic reasons, i.e. that in some cases truth expressions do not perform an essential role. But he denies that this is always so, and particularly not when the proposition whose truth is affirmed is merely described and not explicitly given (Ramsey 1927, 1990, pp. 38—9). In these cases, he says, 'we get statements from which we cannot in ordinary language eliminate the words "true" and"false"' (my emphasis, loc. cit., p. 39). It is to be expected that after these few indications the connection between Ramsey and the redundancy theory of truth is severed. Ramsey never argued for the eliminability of the truth predicate, and the same can be said of any other prosententialist for whom truth is a valuable maker of (complex) prepositional variables. And now let us consider the correspondentist intuition. Contrary to what happens as regards the redundantist intuition, Ramsey always considered himself a proponent of a weak version of a correspondence theory of truth. Thus, it would have been historically more justifiable to have attached the name of Ramsey to the fate of correspondentism rather than to that of redundantism. In 'The Nature of Truth' (1991, p. 11 and ff.) he guesses that his position will be interpreted as a sort of correspondence theory. In this case, he warns against the well-worn criticisms usually aimed at correspondence. His view, he accepts, might be interpreted as a sort of correspondence theory, though without the problems that arise from defining truth directly in terms of a relation between two poles. Being aware of the philosophical difficulties with which the correspondence theory of truth is beset, Ramsey says: 'But the prospect of these difficulties need not distress us or lead us to suppose that we are on a wrong track in adopting what is, in a vague sense, a correspondence theory of truth. For we have given a clear definition of truth which escapes all
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these difficulties by not appealing to a notion of correspondence at all' (1991, pp. 11—12). An explicit endorsement of this correspondence theory 'in a vague sense' is made in the Introduction (1991, p. 3) where he says: 'Truth is an attribute of opinions, statements, or propositions; what exactly it means we shall discuss later, but in a preliminary way we can explain it in accordance with fact.' Being faithful to the common usage of words, no reasonable proposal about the meaning of'is true' can disdain the feeling that the sentence 'Snow is white' expresses (in appropriate contexts) a true proposition because snow is, in fact, white. This is basically the correspondentist intuition. And, as it is so reasonable, it should be accounted for by any acceptable theory of truth. The prosentential account does not reject the feeling but rather takes it on board and offers a detailed explanation of its force from a logical point of view. It is also a historical curiosity that the first time that the word 'prosentence' appeared in print was in Prior's article 'Correspondence Theories of Truth' written for the Encyclopaedia of Philosophy, edited by Paul Edwards (1967, p. 229). If truth is a second order predicate which is an instance of the identity operator, the logical role of truth is to mark the repetition of a propositional argument, in Williams's sophisticated version. In Grover's explanation the truth predicate helps to construe prosentences that inherit the content of other propositions. In both cases, and in any prosentential stance, 'it takes two to make a truth', using the felicitious expression of Austin (1950, p. 154, n. 13), a renowned correspondentist. If a prosentence, i.e. a propositional variable, does not possess a content in itself but, referring to something (a sentence, a proposition, a fact, a belief, etc.) prominent by the context or else (in the quantified case) it has instances that are genuine propositions, truth ascriptions require the two items to be well formed. The first item is a genuine proposition endorsed, considered, or merely entertained by somebody, the second item is the level of the prosentence at which the genuine proposition is referred to by means of a proform that in the appropriate context inherits its content. This is the sense in which the correspondentist intuition is assumed by the prosentential account. What is inaccurate in some versions of the correspondence theory is the logical category given to the truth operator. The grammatical predicate 'is true' does not express a property of things, i.e. it is not a first order predicate, and so the left side of Tarskian sentences does not have a subject-predicate logical form. The kind of thing that we do when we say that the table in front of me is made of wood is not the kind of thing that we do when we say that the Big Bang theory is true. In prosentential theories the truth predicate is a logical operator, a second or higher order operator, whose arguments are logical entities: propositions or predicates. Thus it might be counted among the logical constants: connectives, quantifiers, identity (in Williams's account), the belief operator, all of which are second or higher order intralinguistic functions.
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Neither is the truth predicate a first order binary relation between language (or mind) and the world, as is typically maintained by many correspondentists. The history of philosophy has displayed the myriad difficulties bound to the task of offering a worked-out treatment of the relation itself and its two relata. Ramsey refers to these difficulties in the text quoted above when placing his definition apart from the set of correspondence theories. Fortunately, there is no need to answer these difficulties, because they vanish as soon as the logical category of the truth predicate is recognized, and with it, a logically correct account of how it works is provided. Let us end this section with two texts of Ramsey's about correspondence, both from the 'Appendix to Chapter 1' (1991): Indeed of the three leading types of theory, the Correspondence Theory, the Coherence Theory and Pragmatism, only the first agrees with us on the main issue that a belief that A is B is true if and only if A is B, and our view belongs undoubtedly to the class of correspondence theories, although we have not yet used the word correspondence, (op. cit., p. 18) And on the next page he says: [TJhis talk of correspondence, though legitimate and convenient for some purposes, gives, in my opinion, not an analysis of truth but a cumbrous periphrasis, which is misleading to take for an analysis. To believe truly is to believe that p and p., and there is no need [but many disadvantages in restating] to recast this definition in terms of correspondence ... (op. cit., p. 19)
Conclusion Ramsey's treatment of truth has been adopted several times during the twentieth century, independently of Ramsey's work. The paper in which Ramsey introduces the term 'prosentence' was not published until 1991 and his (1927) 'Facts and Propositions' has been interpreted as a defence of a redundancy theory of truth. But prosententialism has been rescued by the brightest minds of the last decades and explicitly developed and endorsed by Grover, Camp and Belnap and also by Williams. Recognizable prosententialist accounts are found in Prior and Strawson, and with a higher or lower degree of accuracy have been pointed at by Tarski and Austin, and now by Horwich, not to mention Aristotle. The view is not completely worked out in Ramsey's writings but all that Ramsey says about the nature of truth dovetails perfectly into any prosentential setting.
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To conclude, I would like to remark again that Ramsey never dismissed the truth predicate as redundant. On the contrary, he saw himself as a defender of the correspondentist intuition and a follower of Aristotle. He offered a correct account of how the predicate 'is true' works in natural languages and traced its connection with the prepositional variables of artificial calculi. He also introduced for the first time the word 'pro-sentence' and analysed the truth predicate taking advantage of the analogy with pronouns. His position is thus a prosentential account, only drawn into the philosophical arena around 1976, independently of Ramsey's writings and fifty years after it was handled by the genius of Cambridge.
Endnotes I am indebted to my colleagues J. J. Acero, M. J. Garcia-Encinas, Manuel de Pinedo and Neftali Villanueva for illuminating comments on earlier drafts of this chapter.
References Austin, J. (1950), 'Truth'. In Blackburn, Simon and Simmons, Keith (eds): Truth. Oxford: Oxford University Press, 1999, pp. 149—62. Dokic, J. and Engel, P. (2002), Frank Ramsey: Truth and Success. London: Routledge. Edwards, P. (ed.) (1967), The Encyclopaedia of Philosophy. London and New York, Collier Macmillan. Grover, Camp and Belnap (1975), 'A prosentential theory of truth'. Philosophical Studies, vol. 27, pp. 73-125. Also in Grover (1992). Grover, D. (1992), A Prosentential Theory of Truth. Princeton, NJ: Princeton University Press. Horwich, P. (1998), Truth. Oxford: Clarendon Press. Kneale, W. and Kneale, M. (1962), The Development of Logic. Oxford, Clarendon Press. Loar,B. (1980), 'Ramsey's theory of belief and truth'. InMellor (1980). Mellor, D. H. (1980), Prospectfor Pragmatism. Cambridge: Cambridge University Press. (ed.) (1990), Philosophical Papers: F. P. Ramsey. Cambridge: Cambridge University Press. Ramsey, F. (1925), 'The Foundations of Mathematics'. In Mellor (1990), pp. 164-224. (1927),'Facts and Propositions'. In Mellor (ed.) (1990), pp. 34-51. (1991), 'The Nature of Truth'. In Rescher and Majer (eds) (1991), pp. 8-20. Recanati, F. (2000), Oratio Obliqua, Oratio Recta. An Essay on Metarepresentation. Cambridge, MA and London: The MIT Press.
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Rescher, N. and Majer, U. (eds) (1991), On Truth: Original Manuscript Materials (19271929) from the Ramsey Collection at the University of Pittsburgh. Dordrecht: Kluwer Academic Publishers. Russell, B. (1910), 'The Nature of Truth and Falsehood'. In Russell (1992), pp. 115-24. (1992), Logical and Philosophical Papers 1909—13. London and New York: Routledge. Sahling, N.-E. (1990), The Philosophy ofF. P. Ramsey. Cambridge: Cambridge University Press. Tarski, A. (1935): 'The concept of truth in formalized languages'. In Logic, Semantics, Metamathematics, Indianapolis: Hackett Publishing Company, 1983. Williams, C.J. F. (1976), What Is Truth? Cambridge: Cambridge University Press. (1989), What Is Identity? Oxford: Clarendon Press. (1992), Being, Identity and Truth. Oxford: Oxford University Press. (1995): 'The Prosentential Theory of Truth'. Reports on Philosophy, No. 15, pp. 147-54.
6
Ramsey's big idea1 Colin Hows on
The theory of probability is in fact a generalisation of formal logic Frank Plumpton Ramsey, 'Truth and Probability'
Introduction Frank Ramsey was a prodigious talent. Dying in his 27th year, he bequeathed the world a host of seminal work, in mathematics, philosophy, logic and decision theory, much of whose significance became appreciated only many years after his death. Had he survived longer, he might well have been a British John von Neumann; as it was, his contributions are fundamental and groundbreaking. I want to look at just one of them, his famous paper 'Truth and Probability', which despite its informality is now generally acknowledged to be a revolutionary contribution to philosophy and science. In it, Ramsey singlehandedly reinvented the theory of utility as an application of the mathematical theory of measurement, which is how it has been understood ever since, and he was also the first to show how epistemic probability can be located within the same theoretical milieu of utility, where for most people it has remained ever since. He also anticipated de Finetti's celebrated justification of the probability axioms in terms of invulnerability to what is called a Dutch Book. As if this was not enough, Ramsey advocated an entirely novel view of the laws of probability as logical consistency constraints. To appreciate the significance of these moves they need to be placed in a wider, and longer, perspective. Ramsey's primary preoccupation in the 'Truth and Probability' paper was probability, and in particular epistemic, or what is now more usually called Bqyesian, probability. Epistemic probability had been the subject of keen discussion for well over two centuries before Ramsey wrote. Its importance lay in supplementing the theory of deductive inference to provide a general theory of logically valid reasoning comprising both certain (deductive) and uncertain (probabilistic) inference. We already have the key elements of the discussion: logic, valid inference, probability. While no wholly satisfactory (by today's standards) account of deductive logic existed in the seventeenth, eighteenth and even most of the nineteenth
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centuries, its nature was nevertheless held to be relatively uncontroversial. Although very restricted, Aristotelian syllogistic had by 1800 already enjoyed a relatively undisputed reign for two millennia, which continued until it was finally superseded at the beginning of the twentieth century. Probability was a different matter: it was a new discipline, emerging famously from a gaming problem 'pose a un austere janseniste par un homme de monde' (Poisson) in the 1660s. Mathematically and conceptually it quickly grew away from its mundane origins, and already by the later years of the century Leibniz predicted that it would become 'a new kind of logic' (Nouveaux Essais), not like deductive logic, guaranteeing the truth of a conclusion from the truth of premises, but showing how data can in suitable circumstances endow non-deductive inferences with degrees of certainty. Aristotelian syllogistic, as Bacon had famously pointed out, was in no way a logic of factual discovery, being able to tell us only what, in a sense, we know already in the premises. That this sort of limitation did not apply to the new logic of probability, on the other hand, was immediately apparent to Leibniz (and also to the authors of the Port Royal Logic, Book IV of which concerns probable reasoning; the title in the original Latin — Ars Cogitandi - mimics that of Bernoulli's monograph, and the work was still used in Oxford in the nineteenth century). When suitably articulated, it should be able to tell us to what extent we were entitled to place confidence in all sorts of claims that deductively transcend our immediate information: for example, about innocence and guilt in the light of testimony, and, more dramatically, about the likely truth-values of predictive hypotheses about the future course of events based on evidence from the past. It was, therefore, of potentially enormously great practical significance for science, law, and decision-making in general. James Bernoulli summed up the contemporary view (to a great extent that of Leibniz himself, with whom Bernoulli maintained an extensive philosophical correspondence) in his manifesto-like Part IV of the Ars Conjectandi: To conjecture about something is to measure its probability; and therefore, the art of conjecturing or the stochastic art is denned by us as measuring as exactly as possible the probabilities of things with this end in mind: that in our decisions or actions we may be able always to choose or follow what has been perceived as being superior, more advantageous, safer, or better considered; in this alone lies all the wisdom of the philosopher and all the discretion of the statesman. (Chapter II, p. 13) It was not until the second half of the eighteenth century, however, that the promise became realized in an important scientific advance, and one relating to some unfinished business from the Ars Conjectandi. There James Bernoulli had proved that, subject to the so-called iid conditions (independence and
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identical distribution from trial to trial), the probability that the difference between a fixed probability of 'success' and the sample average s is less than some preassigned magnitude tends to 0 as n, the number of trials, tends to infinity. For many people, including Bernoulli himself, who had been looking for a way of determining probabilities a posteriori, this did not quite do the job: in today's jargon, Bernoulli made x the parameter and s the random variable whose probability distribution depends on x and n. What was needed, or so it seemed to many, was to 'invert' Bernoulli's result, i.e. to give a formula for the probability that x, itself considered as a random variable, should lie within any given bounds given the data s. This is what the English clergyman and mathematician Thomas Bayes succeeded in doing. In a posthumously published Memoir (1763), he gave a posterior probability distribution for x as a function of s and n. However, to obtain his result Bayes found that he had to use a uniform, or constant, prior probability distribution over the range of values x, i.e. the closed unit interval. It seems that Bayes himself was not entirely happy about the justification for this; his discussion occupies a Scholium, in which he appears to advance something like the principle that if your background information is neutral between the possible values of x, then that neutrality should take the form of a uniform prior density. Bayes's own appearance of tentativeness in adopting this principle (if indeed he did adopt it; commentators are divided on this point), later called by Keynes the Principle of Indifference, seemed subsequently vindicated by a disturbing consequence of the Principle, albeit one relatively slow in coming to light. When it did, however, it seemed increasingly to cast doubt on the entire Bayesian methodology. The consequence is this. If you know nothing about x, for 0 < x < 1, then you presumably know nothing about f(x), where^is any continuous invertible tranformation of x, e.g. x2. But x and x2 cannot both be uniformly distributed. It seems to follow that no probabilistic notion of complete epistemic neutrality can be well-defined. Not only that: x's possible values conveniently form a bounded interval. But there are other statistical parameters of equal if not more interest which do not, and for these no uniform distribution is possible which obeys the laws of probability. Yet Bayes's theorem tells us that some prior distribution must be assumed if a posterior probability is to exist. These consequences were slow to become appreciated, however, and after Laplace had rediscovered Bayes's result and adopted the Principle of Indifference with none of Bayes's misgivings (he used it famously to define probability as the ratio of the number of favourable to the number of all possible outcomes), the theory of posterior distributions based wherever possible on uniform prior distributions became the dominant methodology for inductive inference. Nevertheless, by the early years of the twentieth century the objectionable features alluded to, particularly the vulnerability of the Principle of
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Indifference to transformational problems, were regarded as sufficiently disturbing to suggest that something might be fundamentally wrong. We have reached 1921, the year of publication ofj. M. Keynes's Treatise on Probability. This book was the inspiration of Ramsey's paper, 'Truth and Probability', which starts with a highly critical review of the main ideas of the Treatise, and uses these reflections to motivate Ramsey's own, very different theory. In the Treatise, Keynes had advanced an explicitly logical theory of probability (from now on, whenever I use the word 'probability' it is the epistemic interpretation I am referring to). Keynes believed that all ascriptions of probability are conditional, implicitly referring to a conjunction of propositions, q say, expressing the total of the knowledge the agent possesses by direct acquaintance, on which the conjectured proposition,/) say, whose probability is sought is conditioned. Not only are these probabilities conditional: according to Keynes any such conditional probability relation P(p \ q) = r, denotes a logical relation of partial entailment between q and/?, where the strength of the entailment is expressed by the real number r. In the limit, where q deductively entails/?, we have of course P(p \ q} = 1, while if q is deductively inconsistent with/?, P(p \ q) — 0. All other probabilities are confined within these two limits (where they are defined: Keynes held that probabilities could be expressed by real numbers only in quite specific circumstances; I shall come to this shortly). Keynes also subscribed to the view, common at the time, that logic was a system of canons of rational thought, and that its relations correspondingly prescribed how the rational person should reason. Accordingly, he held that, being logical, P(p \ q} determines the numerical degree of rational belief which the agent, knowing only q, should entertain in/?. This was not the only traditional motif of the Treatise: there was another. I remarked that for Keynes not all probabilities could be expressed numerically. Clearly, in the extreme cases of deductive entailment, numerical values of 1 and 0 can, and indeed must, be assigned. For all other, non-deductively based, numerical assignments Keynes believed the Principle of Indifference was indispensable: In order that numerical measurements may be possible, we must be given a number of equally probable alternatives ... The Principle of Indifference asserts that if there is no known reason for predicating of our subject one rather than another of several alternatives, then relatively to such knowledge the assertions of each of these alternatives have an equal probability. (1921,pp.41,42) Thus, if there are n such alternatives, the probability axioms determine that each receives the probability jk First, however, the Principle had to be sanitized. Keynes himself listed the main inconsistencies the Principle was known to generate: the specific density/
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specific volume paradox, Bertrand's paradox and others, and to these added some of his own, like the 'book paradox' (1921, pp. 46-7). Keynes believed that these anomalous results stemmed from a failure to partition the class of possibilities admitted by q into 'indivisible alternatives' relative to p (1921, p. 60). This admittedly seems to answer the difficulty of the 'paradox' generated by applying the Principle simultaneously to the partitions (red, not red), (green, not green), and (blue, not blue}, generating the inconsistent assignment P(red) =/ > (green) =P(blue) = | and hence P(red or green or blue) | (this is Keynes's book paradox, the colours being the possible colours of the cover of a book), since 'not green', 'not red' and 'not blue' are composite; when they are analysed into disjunctions of primary-colour primitives the inconsistency disappears. Keynes's proposal as it stands does not seem capable of solving the 'geometric' paradoxes, like Bertrand's, however, which involve uncountable sets of'indivisible' points; assigning each of these probability zero does not determine the distribution over non-degenerate intervals. Keynes's treatment of this problem was to regard the continuous case as a passage to the limit of a discrete one, dividing a bounded interval up into n equal subintervals, and regarding all of these as equally likely, for all finite values of n. Unfortunately, as we have seen, two continuous variables x and 7 of bounded variation may be related by a non-linear one-to-one continuous bijection/Tor which equal subintervals of x do not transform into equal intervals ofjy. But the fundamental problem with Keynes' reconstruction of the Principle of Indifference afflicts the discrete, and even finite, case as much as it does the continuous: it is that a state indivisible relative to one classification scheme need not be with respect to another. We can see this clearly with the help of a simple example. Suppose L\ and L% are first order languages with identity, each having one one-place predicate symbol Q. Suppose also that LI has no individual names and L% has 2. Letj^ be the sentence 3#Q,(#) and q the sentence 3x3y(x ^y & Vz(z = xvz =J>)) for some choice of variables x, y and z\ i.e. p states that at least one individual has Q^and q that there are exactly two individuals. Note that p and q are common to both languages. However, if as before we take P(p \ q} to measure the proportion of q's models which are models of p, then identifying isomorphic models we find that with respect to L\, P(p \ q) = |, while with respect to L% P(p \ q} = | (on the simplifying assumption that individuals with distinct names are distinct). It might be objected that one should always adopt the language capable of the finest-grained distinctions, in this case allegedly LI. There are two things wrong with this answer. First, it begs the question: why 'should' one? - logic by itself cannot justify any such preference. Second, it presupposes that L% necessarily makes finer distinctions than the one without, and this — surprisingly perhaps — is not true. Paired bosons, for example, are indistinguishable
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according to quantum mechanics (and their quantum statistics match those for L\ (Sudbery 1986, pp. 70-4)), as are, less exotically, pounds in bank accounts: it makes no sense to say that there are four distinct ways in which two pounds can be distributed between two accounts. We are now ready to turn to Ramsey's paper, which takes as its point of departure an extended critical discussion of Keynes's theory, and ends with Ramsey proposing a radically different theory of logical probability, in which the Principle of Indifference plays no role whatever.
'Truth and Probability' Ramsey commences with a destructive criticism of Keynes's idea that there are logical relations of probability between pairs of sentences, perceivable, according to Keynes, by a faculty of intuition better or worse developed depending on the individual. Ramsey's objections to these claims are presented in a much-quoted passage: let us now return to a more fundamental criticism of Mr Keynes' views, which is the obvious one that there really do not seem to be any such things as the probability relations he describes. He supposes that, at any rate in certain cases, they can be perceived; but speaking for myself I feel confident that this is not true. I do not perceive them, and if I am to be persuaded that they exist it must be by argument; moreover I shrewdly suspect that others do not perceive them either, because they are able to come to so very little agreement as to which of them relates any two given propositions. All we appear to know about them are certain general propositions, the laws of addition and multiplication; it is as if everyone knew the laws of geometry but no-one could tell whether any given object were round or square; and I find it hard to imagine how so large a body of general knowledge can be combined with so slender a stock of particular facts. (1926, pp. 161, 162) This is slightly unfair, though Keynes must take much of the blame for emphasizing the role of perception. In fact, perception plays a negligible role in his theory as he goes on to present it: as we have seen, he invokes the Principle of Indifference to provide numerical values where he thinks they can be provided, and the condition that he lays down for its application, flawed though it is, does not depend on powers of perception, for it is a purely syntactic one. Surprisingly, Ramsey does not stress the well-known paradoxes afflicting that principle as a major objection to Keynes's theory; yet the fact is that Keynes's theory rests squarely on the Principle.
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Ramsey's belief that there are no logical relations between arbitrary pairs of sentences nevertheless seems well-founded. Perhaps surprisingly, he still believed that there was an authentically logical interpretation of the laws of probability: they were, he argued in the later, constructive, part of his paper, not laws of partial entailment but consistency constraints on the distribution of partial belief. This was a completely novel departure from the orthodox view that the logical nature of probability (assuming there was one) consisted in its being a sort of generalized deduction. In the remainder of this chapter I shall sympathetically examine Ramsey's claim — what I have called in the title of this chapter his Big Idea - and make what I believe is a constructive amendment to it.
Partial Belief Though Ramsey repudiated the substance of Keynes's theory, he did not reject all of it, and the part that he did not, he built on in a quite different way to Keynes. Recall that Keynes had characterized probability as a degree of rational belief, and a degree of belief, rational or otherwise, is a partial belief. Ramsey accepted Keynes's idea that epistemic probability described states of partial belief, and that in some sense probability is logical. He departed from Keynes, first, by regarding the partial belief in question as that of an actual human reasoner, like you or me, and not as that of some ideal reasoner; and second, by taking the logical aspect of probability to consist not in the identification of some logical relation between arbitrary pairs of sentences, but in representing a standard of logical consistency for individuals' degrees of beliefs. Such a project, he remarked, must start by investigating the nature of partial belief itself, and its measurement: The subject of our enquiry is the logic of partial belief, and I do not think we can carry it far unless we have at least an approximate notion of what partial belief is, and how, if at all, it can be measured. ... We must therefore try to develop a purely psychological method of measuring belief. (1926, p. 166) The gap between Ramsey's and Keynes's theories has now become a gulf. Probability relations for Keynes were necessarily those of conditional probability and are logical in nature; for Ramsey degrees of belief can be either conditional or unconditional, and are psychological. Ramsey's psychology was famously behaviouristic, at any rate as far as belief is concerned. He takes belief to be an empirically determinable quality capable of measurement, and considers two possible methods. The first is
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introspection leading to reports of intensities of belief in various propositions. This he dismisses rather summarily, on the ground that The beliefs which we hold most strongly are often accompanied by practically no feeling at all; no one feels strongly about things which he takes for granted. (1926, p. 169) But this is knocking down a straw-man. Ramsey's attack merely exploits a pun, between feeling in the sense of some emotional surge, and feeling in the sense of the report of an attitude. I believe that by introspection I can accurately report a feeling of complete certainty about something, but I do not mean by this anything except that I have recorded a definitive judgement to that effect, in the same way that I might record a definitive judgement that something is red. But behaviourism, and a corresponding reluctance to take reports of mental events seriously, were strongly the fashion when Ramsey wrote, and it is to a behaviouristic alternative that he turns, where strength of belief is identified via the actions to which it gives rise: We are driven therefore to the second supposition that the degree of a belief is a causal property of it, which we can express vaguely as the extent to which we are prepared to act on it. (ibid.} The rest of his discussion is concerned with refining this idea to the point where a satisfactory method of measurement becomes possible. First, Ramsey points out that he is considering what he calls dispositional beliefs, that is to say beliefs which are not necessarily conscious when planning actions which would be guided by those beliefs in the relevant circumstances: for example, his own belief that the earth is round (p. 172). He considers 'the old-established way' of measuring such beliefs, which is to propose a bet to the individual whose beliefs are being assessed, and determining the lowest odds that they will accept. While such a method will eventually become the basis of Ramsey's theory of belief-measurement, the proposal as it stands will not do. First, as Ramsey notes, people may be unwilling to bet in any case, and second, even if they are, there is the problem of the diminishing marginal utility of money: the amount any normal person would be willing to forfeit in return for $x if a proposition^ is true and SO if not will not increase as a fixed proportion of x, but exhibit a flattening-ofF effect as x grows larger. Ramsey's strategy for circumventing these problems is now usually regarded as inaugurating the modern theory of rational decision. It was to measure gains and losses not in terms of monetary rewards, but in terms of their value to the agent, or as we now call it, their utility. In one sense, utility was nothing new. In the mid-eighteenth century, the celebrated St Petersburg
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Problem was becoming increasingly widely regarded as showing that monetary expected values were an unreliable criterion of probability (the St Petersburg problem concerns the pricing of an offer to pay 2" ducats if the first toss of a repeatedly tossed fair coin occurs at the nth throw; it is easy to see that the expected value of the game is infinite, but clearly it would be foolish to pay even a moderate sum for it). In his highly influential discussion of the problem (1738), Daniel Bernoulli had proposed measuring utilities proportionally to the logarithm of the monetary rewards: the logarithm has the required flattening property, and the particular measure that Bernoulli recommended seemed to give intuitively acceptable results. But there were problems. One is that a logarithmic measure does not behave well at extreme values, and in effect merely replicates the original problem: suppose that instead of paying 2" ducats the St Petersburg game pays that amount raised to the power 2. The expected value in Bernoulli's utility is again infinite (this particular difficulty was pointed out by Bernoulli's contemporary, Cramer). The problem is, of course, that the logarithm is unbounded. It is not difficult, mathematically, to define a bounded utility function, but there are more fundamental problems with this theory: in particular, its very limitation to actions under uncertainty whose outcomes are purely monetary, and the lack of any general account of probability outside that employed in classical games of chance, which was based on the Principle of Indifference. There was no combined theory of utility and probability, in other words, which could furnish the basis of a completely general theory of choice in conditions of uncertainty. So matters stood until Ramsey's paper revolutionized the subject by showing that both a numerical measure of utility and a corresponding probabilistic measure of uncertainty were, in a sense which can be made precise, merely artifacts of a suitably constrained preference ranking. We have seen that Ramsey regarded beliefs as manifested in behavioural display, of which indicating preferences among uncertain options like bets is a type. The next move in his paper was to show, by means of a so-called representation theorem, that, suitably constrained, the preference-ordering uniquely determines an almost-unique numerical measure of utility, from which a probabilistic measure of degree of belief can be constructed in terms of utility-valued odds. The following is a sketch of how he achieves this result.
Ramsey's System Ramsey's system contains two primitive relations, of binary indifference and binary preference, the latter denoted by <, defined on the class of worlds together with conditional options of the form [7 if/?], [a ifp, (3 if not-/?], [8 if p&q, K if (not-p}&.q, X ifnot-q] etc., where/), q, ... are any propositions. Any
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world a can in principle be assigned as the 'reward' for the truth ofp independently of any considerations of causal plausibility (a fact which has made some commentators critical of Ramsey's theory). < is supposed to be a strict ordering. One further element is required in order for him to define the idea central to his theory of utility, that of the difference in value between one pair (a, /?) of worlds being the same as that between another, (7,8), and symbolized aj3 = 7#. This further element is that of an ethically neutral proposition believed to degree ^. An ethically neutral proposition is one, roughly, between whose intrinsic truth and falsity the agent is completely indifferent. The existence of such propositions might be thought to be a moot point, though it is certainly possible to approximate them: a statement describing the outcome of the toss of a coin, considered independently of any possible consequences, is one example. The notion of an ethically neutral proposition is thus an idealization, but it is just one of many in Ramsey's theory: the assumed total preference ordering, the axioms which ensure that there are enough things in its domain as there are real numbers, with a similar continuity structure, and so forth, are all very considerable idealizations; even the psychology on which Ramsey's theory of belief-measurement rests he acknowledges to be false: I mean the theory that we act in the way we think most likely to realise the objects of our desires, so that a person's actions are completely determined by his desires and opinions. This theory cannot be made adequate to all the facts, but it seems to me a useful approximation to the truth particularly in the case of our self-conscious or professional life, and it is presupposed in a great deal of our thought, (p. 173) Ramsey's relaxed attitude to mathematical idealization stemmed from a view of it as basic to the advanced sciences, and 'if it is allowable in physics it is allowable in psychology also' (p. 168). The view is a plausible one, but this is not the place to discuss it at length; let us simply assume that there is an approximately ethically neutral proposition, call it p, and see what it means to say that/; is believed to degree ^. According to Ramsey, p is believed to degree ^ by a subject if they are indifferent between [a ifp, (3 if not-/>] and [/5 ifp, a if not-/?], but have a preference between /3 and a, for all such pairs (a, j3). We can now define the difference in value between worlds a and j3 to be equal to that between 7 and 8, just in case the subject is indifferent between the options [a ifp, 8 if not-/>] and [/3 if p, 7 if not-/?], where p is any ethically neutral proposition believed to degree |. Ramsey claims that this 'roughly' expresses the familiar idea that an agent is prepared to take either side of a bet, at the same stakes, on a proposition which they believe to be as likely as not. It expresses this idea only roughly, however,
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because it is subject to a very important qualification: that the stakes are measured in terms of value, with the measure obeying the expected utility principle. The latter is actually implicit in the definition above: writing a — (3 for a(3, etc., we infer from a(3 = 78 that a + j3 = 7 + 8 and hence that ka + kf3 = £7 + kS for any constant k. Conversely, equating the values of the two options above according to expected value gives (^)a + (|)/3 + (7^)7 + (|)<5. We shall see shortly that the expected utility principle is also presupposed in Ramsey's defintion of degree of belief for arbitrary propositions. Ramsey now lays down seven axioms constraining equality of value-difference between pairs of worlds (in fact Ramsey has eight axioms, but axiom 1 merely states that there exists an ethically neutral proposition believed to degree ^). These suffice to construct a numerical scale for measuring value; more specifically, they allow equal differences of value in Ramsey's to be faithfully represented by the differences between correlated real numbers: These axioms enable the values to be correlated one-one with real numbers so that if a1 corresponds to a, etc. a/3 = 7<S = if a1 - /51 = 71 - 61
(1926, p. 179) Writing a1 as/(a), it follows immediately that the representation is unique up to positive affine transformations, i.e. transformations of the form of + b where a > 0, which in turn implies that iff and/' are any two 'correlating' functions then
f
/(a)
-/(/?) =./»
/(7)
-/(*)( /'(7)
-/(/?) -f(S)
In words, the representation determines an interval scale of measurement. The use of representation theorems like this to show that real numbers under suitable arithmetical operations faithfully represent a rich enough algebraic structure was completely new to discussions of probability at the time Ramsey wrote, though it was already an established focus of mathematical research, eventually to become the discipline called measurement theory. The celebrated mathematician David Hilbert proved the first important representation theorem, showing that suitable qualitative axioms for geometry determine an algebraic structure homomorphic to the real numbers under corresponding arithmetical operations (1899). Ramsey never gave a formal proof of his representation theorem, but as Bradley, who seems to have been the first to notice it, points out (2001), a very similar result had been stated and proved in an abstract setting twenty years earlier by the German mathematician Holder (Ramsey read the continental mathematical journals and
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Bradley conjectures, plausibly, that he would have been aware of Holder's work). Holder had axiomatized, a la Hilbert, the notion of an algebraic difference structure and proved a corresponding representation theorem. An algebraic difference structure is an ordered set (A x A, <) such that (1) (2) (3) (4) (5)
< is a complete, reflexive and transitive relation on A. (a,b) < (c,d) => (d,c) < (b,a), for alia, b, c, din A. (a, b) < (a', b') and (b, c) < (b1, c'} =*> (a, c) < (a', c1} for all a, b, c in A. Fora.lla,b,cinA,(a,b}<(b,c}<(b,a)=>3x,x!inA(a,x} = (c,d) = (x',b). Only finitely many equal subintervals are contained in a bounded interval (this is called an Archimedean condition).
Holder's representation theorem states that there exists a real-valued function /on ^4 such that for all a, b, c, din A, (a,b) < (c,d] =>f(a} -f(b)
which we have to assume is independent of any particular a, ft, 7 (1926, p. 179). That granted, we know from the representation theorem that the quotient is independent of the particular u chosen, and so we have an invariant measure of partial belief. Note that the belief so measured is strictly partial: the definition applies only to degrees of belief short of certainty, for as Ramsey notes, if you are certain ofp then you will be indifferent between a and [a ifp, j3 if not-p], for every (3 .
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It is not difficult to show, on the basis of Ramsey's axioms, that the quotient above determines an unconditional probability function, call it P. Furthermore, P(p) so defined amounts roughly to defining the degree of belief in p by the odds at which the subject would bet onp, the bet being conducted in terms of differences of value as defined. (1926, p. 180) To see why, suppose that a, /3,7 are small money sums, which can be assumed to be proportional to utility (Ramsey makes something like this assumption in his proof of the multiplication theorem), with 7 < a < {3. Then indifferenc between a and [(3 if p, 7 if not-/?] signifies a willingness to exchange a for that conditional option, with a net payoff of (3 — a. ifp is true and 7 — 0; ifp is false. This means that the subject is in effect willing to bet onp at odds a — 7: (3 — a (since 7 — a is gained ifp is false, minus that quantity, i.e. a — 7, is given up: the sign signifies the direction of the transfer). Odds, which vary between 0 and infinity, are normalized (to give what is called the betting quotient) by the transformation odds/(l +odds); if we do this then we obtain the normalized odds a — 7: J3 — 7, i.e. just the quantity Ramsey uses to define the subject's degree of belief in p, rendered in terms of the utility, or value, measure u. It should be clear that Ramsey's use of a betting analogy does not stop his theory being of quite general application; though as he also points out, any choice among uncertain options can be regarded as a type of bet (1926, p. 183). But there is a more fundamental relationship implicit in the definition ofP(p), one we are already familiar with, and which is revealed by working backwards. Suppose that the mapping u has been extended to the full domain of the preference-indifference relation, so that it is also defined on all conditional options as well as worlds. Then indifference between a and [(3 ifp, 7 if not-p] is equivalent tou(a) = u([/5 ifp, 7 if not-/?]). By Ramsey's definition, we have P(p) = (u(a) — #(7)) -r (u(j3 ) — "(7)). Rearranging terms and cancelling, we quickly find that u([fl if p, 7 if not-p] ^P(p}u(j3 ) + P (not-p) u(j); i.e. we have the expected utility principle again. Its role is absolutely fundamental in Ramsey's system, as he explicitly acknowledges: It should be remembered, in judging my system, that in it value is actually defined by means of mathematical expectation in the case of beliefs of degree |, and so may be expected to be scaled suitably for the valid application of the mathematical expectation in the case of other degrees of belief also. (1926, p. 183) The degree of belief function Ramsey defined above is an unconditional one, but he also shows how to define 'a very useful new idea' (p. 180), that of a
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conditional degree of belief in p given q, as follows. Suppose the subject is indifferent between the options (1) [a if q, ft if not-q] and (2) [7 if p&q, 8 if (not-p}&q, (3 if not-*/]. Then their degree of belief in p given q is defined to be equal to
u(a) — u(8] u(i) - u(8] where again this is assumed to be independent of the particular choice of a, {3, 7 and 8. Call this quotient P(p \ q} (this is not Ramsey's notation). Assuming the admissible utility functions extended to conditional options, we can equate the utilities (the choice of representative function we know to be unimportant) of the options (1) and (2) above, whereupon a little manipulation and cancelling yields the multiplication law P(p \ q)P(p) = P(p&q} (Ramsey's proof (p. 181) does not make this assumption, though it does assume that values add like numbers). As in the case of Ramsey's definition of an unconditional degree of belief, the definition of a conditional degree of belief can be understood as determining odds in a type of bet, only in this case one which only goes ahead if q is true; such a bet is called a conditional bet. For suppose that the subject is willing to exchange (1) for (2). As we saw earlier, exchanges in order to receive conditional options can be seen as generalized bets, and in the event of q's falsity the net gain is as if no exchange at all had taken place. Restricting the payoff table to just the 'q true' part of the joint truth table for p and q, we see that it represents a bet ofp with betting quotient
u(a) — u(8) «(7) - u(6) i.e. the definition of P(p \ q). Thus/*(j& | q) roughly expresses the [normalized] odds at which he would now bet on p, the bet only valid if q is true. Such conditional bets were often made in the eighteenth century. (1926, p. 180) Even so, the definition of conditional probabilities in terms of conditional betting quotients is subtle and calls for considerable care in interpretation. Even Ramsey himself was not consistent in this. He first (correctly) emphasizes that P(p | q) is not to be interpreted as the subject's degree of belief inp were they to learn q, since knowledge of q might for psychological reasons profoundly alter his whole system of beliefs (ibid.}
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Yet only a little later he remarks that obviously if p is the fact observed, my degree of belief in q after the observation should be equal to my degree of belief in q given/) before, or by the multiplication law to the quotient of my degree of belief in pq by my degree of belief in/). When my degrees of belief change in this way we can say that they have been changed consistently by the observation. (1926, p. 192) In today's terminology, Ramsey is saying that the only consistent way of updating jour beliefs on receipt of further factual information is by conditionalization. But there is a problem: if the conditional probability of q given p is not the degree of belief in q which the subject would have should they learn that p is true, as Ramsey claims in the first of the quotations above, then it is difficult to understand why, as he claims in the second, consistent belief-change should equate to degree of belief in q after p is observed with the conditional degree of belief in q given p before p is observed. Why should changing your belief function by conditionalization be a condition of consistency? In fact, it is not, and in suitable circumstances will even generate inconsistency. It is easy to show this. Suppose that q is the assertion that the subject's degree of belief in p tomorrow will be r, where r is some number less than 1. Suppose that now the subject assigns a non-zero probability to | q) cannot as a general rule be interpreted as the degree of belief in/) were the subject to come to know q for certain. However, Ramsey's own explanation why not because 'knowledge of q might for psychological reasons profoundly alter his whole system of beliefs' — is also somewhat off the mark: as we have seen, there are in addition purely logical reasons why such an interpretation cannot be sustained. By a remarkable coincidence the Italian mathematician Bruno de Finetti, at that time unacquainted with Ramsey's work, and in a paper that has also become regarded as revolutionary, like him proposed to measure a subject's conditional degree of belief in terms of a willingness to take either side in a conditional bet. However, in de Finetti's account the payoffs are ordinary money, the amounts being assumed to be sufficiently small to avoid the familiar problems associated with people's propensities to bet at a given rate being dependent on vagaries of fortune and personal disposition. Ramsey had considered this strategy, only to reject it:
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if money bets are to be used, it is evident that they should be for as small stakes as possible. But then again the measurement is spoiled by introducing the new factor of reluctance to bother about trifles. (1926, p. 176) In fact, it is impossible to bound the stakes because in the proof of the multiplication law the stake on a conditional bet onp given q determined by bets on p&q and q is the quotient of the stakes on the last two (Ramsey's proof of the multiplication law makes this clear (1926, p. 181)). De Finetti himself later conceded that only a utility measure avoids the objections (1964, p. 102). There was another point on which de Finetti's and Ramsey's papers agreed, and one which has become celebrated in the Bayesian literature (and beyond). Ramsey adumbrated it with the remarks that These are the laws of probability, which we have proved to be necessarily true of any consistent set of degrees of belief. Any definite set which broke them would be inconsistent in the sense that it violated the laws of preference between options, such as that preferability is a transitive asymmetrical relation, and that if a. is preferable to (3, (3 for certain cannot be preferable to ex. ifp, /3 if not-/?. If anyone's mental condition violated these laws, his choice would depend on the precise form in which the options were offered him, which would be absurd. He could have a book made against him by a cunning better and would then stand to lose in any event. (1926, p. 182) A choice of stakes which results in certain loss is called by bookmakers a Dutch Book. Ramsey is claiming, therefore, that anyone whose betting quotients violated the laws of probability is in principle Dutch Bookable. That result is accordingly often called the Dutch Book Theorem, and is still frequently urged as a justification of the probability axioms. Ramsey himself did not give an explicit proof of the result, and the quotation above does no more than indicate in very general terms the way a proof might proceed. Ramsey's typical failure to present explicit and rigorous proofs of the key innovations and results, including this one, did him a lasting disservice, for it meant that the credit for them generally went elsewhere. This is certainly true in the case of the Dutch Book Theorem: the credit went to de Finetti, who proved it in his classic paper (1937). On the other hand, demonstrating that someone violating the laws of probability is in principle Dutch Bookable was not a primary objective of Ramsey's discussion: vulnerability to a Dutch Book is merely one symptom of a violation of'the laws of preference'. Nevertheless it still implied inconsistency. For de Finetti also, being Dutch Bookable manifested a type of inconsistency. True, it is not called 'inconsistency' in the standard English translation of his 1937 paper (1964) but 'incoherence', a literal translation of the original French 'incoherence'.
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'Incoherence' also means 'inconsistency', however, and it is reasonably clear from de Finetti's text that this rendering would have been more faithful to his intentions. But Ramsey went conceptually far beyond de Finetti in making explicit what he thought was an intimate connection with logic., conceiving inconsistency in his sense of violating 'the laws of preference' as a species of logical consistency. Indeed, he regarded what he had been doing as actually nothing less than logic: We find, therefore, that a precise account of the nature of partial belief reveals that the laws of probability are laws of consistency, an extension to partial beliefs of formal logic, the logic of consistency. (1926, p. 182) The view of the laws of probability as rules of logical consistency has profound implications for an understanding of the Bayesian theory and its role as a foundation for uncertain inference, and Ramsey saw them clearly. In the first place, there is no suggestion that the rules of probability are anything but rules of consistency for any subject's personal degrees of belief, not necessarily rational ones: They do not depend for their meaning on any degree of belief in a proposition being uniquely determined as the rational one; they merely distinguish those sets of beliefs which obey them as consistent ones. (1926, p. 182) So much for Keynes. Second, this logic of pure consistency does not furnish an ampliative logic of induction: We can divide arguments into two radically different kinds, which we can distinguish in the words of Peirce as (1) 'explicative, analytic or deductive' and (2) 'ampliative, synthetic or (loosely speaking) inductive' ... Logic must then very definitely fall into two parts: ... we have the lesser logic, which is the logic of consistency, or formal logic; and the larger logic, which is the logic of discovery, or inductive logic. ... The theory of probability is in fact a generalisation of formal logic. (1926, p. 186) Third, though it follows immediately from these remarks, in respect of the power of valid probabilistic reasoning we have an exact analogue of deductive logic, in which all valid inference is also non-ampliative, doing nothing more than transforming, and possibly diminishing, the information implicit in the premises. Fourth, there are no synthetic logical principles in this probabilistic logic. In particular, in the logic of consistent partial belief the Principle of Indifference is summarily dethroned:
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The Principle of Indifference can now be altogether dispensed with; we do not regard it as belonging to formal logic to say what should be a man's expectation of drawing a white or a black ball from an urn; his original expectations may within the limits of consistency be any he likes; all we have to point out is that if he has certain expectations he is bound in consistency to have certain others. This is simply bringing probability into line with ordinary formal logic, wjhich does not criticize premises but merely declares that certain conclusions are the only ones consistent with them. (1926, p. 189) The methodological significance of these claims is considerable (I discuss them at length in my book on the problem of induction (Howson 2000)). In particular, they imply that there can be no logical inductive ascent using the formal theory of probability, as Leibniz and all the promoters of the probability C logic inclusion had hoped, up to Keynes and even beyond: We all agree that a man who did not make inductions would be unreasonable: the question is only what this means. In my view it does not mean that the man would in any way sin against formal logic or formal probability. (1926, p. 197) Perhaps surprisingly in the light of these remarks, Ramsey was not by any means a Humean sceptic. His antidote to Hume's astringent, indeed withering, philosophical potion is the observation that while inductive arguments may have no logical basis, that does not make their employment unreasonable. On the contrary: We are all convinced by inductive arguments, and our conviction is reasonable because the world is so constituted that inductive arguments lead on the whole to true opinions. (1926, p. 187) This is the doctrine later to be called Reliabilism (Ramsey was also the first philosophical naturalist Induction is such a useful habit, and so to adopt it is reasonable. All that philosophy can do is to analyse it, determine the degree of its utility, and find on what characteristics of nature this depends, (p. 198)) But inductive scepticism is not so easily defeated, and twenty years after Ramsey wrote those words Nelson Goodman (1946) showed that, without the sort of qualification that made them essentially question-begging, they were vulnerable to the paradox which has now become known as Goodman's
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Paradox: by redescribing the data using suitably defined predicates — 'grue' was Goodman's famous example - we can apply the same inductive rule of generalization from observed instances in mutually inconsistent ways. It is therefore demonstrably false that 'inductive arguments lead on the whole to true opinions'.
Conclusion In the compass of a relatively brief and informal paper Ramsey revolutionized an entire field of research, setting the ground-rules for what has become the mainstream, if not orthodox, way of developing probability and rational choice theory as an application of measurement theory. And all this was done by way of a critical review of Keynes's Treatise on Probability and an attempt to define a logical account of probability less subject to the radical objections Ramsey brought against Keynes's; the revolutionary developments his investigations yielded were merely a by-product of this concern. However, I think it must be said that Ramsey is only partially successful in establishing that account. His claim to have shown that the rules of probability are logic rests on his arguing that they are conditions of consistency. But logical consistency concerns relationships between propositions, whereas for Ramsey, as we have seen, consistency in his sense is simply obedience to 'the laws of preference between options'. It is on this point that I believe that Ramsey's logical view of probability breaks down: whatever species of consistency Ramsey might be correct in appealing to here, it is not a logical one in the sense in which we understand logic today, or even in his own time for that matter. Few if any people would now regard the theory of constrained preference as a part of logic: it is, rightly, viewed as a theory of rational choice. Of course, we all speak loosely sometimes, and talk about the logic of choice, just like talk about the logic of special relativity, is usually harmless. But strictu senso neither of these is genuinely logic. There is nevertheless something very compelling about Ramsey's idea of the laws of probability as rules of consistency for the distribution of partial beliefs. Indeed, I believe that this is the most natural way to view the Bayesian theory. But can we amend Ramsey's account to yield a genuinely logical foundation for it? It can be done, I think, though at the cost of abandoning a central tenet of Ramsey's account, his psychological behaviourism. For if belief is reduced to revealed preference then it is a natural step to seeing consistency of belief simply as consistency of preferences. On the other hand, giving up behaviourism is not difficult now, since that particular dogma has long ceased to exercise the compulsion it used to. Moreover, there is an alternative way of eliciting numerical estimates of probability, one with a long tradition behind it, which
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involves no element of psychology whatever. From the work of the pioneers right up to Carnap's monumental treatise (1950), there has been a widespread view that judgements of probability are expressed by a type of betting quotient: not the subject's actual betting quotients, which as we know may depend on extraneous considerations, but their fair betting quotients. These can be elicited via the following experiment. Suppose two individuals, A and JB, are about to bet on the truth of a proposition p, with the winner taking all of some stake S provided by some third party. Before the truth-value of p is revealed, the bet is stopped, and S is to be divided equitably between A and B in the proportion xS to A, (1 — x)Sto B. A natural criterion of equity is that the proportion be judged solely in the light of how likely the subject thinks/? is, and the value of x they choose we may accordingly call their probability of p. This may call for a little skill in expressing a view numerically, and to assume that an exact real number will always be forthcoming is a very considerable idealization. At this point, however, we invoke Ramsey. If what was good enough for physics was good enough for him then it is good enough for us, so let us not worry too much about that. The connection with betting odds now follows naturally: the ratio x: 1 — x naturally determines the subject's fair odds onp, and the betting quotient associated with those odds is just x itself. Thus x is the subject's fair betting quotient on p. Note that St Petersburg-type problems are by-passed: to bet at odds 1:2"— 1 that the first head will occur at the nth toss, for all n = 1,2, 3,..., k, for any finite k, though possibly imprudent, is not unfair even according to casino standards (as Condorcet first pointed out (see Todhunter 1886, p. 393)). A deductive analogy suggests how to proceed further. Consistency in deductive logic is usually thought of as a predicate of sets of sentences. An equivalent way of looking at it is as a predicate of truth-value distributions, and as a matter of fact one deductive system for propositional logic is explicitly a system for testing the consistency of truth-value assignments to arbitrary (countable) sets of sentences. This is the method of signed semantic tableaux, as set out in Smullyan 1968 (pp. 15-20), in which truth-values are assigned in some way to a set of sentences, and the tableau rules determine how they get distributed over propositional subformulas (there is a natural extension to first order logic, but the analogy between propositional logic and probability theory is of primary interest here). An assignment of truth-values to a set of sentences in a propositional language L is consistent just in case it has a single-valued extension to all the sentences in L, i.e. the initial assignment is solvable over the entire set SL of sentences in the propositional language L, subject to the constraints of the usual truth-table rules for the connectives. Thus deductive consistency is a subspecies of the general mathematical concept of consistency as the solvability of sets of equations: in this case, solvability subject to the general constraints on truth-value distributions specified in a
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classical truth-definition. We can regard probabilistic consistency as yet another subspecies, in this case the solvability of assignments of fair betting quotients subject to the general constraints on such distributions. What should these be? To say 'the laws of probability' is begging the question: the constraint has to come out of an analysis of the notion of fairness, just as in the deductive case it comes from an anlysis of the notion of truth. Well, we know in general terms what a fair betting quotient is on a single proposition. By analogy with truth-definitions, however, we need some condition relating these in suitable compounding contexts. The following, call it C, is highly plausible. A set of simultaneous bets on k propositions will in certain circumstances determine a bet on another proposition/? with unique betting quotient x (it is not difficult to see that the sum of two bets on propositions^ and q, each with stake Tand betting quotients x andjy, are equal to a bet on the disjunction pvq with betting quotient x -\-y and stake T}. It seems intuitively compelling that if the k bets are at the agent's fair betting quotients, then x should be the fair betting quotient on p. Cdoes seem very plausible. Moreoever, all of Ramsey's conditions of consistency are met by it: C reflects the fact that a choice among conditional options should not 'depend on the precise form in which the options were offered' (p. 182); it implies that betting quotients satisfying it are not Dutch Bookable; and finally, the coup, it implies that Cforces consistent betting quotients to obey the laws of probability: it is not difficult to prove that an assignment K of fair betting quotients to a set of propositions is solvable over a cr-algebra 91 of propositions, subject to the constraint C, if and only if Kis the restriction of a countably additive probability function defined on 9? (Howson 2000, chapter 7, pp. 130-2). Given that the conditions determining the properties of a probability function are specified axiomatically, this theorem establishes an equivalence between a semantic condition defining fairness with respect to sets of fair betting quotients (C), and a syntactic one, deductive consistency with the probability calculus; thus the axioms of probability play a role analogous to the logical axioms or rules for any deductive system for propositional logic. As such the result is naturally seen as a type of soundness and completeness theorem for a logic of probability values, analogous to the soundness and completeness theorems for the (propositional, and indeed first order) logic of truth values. We can also do some model theory, defining the notion of a probabilistic model in a way analogous to a propositional one (as a single-valued extension of some initial assignment), and a notion of probabilistic consequence: an assignment K is a consequence of another, A"', just in case every model of JT is a model of K'. A probabilistic version of the strong completeness theorem for propositional logic follows easily from the theorem above: K is a consequence of K1 just in case there is a derivation of A" from K' together with the probability axioms.
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It all seems very natural, though what Ramsey himself would have thought of it, or any other alternative way of developing subjective probability as a logic of consistency, I do not know. I myself think it provides a more compelling justification for invoking the terminology of logic, but the inspiration for it nevertheless remains Ramsey's own work in that remarkable paper. Endnotes 1.
I am very grateful to Richard Bradley for his helpful comments on an earlier draft of this chapter.
References Bayes, T., 1963. 'An Essay Towards Solving a Problem in the Doctrine of Chances', Philosophical Transactions of 'the Royal Society of 'London', 53, 370-418. Bernoulli, D., 1738. 'Specimen theoriae novae de mensura sortis', Commentarii academiae scientiarum imperialis Petropolitanae, Vol. V, 175—92. Bradley, R., 2001. 'Ramsey and the Measurement of Belief', Foundations ofBayesianism, eds D. Corfield and J. Williamson. Dordrecht: Kluwer, 263-90. Carnap, R., 1950. Logical Foundations of Probability. Chicago: University of Chicago Press, de Finetti, B., 1937. 'La prevision: Ses lois logiques, ses sources subjectives', Annales de rinstitut Henri Poincare, 7, 1—68. 1964. English translation of the above in Studies in Subjective Probability, eds H. Kyburg and H. Smokier. New York: Wiley, 93-158. 1981. 'The Role of "Dutch Books" and "Proper Scoring Rules" ', British Journal for the Philosophy of Science, 32, 55-6. Goodman, N., 1946. Fact, Fiction and Forecast. Cambridge: Harvard University Press. Hilbert, D., 1899. Grundlagender Geometric. Leipzig. Howson, C. and Urbach, P., 1993. Scientific Reasoning: The Bayesian Approach (2ed.). Chicago: Open Court. Howson, C., 2000. Hume's Problem: Induction and the Justification of Belief. Oxford: The Clarendon Press. 2003. 'Logic and Probability', 'Journal of Applied Logic', 1, 151-65. Keynes, J. M., 1921. A Treatise on Probability. Page references are to the Collected Papers, ed. D. Moggridge, 1973, published by Macmillan. Krantz, D. H., Luce, R. D., Suppes, P. and Tversky, A., 1971. Foundations of Measurement. Vol. 1. New York: Academic Press. Smullyan, R., 1968. First Order Logic. New York: Dover. Sudbery, A., 1986. Quantum Mechanics and the Particles of Nature. Cambridge: Cambridge University Press. Todhunter, I., 1866. A History of the Mathematical Theory of Probability from the Time of Pascal to that of Laplace. London: Macmillan.
7
Ramsey's removal of Russell's 'axiom of reducibility' in the light of Hilbert's critique of Russell's logicism Ulrich Majer
Introduction It is well known that Ramsey made a serious effort to simplify Russell's ramified theory of types by distinguishing two kinds of paradoxes: purely logical paradoxes and what he called epistemological paradoxes. Using this novel distinction he hoped to avoid the apparently necessary but logically strange 'axiom of reducibility' by introducing a new notion of 'predicative function' into Russell's Principia Mathematica (PM). (Russell too uses the notion of'predicative function' but Ramsey stresses that his notion is quite different from Russell's. See Ramsey 1978, p. 189.) By these means Ramsey hoped to avoid the 'ramification' of Russell's theory of types and, hence, dispense with the 'axiom of reducibility'. Whether this proposal was successful from an epistemological point of view, or only a technical monstrosity, I will investigate in the following considerations. My point of departure is, of course, Russell's ramified theory of types as presented in the first edition of Principia. Within the frame of this theory the axiom of reducibility seems inevitable, at least if one wants to base analysis on a pure logical foundation as Russell, still in the spirit of Frege's logicism, undoubtedly did. In the course of my considerations I'll take advantage, however, of some lectures of Hilbert from 1917 to 1920, in which he investigates Russell's Principia and his ramified theory of types. These lectures were never published1 and, hence, are not known today except to a handful of historical experts. Nonetheless, these lectures entail among other things an extremely interesting analysis and reconstruction of Russell's theory of types in general, and of his axiom of reducibility in particular. In the first lecture on the 'Principles of Mathematics' in 1917/18 Hilbert presents a 'functional-calculus' which he uses again in the lecture 'Problems of Mathematical Logic' of 1920, in which he sets out, for the first time, to criticize Russell's approach in PM. I will use this functional calculus in my further considerations. In the first part of this chapter I will present an outline of Hilbert's analysis of PM. In this way, I hope, it will become clear why the axiom of reducibility is only an artificial means to 'overcome' some of the restrictions on concept
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formation, which Russell incorporated in his ramified theory of types in order to banish the paradoxes of set theory and the other known paradoxes, like Richard's or Burali-Forti's paradox. Such a liberalization of the rules turned out to be required for the reconstruction of analysis within the theory of types. Hilbert criticizes Russell's procedure in PM as 'incoherent'. In the second part of the chapter I will investigate whether Ramsey's proposal of dispensing with the axiom of reducibility by inventing a new notion of 'predicative functions' really escapes Hilbert's critique of Russell's procedure in PM as 'incoherent', or whether the proposal runs into similar difficulties, because Ramsey needs for his solution even stronger assumptions than Russell's axiom of reducibility.
Hilbert's analysis of Russell's Principia Mathematica and his critique of the axiom of reducibility On 11 September 1917 Hilbert presented a lecture to the Swiss Mathematical Society in Zurich, entitled 'Axiomatisches Denken', in which he recommended his so-called axiomatische Methode as a universal method of research, applicable not only in mathematics but also in science. This lecture was published in 1918; it entails (besides the much later published book Gmndzjige der Theoretischen Logik) the only trace of Hilbert's extensive occupation with Russell's PM in the period 1913-17. Toward the end of this very programmatic paper we find an interesting but at the same time rather cryptic and irritating remark: Dieser Weg (die Logik selbst zu axiomatisieren und nachzuweisen, dafi Zahlentheorie sowie Mengenlehre nur Teile der Logik sind) ist schlieBlich am erfolgreichsten durch den scharfsinnigen Mathematiker und Logiker Russell eingeschlagen worden. In der Vollendung dieses groBziigigen Russellschen Unternehmens der Axiomatisierung der Logik konnte man die Kronung des Werkes der Axiomatisierung iiberhaupt erblicken. (Emphasis in original) When I read this remark for the first time, I was deeply irritated for two reasons: First, I asked myself, has Hilbert - the leader of the 'formalistic' school with respect to the foundations of mathematics - become a partisan of the FregeRussell camp of logicism? Was he not any more the most severe critic of the logicistic programme to reduce arithmetic to pure logic? Had he changed his mind?
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Second, in which sense is Russell's PM the coronation of the axiomatic programme? Was Russell - his logicistic ambitions notwithstanding — not a 'secret' admirer of Poincare's constructive approach not only to mathematics but to science in general? And how could Hilbert — in the presence of Weyl, who had divorced himself from Hilbert's axiomatic point of view and sided in his (recently published) book Das Kontinuum with Husserl's genetic approach — praise Russell's PM as the coronation of the axiomatic point of view? Was that not frivolous? The answer to both questions is entailed in the unpublished lecture from summer 1920, already mentioned, about 'Probleme der mathematischen Logik'. In this lecture Hilbert developed a number of impressive reasons for his 'new', i.e. his proof-theoretical approach to the foundations of mathematics. The most serious reason is, of course, the absolute necessity to banish the danger of paradoxes and inconsistencies.2 But this is not the only reason. Beside it there are other reasons, which are connected with the subsequent question, 'how' to banish the danger of paradoxes and inconsistencies. In 1920 only two proposals3 were on the table that could be taken seriously as possible candidates for a foundation of mathematics.4 First, Russell's PM with its strong logicistic tendencies, and second, Weyl's book Das Kontinuum with its genetic procedure to the foundations of mathematics. Hilbert had studied Russell's PM over several years and devoted a large part of his lecture 'Prinzipien der Mathematik' (1917/18) to a precise presentation and thoroughgoing discussion of Russell's theory of types. Regarding Weyl's book Das Kontinuum we have no written documents which show that Hilbert studied it in detail, but one point is pretty sure: Hilbert must have grasped its intentions immediately, because Weyl had 'habilitated' in 1910 in Gottingen with a lecture on the definitions of mathematical concepts, in which he had already formulated the rigorous principles of concept formation, which form the point of departure for his 'critical investigations of the foundations of analysis' seven years later — such the subtitle of his book. For reasons which I will explain later, Hilbert could not accept any of the two approaches, neither Weyl's genetic nor Russell's constructive approach. And these in turn were for him important reasons — very important reasons let me stress - to develop a new alternative approach to the foundation of mathematics. The lecture 'Probleme der mathematischen Logik' of summer 1920 is the first (preserved) document of this new approach. Having this in mind, it is relatively easy to answer both questions. Regarding the first question: has Hilbert become a partisan of the FregeRussell camp of logicism? The answer is, of course, No! The development of his views took precisely the opposite direction. Roughly until 1918 Hilbert had considered Russell's PM as the most promising approach to a simultaneous foundation of logic and mathematics. (Notice, Hilbert did not 'stop'
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the publication of his lecture 'Axiomatisches Denken' in 1918.) But toward the end of 1918 (or so) he changed his mind and became more and more sceptical whether Russell's PM was a reasonable starting point to a firm and unquestionable foundation of mathematics. The reasons for this unexpected change are very interesting and enlightening. They can be best understood if we turn to the answer of the second question: why, for heaven's sake, did Hilbert characterize Russell's PM in his lecture 'Axiomatisches Denken' as 'the coronation' of the axiomatic approach? The answer is quite simple: Hilbert must have noticed at some point7 that Russell — his constructive confessions notwithstanding — had left Poincare's constructive approach and secretly converted to the axiomatic point of view in the sense that Russell had apparently no scruples to claim the existence of certain entities without possessing a proof (or something like this) for the existence of these entities. This imprecisely what Russell asserts in the axiom of reducibility, as I will show in a moment. In this (and only in this) sense 'could the completion [of Russell's PM] be taken as the coronation of the axiomatic point of view'. But this does not mean - the subjunctive in the quotation is important here — that Hilbert really agreed with Russell's axiomatic procedure, as practised in the axiom of reducibility. On the contrary, already in 1920 Hilbert begins to criticize Russell for his axiom of reducibility as an arbitrary unjustified means to accomplish the foundations of mathematics and, consequently, begins to develop a completely different route (than Russell's logicistic approach) with respect to the foundation of mathematics. Now let me explain a bit more closely what Russell really did in PM and what he achieved by means of the axiom of reducibility with respect to the foundations of mathematics, and last but not least, where he failed in Hilbert's view. The first point one has to take notice of is the remarkable circumstance that Hilbert sees a close relationship between Russell's and Weyl's approach to the foundations of mathematics. Although their methods are different, their approach is fundamentally the same: both aim at a reduction of mathematics to logic, and, still more important, both try to achieve this aim by using logical predicates instead of sets, sets of sets and so on in the sense of Zermelo's set theory. Having discussed the latter Hilbert turns to Russell and Weyl: Das alte Problem, die gesamte Mathematik auf Logik zuruckzufiihren, gewinnt durch das Zermelosche Axiomensystem, in welchem die Grundlagen fur die gesamte Analysis so iibersichtlich zusammengefaBt sind, eine starke Anregung und Belebung. Und der Versuch einer ^uruckfuhrung auf die Logik scheint besonders deshalb aussichtsvoll, weil zwischen den Mengen, welche ja die Gegenstande in Zermelos Axiomen bilden und den Pradikatender Logik ein enger Zusammenhang besteht. Namlich die Mengen lassen sich auf Pradikate zuriickfuhren.
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Diesen Gedanken haben Frege, Russell und Weyl zum Ausgangspunkt genommen bei ihren Untersuchungen iiber die Grundlagen der Mathematik. Als sachgemaB erscheint uns folgende AufFassung, welche insbesondere von Russell und ungefahr auch [so] von Weyl vertreten wird: Die Aussagen, in denen von Mengen (Klassen, Gesamtheiten) die Rede ist, sind nur Umformungen von Pradikationen; namlich statt z.B. zu sagen: ,,die Rose ist rot" sagt man ,,die Rose gehort zur Menge der roten Dinge". Und nun wird mit,,Menge der roten Dinge" formal wie mit einem Gegenstande operiert. In order to appreciate the subtlety of Hilbert's consideration in the quotation one has to become clear about two crucial points. First, the use of sets instead of predicates offers (some minor difficulties notwithstanding) three decisive advantages: (a) Zermelo's set-theory is, as far as we know, consistent, at least it avoids all the known paradoxes; (b) the plurality of intentionally different, but extensionally equivalent, i.e. identical predicates such as 'shortest line between two points', 'line with curvature 0' and 'line with complete rotational symmetry (relative to a pair of points on the line)' can be completely avoided; (c) the hierarchy of predicate-logic can be ignored within set theory; it becomes superfluous. These advantages Hilbert had in mind with his remark about Zermelo's axiom system that 'in it the foundations of analysis are summarised in a very lucid way'. So much to the first point. The second point is then a 'natural consequence' of the first. If somebody prefers predicates over sets (why doesn't matter) he has to recover somehow the advantages of Zermelo's set theory, because otherwise he is risking running again into the paradoxes of 'naive' set theory and the many other difficulties, which Frege had to overcome in his Grundgesetze der Arithmetik. But if Zermelo's set theory has in fact these advantages, then the simple question arises: why should anyone prefer predicates over sets? Why should anyone, who has an interest in the foundations of mathematics, prefer the language of predicate logic over Zermelo's set-theory? And furthermore, why does Hilbert consider this question at all? Why does he not straight out cling to Zermelo's set-theory? The answer to both questions seems to be this: Hilbert does not regard logicism as a priori absurd; he seems to consider the idea that mathematics is a branch of logic at least as a possible one, which would clarify and simplify the foundations of mathematics considerably. But, of course, the idea alone isn't sufficient; it has to be shown that it works, i.e. that analysis in its totality can be recovered within pure logic. This in turn can't be done without a corresponding research programme, which shows how the successes in the
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foundations of mathematics achieved by set theory can be transformed into successes of recovering analysis within predicate logic. Now, the only two research programmes of this kind, available in 1920, were Russell's Principia and Weyl's Kontinuum. And this is the very reason why Hilbert discusses them: he wants to become clear how far the logicistic programme can be realized and where it, eventually, comes to a halt or insurmountable difficulties. The leading idea of the two programmes was — up to a certain point — the same. Both regarded predicates as (epistemologically) prior to sets, or to put the same point the other way around: Russell and Weyl regarded propositions about sets only as linguistic transformations of predications in the logical sense: instead of saying 'the rose is red' the set theoretician says 'the rose belongs to the set of red things' and takes then 'the set of red things' formally like an object in the logical sense. But this step is, according to Russell and Weyl, neither necessary nor natural. Hence, both avoid the notion of set quite consciously as a basic notion, and try instead to 'reduce' sets to predicates wherever it seems possible, and to recover analysis in the language of predicate logic. So far they agree, but then their routes diverge. Whereas Weyl in the Kontinuum follows a strict constructive or genetic point of view regarding the formation of complex judgements and concepts from simple ones in order to avoid any trace of'impredicativity' and other kinds of circularity, Russell's procedure in the first edition of PM is less restrictive. How he proceeds precisely in detail will be discussed later. At this juncture only the result of the different procedures is of interest: whereas Weyl has to give up considerable parts of analysis, e.g. the sentence of the existence of an upper limit in any bounded set, what Weyl frankly admits,9 Russell can apparently recover the whole of analysis in his PM. The adverb 'apparently' has been inserted quite deliberately, because Russell has to 'pay a price' for this success, and we have to ask whether this price is adequate or eventually too high, if we take all aspects together. This brings me back to the question: how does Russell construct analysis in PM? How does he 'recover' the advantages of set theory in the language of predicate logic? And more specifically: how does he 'reduce' sets and sets of sets to predicates? The principal aim of Russell's PM is, of course, to present a pure logical foundation of analysis, which at the same time avoids the paradoxes of naive set theory,10 which have ruined the final success of Frege's Grundgesetze. In order to achieve this aim Russell introduces a 'formal restriction' in the formation of propositions and concepts from functions and arguments, which can be stated shortly as follows: The order-number of a function must always be greater than the order numbers of each of its arguments. And objects in the proper sense - the individuals of the lowest level of arguments - have the order-number 0. Consequently, the elementary predicates (the truth-functions of the lowest level) have the
order number 1, predicates of predicates (of first level) the order-numb order number 1, predicates of predicates (of first level) the order-number 2 and so on. This rule comes up to the following restriction: no predicate, no truth-function can be its own argument, or to express it inversely: no predicate can be applied to itself as argument. In this way Russell inhibits the generation of concepts like 'the set of all sets, which do not contain themselves', that lie at the bottom of the paradoxes of set theory. The resulting system is the well known (simple) theory of types or, as Hilbert calls it, a 'Stufenkalkul' (functions of functions). With this restriction in mind let's see how Russell recovers Zermelo's set theory in the theory of types. As a first simple example we take the notion of power-set, i.e. the set of all subsets, and ask how it can be expressed in the language of predicate logic. The first step is easy. If m and n are two arbitrary sets, p and q their corresponding (i.e. extensionally equivalent) predicates, then the proposition 'm is a subset of n' is logically equivalent with the proposition 'from p follows q(x)), which means that for every element x, for which p(x) is a valid proposition, also q(x) is a valid proposition. The next step in recovering the notion of power-set is more tricky. If we fix q (respectively n, because we want to form the notion 'all subsets of n'} we can regard the proposition Vx(p(x] —> q(x}} as a proposition P(p] about the predicate p, which asserts a certain property P about the predicate p, namely that 'from the validity of p that of q follows'. This property P of the predicate/? is an example of a second-order predicate (ifp is of first order). To this second-order predicat To this second-order predicate ment x of the basic domain; and the latter is the case for all and only those predicates/) [p now taken as variable of second order] whose extensions lie within or are a subset of the extension of q. So far then we have recovered the notion 'set of all subsets ofn' within predicate logic by the definition of the second'set of all subsets ofn' within predicate logic by the definition of the seconpoint of view. But if we now try to recover in a similar way the notion of the 'unification of sets', as stated in Zermelo's axiom 6, we run into difficulties. Let me explain 'how and why'. Let M be a set of sets, whose elements are the sets m, m', m",... (In other words Mis the unification of the sets m, m', m",...) To every element of M corresponds a predicate/?,p f ,p",... As in the previous case we can characterize M by a predicate of predicates P(p), which characterizes the predicates p(x], or rather their extensions, belonging to M. If we now ask, which [individual] objects belong to the unified set M, the answer is: every element of the elements of M, in other words, an object a belongs to M if and only if it is an element of an element of M, i.e. an element of at least one m1. This condition (of being an 'elementary' member of the unified set M} can again be expressed in the
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language ofPM by means of the higher order predicate P(p)', it goes like this: a is an element of M if and only if a possesses one of those predicates p(x), which in turn possess the property expressed by the predicate P(p]. In other words: a belongs to M if there exists a predicate/) (x), which applies to a and whichitself belongs to the extension ofP(p); symbolically we write this as: belongs to M if there exists a predicate/)
(1)
At first glance the 'definition' of M by the right side of the equivalence (1) seems quite acceptable. But a moment's reflection shows that this is not the case, at least not if we take the idea of logicism seriously, because this means that we have to reduce the notion 'unified set of sets' to predicate logic, pure and simple. But (1) is not of this form, it entails a difficulty, which becomes evident if we ask: what does it mean 'there exists a predicate^'? Usually we relate the expression 'there exists a so and so' to a domain of objects, given independently., that is in advance of any particular existence claim. But the domain of predicates is not and cannot be regarded as 'given in advance'. The predicates in the domain of quantification have first to be formed or 'constituted' by means of certain well-defined logical operations. And these logical operations have to be stipulated in advance. Only if that has been done, only after the 'rules of constitution' for predicates have been given in advance, is the domain of predicates well determined. Only then are we legitimated to make existence claims with respect to the domain of predicates. From this reflection it becomes clear that a reference to the domain of predicates is not permitted in the 'rules of constitution' themselves. This would be an open 'circulus vitiosus'! But this is precisely what we have done in the definition of M by claiming the existence of a predicate p, without having stipulated 'in advance' the 'rules of constitution' for predicates. This means: we have no well-defined domain of predicates and, consequently, the definition of M by (1) is undetermined. The result of this whole consideration is this: we are, in principle, not permitted to use existence claims with respect to predicates for the definition of predicates belonging to the same domain of predicates to which we referred in the existence claim. Consequently, we cannot reduce the notion of a unified set to an existence claim like Ep(x) (p(oi) and P(p}}, as we did in (1). Precisely this fundamental difficulty 11 was also recognized by Russell and Weyl, and it had, as we know from the wisdom of hindsight, a enormous influence on their work. But of what kind was this influence? What was their reaction to the difficulty? As already mentioned their reactions were rather different. Whereas Weyl maintained a constructive position regarding the 'rules of constitution' for predicates in order to avoid the circularity of'impredicative' definitions like (1), and, as a consequence, sacrificed a considerable part of analysis, Russell took another, almost opposite route.
Ramsey's Removal of Russell's 'Axiom ofRedudbility' His main intention was to save analysis in its entirety and in order to do so he had to substitute the impredicative definition of M by another 'predicative' definition, which does not refer to the undefined domain of all predicates and, hence, is not subdued to the same objection of circularity as the first proposal. The basic tool by which Russell hoped to achieve this goal was his axiom of reducibility according to which the second-order predicate P(p] could be substituted by an extensionally equivalent predicate of the lowestlevel. Unfortunately, however, Russell could not present such a predicate, but was nonetheless convinced that it existed, and exactly for this reason he introduced the axiom of reducibility, which asserts just this: the existence of an extensionally equiva-. lent predicate of the lowest level for every predicate of higher levels. I claim now that it was precisely this defect of Russell's approach, which Hilbert had recognized as defect at some point between his lecture in Zurich 1917 (where he had already welcomed Russell in the club of the axiomatic approach) and his lecture in summer-term of 1920, where he criticizes Russell's approach for the first time as 'defective'. The failure of both approaches, Russell's and Weyl's, became for Hilbert then a decisive reason to look for a different approach regarding the foundations of mathematics. Because this assertion is important for my whole argument, let me quote the relevant passages in full. First, Hilbert summarizes his reflections with respect to the intended definition of M. Es ergibt sich somit, dafi wir das aufgestellte Pradikat Ep(x}(p(a} + P(p]} zur Definition der Vereinigungsmenge nicht verwenden konnen. Hier liegt also eine grundsatzliche Schwierigkeit vor, und diese wird auch von Russell und Weyl bemerkt. Beide sehen sich dadurch zu einer Resignation veranlasst. In der Art ihres Verzichtes ist aber ihr Standpunkt entgegengesetzt. Next, Hilbert characterizes Russell's procedure in PM in the following way: Russell geht von dem Gedanken aus, dass es genugt, das zur Definition der Vereinigungsmenge unbrauchbare Pradikat durch ein sachlich gleichbedeutendes zu ersetzen, welches nicht dem gleichen Einwand unterliegt. Allerdings vermag er ein solches Pradikat nicht anzugeben, aber er sieht es als ausgemacht an, dafi ein solches existiert. In diesem Sinne stellt er sein ,,Axiom der Reduzierbarkeit" auf, welches ungefahr folgendes besagt: ,,Zu jedem Pradikat, welches durch (ein- oder mehrmalige) Bezugnahme auf den Pradikaten-Bereich gebildet ist, gibt es ein sachlich gleichbedeutendes Pradikat, welches keine solche Bezugnahme aufweist." - Hiermit kehrt aber Russell von der konstruktiven Logik zu dem axiomatischen Standpunkt zuriick.
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Regarding Weyl's position he gives a somewhat more sympathetic description: Weyl hingegen lasst sich auf eine Annahme wie die Russellsche nicht ein, und da er keine Moglichkeit sieht, die Ersetzbarkeit jenes zur Definition der Vereinigungsmenge unbrauchbaren Pradikates durch ein hierfiir geeignetes Pradikat zu beweisen, so lehnt er das Schlussprinzip, welches durch das Axiom von der Vereinigungsmenge formuliert wird, iiberhaupt ab. - Das Weylsche Verfahren bleibt also einheitlich bei der konstruktiven Methode, es fuhrt aber auch nicht zu einer Begriindung der in der Analysis iiblichen Schlufiweise. Finally, he presents his resume regarding the prospects of the logicistic programme to reduce set theory (and with it analysis) to logic, which he in turn then takes as an important argument for his new approach to the foundations of mathematics: Das Ziel, die Mengenlehre und damit die gebrauchlichen Methoden der Analysis auf die Logik zuriickzufuhren, ist heute nicht erreicht und ist vielleicht iiberhaupt nicht erreichbar. Because Hilbert's considerations regarding the 'axiom of reducibility' may look a bit sketchy, it is important to note that he was completely clear about the role and content of the axiom of reducibility. Already in the context of Cantor's proof for the existence of uncountable sets, the theory of types without the axiom of reducibility (Stufenkalkiil) is too restrictive: the required 'diagonalization' cannot be carried through, because the domain of the number predicates has to be restricted to a certain level n, whereas the predicate constructed by the diagonalization is of level n + 1 and, hence, as such does not belong to the original domain of predicates. This difficulty is lifted by the axiom of reducibility insofar now (according to the axiom) to every predicate P of level n an extensionally equivalent predicate of level 1 exists, such that the diagonalization can be carried through within the first level. Of course, this solution is rather strange, because it supposes that the predicates and relations of the first level exist in themselves, independent of any construction or definition. Although Hilbert is completely aware of this strange kind of realism., he accepts nonetheless the axiom of reducibility in 1917/18, because he thinks that the axiom is unavoidable, at least, if one is not prepared to give up the idea of'logicism'. But in 1920 he has changed his mind and rejects the axiom of reducibility as a mere cure of symptoms. The prime reason for this change is, of course, that he had developed meanwhile an alternative strategy for the foundations of mathematics, namely proof theory, which seemed more effective than the reduction of set theory and analysis to predicate-logic.
Ramsey's Removal of Russell's 'Axiom of Reducibil Now let's turn to Ramsey and explore his attempt to come to terms with Russell's theory of types and the apparently inevitable axiom of reducibility. Was his proposal in the 'Foundations of Mathematics' to get rid of the axiom of reducibility by means of so called 'predicative functions' successful or has it — in the light of Hilbert's analysis of Russell's procedure in PM — to be criticized in a similar way and for similar reasons as Russell's axiom of reducibility? For the sake of clarity it seems reasonable to divide this complex question into three sub-questions. The first question is, of course: did Ramsey address the same problem as Russell and Weyl? Or to express it more precisely, was his understanding of the foundational problem of analysis the same as Russell's and Weyl's, as we explained it in terms of Hilbert's analysis of the problem? If the answer is 'Yes' - and we will see that this is the case, at least in principle, if we put inessential differences aside - then the next question we have to ask is, of course: what exactly is his solution to the foundational problem, such that he can 'dispense with' the axiom of reducibility? If this has been answered sufficiently, we come to the last and decisive question: would Hilbert have accepted Ramsey's solution, or would he have rejected it for similar reasons as he had rejected Russell's solution by the axiom of reducibility?
Why Ramsey's 'simple' theory of types is not simple Addressing the first question, whether Ramsey shared Russell's and Weyl's view that the foundation of analysis becomes a problem, if one restricts the admissible means of concept formation to pure logic, one can see rather quickly by checking through Ramsey's 'Foundation of Mathematics' that he in principle holds a similar view, although his diagnosis of the 'causes' and, hence, his solution of the problem differs significantly. Having explained to the reader that he, Ramsey, shares Wittgenstein's view that the propositions of mathematics are nothing but tautologies, he comes to the central problem of this view: I shall first try to explain the great difficulties which a theory of mathematics as tautologies must overcome . . . They spring from a fundamental characteristic of modern analysis which we have now to emphasize. This characteristic may be called extensionality, and the difficulties may be explained as those which confront us if we try to reduce a calculus of extensions to a calculus of truth-functions. Here, of course, we are using 'extension' in its logical sense, in which the extension of a predicate is a class, that of a relation a class of ordered couples; so that in calling mathematics extensional we mean that it deals not with predicates but with classes, not
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with relations in the ordinary sense but with possible correlations, or 'relations in extension', as Mr Russell calls them. (Ramsey, 1978, p. 165) If one exchanges in the quotation 'sets' for 'classes' and substitutes Zermelo's set theory for 'calculus of extension' as well as predicate logic for 'calculus of truth-functions', one has almost literally Hilbert's description of the general problem of the reduction of set theory to pure logic, which is an essential precondition for a successful foundation of analysis by pure logical means. So far we have an almost perfect agreement regarding the general, however, still unresolved problem of a pure logical foundation of analysis. But how do things stand, if we look closer at the details of the problem? Where do the insurmountable problems concretely occur, according to Ramsey, if we try to realize the intended reduction? In order to answer this question we have to inspect what Ramsey says about Russell's attempt at a concrete solution of the problem by means of the ramified theory of types and the axiom of reducibility. This is Part II of Ramsey's essay, in which he explains why, in his view, Russell's attempt of a logical foundation of analysis failed. I will now turn to this part but not without mentioning that Ramsey agrees with Hilbert that the first steps in the reduction of sets to predicates are unproblematic: 'That the class of -0's includes the class of 0's means simply that everything which is a (f> is a if), which, as we have seen above is a truth-function' (ibid., p. 167). The real problems first emerge later and it is here that Hilbert's and Ramsey's diagnoses begin to diverge. Ramsey identifies 'three important defects' in Russell's PM, of which we have to consider only the second because it is directly connected with the theory of types, more properly speaking, with its 'ramified' form as a misguided attempt to avoid the paradoxes of set theory and the other known paradoxes. But why is the ramified theory of types a 'misguided' attempt to avoid the paradoxes? The short answer 'because it makes the axiom of reducibility inevitable' is obviously question-begging, because what we really want to know are not the consequences, good or bad, that an adoption of the ramified theory of types has, but the reasons why we should not adopt it in the first place. The main reason for not adopting the ramified theory of types is that it rests on an inappropriate diagnosis of the paradoxes. According to Ramsey's view the paradoxes should not be lumped together in one group, as Russell did it when he implicitly supposed that they all spring from the same mistake — the vicious circle of impredicative concept formation — but being separated into two groups, the logical and the epistemological paradoxes, each of which requires a different treatment. The second defect in Principia Mathematica represents a failure to overcome the contradictions discussed at the end of Chapter I (Ramsey 1978, p. 175).
Ramsey's Removal of Russell's 'Axiom of Reducibil These contradictions it was proposed to remove by what is called the Theory of Types, which consists really of two distinct parts directed respectively against the two groups of contradictions. These two parts were unified by being both deduced in a rather sloppy way from the 'vicious circle principle', but it seems to me essential to consider them separately (ibid., p. 175). I will not dispute Ramsey's distinction of the two groups of paradoxes but simply accept it for the sake of argument and explain instead immediately the three main points, which are relevant for our further discussion: (1)
(2)
(3)
Why has the second group of paradoxes, the epistemological ones, led Russell to the 'ramification' of the theory of types into a further hierarchy of orders within the hierarchy of types. How can the 'ramification' of the theory of types into orders be avoided? By which means, thinks Ramsey, can he turn the ramified theory of types into a simple 'unramified' form? Last but not least, what is the 'prize' for Ramsey's solution (in distinction to Russell's axiom of reducibility)?
Before I address these questions I should perhaps make clear that the distinction of predicates (propositional-functions) into different levels or types is not only in Russell's but also in Ramsey's view logically justified and inevitable in order to avoid the logical paradoxes of the first group. The contradictions of group A are removed by pointing out that a prepositional function cannot significantly take itself as argument, and by dividing functions . . . into a hierarchy of types according to their possible arguments. . . . This part of the Theory of Types seems to me unquestionably correct.... thus there are functions of individuals, functions of functions of individuals, functions of functions of functions of individuals, and so on. (ibid., p. 175) With respect to the second group of paradoxes the logical situation is, however, according to Ramsey's view basically different: the distinction of predicates into different orders is neither logically justified nor inevitable in order to avoid the paradoxes of the second group. Before I discuss how Ramsey came to this rather surprising view, I have, however, to answer the first question: what reasons did Russell have to distinguish between predicates of different order within the same type, what motivated him to make the distinction of orders in addition to the distinction of types? The short answer is, of course, that the distinction was made precisely in order to avoid the epistemological paradoxes. But this answer, although correct, does not really explain the peculiar reasons, which lie at the bottom of the epistemological paradoxes. The interesting point is now that Ramsey seems to
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share Hilbert's view that Russell was driven to make the distinction of orders [in addition to that of types] just because he believed that the paradoxes of the second group were created from a violation of the 'vicious-circle principle'. In other words Russell was convinced, according to Ramsey, that the source of the epistemological paradoxes was an 'impredicative' definition of concepts, i.e. a definition of predicates, in which the predicate to be defined already belongs itself to the totality of predicates, by which the predicate is to be defined. In order to inhibit such 'impredicative' definitions Russell introduced an ordering of the predicates (of every type) in such a way that a recourse to the totality of all predicates of a certain type became impossible. For example, a recourse to all predicates of the first type, i.e. of individuals, is impossible, because there are always predicates of'higher' order (within the first type), which do not belong by means of Russell's distinction to the set of first order predicates from which we started. In this way Russell was able to block impredicative definitions and, hence, the epistemological paradoxes arising from them. So far at least Ramsey and Hilbert agree in their diagnosis of Russell's PM. But there remains an important difference to which I now turn. We have already seen that Hilbert agrees with Russell and Weyl that the problem of impredicative definitions is a very serious one, which must be solved in one or another way, if one believes in 'logicism' as a reasonable aim. Ramsey, however, seems not to share this opinion, at least not that impredicative definitions/w se must be avoided, because they lead necessarily to contradictions. On the contrary, he argues that Russell's solution of the paradoxes was misguided, just because he attributed the second group of paradoxes to impredicative definitions. This was a fundamental error according to Ramsey's view: not only are the real reasons for the second group of paradoxes certain 'ambiguities of meaning', but the false diagnosis of the causes of the paradoxes has also led (via its therapy by means of the ramification of types into a 'hierarchy of orders') to inacceptable consequences: Thus this theory of a hierarchy of orders of functions of individuals escapes the contradictions; but it lands us in an almost equally serious difficulty, for it invalidates many important mathematical arguments which appear to contain exactly the same fallacy [of impredicative definitions] as the contradictions. In the first edition ofPrincipia Mathematica it was proposed to justify these arguments by a special axiom, the Axiom of Reducibility, which asserted that to every non-elementary function there is an equivalent elementary function. This axiom there is no reason to suppose true; and if it were true, this would be a happy accident and not a logical necessity, for it is not a tautology. . . . Such an axiom has no place in mathematics, and anything which cannot be proved without using it cannot be regarded as proved at all. (ibid., p. 178-9)
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Obviously Ramsey comes to similar critical conclusions as Hilbert regarding the validity of the axiom of reducibility but, of course, for totally different reasons. Whereas Hilbert accepts the hierarchy of orders as one of two possible means13 to avoid the circulus vitiosus of impredicative definitions and, as a result, also the paradoxes, Ramsey rejects the hierarchy of orders of functions and looks instead for another less ad hoc solution than Russell's ramified theory of types and his axiom of reducibility. This becomes particularly clear in his concluding remark regarding the second defect of Russell's PM. The principal mathematical methods which appear to require the Axiom of Reducibility are mathematical induction and Dedekind section. Mr Russell has succeeded in dispensing with the axiom [of reducibility] in the first case, but holds out no hope of a similar success in the second. Dedekindian section is thus left as an essentially unsound method, as has often been emphasized by Weyl, and ordinary analysis crumbles into dust. That these are its consequences is the second defect in the theory ofPrincipia Mathematica, and, to my mind, an absolutely conclusive proof that there is something wrong. For as I can neither accept the Axiom of Reducibility nor reject analysis, I cannot believe in a theory which presents me with no third possibility. (ibid., p. 180) This brings me to the second and third question: what is Ramsey's solution and what is eventually the prize for his solution. In order to cut a long story short I will concentrate on the core of his solution and not explain why the solution is a solution, i.e. how it serves to avoid the paradoxes of the second group. Before explaining any details let me emphasize that Ramsey's solution is inspired not by one but by two leading ideas; the one is genuinely philosophical, the other a simple but ingenious technical idea. Let me begin with the philosophical idea. It can be paraphrased by the mnemonic claim: there exist more things (between heaven and earth) than we can name, describe or denote in some other, more or less indirect way. Although this claim does not sound very 'philosophical' it turns out to be a very powerful philosophical guide in tandem with the technical idea, as we will see in a moment. The technical idea can be characterized by the proposition that there is no logical reason to restrict the number of argument-places in a truth-function to a finite number of arguments, or expressed positively: the number of arguments in a truth-function can be infinite and has to be taken in a logical calculus of functions to be infinite. Of course, this statement sounds very strange and a large number of serious epistemological objections come immediately to mind. But before we consider the most important of these objections, let me be fair to Ramsey's truly
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original ideas and sketch how his proposal works. To do so demands a number of steps. Starting from the classical notion of an 'atomic proposition', which is either true or false, we first define what a 'truth function' in Ramsey's sense is. It is a truth-function quite in the usual sense having truth-values of atomic propositions as arguments, but with one important difference: its number of possible arguments is not restricted to being finite, it can have an infinite number of arguments, i.e. an infinite number of atomic propositions, whose truth-values uniquely determine its truth-value. Of course, this definition provokes the questions: what is it good for?; what is its significance? We never form propositions out of infinite chains of atomic propositions. I will not discuss this objection now, but only explain why this objection does not bother Ramsey very much. The answer is entailed in the 'philosophical' idea: there may exist such infinite propositions, whether we can express them or not does not matter; their existence depends only on their objective being. Indeed Ramsey points out that we adopt a likewise 'objective' position in similar cases: their objective being. Indeed Ramsey points out that we adopt a likewis of the form ';*;' whether or not we have names for the values of x. General propositions must obviously be understood as applying to everything, not merely to everything for which we have a name, (ibid., p. 185) The next consideration is the decisive one; it leads from the notion of an 'atomic function of individuals' to the concept of a 'predicative function' in a new, more general sense, not to be confused with the old notion in Russell's PM. First, what is an 'atomic function of individuals'? '[It is] the result of replacing by variables any of the names of individuals in an atomic proposition; . . . The values of an atomic function of individuals are thus atomic propositions' (ibid., p. 189). Next we have to extend 'the idea of a truth function of propositions' (as above defined) to propositional functions (of atomic as well as composed propositions). Because the definition is rather complicated I sketch only its idea: the idea is to include functions (of atomic propositions) 'among the arguments of any truth-function'. Let me give an example: if i (^jjv) 5 02 (^j) with two free variables x,y, then to maintain that if)(x,y) is a truth-function of the functions (j)\ (x,y),
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111
A predicative'function of individuals is one which is any truth-function of arguments which, whether finite or infinite in number, are all either atomic functions of individuals or propositions, (ibid., p. 190) This procedure can be extended in an analogous manner to functions of functions of individuals, and so on. For example, 'an atomic function of (predicative) functions of individuals and of individuals can only have one functional argument, but may have many individual arguments'; it is a truth-function whose arguments are all either propositions or atomic functions of functions of individuals and of individuals. 'In this way we can proceed to define predicative functions of functions of functions and so on to any order' (ibid., p. 191). Now it's time to ask: (1) what did Ramsey achieve by inventing the new notion of'predicative function'; and (2) what did he pay, perhaps even sacrifice, for the success? The first question can be answered rather easily; there are two main points: (A) The set of predicative functions (of individuals) is well defined and. hence, a quantification over predicative functions is no longer logically ambiguous. Take for example the proposition (0)/(); the range of values of the variable is the set of all predicative functions of individuals $\,<$>i, ... and, since the domain of possible propositions for each predicative function i, 0 2 > . . . is well defined, the proposition ()/() has a definite significance.15 (B) There is no need for an axiom of reducibility because all functions of individuals (in the sense of PM) are predicative functions and, for this reason, belong to one and the same domain of predicative functions. These are remarkable achievements of Ramsey's theory of predicative functions. Nonetheless we have to ask: what is the price of this success? I will consider this question from two different points of view, an internal one, sharing Ramsey's confidence in logicism, and an external one that is not related with logicism; as representative of the second position I take, of course, Hilbert and ask: what would he have objected to Ramsey's proposal, if he had been acquainted wit There are two principal problems with Ramsey's theory of predicative functions, even from his own point of view. First, the already mentioned concept of'truth-function with infinitely many arguments' and, second, the problem that the definitions of'predicative functions of functions of individuals' are - their nomination not withstanding - 'impredicative' in Russell's sense and, consequently, violate the 'vicious circle' principle. In fact, Ramsey discusses both problems, but, small wonder, does not acknowledge them as serious objections. On the contrary, he tries to destroy them by some counterarguments, which are worthwhile to reflect on for a moment before we turn to Hilbert and consider how he might have objected.
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As already indicated, the first problem which arises from the idea of'truthfunctions with infinitely many arguments' Ramsey tries to counter-balance by the philosophical idea that there exist more things than we have names for to denote. We must now consider the significance and possible value of this idea is. In one sense the proposition expressing the philosophical idea is trivially true. Once we have convinced ourselves first that there actually exist infinitely many numbers and second that our means to denote them individually are strictly finite then, of course, the proposition is correct. That is just the argument underlying Richard's paradox, and insofar the proposition is question begging regarding the problem: how do we know that there exist infinitely many numbers? On the other hand, the proposition is far from obvious, if we do not assume that mathematical entities like numbers, functions, sets, etc. really exist, but are symbolic constructions. This does not mean to deny their 'objectivity', but the whole idea of a language transcendent recognition of mathematical objects is somehow obscure. Ramsey also seems to recognize this because he states it as a fact 'that the expressions "function of functions" and "function of individuals" are not strictly analogous; for, whereas functions are symbols, individuals are objects' (ibid., p. 187). Nonetheless, he defends his idea of'truth-functions with infinitely many arguments' with the remark: Admitting an infinite number [of arguments] involves that we do not define the range of functions as those which could be constructed in a certain way, but determine them by a description of their meaning. In this way we shall include many functions which we have no way of constructing. (ibid., p. 187) This seems to me incoherent, to say the least, and supposes a kind of realism (or platonism, if you like) to which I cannot agree. The same kind of realism occurs again in Ramsey's answer to the second problem regarding the circularity of impredicative definitions. Considering the objection that it would be a vicious impredicative definitions. totality of the 0's', Ramsey replies that this is not really a vicious circle, but only a 'circuitous process [having] nothing vicious about it'. In other words, Ramsey accepts impredicative definitions, but denies that they are circular in any vicious sense. Such a view is only coherent, I think, if one believes in the reality of mathematical functions and objects as one believes in the reality of cities and houses. Indeed, this is what Ramsey maintains: the impredicative definition ofafunction of functions as F(x) — ()f('s is 'merely to describe it in a certain way, by reference to a totality of which it may be itself a member, just as we may refer to a man as the tallest in
Ramsey's Removal of Russell's 'Axiom of R a group, thus identifying him by means of a totality of which he is himself a member without there being any vicious circle' (ibid., p. 192) So much for the internal difficulties of Ramsey's proposal. There are, of course, many more problems that I didn't touch on. Because most of them fall outside the internal point of view, let's turn to Hilbert and ask: would he have accepted Ramsey's simple theory of types or repudiated it in the end just like Russell's proposal? The answer is, of course, necessarily somewhat speculative, but I try to be honest. In The Foundations of Mathematics Ramsey rejects Russell's constructive point of view, not to speak of Weyl, and favours instead a language-transcendent view, as I'll call it, that 'include[s] functions which could not even be expressed by us at all, let alone elementarily, but only by a being with an infinite symbolic system'.17 Although Hilbert was no 'constructivist', I am convinced he would have sharply disagreed with this language-transcendent aspect of Ramsey's view and this for at least two reasons: the first concerns the problem of impredicative definitions. Although Hilbert did not share Russell's and Weyl's constructive point of view, he certainly regarded the problem of impredicativity as a real problem and did not dismiss it simply as linguistic pseudo-problem, as Ramsey does, when he claims that the problem is generated merely by 'our inability to write propositions of infinite length, which is logically a mere accident' (ibid., p. 192). This brings me to the second reason: Ramsey's treatment ofinfinity. In Ramsey's simple theory of types the infinite occurs not as & potential infinity of individual objects like the natural numbers but as an actual infinity of (atomic) propositions, which are not 'constructed' by us, but which exist perse; whether we can express them or not does not matter. The problem is ultimately to fix as values of/((#)) some definite set of propositions so that we can assert their logical product and sum. (In PM they are determined as all propositions which can be constructed in a certain way.) My method is to disregard how we could construct them, and to determine them by a description of their senses or import; and in so doing we may be able to include in the set propositions which we have no way of constructing, just as we include in the range of values of (j)x propositions which we cannot express from lack of names for the individuals concerned (ibid., p. 188-9). Again I am pretty sure that Hilbert would not agree, and this not only because it is fundamentally unclear what 'propositions' are 'which we cannot express' (note, that even axioms have to be expressed if they shall be of any practical use) but also because Hilbert does not share the realistic or platonic view that the infinite is something 'actually existing', let alone that it is 'actually given' to us just like a finite totality. Speaking of the infinite as something actually existing is, first at all, only a mode of'idealization', an 'ideal element',
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as Hilbert calls it. And this mode of speaking is only legitimate, if we canprove that the introduction of the infinite as a completed totality, as an ideal element into a theory, does no harm, i.e. does not lead to a contradiction within the theory in question. Let me come to the conclusion. I hope the foregoing considerations have made it sufficiently clear that Hilbert would not have accepted Ramsey's simple theory of types as a faithful solution of the problem of impredicative definitions and, hence, not as a reasonable way out of the fundamental difficulties which any attempt of pure logical foundation of mathematics has to face. Let me state it contra-factually. Even if Hilbert had been acquainted with Ramsey's simple theory of type this would not have changed his course of research regarding a new foundation of mathematics, namely moving away from logicism towards a theory of proof.
Endnotes 1. Although literally true, this statement is a bit misleading insofar as the lecture, 'Principles of Mathematics', of 1917—18 was published ten years later in a disguised form. What most people do not know is that the book, Grundzuege der TheoretischenLogik, Berlin, Springer, published by Hilbert and Ackermann in 1928, is based from the first line on the lecture of 1917—18. 2. I regard 'paradoxes' as a special instance of logical contradiction or inconsistency: a paradox is a contradiction, in which both assertions (of a pair of contradicting assertions) seem to be true, or (to express it more carefully) seem to have a proof, or, if not a proof, then at least convincing reasons of the same weight in favour of their truth. 3. This is not quite correct. There was, of course, Zermelo's set theory. But for reasons beyond the scope of this chapter, Hilbert did not regard set theory as a foundation of mathematics. 4. 'Mathematics' means in the following always arithmetic inclusive analysis; otherwise only'arithmetic'is used. 5. In spite of a gap in the documents between the end of the winter term 1917-18 and the summer term of 1920 it is difficult to determine Hilbert's change of mind more precisely. 6. This change is 'unexpected', of course, only from an earlier perspective, in particular from the perspective of the lecture 'Prinzipien der Mathematik' of 1917-18. Having discussed Russell's simple theory of types at great length Hilbert closes this lecture with the following optimistic remark: So zeigt sich, dass die Einfuhrung des Axioms der Reduzierbarkeit das geeignete Mittel ist, um den Stufenkalkiil zu einem System zu gestalten, aus welchem die Grundlagen der hoheren Mathematik entwickelt werden konnen. 7. The precise point is difficult to determine. Anyway, it must have been before he presented his lecture in Zurich.
Ramsey's Removal of Russell's 'Ax 8.
See Weyl's Kontinuum, §2, pp. 4—8 but also his Philosophy of Mathematics and Natural Science, chapter I. 9. See Weyl's Kontinuum., §6, pp. 23—4. 10. And, of course, all other known paradoxes such as Richard's paradox, etc. 11. It is, of course, only a difficulty for the logicist, who wants to reduce set theory to pure logic, not for the axiomatist, who does not believe in logicism and, hence, does not insist on a complete reduction of set theory to pure logic. 12. 'Tautologies', of course, in a special sense, explained by Ramsey in accordance with Wittgenstein's view in the Tractatus. 13. The other method is Weyl's constructive procedure, which excludes impredicative definitions of predicates. 14. By 'atomic proposition' Ramsey means any proposition, free of logical connectives and 'expressed by using names alone', which means free of any variables, whether bound or unbound. 15. Corresponding arguments can be given for more complex predicative functions l i k e f ( ( j ) ( z } , x hence, itself predicative if the instances are. 16. There is no hint in Hilbert's work, as far as I can see, that he was acquainted with Ramsey's work. 17. See Ramsey (1925), in (1978), p. 193. This is by no means the only utterance of this kind, the whole essay is spoiled with remarks like this.
References Hilbert, D. (1917/18). Prinzipien der Mathematik. Unpublished typescript. Library of the Mathematical Institute, Universitat Gottingen. (1920). Problems der mathematischen Logik. Unpublished typescript. Library of the Mathematical Institute, Universitat Gottingen. Ramsey, F. P. (1978). Foundations: Essays in Philosophy, Logic, Mathematics and Economics. Mellor, D. (ed.), London: Routledge and Kegan Paul. Weyl, H. (1918). Das Kontinuum. Kritische Untersuchungen uber die Grundlagen der Analysis. Leipzig.
8 Ramsey and pragmatism: the influence of Peirce Christopher Hookway
Although there are a number of different and important questions about Ramsey's relations to pragmatism, they fall into two broad groups. First, we can ask about Ramsey's role of transmitting the ideas of the classical American pragmatist tradition to philosophers in the British tradition of analytical philosophy. How many of Ramsey's ideas were influenced by the writings of the pragmatists? How far did Ramsey contribute to the impact of pragmatist ideas upon the later philosophy of Wittgenstein? Second, we can ask how far ideas drawn from Ramsey's work have provided tools for contemporary pragmatists. Of course, the two questions are related: ideas which reflected the influence of the earlier pragmatists may loom large among those that have influenced more recent ones. I shall concentrate more upon the first set of issues than the second, but I shall use some answers to the second as guidance in exploring how we should answer the first. Questions in the first group can themselves be subdivided. We can examine the effects on Ramsey's work of his reading of thinkers commonly described as 'pragmatist': most important here is his reading of some of Peirce's early writings and his exposure to his work on the theory of signs. And we can also look at his remarks about 'pragmatism' and to his exposure to pragmatism as a philosophical position. This distinction is important because some of the themes from Peirce's writings which influenced Ramsey may not be views which Peirce himself would have described as characteristic of his pragmatism. And when Ramsey reports that his 'pragmatism' comes from his reading of Russell, this may raise questions about just how much continuity there is between Ramsey's pragmatism and that of Peirce and James. 'Pragmatism' Where does Ramsey discuss pragmatism? In 'Facts and Propositions', after acknowledging that everything he says is due to Wittgenstein 'except the parts which have a pragmatist tendency', he concludes: My pragmatism is derived from Mr Russell; and is, of course, very vague and undeveloped. The essence of pragmatism I take to be this, that the
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meaning of a sentence is to be determined by reference to the actions to which asserting it would lead, or, more vaguely still, by its possible causes and effects. Of this I feel certain, but of nothing more definite. (1927, Ramsey 1990, p. 51) And at the end of Truth and Probability., commenting on his defence of the rationality of relying upon induction, he concludes: This is a kind of pragmatism: we judge mental habits by whether they work, i.e. by whether the opinions they lead to are for the most part true, or more often true than those which alternative habits would lead to. Induction is such a useful habit, and so to adopt it is reasonable. All that philosophy can do is to analyse it, determine the degree of its utility, and find on which characteristic of nature this depends. An indispensable means for investigating these problems is induction itself, without which we should be helpless. In this circle lies nothing vicious. (1926, Ramsey 1990, pp. 93-4) These passages make slightly different points: the former treats pragmatism as a tool for clarifying the meanings of sentences; and the second identifies it with a role for assessing the reasonableness of'habits' or, more generally, rules of inference. Peirce usually uses 'pragmatism' to refer to the former. James's introduction of the term in Pragmatism has a similar character although, as when he says that pragmatism is a theory of truth, he also uses it in the other looser sense. What they have in common is a kind of consequentialism: the meaning of a sentence is a matter of what its acceptance would commit us to in the way of actions and expectations; and the habit of inference is judged by the consequences of accepting it. The most striking superficial difference is that, where the former deals in sentences, the latter is concerned with habits. But this may not be an important difference: Peirce's discussions of pragmatism frequently suggest that we may clarify a sentence by identifying the habits (of inference, action and expectation) that we acquire when we accept it as true. Belief and action During the late 1870s, Peirce published a series of papers under the running title 'Illustrations of the Logic of Science' (1992, pp. 109ff). The first two papers in this series, 'The Fixation of Belief and 'How to Make our Ideas Clear', introduce a framework for thinking about issues in logic and epistemology which shaped the development of pragmatism. James subsequently
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referred to these papers as the first public manifestation of pragmatism, and John Dewey's Logic: The Theory of Inquiry (1938) develops the account of inquiry that is presented in the first of these papers. Although the word 'pragmatism' is not used in these papers, the second presents the pragmatist principle for clarifying the content of hypotheses and concepts. This paper also offers an early formulation of Peirce's ideas about truth, and 'The Fixation of Belief develops ideas about belief, doubt and inquiry which, again, have influenced most subsequent pragmatists. These papers were included in the anthology of Peirce's writings that Ramsey knew and referred to in 'Truth and Probability'. Thus it will be useful to organize the first part of our discussion around what he would have learned from these papers. In 'The Fixation of Belief', inquiry is identified as a process through which doubt is replaced by settled belief and Peirce asks what rules we should follow in carrying out such inquiries. We should begin by asking what beliefs. Peirce's first claim is that 'Our beliefs guide our desires and shape our actions The feeling of believing is a more or less sure indication of there being established in our nature some habit which will determine our actions' (1992, p. 114). In the second paper, we are reminded that acquiring a belief 'involves the establishment in our nature of a rule of action, or say, for short, a habit. Peirce concludes that 'The essence of a belief is the establishment of a habit, and different beliefs are distinguished by the different modes of action to which they give rise. If beliefs do not differ in this respect... then no mere difference in the manner of consciousness of them can make them different beliefs' (1992, pp. 129—30). According to Peirce, 'pragmatism is scarce more than a corollary' of this view of belief which reflected the influence of Alexander Bain upon his thought. This is clear from the way in which, in 'How to Make our Ideas Clear', Peirce proposes that the most explicit clarification of a propositions is one that makes manifest exactly which rule of action, which habit, the belief involves. Inquiry begins when one of our beliefs fails: It is certainly best for us that our beliefs should be such as may truly guide our actions so as to satisfy our desires; and this reflection will make us reject any belief which does not seem to have been so formed as to insure this result. But it will only do so by creating a doubt in the place of that belief. With the doubt, therefore, the struggle begins, and with the cessation of doubt, it ends. (1992, p. 114) This means that a belief which survives the process of inquiry is one that has, so far, enabled us to satisfy our desires. Inquiry selects for beliefs which are successful in guiding action; if a belief would be successful in guiding action in all circumstances, then, in the course of inquiry, it should never be doubted, never
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have been abandoned. (Of course, this is a first crude formulation of a complex idea. For example, account would have to be taken of the fact that our actions generally reflect the influence of many beliefs and difficult decisions must be made about which to blame when things go wrong. I shall ignore these complexities here.) Most of these themes are present in Ramsey's writings and it is plausible that it was his reading of Peirce that led him to think of belief in this manner. After documenting this, I want to explore the ways in which Ramsey's development of this perspective went beyond what is found in the earlier pragmatist tradition. We can start from a well-known passage from 'Facts and Propositions': It is, for instance, possible to say that a chicken believes a certain sort of caterpillar to be poisonous, and mean by that merely that it abstains from eating such caterpillars on account of unpleasant experiences connected with them. The mental factors in such 'belief would be parts of the chicken's behaviour, which are somehow related to the objective factors, viz the kind of caterpillar and poisonousness. An exact analysis of this relation would be very difficult, but it might well be held that in regard to this kind of belief, the pragmatist view was correct, i.e. that the relation between the chicken's behaviour and the objective factors was that the actions were such as to be useful, if, and only if, the caterpillars were actually poisonous. Thus any set of actions for whose utility p is a necessary and sufficient condition might be called a belief that p., and so would be true if/>, i.e. if they are useful. (1927, Ramsey 1990, p. 40) Nils-Eric Sahlin suggests that this passage indicates how Ramsey thought that a Peircean analysis of belief could be carried out (Sahlin 1990, p. 70). Ramsey himself treated the example as a special case which would not provide a model for thinking about beliefs which are 'the most proper object for logical criticism', namely those whose mental factors include 'words, spoken aloud or to oneself or merely imagined, connected together and accompanied by a feeling or feelings of belief or disbelief (Ramsey 1990, p. 40). The contrast here, then, is between what we can call 'animal beliefs' (these consist solely in habits of behaviour) and beliefs whose manifestation consists in a sort of'inner sentence' or inner representation. The latter possess logical structure and the conscious deployment of'words': they are manifested, we might say, in judgements or assertions. Since our belief can be manifested both in what we say (or judge) and in what we do, it is possible for there to be a gap between these manifestations: we can irrationally fail to do what, given our judgements or assertions, we ought to do. (Just as we can irrationally fail to believe what, given our other beliefs and judgements, we ought to believe.)
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Much of the time, when Ramsey is discussing belief and judgement, he is concerned with these conscious manifestations, with items that possess a more complex and sophisticated sort of logical structure than mere animal habits of action. In that case, attending to how beliefs are manifested in action will teach us little about what (non-animal) beliefs are, about their inner structure and constitution. However, the relations between belief and action do take centre stage in some of his most influential writings on this topic. This raises a question about just what the relations are between conscious judgements and 'habits of action'. Although 'The Fixation of Belief' can give the impression that Peirce treats animal belief as the paradigm examples of beliefs, that he treats all beliefs as mere habits of action, this appearance is misleading. When we study his writings on the theory of signs, we find that judgements involve the manipulation of iconic representations of reality, logical diagrams of states of affairs which are anchored to external things through the use of indexical signs such as demonstratives, and whose iconic character means that we can extend our knowledge through conscious inference. We experiment upon our iconic presentations, making substitutions that are licensed by logic, and thus arrive at new representations which bring to bear the information we possess to make new judgements. How far Ramsey was aware of these writings of Peirce's is not clear. But it is manifest that both want to defend an account of belief which combines two elements: Conscious beliefs (at least) involve representations of reality which ar made up of things like words and which display a logical structure which suits them for use in inference. Conscious beliefs (at least) involve representations of reality determination of action which makes it appropriate to regard them as embodying habits of action. In his later work, after around 1903, Peirce attempted to defend his pragmatism by showing that, for certain purposes, the most useful kind of clarification of the content of a belief is one that makes explicit the habits of expectation we acquire when we come to hold it. Tracing the relations between propositions and habits of action and expectation has a central role in explaining the sort of understanding of propositions we require if we are to carry out inquiries responsibly and effectively. Thus he hopes to combine the association between beliefs and habits with a view of beliefs as possessing logical structure and as composed of'words'. Even if Ramsey was sceptical that the connection between belief and action would provide a clue to the proper analysis of the nature of belief and judgement, he was soon using it for a different purpose. In Truth and Probability., he addresses the question how we should measure the strength or degree of
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belief. And after criticizing the suggestion that the degree of a belief'is something perceptible by its owner', he turns to a second idea, that 'the degree of a belief is a causal property of it, which we can express vaguely as the extent to which we are prepared to act on it'. The difference between believing more or less firmly lies in 'how far we should act on these beliefs' (Ramsey 1990, pp. 65-6). I shall leave it to other authors to explore the details of Ramsey's account of degree of belief and his development of a system of decision theory. It is worthwhile to note here that this is a natural, and almost inevitable, development of the pragmatist account of the connections between belief and action that, so far as I can see, the pragmatists did not, themselves, develop at all. As we saw, Peirce claimed that it was of the essence of a belief that it established a rule or habit of action. As Russell and others have noted, this claim is potentially misleading. It seems likely that many true beliefs are useless, not all beliefs lead to action (see Dokic and Engel 2002, p. 8, fn). I may simply have no occasion to act on my beliefs that it just stopped raining or nineteen is a prime number. This may be a serious objection to the view that a belief is a set of actions, as Ramsey's quotation above may suggest. Neither Peirce nor Ramsey is committed to this view. Peirce holds that 'belief does not make us act at once, but puts us into such a condition that we shall behave in a certain way, when the occasion arises'. And, in similar spirit, Ramsey insists that 'a belief is not an idea that does actually lead to action, but one which would lead to action in suitable circumstances; just as a lump of arsenic is called poisonous not because it has or will kill anyone, but because it would kill anyone if he ate it' (1926, Ramsey 1990, p. 66; and see Dokic and Engel 2002, p.8). Beliefs are 'habits' or dispositional states. What is significant is that Peirce showed no inclination to work out a theory of the regularities governing in which circumstances a given belief will determine an action. Whether we act on a belief appears to depend upon what we desire, upon the strength of those desires, upon our other background beliefs and upon their strength. And, as Ramsey saw, in order to work out such regularities, we need to work out a theory of the degrees or strength of belief.
Fallibilism, degrees of belief and the explanation of behaviour The main difference between Ramsey and Peirce here is easy to describe. As Ian Hacking records in The Emergence of Probability, ever since scholars began to take the notion seriously, probability has been 'Janus-faced': On the one side it is statistical, concerning itself with stochastic laws of chance processes. On the other side it is epistemological, dedicated to
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assessing reasonable degrees of belief in propositions quite devoid of statistical background. (1975, p. 12) The statistical notion tends to be objective: it is employed when we talk about the probability of a particular coin coming up heads when tossed, or of an overweight person who smokes twenty cigarettes a day living to the age of seventy, or a sample of radioactive uranium decaying within a certain period, and so on. The other is employed when we judge that, given our evidence, the probability that it will rain tomorrow morning is .7, or that the experimental results suggest that a theory is probably true. John Venn described this contrast by distinguishing 'materialist' and 'conceptualist' conceptions of probability; and Carnap distinguished 'statistical' from 'inductive' probabilities. Ramsey's Truth and Probability (1990, pp. 52ff) is a locus classicus for the development of a systematic understanding of the epistemological, conceptualist, inductivist conception. Peirce rejected Venn's conceptualism and used his pragmatism to defend the view that only the objective, statistical, materialist conception can be defended. Writers differ about the relations between the two. Some write as if there is just one fundamental conception of probability (either statistical or inductivist) and propose to explain the other in terms of this fundamental one. Others discern a family of equally fundamental concepts, unified by the fact that they exploit the formal structures described in the probability calculus, and trace a variety of formal connections between them. Ramsey takes degrees of belief seriously and he identifies them with subjective probabilities or with degrees of credence. In 'The Probability of Induction', a paper from the 1877—8 series that Ramsey knew, Peirce seems to take a similar view: [I]t is incontestable that the chance of an event has an intimate connection with the degree of our belief in it. Belief is certainly something more than a mere feeling; yet there is a feeling of believing, and this feeling does and ought to vary with the chance of the thing believed, as deduced from all the arguments. Any quantity which varies with the chance might, therefore, it would seem, serve as a thermometer for the proper intensity of belief. (1992, p.. 158) However, as the presence of the word 'chance' here might indicate, there is a major difference in the two positions. Already in a review of John Venn's The Logic of Chance, published in 1867, Peirce rejected any use of'degree of credence' in the analysis of probability as an unwarranted form of psychologism. Indeed:
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In order that the degree of credence should correspond to any truth in the long run, it must be the representation of a general statistical fact - a real objective fact. And, then, as it is the fact which is said to be probable, and not the belief, the introduction of'degree of credence' into the definition of probability is as superfluous as the introduction of a reflection upon a mental process into any other definition would be. (Peirce 1984, p. 100) So, for Peirce, the primary kind of probability is an objective statistical notion, one that he sometimes called 'materialistic'. It is possible that degrees of credence should be (and maybe are) sensitive to certain objective probabilities. But that is the only kind of probabilistic input into the measurement of degrees of belief that can be tolerated. The mature Peirce treats probabilities as propensities and the younger Peirce treats them as limiting frequencies. Ramsey rejects this form of realism about probabilities. Many of their differences stem from this fundamental difference. This is clear from the argument of'The Probability of Induction'. Following the passage cited above, Peirce produces an argument that degrees of belief should be measured in accordance with the 'logarithm of the chance'. We do not need to explore his reasons for this view, which depend on the desire to compute degrees of belief by addition rather than multiplication and on the desire to extend Fechner's views about the degrees of intensity of sensations to the degrees of other mental states. What is interesting is his observation that these claims constitute an argument in favour of the conceptualistic view of probability, and his reaction to this (1992, p. 159).
Truth In 'Facts and Propositions' (1927, included in Ramsey 1990), Ramsey undertakes to show that 'there is really no separate problem of truth but merely a linguistic muddle': Truth and falsity are applied primarily to propositions. The proposition to which they are applied may be either explicitly given or described. Suppose ... that it is explicitly given; then it is evident that Tt is true that Caesar was murdered' means no more than that Caesar was murdered, and Tt is false that Caesar was murdered' means that Caesar was not murdered. (Ramsey 1990, p. 38) The extension of this idea to the analysis of cases where the proposition is described is familiar and need not detain us here. The moral that Ramsey
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draws from his discussion is twofold: the problems we face when giving a formal account of truth are not problems as to 'the nature of truth and falsehood, but as.to the nature of judgment or assertion' (p. 39). Indeed, he continues, 'It is, perhaps, also immediately obvious that if we have analyzed judgment we have solved the problem of truth.' There may be interesting issues about the role of the elements of judgements, assertions, beliefs, etc. in enabling us to represent or deal with our surroundings. But once we have solved these problems, there is no additional problem about truth. Let us compare this with a well-known passage from Peirce's 1877 paper 'The Fixation of Belief. Peirce's paper is the first of a series in which he aims to identify the most fundamental norms (or 'guiding principles') which govern our processes of belief formation and inquiry. In order to do this, he seeks to identify the main conceptions we must use in posing the question — in posing what he sometimes calls 'the logical question'. As he insists, inquiry is an activity which seeks to remove the irritation of'doubt' — to solve some problem we face or answer a question that disturbs us - and to come to believe some proposition which provides the solution to our problem or the answer to our question. He tells us that 'the sole object of inquiry is the settlement of opinion' and continues: We may fancy that this is not enough for us, and that we seek, not merely an opinion, but a true opinion. But put this fancy to the test, and it proves groundless; for as soon as a firm belief is reached we are entirely satisfied, whether the belief be true or false. And it is clear that nothing out of the sphere of our knowledge can be our object, for nothing which does not affect the mind can be the motive for a mental effort. The most that can be maintained is, that we seek a belief that we shall think to be true. But we think each one of our beliefs to be true, and, indeed, it is mere tautology to do so. (1992, p. 115) Where Ramsey says that 'It is true that Caesar was murdered' 'means no more than' that Caesar was murdered, Peirce says that it is 'mere tautology' that anyone who believes that Caesar was murdered believes that it is true that Caesar was murdered. I appreciate that these claims are not the same: Peirce does not say that the belief that it is true that Caesar was murdered and the belief that Caesar was murdered are the same belief, whereas Ramsey may be committed to holding that. But the similarities go beyond this, for Peirce seems to hold that the concerns that lead us to 'fancy' that our object or aim in inquiry is truth can be fully accommodated by taking seriously that our aim is genuine belief or judgement. The analysis of what follows from holding that our aim in inquiry is settled belief will suffice to enable us to answer the questions that we sought to pose by asking what rules we should follow in order to ensure that our beliefs are actually true. The latter adds nothing.
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What is additionally interesting is that both Peirce and Ramsey do succeed in saying something more substantial about truth. In the second paper of the series in which 'The Fixation of Belief' was the first, Peirce developed his pragmatist principle for the clarification of ideas and, in one of its first applications, offered the following cryptic formulation of a clarification of truth: The opinion which is fated to be agreed to by all who investigate, is what we mean by the truth, and the object represented in this opinion is the real. (1992, p. 139) What this amounts to is not immediately clear - the examples suggest that this convergence in opinion will occur only among those who try to solve this particular problem, to answer this specific question. Fortunately the details are not important here. The suggestion seems to be that it forms part of our everyday conception of truth that if a proposition is true, then someone who inquires into it will be prevented from believing it only through lack of evidence, possession of misleading or inadequate evidence, or through mishandling the evidence or other considerations that are available. We must adopt normative standards that reflect the 'fundamental hypothesis' that: There are real things, whose characters are entirely independent of our opinions about them; those realities affect our senses according to regular laws, and, though our sensations are as different as our relations to the objects, yet, by taking account of the laws of perception, we can ascertain by reasoning how things really are, and any man, if he have sufficient experience and reason enough about it, will be led to the one true conclusion. (1992, p. 120) Although Peirce believed that he could derive this conclusion, what is important is that this was not derived from any special consideration of the nature of truth. In 'The Fixation of Belief, he derived these conclusions by reflecting on the nature of rewtitiwqo and dosut unless these cnasosdffditi are met he seems to have thought, reflective belief will inevitably give rise to doubt. It reflects an account of the commitments we incur when we believe something (or, in later work, when we assert or judge it). So we can say something substantive about truth; but there is no independent problem of truth. The account of truth 'falls out' of an account of rational belief formation. Something similar occurs in Ramsey's work. Many scholars ascribe to him what is called a 'success theory' of truth, based on passages such as: For a belief that A is B, means... a belief which is useful if and only if A is B, i.e. if and only if it is true; and so, conversely it will be true if and only if it is useful. (Ramsey 1991, p. 91)
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As Dokic and Engel explain, this commits him to such claims about truth as: aTrue beliefs are those that lead to successful actions whatever the underlying motivating desires. aTrue beliefs are those that lead to successful actions based on that belief whatever the underlying motivating desires. (2002, pp. 45-6) And this ('Ramsey's principle'), once more, is a claim that follows from reflections on the rationality of belief and action, and on the correctness of belief, rather than upon an independent investigation of traditional problems about truth. Indeed, the links between Ramsey's principle and Peirce's theory of truth as convergence are close. According to Ramsey, true beliefs are those that lead to successful actions; and according to Peirce they lead to successful experiment and expectation. If a belief leads to unsuccessful action and we cannot blame this on some erroneous background knowledge, that provides a reason to replace that belief by doubt; and if it leads to unsuccessful experiment or to disappointed predictions, then, once again, the belief should give way to doubt unless we can blame the failure on our background knowledge. The structure of Ramsey's linkage of true belief and successful action is much the same as Peirce's linkage of true belief and long-run stable convergence in opinion. Moreover, Ramsey has offered a limited endorsement of Peirce's claims about truth and convergence in his unpublished writings on 'Law and Causality'. Referring to systems of scientific laws, he wrote that: We do ... believe that the system is uniquely determined and that long enough investigation will lead us all to it. This is Peirce's notion of truth as what everyone will believe in the end; it does not apply to the truthful statement of matters of fact, but to the 'true scientific system'. (1929, Ramsey 1990, p. 161) Many ordinary facts may be lost forever and we will not arrive at a stable consensus upon whether they occurred. Systems of universal law are another matter: they will continue to operate into the future and, if they are not correct, we can expect that properly conducted inquiry will reveal this to us. Some conclusions We have noted a number of parallels between the views of Peirce and Ramsey about inquiry, belief and truth. How far these reflect the impact upon Ramsey of his reading of Peirce's writings, and how far we are witnessing another
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instance of ideas whose 'time has come' being discovered and rediscovered by a number of different thinkers, is something I shall not try to settle here. The most striking difference between the two is their very different stands on the nature of probability. Why — given the similarities elsewhere in their views - was Peirce so dismissive of subjective or conceptualist accounts of probability, and Ramsey so ready to embrace them? And this question is particularly puzzling when we note how Ramsey's work on decision theory seems to be such a natural development of Peirce's claims about beliefs and their relations to action. My suspicion is that this question is of more interest for Peirce scholars than it is for the student of Ramsey. The answer lies in themes elsewhere in Peirce's work — his extreme realism which led him to take real objective probabilities seriously, his anticipation of a Popperian view of the history of science which emphasized the role in scientific progress of a willingness to seek refutations of one's views as a spur to arriving at new ways of making precise our vague pictures of how nature works, and his 'common-sense' mistrust in appeals to laws and theories as guides to dealing with practical problems, problems of how to act. Peirce seems simply to have been uninterested in finding a systematic theory of how beliefs guide actions. It took Ramsey's philosophical insights to show how important this is, and to begin to carry out the project.
References Dewey, J., 1938. Logic: The Theory of Inquiry. Boston: Holt, Rinehart and Winston. Dokic, J. and Engel P., 2002. Frank Ramsey: Truth and Success. London: Routledge. Hacking, I. M., 1975. The Emergence of Probability. Cambridge: Cambridge University Press. Peirce, C. S., 1984. Writings of Charles S. Peirce: A Chronological Edition, Vol. 2. Indianapolis: Indiana University Press. 1992. The Essential Peirce, Vol. 1, ed. Houser, N. and Kloesel, C. Indianapolis: Indiana University Press. Ramsey, F. P., 1990. Philosophical Papers (ed. D. H. Mellor). Cambridge: Cambridge University Press. 1991. On Truth, ed. Rescher N. and Kajer, U. Dordrecht: Kluwer. Sahlin, N.-E. 1990. The Philosophy ofF. P. Ramsey. Cambridge: Cambridge University Press.
9
Ramsey and the prospects for reliabilism1 Daniel Quesada
It is generally agreed that simply holding a true belief does not amount to having knowledge. In Theaetetus Plato discussed what is lacking in between (see 201a-201d), and it is not unnatural to interpret him as telling us to look for a. justifying reason for the beliefs that we hold. However, if we look at his example of judges issuing what might be inappropriate verdicts, neither is it too far-fetched to extract the idea that the problem arises because they do not arrive at their verdicts in the right way. The interaction of these two kinds of requirements, as they are understood in contemporary thought, has played an important role in recent epistemology. But the idea of there being a 'right way' to arrive at beliefs, and looking at this as a possible requirement for knowledge, seemed to be lost to philosophy for a long time, until Frank Ramsey suddenly put it forward in his extremely short 1929 piece 'Knowledge'.
Ramsey's definition of knowledge Ramsey characterizes knowledge as belief which is '(i) true, (ii) certain, (iii) obtained by a reliable process' ('Knowledge', in Philosophical Papers, p. 110).2 We may think that the two last elements of this general characterization are the most properly epistemological, and so concentrate on them. Unfortunately, Ramsey does not elucidate the requirement for a belief to be certain. One plausible possibility is that he meant any belief of whose truth one is sure, or a full belief. Nils-Eric Sahlin, who in the chapter devoted to Ramsey's theory of knowledge commends this alternative, claiming that Ramsey 'must have been influenced by what Russell writes about knowledge in The Problems of Philosophy' (Sahlin 1990, p. 90), and he quotes Bertrand Russell's general characterization at the end of Chapter 13 in this classic work: What we firmly believe, if it is true, is called knowledge, provided it is either intuitive or inferred (logically or psychologically) from intuitive knowledge from which it follows logically. What we firmly believe, if it is not true, is called error. What we firmly believe, if it is neither knowledge nor error, and also what we believe hesitatingly, because it is, or is derived from,
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something which has not the highest degree of self-evidence, may be called probable opinion. (Russell 1986, p. 81) Sahlin does not say why he thinks that Ramsey must have been influenced by Russell on the point in question. But if we were only to judge by this influence, we should perhaps be less sure of the proposed interpretation, for it is not clear that influence from The Problems of Philosophy should run in the direction suggested by Sahlin.3 As it happens, throughout the book Russell reserves the word 'certain' for epistemological safety. Being himself in the grip of Cartesian fundamentalism, Russell is on the look-out for self-evidence and certainty. To be absolutely certain of something, in Russell's sense, is roughly to have justification of a kind such that it is impossible to be wrong. Other degrees of certainty should then correspond to inferior kinds or lesser degrees of justification. Certainty in this sense would correlate to firmness of belief only in rational people or people who are correctly able to assess their justification. Thus, if influenced by Russell, it might be that these nuances of meaning are also present in Ramsey's use of'certain'. Moreover, there is a relevant possibility that seems to have slipped from Russell's characterization. For, in Russell's view, what should be said of the case in which I hesitatingly believe that p, but the reason of my hesitance is that I have not realized that p logically follows from some self-evident beliefs? If this were also a case of knowledge, the unchanging association of knowledge with firm belief would be broken. Be that as it may, it is Ramsey's next requirement that is the most interesting. As we will see, this condition deviates from more traditional tripartite characterizations of knowledge. And in this case we do at least have some comments by Ramsey himself, even if tantalizingly brief and fragmentary. Immediately after giving his definition of knowledge and thus enunciating the reliability requirement, Ramsey expresses serious qualms about the adequacy of the word 'process' to express this requirement properly: But the word 'process' is very unsatisfactory ... Can we say that a memory is obtained by a reliable process? I think perhaps we can if we mean the causal process connecting what happens with my remembering it. We might then say, a belief obtained by a reliable process must be caused by what are not beliefs in a way or with accompaniments that can be more or less relied on to give true beliefs, and if in this train of causation occur other intermediary beliefs these must all be true ones, (ibid., p. 110) I see my keys on my bedside table. Taking this at face value, I form a belief about where my keys are (if someone asked me now, I would answer that they are on that table). At a later moment the need arises for me to pick up the keys. I then remember that they are on-the table. Presumably the process: keys
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lying on the table-perception-belief-memory is a reliable process in Ramsey's intended sense. The intermediary belief (that the keys are on the bedside table) must be true. Similarly, if the process is one of inference, 'reliable process', Ramsey tells us, should not only involve a correct logical method, but true premises (ibid.}. Ramsey, however, is not happy with this, since it is more doubtful that the word 'process' can be made to fit in other cases. Take any of the examples he explicitly mentions: telepathy, female intuition, or impressions of character. Is the word 'process' applicable here? If we believe that, for example, in the case of telepathy, there is such a process, the question of whether telepathy can give us knowledge turns on the question of whether such a process 'can be relied on to create true beliefs in the telepathee (within some limits ...)' (ibid.). But maybe we do not believe there is such a process (perhaps because we deny or doubt that telepathy gives us knowledge). The question of whether telepathy is knowledge or gives us knowledge may then be the question of whether the feeling of being telepathed can (more or less) guarantee truth. Hence, perhaps the condition for knowledge should be put in a more general form than is possible by using the word 'process'. 'Perhaps we should say not (iii) obtained by a reliable process, but (iii) formed in a reliable way,' says Ramsey (ibid.). Whatever the proper way of formulating the reliability requirement is, Ramsey points out that we say we know 'whenever we are certain, without reflecting on reliability' (ibid.). However, a normative condition on being certain is imposed by reflecting on reliability: '... if we did reflect then we should remain certain if, and only if, we thought our way reliable'. This holds whether we can be said to know whether such a way is reliable or not. In either case, we should adjust certainty in belief to assumed reliability as soon as we reflect that reliability is (may be) present. But Ramsey does not require a link to exist between reliability as such on one hand, and conviction or justified conviction on the other. Thus, our obtaining the belief at issue in a reliable way is neither a sufficient nor a necessary condition for 'being certain'. And, even if knowledge of reliability (or belief in it) suffices for a high degree of conviction or justified conviction, nothing is said about whether it is necessary for either of those two things. Moreover, there is no sign that knowledge of reliability (or belief in it) is regarded by Ramsey as necessary for knowledge. Dokic and Engel (2002), who discuss the matter, arrive at the conclusion that it is not so regarded (see pp. 28-9 of their book). There is some room, however, for discussion about whether or to what extent Ramsey would require ever, for discussion about whether or to what extent back as implicitly expressing the complaint that we do not reflect on reliability often enough, but it would be more strained to read it as hinting that we should require reflection on reliability, although perhaps even this reading is not too far-fetched.
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Be this as it may, one further matter for reflection concerns the issue of what is involved in thinking a certain way reliable. What Ramsey says about this is that 'to think the way reliable is simply to formulate in a variable hypothetical the habit of following the way' ('Knowledge', p. 110). A 'variable hypothetical', in Ramsey's terminology, is a conditional statement of the form 'All Fs are Gs' which expresses a general belief regarded as a rule of action. According to this, Ramsey is conceiving of a person who thinks that a certain way or process is reliable as somebody who is aware of a certain habit or rule of action. It does not seem that we can go further than this in interpreting Ramsey on this matter (but see Dokic and Engel 2002, pp. 29-30, for further discussion). Nevertheless, I believe that something like my proposal for characterizing awareness of reliability in the case of perception, which will be discussed at the end of this chapter, does not lie far away from what Ramsey has in mind here. The issue about the right formulation of the reliability condition, which surrounds the word 'process', and which is explicitly raised by Ramsey, as well as issues about the relationship between reliability, knowledge of reliability, justified or rational belief and knowledge, which are at least suggested by his observation about the relationship between reflection on reliability and 'being certain', was to occupy later epistemologists. It was around these questions that controversial issues in the discussion of reliability would gather.
Ramsey, Russell and the Gettier cases How did Ramsey arrive at his view? There are some clues in Russell's Problems of Philosophy (Problems, henceforth) that suggest a plausible path. Thus, at the beginning of Chapter 13, Russell discusses an important case type as follows: . . . a true belief cannot be called knowledge when it is deduced by a fallacious process of reasoning, even if the premises from which it is deduced are true, (ibid., p. 76) But, of course, as this text implies, it cannot be called knowledge either if the premises are not true. Now, as anticipated above, Ramsey explicitly takes cases of inference into account at the beginning of his brief discussion of the definition of knowledge. This happens immediately after a warning that the word 'process' is unsatisfactory, and just before discussing memory cases, as follows: But the word 'process' is very unsatisfactory; we can call inference a process, but even then unreliable seems to refer only to a fallacious method not to a false premise as it is supposed to do. Can we say that a memory is obtained
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by a reliable process? I think perhaps we can if we mean the causal process connecting what happens with my remembering it. ('Knowledge', p. 110) It is easy to take this as Ramsey's first giving what should be a reasonably unproblematic case (type) of the application of the word 'process', and then complaining that even in this case we should be prepared to apply the term to more than what most easily comes to mind, that is, to the premises as well and not just to the 'method' (as indeed happens to some extent in the quoted text by Russell). I suggest that Ramsey's believing reliability to be a necessary condition for knowledge comes from his reflecting first on what is common to two (or perhaps three) definite kinds of cases of knowledge failure. All of these are cases of true beliefs, and the two which might have turned out to be most significant are very different from each other. These cases, acting as the 'base line' for his definition, would be exactly those that Russell proposes at the beginning of Chapter 13 in Problems. According to my reconstruction, Ramsey would have reflected further on the possibility of extending the conclusion reached in the analysis of these first cases - case-types - both to ordinary cases of knowledge — such as memories — and to more far-fetched cases (like telepathy, female intuition or impressions of character) where the attribution of knowledge is indeed problematic, thus using the latter as a test for his definition proposal.4 One of the main cases is that of fallacious inference. This is the only one to which Russell explicitly applies the word 'process', and we have already seen Ramsey voicing his qualms about this. The other most relevant case — occurring in Russell's text just before the inference case — is, of all cases... a Gettier case! It runs as follows: If a newspaper, by an intelligent anticipation, announces the result of a battle before any telegram giving the result has been received, it may by good fortune announce what afterwards turns out to be the right result, and it may produce belief in some of its less experienced readers. (Problems, p. 76) To my mind, the case is described somewhat confusedly. 'Intelligent anticipation' suggests that the results of the battle could be somehow reasonably anticipated, while 'good fortune' might suggest a chance hit. Moreover, and more relevantly, I do not find it natural to restrict belief in what the newspaper announces to 'some of its less experienced readers'. I do not know about Russell's times (is it an ironic comment about the press on Russell's part?) but if a reputable newspaper announced the results of a battle 'by an intelligent anticipation', most people would be taken in (unless there are general grounds for thinking that the press is being influenced by military propaganda).
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The possibility of treating the case as a Gettier example avant la lettre depends on all this. If the newspaper is usually trustworthy, we may reasonably believe - indeed, take it for granted — that what it says about the battle comes from a report, and from this justified but false belief infer the result of the battle, which, thus inferred, we may quite reasonably take to be justified as well. What we have here is a justified true belief which is not a case of knowledge. Moreover, since it is inferred from a false belief, it shows the typical structure of a Gettier case (see Gettier 1963). What has gone wrong in this case? Well, the belief in the results of the battle does not originate in the events constituting the battle. It is not caused, as Ramsey puts it, 'by what are not beliefs'. The case could have suggested to Ramsey that Russell's analysis of knowledge does not work, just as the Gettier examples convince us that the true-justified-belief analysis is somehow defective. Let us look at this in detail. The thought that something is seriously wrong with Russell's analysis may come from comparing Russell's treatment of the announcement-of-the-resultof-the-battle case with another that occurs a little later in Russell's text; the first case, in fact, to count as knowledge. It runs thus: Take, for example, the beliefs produced by reading. If the newspapers announce the death of the King [George V at the time], we are fairly well justified in believing that the King is dead, since this is the sort of announcement which would not be made if it were false. And we are quite amply justified in believing that the newspaper asserts that the King is dead (...) it would be absurd to say that the reader does not know that the newspaper announces the King's death, (ibid., p. 77) The case is used by Russell to justify understanding derivative knowledge more widely than may initially be thought. Russell is treating as 'intuitive knowledge' the 'knowledge of the existence of sense data derived from looking at the print which gives the news'. From here, an adult 'passes at once to what the letters mean', and in this case belief about what the newspaper says or means should also count as (derived) knowledge. Thus, in general, a belief that validly follows from 'intuitive knowledge' counts as derived knowledge even if the subject, who could perform the inference, does not in fact do so. Although Russell's concern about the example concentrates on knowledge held by the adult subject about what the newspaper says or means, it is clear from Russell's comments that he also regards belief in the state of affairs, or the fact of the King's death, as a case of knowledge. It is a case of knowledge derived from the knowledge of the meaning of the words in the newspaper, plus further derived knowledge: the knowledge that 'this is the sort
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of announcement which would not be made if it were false'. Thus, Russell's definition of knowledge applies to this case. But why is it supposed not to apply to the announcement of the battle? Assume, for the sake of argument, that the difference between the two cases does not lie in any relevant difference in the trustworthiness of the respective events (result of the battle, death of the King). Then, by parallel reasoning, the announcement-of-the-result-ofthe-battle case should also count as knowledge according to Russell's definition. But it should not count as a case of knowledge. Hence, Russell's definition must be wrong. The relevant difference is in the origins of the two beliefs. This, I suggest, is what attracted Ramsey's attention. In contrast to the death-of-the-King case, in the battle case the belief does not originate in 'what are not beliefs'. Now, notice further how this thought can be driven or enhanced by the thought that, similarly, an inference which is faulty because some premise is wrong does not originate, we might think, in 'what are not beliefs', that is, in the facts of the matter. The idea explicit in Ramsey's text that knowledge needs a (causal) process originating in 'what are not beliefs' lies extremely close to this. For the sake of completeness, let us now turn to a brief discussion of Russell's remaining initial knowledge failure case. As a matter of fact this is introduced by Russell in the chapter preceding the other two. It runs thus: If a man believes that the late Prime Minister's last name began with a B, he believes what is true, since the late Prime Minister was Sir Henry Campbell Bannerman. But if he believes that Mr. Balfour was the late Prime Minister, he will still believe that the late Prime Minister's last name began with a B, yet this belief, though true, would not be thought to constitute knowledge. (ibid., p. 75) On the face of it, this case seems similar to the battle case (whose presentation immediately follows it in Russell's book), in that in both it is by some sort of'accident' that the subject turns out to hold a belief which is right. But, in comparison with the battle case, this new case is under-described. We cannot even begin to speculate here on whether it is the connection with the proper state of affairs — 'what are not beliefs' — that was not made; or rather, it somehow was, but the causal chain is not the right one. This is the reason for my hesitating to regard this case as one from which Ramsey could have inferred much about knowledge. Notice that we do not know anything about the reason why the subject (wrongly) believes that Mr Balfour was the late Prime Minister in the first place. If we knew, it might turn out that it was reasonable for him to hold that belief in such a way that we could regard him as being justified. If so, we would once again be in the presence of a Gettier case. But as it is we do not know.
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Reliabilism and accidental beliefs Having obtained his account of knowledge from at least one standard Gettier case, according to my reconstruction, we might expect the analysis to have no problems with common or garden varieties. But here the limitations of a general idea of either a reliable process or a reliable way are shown. It is true that Ramsey's account can be regarded as intuitively delivering the right predictions in all the cases considered by Russell, the clear case of a Gettier example included. But consider this further case. In optimal conditions a woman sees a blue car jumping a red light and thus causing an accident. For some reason, she absolutely refuses to report witnessing the accident and testifying. But while discussing the seriousness of the case with her husband, they arrive at a solution of sorts: he will report the incident instead, pretending that it was he himself who saw the car jump the light and cause the accident. As he is generally known to be a trustworthy person and is also able to answer all the questions put to him by both prosecution and defence very satisfactorily, the jury justifiably believes his testimony. The question is, should we count this as a case in which the members of the jury really know that the blue car jumped the red light, thus causing the accident? Or simpler still, should we count it as a case in which they know that someone saw the accident? Sahlin does not think so, and we may agree with him on this (Sahlin 1990, p. 87). We would then have a standard Gettier case again: true justified belief which does not count as knowledge, a belief moreover obtained by inference from a (false) belief which we are justified in holding (the belief at issue being that the man has witnessed the accident). Sahlin claims that Ramsey's analysis as formulated is sufficient for reaching the negative verdict on the case, but I think that it is reasonable to entertain some doubts about this. Sahlin's reason is that 'we have a case in which the causal connection between the wife's observation and the judge's belief is lacking' (ibid., p. 93). But do we? Has not the wife's observation caused her belief about the accident, and has not this belief in turn caused the belief about the same event in her husband? And has not this belief in turn caused the husband to testify? (He would not have testified at all if he was not fully convinced that the accident took place just as his wife has told him.) And is it not this testimony that has caused the beliefs about the accident in the jury members? It may be answered that the testimony has caused the belief in the members of the jury via causing the belief in them that the man giving the testimony witnessed the accident. And this belief is wrong. At this point — the reply continues — the chain is broken; Ramsey's requirement that 'intermediary beliefs . . . must all be true ones' is not fulfilled. This observation may be right. But the case at least succeeds in raising questions about the individuation of the processes or 'ways' by which beliefs may be obtained.
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We have already seen some reasons why Ramsey himself was uncertain about the formulation of the reliability condition. The point is now that no matter whether we choose a formulation centred on processes leading to beliefs or a formulation centred on ways of acquiring beliefs, reliabilism faces the general problem of individuation of such processes or ways: which beliefs are acquired through the process or in the way at issue? Moreover, it also faces the problem of spelling out what the reliability of such processes or ways consists in. And, since different processes or ways intervene in the acquisition of beliefs about different matters, finally there is the problem of matching processes or ways on the one hand and matters on the other: which kinds of beliefs are reliably acquired through a certain process or in a certain way? The promise of the reliabilist idea for our understanding of knowledge is so great that it is not surprising that there have been many attempts to surmount those problems (or to circumvent them). In this chapter I am concentrating on the first problem, with the others making only occasional appearances. Why is it widely thought that reliabilism, at least as a necessary condition for knowledge, is such a good idea? The answer, to a great extent, must be that knowledge is correct belief, yes, but not belief that turns out to be accidentally produced or accidentally correct, and the idea of a reliable process or way is seen as potentially capable of shedding light on the relevant sense in which beliefs that are knowledge are not held accidentally. The idea of an 'accidentally elimination' approach to an analysis of knowledge is at work in a great many different attempts that have been made in fairly recent years. Thus, among others, we find Alvin Goldman's appeal to causal connection, Peter Unger's direct appeal to non-accidentality, Fred Dretske's to counterfactuals and David Armstrong's to nomological sufficiency.5 Reliabilism promises, for example, to expose the kind of damaging accidentality which is present in the Gettier cases (or should we say Russell-Gettier cases?). On the other hand, we soon realize that the task of picking out and eliminating only the damaging kinds of accidentality is bound to be difficult, since much of our ordinary knowledge is arrived at accidentally in some clear sense. If I know, for example, that Susan is at the party, it is because I happened to glance at the corner where she was. I was not looking for her at all, and, in fact, I was already leaving the party; thus I could very easily have missed her. If I know that Mary is married it is because I happened to look at the finger on which she was wearing her wedding ring, while I do not ordinarily pay attention to such matters. The accidentality in the case, let us assume, is reinforced by the fact that Mary never wears the ring; she just left it on her finger by accident that time (see Sainsbury 1997, pp. 909-10). Now, while I accidentally arrived at the belief that Susan was at the party in the sense suggested, in a different sense I perhaps acquired such a belief
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non-accidentally. It was through sense perception, or more concretely vision, that the belief was arrived at. And vision (perhaps like sense perception in general) is a reliable way to acquire beliefs. But how are we to conceive the reliability of vision or sense perception in general? One way to do so is to fix on the processes 'beneath the skin', that is, on the psychological segment of the whole process by which a perceptual belief is caused.6 But examples like someone perceiving a barn while driving through an area that for some reason is full of mere barn facades looking realistically like barns (perhaps it is an area where films are shot; see Goldman 1976) show the difficulties which a straightforward approach to this view faces. Although the perceiver is indeed looking at an actual barn, under slightly different conditions —. if, say, she had turned left rather than right at the last junction - she would have found herself unknowingly looking at a barn fagade. This sort of accidentality in the subject's being right disqualifies her perceptual beliefs from counting as cases of knowledge. Difficulties in formulating the reliability idea by attaching reliability to processes or ways may motivate attempts to do without either processes and/ or ways. One well explored candidate for this exploits the idea of tracking truth. Rather than pursuing the idea that knowledge is true belief obtained through a process, or in some way that is likely to produce truths, the idea would now be that beliefs that count as cases of knowledge track truth in the sense that they are produced when their prepositional content is true, and they would not have been produced if their corresponding contents were false, where these conditionals support a counterfactual reading (see Nozick 1981, Chapter 3). The first condition establishes a stronger attunement to the environment than process-reliabilism does, in requiring that in the relevant conditions (for example, of attention, enough light, etc.) the agent should not take the object to be of a different kind to what it really is. Thus, a subject does not know that the church fagade in front of her is of renaissance style even if she believes it is, and her belief is true if she has a tendency to classify similar facades as, say, baroque. The latter condition immediately eliminates the case described above of the perceptual belief about a barn being present, since, were the object the agent is actually looking at a barn facade instead of a barn (she drove to the left at the junction), she would still think it was a barn that lay over there. Robert Nozick also developed a theory of evidential support for a belief, a thesis or a hypothesis. In his approach, the strength of the evidence for a hypothesis is determined by the probability of the evidentiary fact (the fact constituting the given piece of evidence) given the truth of the thesis (the intuitive idea behind the approach being that, for example, the evidence in favour of the belief that a particular car jumped a red light is stronger the higher the probability of the testifying witness saying that it did is, given that the car
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jumped the red light). In his chapter on Ramsey's theory of knowledge, Sahlin advocates an alternative developed by Martin Edman and Soren Hallden which he considers (rightly, I think) to be closer to the spirit of Ramsey's position. This approach relies on the idea of evidentiary mechanisms, that is, the mechanisms through which evidentiary facts are produced. For example, in the case of a machine giving a reading supposedly indicating the value of a magnitude, the evidentiary mechanism is, obviously enough, the mechanism instantiated in the machine, while the particular reading obtained is the evidentiary fact. Presumably, in the eyewitness case, the evidentiary mechanism is constituted by the seeing, registering and reporting of the accident by the witness who gives the testimony (where the giving of the testimony itself acts as the evidentiary fact). The approach then focuses on the probability that the mechanism functions properly, given the evidentiary fact (the intuitive idea being that, for example, the evidence for the belief that the car at issue jumped the red light is stronger the higher the probability that the witness has seen, registered and reported the accident correctly is, given what has just been said in the witness stand). I find this a thought-provoking idea, and Sahlin claims it has advantages (see Sahlin 1990, pp. 99-100, and the references given there). Yet, Sahlin's proposal of substituting appeal to satisfactorily reliable evidentiary mechanisms for Ramsey's appeal to reliable process (loc. cit.}, even if it is a nice example of a 'beyond the skin' (or, rather, 'not necessarily beneath the skin') approach to reliability, also faces the difficulties to be mentioned presently. The tracking account generally works very well with Gettier cases. However, the sort of'accidentality-elimination' to be found in tracking arguably fails for perfectly correct cases of accidental knowledge like knowing that Susan was at the party or that Mary is married. Since I have no curiosity about Mary's marital status, I would not have formed any belief about it if I had not happened to look at the ring on a day on which she was accidentally wearing it, and this circumstance seems to indicate that the first condition in Nozick's account is not fulfilled in this case. If this is correct, the account incorrectly rules out the case as one of knowledge. Richard Sainsbury has discussed these cases (see Sainsbury 1997, pp. 909—10). It is well known that Nozick's account rules out as knowledge anti-sceptical beliefs such as the belief that I am not a brain in a vat. Many would be prepared to regard these beliefs as cases of knowledge. But the issue is somewhat controversial, and so the putative failure of the account for perfectly correct cases of accidental knowledge pointed out by Sainsbury is truly significant. In fairness to Nozick, however, it must be pointed out, as Sainsbury does, that Nozick himself suggests that appealing to 'methods' (ways) is unavoidable in an account of knowledge, so that the tracking account, as it has been presented, is insufficient by itself. The tracking account would fail in any
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case as a sufficient condition (fulfilling its conditions is not sufficient for knowledge), as is shown by cases like the hypothetical case of clairvoyance mentioned in the next section. In this failure, however, it is on a par with all the reliabilist accounts mentioned thus far. And, in any case, being wrong about necessary conditions must surely be seen as a more serious failure than inadequacy on sufficient conditions. We owe a recent and very direct attempt at adequately excluding accidentality to Sainsbury, namely his Reliability Conditional: if one knows, one couldn't easily have been wrong (see ibid., p. 907). One very good thing about this necessary condition is that it does not rule out good ordinary cases of accidental knowledge (like the cases about Susan or Mary). However, the apparent simplicity of the approach is somewhat misleading. One should investigate, as Sainsbury does, what the 'easy possibilities' involved in the condition are. Furthermore, the issue of the 'methods' or ways of acquiring beliefs still seems to be waiting around the corner. As Sainsbury also points out, this question should be brought in again when explaining why the Reliability Conditional is true (if it is). Nevertheless, in this account, it lies at one remove from an immediate definition of knowledge. Perhaps then this separation can be used to inquire whether the focus of the reliability should not be moved from the beliefs to the believers. Indeed, the Reliability Conditional seems to be compatible with such a shift, as Ernest Sosa has been advocating for quite a few years. Sosa has argued that if we are to characterize knowledge, it is on the subject of a belief and its pertinent faculties or intellectual virtues that explanatory emphasis should be placed, rather than on the process that leads to the belief or the way it is acquired. We do not rate an entirely effective backhand stroke by a completely inexperienced tennis player as skilful, but rather as simple luck. We would not qualify the shot as skilful even in the extreme case where, just by chance, the novice player went through exactly the same movements as a tennis champion. In effect, we focus on the player's ability and prowess, and not on what he or she does with regard to any particular stroke. In much the same way, according to Sosa, we have reason to focus on the cognitive skills of the knower. Sosa takes a broad view of such cognitive skills and calls them cognitive virtues. In this sense, a being that has excellent eyesight and also possesses first-class organs for processing what it sees may be said to undergo reliable vision processes on that count, but if, for some reason, it is subject to internal hallucinogenic interference most of the time, it fails to show cognitive virtue in regard to vision.9 What it lacks is a faculty which, with regard to a field of propositions and under certain regularly obtaining circumstances, tends to arrive at true beliefs in a certain environment. This - the having
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of a reliable faculty in that sense - would constitute enjoying the epistemic virtue of visual perception. As may be seen from the example just given, the cognitive virtues approach introduces a new perspective on accidentality elimination. This is further seen in the contrast with both the tracking and the reliable-process accounts of knowledge, as illustrated by the following imaginary case in which both approaches would give the incorrect verdict. Assume somebody fancies himself an expert on tomato ripeness, but actually suffers from a rare form of colour blindness that precludes him from discerning any shades of red perceived by normal subjects, with the exception of a particular shade. Unaware of his disability, the subject issues judgements of ripeness or unripeness on all sorts of shades, including a wide spectrum of shades of red. Assuming further that the variety of tomatoes in the area always ripen to exactly the particular shade of red that the subject is able to recognize, he turns out to be an extremely reliable predictor of ripeness. For this reason, the requirements of the reliabilist accounts just mentioned would arguably be met; it seems, however, that this sort of case should be excluded as a case of knowledge. Virtue epistemology is aimed at outlining a broader view of reliability that also excludes these kinds of hitting the target accidentally (see Sosa 1997, p. 418). Involving as it does the restriction of each epistemic virtue to specific circumstances and/or reference to an environment (besides the restriction to a particular field of propositions), the account seems to be equipped with the resources to handle the barn case, and to have no special problems with the simple cases of accidental knowledge which are perfectly in order. There remain, however, the problems of delimiting in a principled way the parameters it recognizes (field, circumstances, environment), which are to some extent analogous to the problems of marking out reliable processes or ways correctly.1
Reliabilism and justification Ramsey's original definition of knowledge did not seem to leave room for rational justification as a necessary condition for knowledge. At most, from Ramsey's brief remarks, we could gather the condition that rational attachment to a belief should be calibrated according to reliability, if and when we reflect that the belief has been acquired in a reliable way (see above, at the end of the first section). This leaves open the possibility of regarding as a case of knowledge a true belief which we have acquired in a reliable way, and which we regard as 'certain' (whatever this means for Ramsey), without further ado, that is, without reflection on reliability, or any justification we are prepared to give. Extreme reliabilists like Goldman are ready to follow
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Ramsey here. They, however, have been inclined to formulate their position without renouncing the idea that in such circumstances subjects are justified in holding the belief, even if they are not in any sense prepared to give any justification; that is, even if they are neither aware of any reason to think that the belief is true nor is this something they could become aware of. De facto reliability alone suffices to confer justification. (This view is widely known as externalism in epistemic justification.) This view of justification seems to clash with deep-rooted intuitions about justification. Several thought experiments have been used to demonstrate this, indeed to give grounds for the idea that reliable processes or ways to acquire beliefs are neither sufficient nor necessary for rational justification. I will briefly rehearse the well-known clairvoyance and evil demon cases (see Bonjour 1985, pp.41-5 and pp. 183-5, respectively). Imagine first a purely hypothetical person who has completely reliable clairvoyant powers about certain subject matters under certain usually obtaining conditions, and who has no evidence or reasons either to believe or disbelieve in this cognitive power, or for or against the possibility that he or she possesses it. Focus on any beliefs this person may form that result from his or her power under circumstances under which this power is reliable, but for which the person lacks any evidence. It seems that these beliefs should be regarded as justified from the viewpoint of radical reliabilism, but they are intuitively not thought to be so. (Notice that Ramsey's example of telepathy could be adapted to serve the same purposes as this clairvoyance example; see section 1.) Now consider unfortunate beings that are victims of an evil demon or a perverse team of neuroscientists which makes them believe they are people just as we believe we are, but who in fact are only brains in vats whose experiences are caused by the demon or the team of neuroscientists. Suppose further that whatever presumably reliable processes normal people are assumed to undergo for forming beliefs or memories from those experiences and reasoning from them are still in force in the case of those beings (or brains). Focus now on any beliefs they may hold which are thus obtained. There seems to be a clear sense in which these beings are justified in holding these beliefs (they are 'epistemically blameless'), even if the beliefs have not been acquired by any reliable process. There have been several attempts to respond to charges of non-sufficiency and non-necessity based on examples like these by elaborating versions of (radical or pure) reliabilist justification. However, there has always been a question as to whether these proposals, successful as they may be in avoiding the inadequacies of former versions, are well-motivated rather than merely ad hoc. One of the most recent of such accounts, which is not at the least motivated end of the spectrum, has been offered by Goldman, within the general
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framework of Sosa's epistemic virtues. According to this proposal, we should think about epistemic justification by thinking of an 'epistemic evaluator's' activity as proceeding in two stages. In the first stage the evaluator tries to settle on a list of epistemic virtues and vices simply by determining which of these are reliable and which are not. In the second stage particular beliefs are evaluated by determining whether the way they are acquired or sustained resembles the activity of virtues or that of vices. According to Goldman's proposal, beliefs arrived at by (alleged) clairvoyance should count as being unjustified, since presumably clairvoyance would appear in the list of vices rather than that of virtues. Conversely, beliefs acquired from the experiences caused by the demon or team of evil neuroscientists would bejustified, since belief formation on the basis of experiences presumably appears in the list of virtues. However, in Goldman's view, justification does not depend in a general way on the believers themselves having any inkling about whether they are epistemically virtuous or not. Although the issue is not settled at the present time, either the intuitive force of the sheer claim that rational justification for a belief being held requires awareness or, at least, potential awareness of a reason for thinking that the belief is true, perhaps reinforced by thought experiments which are difficult to deal with in extreme or pure reliabilist views, or the perceived degree of ad-hocness in reliabilist proposals designed to overcome such difficulties, have succeeded in convincing most epistemologists of the weakness of a pure reliabilist theory of justification. In this respect it is useful to look at Sosa's case. To begin with he incorporated the reliabilist element within justification: 'To be epistemically justified in believing is to believe out of intellectual virtue' which 'may be viewed as a subject-grounded ability to tell truth from error infallibly or at least reliably in a correlated field.' But then he came to distinguish between justification and aptness, where it is the latter that mainly picks up the reliabilist element.11 In the end, how bad this should be for a pure reliabilist theory of knowledge depends on the views we may hold on the status of rational justification as a condition for knowledge. Traditional views on the relation between rationally justified belief and knowledge are being revised nowadays, and there is some turmoil about how this revision should be carried out. In his recent book Knowledge and its Limits, Timothy Williamson convincingly argues that knowledge cannot be defined in terms of justified belief. He opts for reversing the explanatory relation: it is knowledge which would account for rationally justified belief. Richard Foley has gone further still in claiming that the whole epistemic tradition has been wrong in thinking that the two notions are conceptually linked (this then includes Williamson too; see Foley 2002). According to Foley, the enterprise of characterizing rationally justified belief constitutes a project (roughly: 'that of exploring what it is appropriate to
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believe insofar as one's goal is to have accurate and comprehensive beliefs') which is entirely different (and independent, it seems) from the project, roughly, 'of exploring what is required for one to stand in a relation of knowledge to one's environment'); see ibid., p. 725. Although I do not argue in its favour, the view on knowledge of reliability for the case or perception which is outlined in the next section is congenial to a further possibility: that even if knowledge and justified belief are conceptually linked, they are not definitionally or reductibly so. That is, we cannot understand any of those concepts without already having some understanding of the other (they form, in effect, a 'local holism'). Be all this as it may, even renowned critics of pure reliabilism admit that its prospects as a theory for knowledge are better than its prospects as a theory of justification. And, of course, what Ramsey proposed was reliabilism as a theory of knowledge, and that is what mainly concerns us here. In this respect we could briefly go back to the clairvoyance case and consider whether Goldman's perspective on this really gives us a reason to think that a treatment such as the one he proposes rules out this case as one of knowledge (or, for that matter, as one of justified belief; but let us concentrate on knowledge). Recall that the case was initially set up as one in which clairvoyance is reliable; thus, presumably, in the hypothetical world in which clairvoyance is reliable, the subject acts out of virtue if he follows its guide. Hence, in this hypothetical world the case should rather be regarded as one of knowledge (assuming we are focusing on beliefs that are true), according to a view which accounts for knowledge in terms of epistemic virtues. Or so it seems. It is for reasons essentially like this that Sosa, the original proposer of the view, takes a completely different line. He is led to attribute the knowing subject with an appropriate epistemic perspective on his abilities and virtues. He does this by proposing a distinction between animal knowledge and reflective knowledge (see e.g. Sosa 1991, pp. 239-44). Knowledge of the first kind would be attained by simple use of epistemic virtues, while knowledge of the second kind would involve additional reflection on its sources. In the final section I explore this new sort of view, which requires a kind of epistemic perspective on the part of the subject. But, as we will see, I do not endorse the way in which Sosa incorporates this element into a theory of knowledge.
Reliabilism and epistemic perspective There is a kind of knowledge - we are now ready to agree — which involves having a basically correct epistemic perspective. I will assume that such a perspective consists roughly of knowing about the reliability of our faculties. One way
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to make this requirement concrete is to accept the following principle, which states a necessary condition (see Cohen 2002, p. 309): KR A potential knowledge faculty K can yield knowledge for S (of the kind at stake), only if S knows K is reliable. Sosa's epistemic perspectivism accepts analogous requirements. In Sosa (1997), this is presented (perhaps hypothetically) as a consequence of accepting a more general principle, the principle of'exclusion, which says that '[i]f one is to know that p, then one must exclude (rule out) every possibility that one knows to be incompatible with one's knowing that/)'. As a matter of fact, when Sosa himself defends something similar to KR he does not do it on such a basis (see Sosa 1997, pp. 425—6). Anyway, I do not think it is necessary to defend a principle such as KR by deriving it from this principle of exclusion. So, how is knowledge about the reliability of our faculties to be attained? It seems as though this will be difficult to get at, for surely in trying to acquire knowledge about the reliability of our faculties we will have to use the very same faculties whose status as reliable knowledge sources is in question. If we still do not know that a faculty K is a reliable source, we do not know whether it can yield us knowledge, according to KR. And if we need to know this to establish that K is reliable in the first place, we seem to be completely stuck. We have mentioned a kind of knowledge (Sosa's 'animal knowledge') for which something like KR does not hold. From the basis of such knowledge it may be possible to somehow arrive at a reasonably correct epistemic perspective. This is what Sosa holds: ... reflective knowledge, while building on animal knowledge goes beyond it precisely in the respect of integrating one's beliefs into a more coherent framework. This it does especially through attaining an epistemic perspective within which the object level animal beliefs may be seen as reliably based. Also: . . . perception and introspection, along with intuition, as well as inductive and abductive reasoning, along with . . . deductive reasoning . . . By use of such faculties... one attains... a broad view of oneself and one's environing world. And, if all goes well, then in terms of this epistemic perspective one can feel confident about the reliability of one's full complement of one's faculties. But again, how exactly do we obtain such a perspective? How, to begin with, is it possible to obtain at least part of it? Maybe we are not completely stuck
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after all if we have all that 'animal knowledge' at our disposal. Nonetheless, is there not some kind of circularity involved in a reasoning that uses, say, perception, in establishing the reliability of perception? Sosa agrees that there is, but he questions whether it is a vicious circularity: But is not any such reasoning circular? Yes, circular it does seem to be, 'epistemically circular,' let us say. But is it viciously circular? . . . When we reason in the way alleged to be viciously circular, wherein lies the defect in our reasoning or in the resulting state? (Sosa 1997, p. 424) Let us consider the case of a crystal ball gazer that uses looking into a crystal ball to ascertain the reliability of crystal ball gazing. This person justifies the (alleged) high degree of success in the predictions obtained simply by gazing into a crystal ball, and the reliability of this potential knowledge source by looking into the crystal ball again to see whether the predictions have turned out right. Sosa is prepared to put this crystal ball gazer on a par with the perceiver who appeals to perception in reasoning for the reliability of perception (perhaps he concedes this purely for the sake of the argument); see ibid., pp. 424-5 and p. 427. But would this not be sufficient grounds for being wary of the position? Sosa does not think it is: So far we have been told that we must avoid epistemic circularity because it entails arriving at a generally positive view of one's faculties only by the use of those very faculties. But why should that be frustrating when it is only the inevitable consequence of its generality? So far the answer is only that the superstitious crystal gazer could reason analogously and with equal justification in defense of his own perspective. How damaging is this? (op. cit., p. 427) A sceptic about the circular method put forward by Sosa will hold that if the effort in arriving at 'a generally positive view of one's faculties' necessarily involves such a comparison between the crystal ball gazer and the perceiver, this effort is not worthwhile. But Sosa does not think it is damaging because, after all, there is a decisive difference between one and the other, namely, perception is a reliable method, while crystal ball gazing is not: In light of that result, why not distinguish between the gazers and the perceivers in that, though both reason properly and attain thereby coherence and justification, only the perceivers' beliefs are epistemically apt and constitute knowledge? On this view, the crystal gazers differ from the perceivers in that gazing is not reliable while perceiving is. (ibid.}
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According to this position, the crystal ball gazers can be as justified internally as the perceivers, but since this internal — coherentist — justification is not accompanied by de facto reliability, their belief about the reliability of their source of knowledge is not apt, and so does not qualify as knowledge. The perceivers, in contrast, are not only justified in their belief by the circular internal cohesionist method, but they have an apt belief in their reliability, and thus, as this belief is also true, they have a correct perspective, that is, they have knowledge of reliability. As Sosa puts it: ... the theory of knowledge of the perceivers is right, that of the gazers wrong. Moreover, the perceivers can know their theory to be right when they know it in large part through perception, since their theory is right and perception can thus serve as a source of knowledge. The gazers are, by hypothesis, in a very different position. Gazing, being unreliable, cannot serve as a source of knowledge ... the gazers . . . lack knowledge, (op. cit., p. 427) Sosa is in effect applying to epistemic perspective (and especially to belief in reliability) the same difference between justification and aptness that he applies to ordinary, first-level beliefs (see endnote 11 below). This makes a position like Sosa's difficult to classify. Above I have used the labels 'extreme reliabilism' and 'pure reliabilism', which are found in the literature, interchangeably. Sosa's virtue perspectivism certainly does not occupy an extreme reliabilist position (like the position we may attribute to Ramsey or Goldman). This much is clear on account of his recognizing the centrality of epistemic perspective and epistemic reflection. But does he still deserve the label 'pure reliabilist' on account of its appeal to 'aptness' in the last resort? How good is Sosa's solution? To find out, we need to look at his reply to an objection raised by Barry Stroud, which is expressed thus: ... Sosa 'externalist' could say at most: Tf the theory I hold is true, I do know or have good reason to believe that I know or have good reason to believe it, and I do understand how I know the things I do.' I think . . . we can see a way in which the satisfaction the theorist seeks in understanding his knowledge still eludes him . . . he will still find himself able to say only T might understand my knowledge, I might not. Whether I do or not at all depends on how things in fact are in the world I think I've got knowledge of (Stroud 1994, pp. 303-4) Sosa replies to this objection as follows: It is not easy to understand this position, however. If our perceivers believe (a) that their perception, if reliable, yields them knowledge, and (b) that
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their perception is reliable, then why are they restricted to affirming only the conditional, a, and not its antecedent, b? Why must they wonder whether they understand their relevant knowledge? I think we can understand both Stroud's dissatisfaction and Sosa's rejection of it. It seems to me that Sosa is right, in so far as the perceivers could say 'I know that perception is a reliable source of knowledge', and we think they would thereby be making a true claim, whereas the crystal ball gazers, even if able to claim 'I know that crystal ball gazing is a reliable source of knowledge' and to claim this with equal justification (let us suppose), would not be making a true claim. In this sense the perceivers, we think, understand their knowledge and the crystal ball gazers do not. But Stroud's qualms reappear, I believe, when we think about the answers we should give to the very questions Sosa asks in this text. 'Why — he asks - must [the perceivers] wonder whether they understand their relevant knowledge?' Yes, why indeed? But if they have no reason to wonder, it must be because they feel fully justified. And this is exactly what happens with the crystal ball gazers. Why, indeed, should the crystal ball gazers wonder whether they understand their relevant knowledge? 'Why - asks Sosa - are [the perceivers] restricted to affirming only the conditional, a, and not its antecedent, b?' Indeed, they are not so restricted. But neither are the crystal ball gazers. It is this complete parallel, I suggest, that is the source of Stroud's worry. And I think he is right here. I believe an alternative account of how the owners of reliable faculties know about the reliability of their faculties should and can be found. Sosa accepts that '... one cannot possibly know that/? unless one knows that the faculties involved are reliable' (Sosa 1997, p. 426). He thinks, however, that 'this is just the sort of knowledge that we seem able to attain only through epistemically circular reasoning' (ibid.}, where this is understood as reflection on one's own faculties. I will not go so far as to claim that epistemically circular reasoning of this kind lacks any legitimate uses. But I think Sosa is wrong in saying that only through epistemically circular reasoning can we know about reliability. In stating this, I am not thinking of one single account of the reliability of our faculties or cognitive prowess. I feel there may be a different and detailed story to be told about each faculty or knowledge source, or about how we know we are right. Or indeed, that even within each faculty, we may need to distinguish different kinds of cases. As an example of the kind of account I mean, I would direct the reader's attention to Christopher Peacocke's account of introspection in the special case of knowledge of consciously based selfascription of attitudes (see Peacocke 2000, §§5.2-5.4). To finish this chapter I will outline an account of our knowledge of the reliability of basic or simple
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cases of perception which is, I hope, in the spirit of Peacocke's partial account of introspective knowledge. 13 My suggestion is that, as certainly also happens for some sorts of introspection and memory, for some kinds of perception there is also a sort of knowledge that is intermediate between what should properly be called 'animal knowledge' (knowledge that pre-linguistic children share with superior animals) and Sosa's reflective knowledge. What I have in mind here are perceptual beliefs acquired by the perception of observable properties, or more concretely, properties whose exemplification is manifest to beings like us. Properties such as squareness or redness, which we capture with observable concepts which find linguistic expression in terms like 'square' or 'red'. (To avoid possible complications with so-called secondary qualities, I will only explicitly address cases of primary properties.) As normal adults I think we are entitled to our own perceptual beliefs in such basic cases, and that this entitlement ultimately comes from the fact that we are sensitive to the relationship between our own perceptual experiences (Erlebnisse} and instantiations of the observable property at issue. The very nature of the property ensures that we have such sensitivity, since it is a property — squareness say — which beings such as us can detect perceptually, being ourselves aware of its detection. 4 Now, in my view, while sensitivity to the mentioned relation might give a being entitlement of sorts (perhaps in regard to protobeliefs), it definitely does not suffice for knowledge, not even for implicit knowledge, at least not for the kind of intermediate (implicit) knowledge we possess as normal adults with regard to the instantiation of observable properties. At most it gives the kind of knowledge a very young child or a (superior kind of) animal have of their immediate environment - 'animal knowledge', properly restricted. I think we must reserve the word 'knowledge' in a proper sense for cases where a minimum of rationality and objectivity is present. It is for this reason that I suggest the line should be drawn at the stage of possession of the corresponding concepts. Possessing a concept expressible in a language requires differentiation between cases of correct and incorrect application. When a being is aware that being correct or incorrect is something that does not depend of itself, such being has thereby a glimpse of the idea of objectivity. Such beings have then a minimum of rational differentiation between their experiences on the one hand and objective objects and events on the other. At some point a child acquires the concept square, which involves both being able to apply it when having certain experiences (Erlebnisse) and the beginnings of competence in using the word 'square', which, when it is above the stage of a conditioned reflex (that is when there is awareness of being correct or incorrect), involves the required minimum of objectivity. It is of someone who is
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competent in this way that we can rightly say that he or she knows that that [the object in front of them] isF (where ^is the concept of a simple observable property). Thus, certain beliefs - basic perceptual beliefs of the kind in question — constitute knowledge for beings with a minimum of conceptual competence. Possession of the relevant concepts brings with it awareness that (in the relevant cases) one is making transitions from impressions to objective situations. Hence the subject is entitled to think that such transitions are sound, i.e. that the procedure that results in forming a perceptual belief from an impression or experience is reliable. Therefore, I suggest that in beings that possess the relevant concepts the mere (but crucial) sensitivity to the relationship between experience and instantiation of the observable property also becomes the entitlement to the belief in the reliability of the relationship. It is at this point that subjects are entitled to believe in the reliability of their experiences as a means to reach true perceptual beliefs. Perhaps we may also say that subjects implicitly know of the reliability of the transition between experiences and beliefs (in the kind of cases in question). The basis for this claim should be that they are in a state which only lacks conceptual sophistication to be counted as (explicit) knowledge. If we are right to apply the term '(implicit) knowledge' here, perhaps it seems fair to say that entitlement (to the transition) and knowledge come together, rather than saying that the knowledge comes from the entitlement. This and other details are matters for further development. What should be clear, however, is that normal subjects — as opposed to a reflective philosopher who discusses the issue — have no reflective knowledge of their entitlement to believe in the reliability of the transitions from experience to recognition of property instantiation through their application of the corresponding concepts. Subjects do not need to think on the relations which they bear to the conscious states which give them reasons (i.e., which entitles them) to (simple perceptual) objective judgements for being thus entitled. They usually lack any reflective knowledge of their entitlement to the corresponding perceptual beliefs.15 In this view, it might be said that the subject's entitlement to such beliefs flows from his or her entitlement to a belief in the reliability of the transitions at issue by subsumption, as it were. But it would not be true to say that a person, as a subject (as opposed to a theorist) 'knows of how [he or she] is right, on this account, [as a] product of second order reflection upon the credentials of a basic first order belief-forming mechanism' (Brewer 1996, p. 259). As I have just suggested, determining whether it is more correct to say that the subject still does not know how he or she is right, or rather, as I believe, that (implicit) knowledge of how he or she is right is wholly constituted by his or her sensitivity to the forementioned transition plus the modest conceptual capabilities at stake is a matter for further development. In any case, instead
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of reflection upon the credentials of a rule — reflection on reliability — what we have is a minimally rational sensitivity to a regular association. Now, an old element in the idea of apriori justification for a belief (or knowledge) is that it is justification (or, respectively, knowledge) that we get purely from understanding the concepts in the belief content. According to this view then, and on account of the central role that concept possession plays in the subject's being entitled to the belief in the reliability of obtaining perceptual beliefs of the relevant kind from experiences, we may wish to say that subjects are a priori entitled to believe in such reliability. And if we are prepared to apply the word 'knowledge' here, we may by the same token talk of a priori knowledge of reliability.16 In any case, it is not a sort of 'absolute' notion of a priori (entitlement or knowledge) which is at issue here. Such entitlement or knowledge is, in general, defeasible. Claims to entitlement or knowledge can be potentially withdrawn in view of a posteriori evidence, empirical evidence, that is, that goes beyond the empirical evidence necessary for the learning of the concepts. Apriori entitlement or knowledge which is a posteriori confirmable (and which then may become justified belief or full reflective knowledge) or refutable. Moreover, and finally, subjects learn more about reliability as they mature epistemically, that is, as their epistemic perspective becomes more elaborate.
Endnotes 1.
This chapter has benefited from funding from the Spanish Ministry of Science and Technology, through research project BFF2001—2531, and also from funding by the Catalan Government (Generalitat de Catalunya) given to the research group in epistemology and cognitive sciences at the Universidad Autonoma de Barcelona (GRECC), reference 2001SGR-00154. I am grateful to Olga FernandezPrat for numerous clarifying talks about the issues dealt within it, and to Carolyn Black for linguistic correction. 2. 'Knowledge' was written in the year before Ramsey died and published after his death. The references are taken from the reprint in the collection of Ramsey's papers edited by Mellor. Sahlin and Dokic and Engel mention the possibility that the conditions are not meant as a definition in the classical sense (see Sahlin, p. 91, and Dokic and Engel, p. 28, and note 2, p. 87). Indeed Ramsey's introduction to the conditions ('I have always said that a belief was knowledge if it was...') leaves room for doubt about whether what follows is a list of all the necessary conditions, a list of conjointly sufficient conditions, a list of both, or only a list of some of the necessary conditions. I believe the list includes at least all the necessary conditions. Be this as it may, I will continue using the term 'definition' for convenience. 3. One thing at least is clear: Ramsey not only mentions the book, but also gives clues that reveal his acquaintance with the book's contents in his paper on knowledge, as will be seen below.
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4. Sahlin also rather vaguely suggests a similar path (cf. op. cit., 92). 5. Those approaches are found in Goldman (1967), Unger (1968), Dretske (1971) and Armstrong (1973) respectively. 6. Goldman's paper 'What Is Justified Belief?' (1979) may be regarded as the foundation stone of post-Ramsey reliabilism. The subsequent approach, in which epistemological questions are pursued within the framework of cognitive psychology, is developed in Goldman (1986). In opposition to Ramsey's original approach, it has not always been clear that in this approach reliability is meant as a necessary condition for knowledge, and not for epistemic justification (see below). 7. The importance of this example has been magnified in Robert Brandom's constructivist approach to reliability. Brandom argues that, because of the arbitrariness of delimiting the relevant context, the reliability idea cannot be recruited to support a naturalistic project. He illustrates such purported arbitrariness by the hypothetical case in which there is indeed a very small minority of barns as opposed to mere realistic barn facades in the county through which the agent drives, while there is a large majority of genuine barns in the state where that county lies, even if barn facades turn out to be in the vast majority in the country to which that state belongs (cf. Brandom 2000, chapter 3, §§IV and V). However, it is essential to Brandom's example that he is dealing with artifacts. He would not be able to make the point that there is a fundamental difficulty here — as opposed to a potentially significant practical problem - if he were to deal with natural objects, at least without assuming that there are no natural kinds, that is, without begging the question against a realist naturalist reliabilist position. For Goldman's own attempt to deal with the relevant difficulties see Goldman (1986), §3.2. 8. Regarding this denomination, Sosa explains that 'virtue' here is to be taken in a broad sense, still recognizably of Greek origin, according to which anything with a function — whether natural or artificial — has virtues (e,g. a knife); see Sosa (1991), p. 271. Sosa defends his designation against complaints that it is misleading in Sosa (1994), p. 33. 9. See Sosa (1991) p. 242. To my knowledge chapter 16 of this work contains Sosa's fullest attempt to elaborate on many of the details of his position. The sketch in the present text gives only a rough outline. 10. See Sosa (1991), pp. 280-8 for Sosa's handling of such problems, and his treatment of the generality problem in the framework of his 'epistemic perspectivism' (see below for more about this). 11. On the first reaction, see Sosa 1991, p. 242, chapter 13 and the characterization of prima facie justification at p. 239. Also, on the second step, ibid. pp. 289-90, chapter 16. 12. These recent formulations come from Sosa (forthcoming). They are given as quoted in Cohen (2002) from where I have borrowed them. 13. As will be seen, my approach to an account of knowing 'how we are right' for basic cases of perception is sympathetic to Bill Brewer's approach in rejecting what he calls 'reflective internalism', but deviates from it in having non-conceptual content in its background (see Brewer 1996, especially pp. 267-9, and Brewer 1999, especially §6.2). I criticize Brewer's account in Quesada (2001), where I first
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outline a version of the present approach. I have found inspiration for my position in Garcia-Carpintero (2000) (see especially pp. 80-4). For a recent initiative to account for knowledge of reliability without circularity which moves along a different track to mine see Cohen (2002). (I should perhaps warn that I do not use the term 'basic knowledge' in the specific sense Cohen does.) 14. What assures our sensitivity, however, is not what assures explicit knowledge of our sensitivity. And so, even if sensitivity gives us entitlement, it does not give us full knowledge of our entitlement (or an explicit justified belief in it). This knowledge is, so to speak, a sort of much 'higher level' knowledge (and the corresponding claim holds for justification). 15. Peacocke holds that a psychological self-ascription grounded in the subject's being in a certain conscious state requires that the subject possesses the concept of the psychological state in question (see Peacocke, p. 301). In contrast, having neither the concept of a perceptual experience nor the concept of a conceptual belief seems to be necessary for holding a perceptual belief in such a way that it may be correctly said that the subject is entitled to this belief. 16. Such 'strong' conceptions of the a priori are somehow 'in the air'. See John Hawthorne's views in the paper mentioned in the references for a recent defence. A brief explanation of how Hawthorne's views on how the a priori might be applied to the present problem is given in the paper by Stewart Cohen that has been mentioned above, although, if I understand him correctly, Cohen rejects the possibility of a solution to it on these lines (see pp. 320-2).
References Armstrong, David (1973). Belief, Truth and Knowledge, Cambridge: Cambridge University Press. Bonjour, Laurence (1985). The Structure of Empirical Knowledge, Cambridge, MA: Harvard University Press. Brandom, Robert (2000). Articulating Reasons: An Introduction to Inferentialism, Cambridge, MA: Harvard University Press. Brewer, Bill (1996). 'Internalism and Perceptual Knowledge', European Journal of Philosophy, 4, 259-75. (1999). Perception and Reason, Oxford: Clarendon Press. Cohen, Stewart (2002). 'Basic Knowledge and the Problem of Easy Knowledge', Philosophy andPhenomenological Research, 65, 309—29. Dokic, Jerome and Engel, Pascal (2002). Frank Ramsey: Truth and Success, London: Routledge. Dretske, Fred (1971). 'Conclusive Reasons', Australasian Journal of Philosophy, 49, 1-22. Foley, Richard (2002). 'Review of Timothy Williamson: Knowledge andits Limits\ Mind, 111,718-26. Garcia-Carpintero, Manuel (2000). 'Las razones para el dualismo', in Chacon Fuertes, P. and Rodriguez Gonzalez, M. (eds). Pensando la mente, Madrid: Biblioteca Nueva, 27-119.
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Gettier, Edmund (1963). 'Is Justified True Belief Knowledge?', Analysis, 23, 121-3. Goldman, Alvin (1967). 'A Causal Theory of Knowing', Journal of Philosophy., 64, 355-72. (1976). 'Discrimination and Perceptual Knowledge', Journal of Philosophy., 73, 771-91. (1979). 'What is Justified Belief?', in Pappas, G. (ed.), Justification and Knowledge, Dordrecht: Reidel. (1986). Epistemology and Cognition, Cambridge, MA: Harvard University Press. (1992). 'Epistemic Folkways and Scientific Epistemology', in Goldman, A. Liaisons: Philosophy Meets the Cognitive and Social Sciences, Cambridge, MA: MIT Press. Hawthorne, John (2002). 'Deeply Contingent A Priori Knowledge', Philosophy andPhenomenological Research, 65, 247-69. Nozick, Robert (1981). Philosophical Explanations, Cambridge, MA: Harvard University Press. Quesada, Daniel (2001). 'Contenido perceptual y justificacion epistemica', in Acero, J. J., Camos Abril, F. and Villanueva Fernandez, N. (eds). Actas del III Congreso dela Sociedad Espanola de Filosofia Analitica, 73—80. Peacocke, Christopher (2000). Being Known, Oxford: Clarendon Press. Ramsey, Frank Plumpton (1990). Philosophical Papers, ed. D. H. Mellor, Cambridge: Cambridge University Press. Russell, Bertrand (1986). The Problems of Philosophy, Oxford: Oxford University Press (originally published in 1912). Sahlin, Nils-Eric, (1990). The Philosophy ofF. P. Ramsey, Cambridge: Cambridge University Press. Sainsbury, Richard, (1997). 'Easy Possibilities', Philosophy andPhenomenologicalResearch, 67,907-19. Sosa, Ernest, (1991). Knowledge in Perspective: Selected Essays in Epistemology, Cambridge: Cambridge University Press. (1994). 'Virtue Perspectivism: A Response to Foley and Fumerton', Philosophical Issues, 5, 29-50. (1997). 'Reflective Knowledge in the Best Circles', Journal of Philosophy, 94, 410-30. , forthcoming. 'Virtue Epistemology' in Epistemology: Internalism Versus Externalism (co-authored with Laurence Bonjour). Stroud, Barry, (1994). 'Scepticism, "Externalism", and the Goal of Epistemology', Proceedings of the Aristotelian Society, supplementary volume 44, 291-307. Unger, Peter (1968). 'An Analysis of Factual Knowledge', Journal of Philosophy, 65, 157-70.
10 1
Ontology from language? Ramsey on universals Francisco Rodriguez-Consuegra
We usually accept there is a distinction between particulars and universals, particulars being individual objects and universals being general concepts, or abstract 'objects'. Also, we usually accept this ontological distinction to be represented in language by the subject—predicate distinction, subjects being proper names (for objects), and predicates being adjetives and verbs (for properties and relations). Russell defended such a position in some well-known writings, especially that of 1911. In a paper published in 1925, and entitled 'Universals', Ramsey challenged both distinctions through a variety of arguments. The paper has been mentioned quite a few times and even partially discussed in print, but I think it is not really very well-known and those arguments are usually not clearly identified, separated or assessed. This is the goal of this chapter. In the first section I give a summary of each of the four main arguments which I think can be clearly found in Ramsey's paper. The second section is devoted to trying to find general patterns and assumptions in those arguments, then to see if all this can be reduced to a simple, common type of argument and to a few assumptions. This is then inserted into a historical context, where Bradley and Moore come to mind. Finally a third section is devoted to a general assessment of Ramsey's arguments, where his main assumptions are discussed and some criticisms are taken into consideration. The general result is that, although Ramsey's paper is brilliant, his position is not very original and his arguments are rather unconvincing.
Arguments This section is purely expository, so I just offer a summary of the four main arguments against the universal-particular distinction which may be found in Ramsey's paper on universals. The character of the arguments themselves might be controversial, and the paper is very convoluted and the arguments are mixed with discussion of other authors, mostly Johnson and Russell, so I think a section like this is necessary as a useful starting point to the rest of the chapter. Needless to say, this summary somehow simplifies some steps of the arguments. Also, I take the liberty of inserting a few clarifications in brackets, and give names to the arguments for convenience of further reference.
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Symmetry in language The universal—particular distinction depends upon the subject—predicate one. But the subject—predicate distinction depends on the assumption that they are completely dissimilar categories, which represent completely different entities in the world. Yet we can exchange subject and predicate in our statements, so there is no absolute distinction at all, then no distinction between objects and concepts: 'In a sense, it might be urged, all objects are incomplete; they cannot occur in facts except in conjunction with other objects, and they contain the forms of propositions of which they are constituents' (p. 11). This can be illustrated with a particular example of a subject predicate statement: '... in "Socrates is wise", Socrates is the subject, wisdom the predicate. But suppose we turn the proposition round and say "Wisdom is a characteristic of Socrates", then wisdom, formerly the predicate, is now the subject.' Both sentences express the same proposition, i.e. they have the same meaning, so it is rather a matter of grammarians: '... with a sufficiently elastic language any proposition can be so expressed that any of its terms is the subject. Hence there is no essential distinction between the subject of a proposition and its predicate, and no fundamental classification of objects can be based upon such a distinction ... the whole theory of particulars and universals is due to mistaking for a fundamental characteristic of reality what is merely a characteristic of language' (pp. 12-13).
Complex universals It could be assumed that the subject-predicate pattern may apply to 'compound' propositions, i.e. those propositions containing atomic propositions as constituents through the use of logical connectives, as for instance 'Either Socrates is wise or Plato is foolish'. Yet this is not the case. If we take a simpler case, 'aRb', the theory of complex universals will lead us to three different propositions: 'R holds between a and b\ 'a has the complex property of "having R to b" ', and *b has the complex property of "having R to a" '. But there are not three propositions, but one, 'for they all say the same thing, namely that a has R to b. So the theory of complex universals is responsible for an incomprehensible trinity. As senseless as that of theology' (p. 14). The reason for the view that variable propositional functions have a definite meaning to be generally held is linguistic convenience. But there are ways to dispense with that supposed need: '"a has all the properties of />" is the joint assertion of all propositions of the form (f)b . D . <pa, where there is no necessity for (f) to be the name of a universal, as it is merely the rest of a proposition in which a occurs. Hence the difficulty is entirely imaginary' (p. 16). Therefore there is no need to accept
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complex universals and so the supposed distinction between subject and predicate should be restricted, at most, to atomic propositions. Thefelt difference We feel there is an important difference between particulars and universals: while a particular is independent, a universal depends upon something else; i.e. in 'Socrates is wise', 'Socrates' is independent but 'wise' is just a quality of 'Socrates'. The reason for us to see this difference at all can be explained by noticing that, although they both are not names for genuine objects but incomplete symbols (those in need of something else to reach true meaning, according to Russell), 'Socrates' gives only one collection of propositions, but 'wise' gives two. In the first case, by collecting all the propositions where 'Socrates' takes place we get ' Socrates', where (f) is a variable, e.g. 'Socrates is wise', 'Socrates is neither wise nor just', and so on. In the second, we not only get '(f) wise', i.e. the collection of all propositons where 'wise' occurs, but also a narrower collection of the form 'x is wise'. Thus, while 'Socrates is wise' and similars are values of'x is wise', 'Neither Socrates nor Plato is wise' and similars are not values of 'x is wise', 'but only of the different function " is a variable' (p. 20). Is this a real difference between both cases or just an apparent one? The answer explores the distinction between properties and qualities, the simplest kind of properties, to try to reach the two former collections of propositions for the first case as well, the 'Socrates' case. In the end this line of thought would make sense only for genuine objects, but not for 'Socrates', which is a logical construction, or incomplete symbol. The new question then is whether a similar distinction can be made for incomplete symbols. The answer is yes: any incomplete symbol 'a' 'will give us two ranges of propositions: the range ax obtained by completing it in the way indicated in its definition (i.e. in conjunction with another symbol x for reaching a "complete" meaning together); and the general range of propositions in which a occurs at all, that is to say, all truth-functions of the propositions of the preceding range and constant propositions containing cc' (p. 23). And this is essentially the same situation we formerly got for predicates or adjectives, especially if we notice that the distinction is the same as the one existing between primary and secondary occurrences of a symbol. (In a primary occurrence for a symbol, the existence of a referent is asserted, while in a secondary occurrence it is not, which takes place because in primary occurrence the symbol is free of the scope of a wider proposition than the one it belongs to. As Russell wrote in his famous 1905: 'A secondary occurrence of a denoting phrase may be defined as one in which the phrase occurs in a proposition/) which is a mere constituent of
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the proposition we are considering, and the substitution for the denoting phrase is to be effected in//, and not in the whole proposition concerned'.) Thus 'any incomplete symbol is really an adjective, and those which appear substantives only do so in virtue of our failing whether through inability or neglect to distinguish their primary and secondary occurrences' (p. 23). This can be illustrated with Whitehead's analysis of material objects, where they are regarded not as substantives, but as mere adjectives of the events where they are located, as in the proposition 'A is situated in E'. This is done precisely by showing that the primary occurrence of the symbol for an object, a substantive, can be converted into the symbol in the secondary occurrence: 'Thus "A is red" will be "For all E, A is situated in E implies redness is situated in E " ' (p. 24). Therefore, the fundamental distinction for incomplete symbols is not between substantives and adjectives, but between primary and secondary occurrences, so that a substantive is just a logical construction, then a subjective property. Then there is just one thing left: to see whether genuine objects (not incomplete symbols) can be divided into particulars and universals, and this can be seen by looking at logical notation. Symmetry in logic The subject-predicate distinction is underlying the object-function one. But logicians could have developed alternative systems in which there was a complete symmetry between objects and functions, so the distinction depends just on the convenience of logicians. Where 0 stands for complex compounds of logical relationships it is an incomplete symbol, so it cannot be defined in isolation or stand by itself. Yet when '« is a two-termed atomic proposition, "" is a name of the term other than a, and can perfectly well stand by itself (p. 26). (This is supposed to be the case of 'atomic facts', formed by two 'objects' in Wittgenstein's jargon.) Mathematical logic is extensional, and so mostly interested in classes and relations in extension, so the distinction between functions as names and functions as incomplete symbols is irrelevant for logicians. To get complete symmetry between functions which are names and other names, in a way that not only functions determine two ranges of propositions, but also names (see the former argument), we just should develop a new logical notation for functions which are names as follows: '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 (f>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
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would determine two ranges of propositions, for a would determine the ranges qa and (f)a, where q and 0 are variables, and q would determine the ranges qx andfq, where x and fare variables' (p. 28).
Unity In this section I offer some comments about the nature of each of the former arguments, trying to emphasize their main assumptions. Also, I make an attempt to unify those assumptions to see if a common, general pattern can be found for the whole argumentation. Finally, I insert some of these assumptions into the more general framework of classical analytic philosophy. The symmetry in language argument, based on a form of reversion, or transposition, of subject and predicate, seems to depend on a formal way to understand language. This is made in the Bradley style: we do not have to look for grammatical analyses, because we are not interested in sentences but 'in what they mean, from which we hope to discover the logical nature of reality' (p. 13). So it seems to me that for Ramsey the inference from language to reality is acceptable, just it should be made not from the language as it is, i.e. ordinary language, but from ideal language, or even better, from language expressing its genuine logical form. Even arguments based upon space and time were seen by Ramsey to be dispensable from the viewpoint of logical form. Thus, when he briefly considers the argument according to which it could be said that objects can only be in one place, while properties can be in many, he however does not develop a full reply to it (that is why I did not include his 'argument' in the former section). Rather, he just says that this way to see the problem is not leading us to reach 'the essence of the matter', because when two people discuss if a table is an adjective or a substantive, they 'are not arguing about how many places the table can be in at once, but about its logical nature' (p. 9). Complex universals are rejected through a form of reversion as well, by extracting three different propositions from aRb. Then by saying that there is just one, not three, because all three have the same meaning. This is similar to the symmetry in language argument, as it shows that different apparent propositions are one and the same, so it seems to insist that there is no essential difference between subject and predicate. Now predicates are more complex, because they involve relational properties, but the nutshell seems to be the same again. Then Ramsey explicitly resorts to the logical form line: '# has all the properties of b' is really the joint assertion of all propositions of the form '(f)b . D . a', so the original expression vanishes, and the true logical form remains, not giving place to apparent, complex universals any longer. Besides, as long as he implicitly maintains that there is just one true analysis of a
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proposition, and therefore all other analyses are wrong, he is also defending the idea of logical form underlying grammatical appearance. (Anscombe 1959, p. 96, already pointed to Ramsey's belief that one analysis of a proposition excludes all other analyses, yet she did not link that belief to the logical form line of thought.) So finally we have here symmetry and logical form: together they show complex universals to vanish. Thus the complex universals argument can be reduced to the deep belief in logical form underlying linguistic appearance. (Yet the argument can be developed in itself, as shown in Oliver 1992 and Mellor 1992.) In the felt difference argument Ramsey uses first a symmetry line, by showing that subjects and predicates can be both reduced until giving rise to two ranges of propositions. This could be a sort of logical form argument as well, for it is designed to show that names are different from qualities (predicates) just apparently. Then by declaring both to be incomplete symbols it is shown that they are actually adjectives in the same way, in spite of the 'felt' difference. This seems to be clearly based again in logical form, which should be 'seen' underneath the grammatical appearance. Finally logical form is used explicitly, both by resorting to a clearly formal category, the division between primary and secondary occurrence, then by applying it to the Whitehead example: starting from 'A is situated in E', 'A is red' can be transformed into 'For all E, A is situated in E implies redness is situated in E\ so the true logical form is now apparent and the pseudo-problem dissolved. The incomplete symbols line of thought seems to depend upon formalization as well, as can be seen in the concept of incompleteness itself, in the classic, Fregean style of predicates/functions as 'unsaturated entities'. Also, it clearly depends on formalization in the way how Whitehead's manoeuvre is accepted without discussion. In the end we have here symmetry and logical form to show that there is no essential difference between substantive and adjective. I see the symmetry in logic argument to depend heavily on the former symmetry in language one: logicians couldhave developed a logical notation where subject and predicate could behave symmetrically, i.e. where reversion between names of objects and names of concepts were possible. This is facilitated by the fact that the functional language of predicate logic was literally taken from the ordinary language pattern. Therefore the main argument is still the same: symmetry in language. Thus, the deep unity of all these arguments is based upon symmetry of subject and predicate (object and function/concept) and upon the idea of logical form underlying ordinary language. Besides, there is the idea that we should not make inferences from the structure of language to the structure of reality. Yet, what is the relation between symmetry and logical form? One relation could be that apparent asymmetry is possible because we are not aware of the true logical form of the expression considered, so once the genuine logical form
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is clarified, the full symmetry between subject and predicate is unveiled, against the grammatical, superficial form. In the end, Ramsey was convinced about this long before he developed these technical arguments, as can be seen in the Tractarian thesis he maintains in the beginning of the paper, when he says that objects are all essentially incomplete: 'they cannot occur in facts except in conjunction with other objects, and they contain the forms of propositions of which they are constituents.' (p. 11). But no argument is given to the reader to support this very strong 'conclusion'. Also, what is the relation between logical form and the arguments based on a reversion? The logical form arguments are a way to dispense with a given expression, so showing that this expression is not really needed, as we can say essentially the same thing without it. Thus no ontological inference can be made on the basis of that expression. The reversion arguments used in the symmetry line have the same implicit structure: by showing that a reversion is possible, essentially leading to the same meaning, it is shown that the former ontological inferences were unnecessary. The main difference between the two cases is that while in logical form we are supposed to reach the truth, in symmetry we just reach an alternate way to say the same, possibly as wrong as the original one, at least from the viewpoint of the genuine logical form, which is always unique. Finally, what is the relation of all this to the rejection of any languagereality inference? Well, it seems to me that for Ramsey ordinary language cannot be used as a guide for ontology because it is misleading, but through the logical form transcription devices we could manage to make some assumptions about reality based on a somehow regimented, formalized language. Based on this there is no distinction between object and concept, because logical form shows that there is no distinction between subject and predicate, and this is an inference from language to reality, but from regimented language! Thus the end is like the beginning: objects are as incomplete as concepts, so they are as needed of completion in the same way, as can be 'seen' in facts. So it seems that Ramsey is building his whole position on a Tractarian base: everything is an object, there are no concepts prima facie, or perhaps concepts should be taken as apparent results of the combination of objects. I'm not implying that Ramsey is somehow denying the existence of concepts, to affirm the existence of objects, as he is obviously denying the whole objectconcept distinction. Like the first Wittgenstein, he seemed to be convinced that there is just a kind of ultimate constituent of facts in the world, no matter how we call them. This is not the place to discuss whether or not Ramsey was right in his implicit interpretation of Wittgenstein's Tractatus, but Anscombe (1959, pp. 98ff.) took him to be clearly wrong. Curiously enough, Moore defended a similar ontology in his 1899 paper, the main difference being that then Moore believed there are only concepts:
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everything is a concept. However, the main consequence is the same: there is no essential difference between subject and predicate. For Moore (1899), propositions are just complex concepts, and any other distinction between concepts, apart from the simple/complex one, is of no importance. Finally, Moore came to reject this simple ontology because he wanted to accept some additional ways to divide concepts into several classes, and so the original plan was abandoned, and the original object—concept distinction accepted again in the papers he wrote from 1900 onwards. His final position, including the thesis that names cannot be predicated from anything else, can be clearly seen in Moore (1923), published just a couple of years before Ramsey's paper. We can speculate about Moore's having somehow triggered Ramsey's, in the sense that Ramsey may have thought of showing a way for names to be somehow converted into predicates just after having read Moore's paper. In the same way, we can speculate about the Ramsey way to transform names into predicates as having somehow influenced Quine's more famous thesis according to which names can be treated as predicates, e.g. 'Pegasus' as 'Pegasises'. Moore's modified position was used by Russell for his ontology in Principles of Mathematics (1903), where he considered some symmetry arguments very similar to Ramsey's (Russell 1903, §48). However, he finally rejected any strong form of symmetry and based his whole logico-ontological system on the subject-predicate distinction, and all his logic on the concept of prepositional function: a version of Frege's concepts. Even more curiously: for Bradley the distinction was to be rejected as well, and although he used some symmetry arguments (1893, chapter 2), they were always directed against any genuine conceptual distinction between terms, qualities and relations: there must be just some sort of ultimate material in the world, and the subject-predicate distinction is to be rejected. Obviously, Moore tried to preserve some of those ideas in his first 1899 ontology, although he failed to develop it out in a convincing way. (For more details about this see Rodriguez-Consuegra 1999a and 2002, chapters 2, 3 and 5.)
Assessment In the following I will make some comments about what I pointed out to be the two most important ideas underlying Ramsey's set of arguments: (1) the ultimate identity between subject and predicate (substantive and adjective), as involved in the symmetry arguments, either in language or in logic (including the logical form assumption here), and (2) the rejection of any legitimate inference from language to reality. As for symmetry, the fact that we could exchange subject and predicate most of the time (which is by no means clear after all) does not mean they are the
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same. All it should prove is just this: we always need some subject and some predicate. So the distinction seems to survive anyway, at least as a relative one. True, this already would undermine the universal—particular distinction as an absolute one, but I cannot help believing there is something more in it than a mere arbitrary, conventional trick, as it seems to be presented in Ramsey's paper. In the end, if this is so, we can always try to see if the distinction conveys some ontological assumptions under it, and also if these assumptions may have some independent value. On the other hand, when we play with those examples, as for instance with the 'Socrates is wise' one in the symmetry in language argument, we may be forgetting that 'wisdom' and 'wise' are not identical. 'Wisdom' is a substantive and 'wise' an adjective, so we can apply here an old distinction used by Russell to point out a difference between 'relating' relations and relations in themselves: while 'wise' would be a 'predicating' predicate, 'wisdom' would be the predicate in itself, so not actually predicating. Therefore, even if we succeed in showing some symmetry as actually working, we can always argue that there is a natural way for substantives and adjectives to occur in sentences, and that exceptions do not involve any ultimate destruction of a deeper distinction. (For more arguments on the essential asymmetry of subject and predicate see Strawson (1970). Strawson (1954) is also interesting in that it explores some lines of argumentation not considered by Ramsey.) Ramsey was not ignoring the distinction, but trying to show that it is just a grammatical one through the symmetry arguments, leading us to logical form, where the distinction vanishes. Curiously enough, the Russell arguments try to show that, even if we can artificially dispense with the distinction through the symmetry manoeuvres, the true logical form of predicating predicates and predicates in themselves is ultimately different. Thus, the logical form arguments can be used to prove contradictory assertions. Also, the symmetry argument might be formulated in alternate ways, which may be closer to Ramsey's original intentions. For instance, instead of reversing 'Socrates is wise' into 'Wisdom is a characteristic of Socrates', we could reverse it into 'Wisdom is Socratic', by taking advantage of the fact that some names can be converted into adjectives in a natural way; i.e. from 'metal' we obtain 'metallic', and from 'analysis' we get 'analytic'. In the case of Socrates, 'Socratic' usually means a follower of Socrates, rather than something exhibiting properties typical of Socrates, but if we can somehow force ordinary language in the way pointed out by Ramsey in his usual symmetry arguments, why not force it in ways supported by ordinary use? (Strawson 1959, p. 174, already mentioned the 'Wisdom is Socratic' possibility, but he did not develop it in my, or any other, way.) At any rate this is deeply linked to the incompleteness argument, according to which names (objects) are as incomplete as predicates (properties and
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relations), as they need completion in a sentence (in a fact) to reach full meaning (existence, or at least a place in a state of affairs). For if the symmetry argument is right, then there is no especial incompleteness in names or objects; and the reverse is also true: if the essential incompleteness argument is true, then a full symmetry is reached. But we can also assess the incompleteness argument by itself: it seems to me obvious that predicates are incomplete in a very different sense than names. As it has been pointed out before (Strawson 1959, p. 153; Dummett 1973, pp. 61ff.), the incompleteness of predicates is just a prepositional one, but we can imagine more than one sort of'incompleteness' for names, as they can be used in a variety of linguistic constructions not necessarily propositional. That is, the predicative incompleteness suggests by itself an assertive completeness, which is totally absent from names. However, this argument could be objected that, if we convert names into predicates, by any of the possible symmetry manoeuvres, then even this sort of propositional incompleteness could be found in names. So in the end we are left with the symmetry line as the main argument again. MacBride (1998a) can be regarded as a more modern way to attack the incompleteness argument, then to defend the symmetry line, and ultimately the non-distinction between particulars and universals. He efficiently analyses Armstrong's realistic position on universals by providing counter-arguments to that position, especially concerning Armstrong's inability to give convincing arguments that universals and particulars are really different kinds of entities. This is particularly related to Ramsey when MacBride criticizes Armstrong's current conception according to which universals are 'unsaturated' entities, because MacBride does so by pointing out that particulars, like universals, 'occur only as the constituents of states of affairs' even in Armstrong's metaphysics (p. 34). Yet I cannot help seeing the Ramsey connection as a rather loose one, as MacBride does not resort to any of the particular arguments actually used by Ramsey, who is not even mentioned in that paper. Therefore we can say just that, although he is somehow 'inspired' in Ramsey's general rejection of the distinction, his arguments do not constitute an improvement of Ramsey's actual arguments. This has been somehow admitted by MacBride in his 1998b, where he writes that the rejection of the particularuniversal distinction 'cannot be substantiated on the basis of the arguments that Ramsey provides' (p. 203). Thus, for the same reasons, MacBride's attempts to 'improve' on Ramsey's arguments in his 1998b and 2001, in the context of more contemporary discussion on universals, should be regarded rather as very loosely inspired in Ramsey. Thus, although MacBride's work is very valuable in itself, I think it cannot be used to vindicate Ramsey's arguments but, at most, his general, sceptical position about the celebrated distinction. However, as we have seen, scepticism over that distinction was already present in Bradley, the first Moore, and even the first Russell.
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The 'is' of predication was taken by Peano to involve membership, and it was clearly distinguished from inclusion by himself and his followers. In our case it seems that if we say 'Socrates is wise' it is membership that is involved, because we think Socrates to be a member of the set of the wise people. Yet when we try to do the same with 'wisdom', e.g. to convert a property term into a subject, we may instantly be involved with inclusion rather than with membership. If we say 'Wisdom is a quality' or 'Red is a colour', it seems that what we are really implying is that everything exhibiting the first order property is also exhibiting the second order one: every wise entity is also an entity with a certan quality, and every red thing is also a coloured thing. Agreed, we can also say that in these examples both wisdom and red are also members of certain sets (so seeing them as 'objects'), but it is also true that this twofold possibility is not open to 'Socrates', unless we make rather strange manoeuvres, so if this is so then the distinction between substantives and adjectives will be a well-supported one from our usual conceptual system, as it underlies ordinary language. Thus, although probably for Ramsey there was no essential difference between 'Socrates is wise' and 'Red is a colour', as both were examples of instantiation between particular and universal (for those accepting the distinction), the truth is that the mere distinction between different 'logical forms' underlying each of them (membership and inclusion) may lead us to completely different treatments of their philosophical consequences. In the whole symmetry arguments line, for Ramsey the relation of characterizing is exactly equivalent to 'is' (1925, p. 29), so it cannot be used to make a difference between subject and predicate: it is a 'verbal fiction'. But we could use the symmetry argument under the characterization form in a way which cannot be done by using 'is', like this: 'Wisdom characterizes Socrates' and 'Socrates characterizes wisdom'. In this way we can see that the characterizing relation might work even as a symmetrical relation. Yet one problem could be that, while it is clear how wisdom could characterize Socrates, i.e. by being a property of Socrates, it is unclear how Socrates might characterize wisdom. Unless we admit wisdom, as equivalent to a set of individuals (the wise people), to be characterized by those individuals, in the sense that perhaps the 'type' of wisdom could be different according to the particular membership of the set. Of course if we take a purely extensional viewpoint then the whole approach is useless, but in a rather natural way we usually characterize properties by the individuals exhibiting them, as it takes place in the language of colours and other empirical qualities. Dummet criticized the symmetry arguments by writing that, while with properties we get a contrary, as when we say 'wise' and 'not wise' (or foolish), this does not take place with names: we do not say 'non-Socrates' (1973, 6Iff.). (In doing so, Dummett probably followed Anscombe and Geach, as both
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authors had been previously defending a similar position; see Strawson 1970 for quotes and comments.) Sahlin already took care of this criticism efficiently (1990, p. 200), based on the Tractarian thesis that there are no negative properties, just positive ones, so the argument does not seem to work against him, at least if we assume Ramsey to follow Wittgenstein once again. Yet if the symmetry argument is to be taken seriously, another response could have been that, if not wise is a property, then in a sense non-Socrates would be one as well. As we saw above, if we take 'Wisdom is Socratic' as a valid proposition (by reading 'Socratic' as 'metallic', 'solid', or any similar adjective) then not Socratic could be a property as well. This might be also used to meet a criticism by Simons (1991, p. 152), according to which in 'Wisdom is a characteristic of Socrates', the actual predicate is not 'Socrates' but the whole expression after 'Wisdom'. If we use 'Wisdom is Socratic', the criticism is no longer valid, so we need deeper criticisms. As for the logical form line, this is not explicitely mentioned by Ramsey in this paper, but it may well be the general framework of the whole paper, as I said in the former section. The closest statement for a typical logical form theory which can be found in Ramsey's paper is this: 'we are ... interested not so much in sentences in themselves, as in what they mean, from which we hope to discover the logical nature of reality. Hence we must look for senses of subject and predicate which are not purely grammatical, but have a genuine logical significance' (1925, p. 13). As a matter of fact Ramsey wrote in 1929 a short paper on philosophy, where he reduced the role of philosophical analysis to build up a sort of axiomatic system based on logic and definitions. In his own words: Tn philosophy we take the propositions we make in science and everyday life, and try to exhibit them in a logical system with primitive terms and definitions etc. Essentially philosophy is a system of definitions, or only too often a description of how definitions might be given' (1931, p. 263). In writing this he was following rather closely what I have described as Russell's method of constructive definitions, so it is not strange that in 1925 he was trying to apply this method to reconstruct some problematic concepts and distinctions to get a cleaner, new presentation of them. (I have studied Russell's method, as applied to mathematical philosophy, in my 1991. Also, a more general description of the method, as applied to many other fields of philosophy, can be found in my 1999b.) Ultimately, when Ramsey holds that 'Socrates is wise' and 'Wisdom is a characteristic of Socrates' he is assuming that both statements express the same proposition, so they presumably assert the same fact. Therefore, he is also assuming that both linguistic expressions are alternate ways to express one and the same logical form, which should closely correspond to the genuine constituents of that fact, where no objects or concepts could be found. Those true constituents can be unveiled just when we develop a logical way to
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dispense with purely linguistic categories, like subject and predicate, and this is the main goal of the different logical manoeuvres he suggests in his paper. Thus, all the arguments of the paper on universals would presuppose that, in transcribing ordinary language into logical form, we are getting the true meaning, then the true form of facts, then somehow dissolving purely apparent philosophical pseudo-problems, based on grammatical, misleading forms. Yet logical form is language, the language of an especial sort of logic, or logical calculus; a language which is built up by taking advantage of a formal structure previously created with definite, technical purposes, which may survive by itself as a part of mathematics. So if we maintain that there is a strong link between this language and the form of facts, this should be guaranteed through some dedicated arguments. Instead, Ramsey takes it for granted without any explicit argument. At the end of the day, if any inference from language to reality is prohibited, while arguments based on logical form are allowed, we are facing a vicious circle: how could we know the way to extract the logical form from ordinary discourse, which is the only one from which we can draw conclusions as for the form of reality, without knowing in advance how reality is? I think this is a problem underlying the whole logical atomism line, from Frege to Russell and the first Wittgenstein. The symmetry in logic argument is once again based on a formal way to understand language. One problem about this argument is that talk about functional symbols as names of properties is presupposing that properties can be regarded as objects. If they are not objects but concepts, the fact that we modify our way to talk about them cannot lead us to transform them into objects. Apart from pure mathematics, in the context of the relation between logic and language, logic seems to use functions because language is conceptual, and language is conceptual for some reason. As Quine used to remind us, the objects of a theory are the objects this theory talks about. So it makes not much sense to maintain that predicate logic says nothing about reality: it is an instrument to be applied very usefully to reality, which tremendously helps us to understand reality thanks to assuming objects and functions to exist. As it happens in empirical science, where we apply models to reality in order to conceptualize - then understand - it, the same seems to be true with logical models, which are used with the same purpose. Another problem is second order logic, as involved in the argument that we could deal with predicates (functions, relations) as mere objects. By the time 'Universals' was written, second order logic was still obscure and not clearly distinguished from first order logic, as can be seen in Principia Mathematica, so it is easy to understand that Ramsey did not see any problem in converting predicates into subjects. Yet z/"this implies that we can quantify over predicates, then the argument fails: second order logic is a very different animal, and behaves weirdly. For once, in first order predicate logic we cannot quantify
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over predicates, and these are seen in the usual, extensional way as functions and relations, i.e. subsets of the universe. Second order logic allows us to do so, but at a price: the usual, well-established metatheoretical results for first order logic fail for second order logic, and model theory becomes extremely difficult. On the philosophical side we could even see second order logic rather as set theory in logical disguise, as Quine used to say. Yet here I'm neutral on the present controversy about the nature of second order logic. All I'm saying is that Ramsey's argument may be committing him to second order logic, then to all the technical and philosophical problems involved, so the original argument may not be worth maintaining at so expensive a price. A further problem with the symmetry in logic argument is that it seems to somehow involve the Tractatus thesis according to which we do not know anything about atomic propositions, so they cannot be instantiated. Then how could we tell the way to deal with them in logic? Are we talking about actual language or about ideal, formalized language? This seems to me to be a further, vital problem in dealing with logical form as a way to dispense with philosophical problems involved in language, and it was also present in the whole logical atomism project (see above). Anyway, Ramsey seemed to finally believe that it is not impossible to discover atomic propositions by actual analysis (1926, p. 31), yet this would be a further sign of his deep belief in logical form as reached by linguistic analysis! And this, again, should be a problem for his belief that no inference can legimately be made from language to ontology. Finally, what about Ramsey's most important assumption, rejecting inferences from language to reality, as considered in itself, independently from the other arguments? If we are not allowed to draw ontological consequences from language, then applying them to reality, what is left? Is there any other means to speculate about the structure of reality than studying the ontological structure of language, i.e. its ontogical implications? What is ontology? It cannot be based on empirical theories: if so ontology would be like physics or astronomy. If we cannot use language as a guide, how could it be possible for us to try to determine what there is? As Quine used to say, it is not that language creates reality, e.g. what there is does not depend upon our language, just what we can say about what there is does depend upon our language. Thought, in the end, is linguistic, even for the Tractatus, a particularly important source of inspiration for Ramsey. So by resorting to 'pure' thought nothing is gained after all. Agreed, it seems that there is some thought which can be produced independently of language, but ontology is a very complex philosophical theory, so it should depend on language in important, deep ways. Ramsey says: the universal-particular distinction is just a result of the subject-predicate distinction, so in accepting it for reality we are taking a linguistic distinction to be an actual, ontological distinction. As we have seen, his main argument is the symmetry one. Yet the symmetry argument is a
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linguistic argument, so he is anyway taking a trait of language to be a trait of reality! i.e. if subject and predicate are but symmetric then there is no real distinction even in language, so there is no distinction in reality either. True, this was not what Ramsey seemed to have in mind, but it could be said anyway by strictly following the logic he was raising. Language contains the conceptual apparatus with which we approach reality to perceive, conceive and handle it, so we might only find in reality what language allows us to find. Language is very similar to a microscope, or a telescope: it is also an instrument. But language has evolved, so it is to be hoped that it has been approaching reality progressively, until being much more closely linked to it than in the beginning, as happens with technical instruments. Reality shows itself according to the instruments we use to approach it. It is us who, by applying different categories of instruments, obtain different pieces of different information from our research. We must use some instrument to approach reality from the ontological viewpoint, we just cannot do it without any conceptual means. Therefore, if language is the conceptual structure through which we understand reality, so it should be closely related to reality, then there must be something about reality in the structure of language. And this seems to be especially true about the subject—predicate distinction, which could be pointed out to be undeniable from a naturalistic viewpoint, as can be seen in the context of the origin and evolution of language. Concepts seem to be constructions out of reflection because they represent properties of objects, which seem to be prior for ordinary life, then for evolution. It can even be said that the subjectpredicate distinction might well be in our genes (rather than a 'universal grammar'), because it is ready for being an hereditary property of our global human culture, then of our language (Deacon 1997). As a matter of fact, interesting experiments with just 13-month-old infants have shown that they can already differentiate between nouns and adjectives as a way to start categorizing the world (Karmiloff and Karmiloff-Smith 2001, p. 68), so the subject-predicate distinction may well be a trait incorporated in our conceptual apparatus by means of genetic assimilation because it somehow shares with reality something which was objectively important for survival in the past. Therefore the right way might not be from subjectpredicate in language to object—concept in ontology, but from object-concept in behaviour and culture to subject-predicate in language, then in ontology. If this is so, Ramsey was mistaken in believing that the distinction was a contingent one accidentally incorporated into language, without reflecting anything real. I think at least a moderate form of linguistic relativity against cognitivism can be defended. In my 'Cognitivismo y lenguaje' I give more details of a general position against cognitivism, together with a series of arguments based on
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important empirical findings concerning a variety of scientific fields. However, I am no pragmatist. I am not saying that by obtaining efficient, useful results you are somehow guaranteeing the truth of the theories or assumptions you are making or applying. All I am saying is that language is already a part of reality (behaviour, culture, evolution), so it must share some traits with it, mostly because it was first used precisely to conceive and handle reality successfully. There is no other way to build up an ontological system than by regarding language as a conceptual structure to be searched, and this can be clearly seen in most philosophical systems and theories.
Endnotes 1.
I am grateful to Maria Jose Frapolli for her kind invitation to participate in so interesting a volume devoted to Ramsey's ideas. My thanks are also due to Fraser MacBride for having read a former version of this chapter, and providing me with interesting comments and suggestions, which allowed me to introduce numerous clarifications and improvements in the final version.
References Anscombe, G. E. M., 1959. An Introduction to Wittgenstein's Tractatus. London: Hutchinson. Fourth ed., 1971. Bradley, F. H., 1893. Appearance and Reality. Oxford: Clarendon. Deacon, T., 1997, The Symbolic Species. London: Penguin. Dummett, M., 1973. Frege: Philosophy of Language. London: Duckworth. Karmiloff, K. and Karmiloff-Smith, A., 2001. Pathways to Language: From Fetus to Adolescent. Cambridge, MA: Harvard University Press. MacBride, F., 1998a. 'On how we know what there is'. Analysis, 58, pp. 27—37. 1998b. 'Where are particulars and universals?'. Dialectica, 52, pp. 202—27. 2001. 'Four new ways to change your shape'. Australasian Journal of Philosophy, 79, pp. 81-9. Mellor, D. H., 1992. 'There are no conjunctive universals'. Analysis, 52, pp. 97—105. Mellor, D. H., and Oliver, A., 1996. Properties. Oxford: OUP. Moore, G. E., 1899. 'The nature of judgment'. Mind, 8, pp. 176-93. 1923. 'Are the characteristics of particular things universal or particular?'. Included in his Philosophical Papers. London: Allen & Unwin, 1959, Chapter 1. Oliver, A., 1992. 'Could there be conjunctive universals?' Analysis, 52, pp. 88-97. Ramsey, F. M., 1925. 'Universals', Mind, 34, pp. 401-17. In Ramsey 1990, pp. 8-30. (The first reprint was in Ramsey 1931. The most recent one is in Mellor and Oliver 1996.) 1926. 'Note on the preceding paper'. In Ramsey 1990, pp. 31-3. 1931. The Foundations of Mathematics and Other Logical Essays, R. B. Braithwaite (ed.). London: Routledge & Kegan Paul.
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1990. Philosophical Papers, D. H. Mellor (ed.). Cambridge: Cambridge University Press. Rodriguez-Consuegra, F., 1991. The Mathematical Philosophy of BertrandRussell: Origins and Development. Basel, Boston and Berlin: Birkhauser (reprinted 1993). 1999a. 'Bradley, Frege and relatedness'. Bradley Studies, 5, pp. 113-25. 1999b. 'Russell y el analisis filosofico'. Introductory essay in B. Russell, Andlisis filosofico, F. Rodriguez-Consuegra (ed.). Barcelona: Paidos, pp. 13-33. 2002. Estudiosdejilosofiadellenguaje. Granada: Comares. 2004. 'Cognitivismo y lenguaje: ^un paradigma que se hunde?' 2004. 'Cognitioismo y lenguaje: ,;un paradigm a que se hunde?' Dialogos 84, in print. Russell, B., 1903. The Principles of Mathematics. London: Allen & Unwin. 1905. 'On denoting'. In 1956, pp. 39-56. 1911. 'On the relations of universals and particulars'. In 1956, pp. 103-24. 1956. Logic and Knowledge, R. C. Marsh (ed.). London: Allen & Unwin. Sahlin, N.-E., 1990. The Philosophy of F.P.Ramsey. Cambridge: Cambridge University Press. Simons, P., 1991. 'Ramsey, particulars and universals'. Theoria, 57, pp. 150-61. Strawson, P. F., 1954. 'Particular and general'. Included in 1971, Chapter 2. 1959. Individuals. London: Methuen. 1970. 'The asymmetry of subjects and predicates'. Included in 1971, Chapter 5. 1971. Logico-Linguistic Papers. London: Methuen. Wittgenstein, L., 1922. Tractatus Logico-Philosophicus. London: Routledge & Kegan Paul.
11 Ramsey and the notion of arbitrary function Gabriel Sandu
Introduction In his article The Foundations of Mathematics (1925)1 Ramsey was concerned with the nature of the statements of 'pure mathematics' and the way these statements differ from those in empirical sciences. He thought that the answer given to these questions by Hilbert and the formalist school, according to which mathematical statements are meaningless formulas, is unsatisfactory for several reasons, which will not be discussed here. He also expressed serious doubts about the intuitionist programme developed by Brouwer and Weyl. It is the logicist school of Frege, Russell and Whitehead, which attempted to reduce mathematics to a few logical concepts, that came closest to Ramsey's views, although he was not completely satisfied with it, either. Carnap introduced a distinction between two theses of logicism: (a) (b)
The definability thesis: the concepts of mathematics can be derived from logical concepts through explicit definitions. The provability thesis: the theorems of mathematics (in which the mathematical concepts are replaced with their definitions as in (a)) can be derived from logical axioms by logical means alone.2
One reduces, for example, arithmetic to logic in the sense (a) if one defines the natural numbers within a logical system which uses the standard logical connectives, quantifiers and identity. This reduction is also one of type (b), if, in addition, the theorems of arithmetic thus reformulated are derived from logical axioms alone together with standard truth-preserving logical rules. Carnap pointed out that both Frege's and Russell's work constitute a fulfilment of the logicist programme in style (a), but not a reduction in style (b). Although Garnap discussed explicitly the relation between Ramsey and Russell, we found no textual evidence that would show him to be aware of the fact that the aforementioned distinction he himself introduced is to be found already in Ramsey, who wrote: Thus Russell, in The Principles of Mathematics, defines pure mathematics as 'the class of all propositions of the form "p implies q" where p and q are
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propositions containing one or more variables, the same in the two propositions, and neither/? nor q contains any constants except logical constants'.3 After this passage, which shows clearly that Ramsey took Russell to have carried out a reduction in style (a) of mathematics to logic, he went on to argue that, for such a reduction to be complete, one also has to show that the resulting logical statements are tautologies in Wittgenstein's sense, a point which Ramsey credited Russell as having recognized later on in his Introduction to Mathematical Philosophy but that he completely ignored in Principia Mathematica. For Ramsey, a logicist programme which reduces mathematics to a class of propositions which contain only logical constants (including identities), but does not show that these propositions are tautologies, stops halfway in that it only shows that mathematical statements possess complete generality without yet showing that they possess full necessity? It is right then to say that Garnap's distinction between two kinds of reduction of mathematics to logic is an echo of the earlier Ramseyan distinction between mathematical statements which have complete generality and those which are necessary. With an eye on this distinction we can say that one essential part of Ramsey's work in the foundations of mathematics was to improve on the system of Principia so that it retains only those axioms which are tautologies (in Wittgenstein's sense). His rejection of Russell's Axiom of Reducibility is a well-known step in this direction. We know today that this project cannot be consistently reconciled with the reduction of mathematics to logic in style (b). Ramsey's second major project, and this is the main focus of the present chapter, was to reinterpret the resulting logical system in such a way that it did justice to the extensional attitude of the mathematics of his days (Cantorian set theory). In this respect, his attitude towards foundations was completely different from that of Frege and Russell. In fact, we shall see in the last section of this chapter that it is precisely through the notion of function in extension that Ramsey (mistakenly) believed he had reconciled the extensionalist attitude of the mathematics of his days with a reduction of mathematics to logic in style (b).
Frege's logicist programme Frege's logicism programme has begun in his Begriffsschriftiftt (18799), a secondnd order predicate calculus with negation, implication and universal quantification as basic logical operators. The individual variables in Frege's system are to be thought of as ranging over all objects in the universe, concrete and abstract. Frege had already in the Begriffsschrift a function-argument analysis that was extended to a distinction between object., concept, and its extension in Die Grundlagen der Arithmetik (The Foundations of Arithmetic), published in 1884. The
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explicit definition of concepts as functions which map objects into the set of truth-values True and False is introduced in 1891 in his Function and Concept. In this paper and in two subsequent ones, On Sinn and Bedeutung (1892) and On Concept and Object (1892), Frege makes an extensive use of the notion of the extension of a concept. This notion finds its most important use in the Grundlagen, in the definition of a natural number as the extension of a certain concept: The number that belongs to the concept F is the extension of the concept 'concept equinumerous to the concept F', where a concept F is called equinumerous to a concept G if the possibility exists of one—one correlation. Frege assumes that the notion of the extension of a concept is well understood, although he expresses some reservations towards it: We assumed here that the sense of the expression 'extension of a concept' was known. This way of overcoming the difficulty may well not meet with universal approval... I too attach no great importance to the introduction of extensions of concepts.6 In the first volume of Grundgesetze der Arithmetik (Basic Laws of Arithmetic] published in 1893, Frege tried to say more about the extensions of concepts by formulating an explicit criterion for their identity, his famous Axiom V: Two concepts F and G have the same extension if and only if whatever falls under the concept F falls under the concept G, and vice versa. In 1902 at the time when the second volume of Frege's Grundgesetze was at press, Russell showed how this Axiom leads to the so-called Russell's paradox. Here is one derivation of the paradox as described by Boolos:7 Let us define first the concept to be russellian. We will say that an object x is russellian if there is at least one concept F such that (a) (b)
x is the extension of F; and x does not fall under F.
Next, consider the concept-word 'the extension of the concept to be russellian'. Since every concept-word has an extension, letjy be its extension. It now makes sense to ask: Isjy russellian or not? Ify is russellian, then by definition, there is at least once concept F such thatjy is the extension of .Fandjy does not fall under F. But on the other side, from the definition ofj,jy is the extension of the concept to be russellian. Thus y = the extension of the concept to be russellian — the extension of F.
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From this equation and Axiom V it follows that every object which falls under F is russellian and every object which is russellian falls under F. But becausey does not fall under F, it follows thatjy is not russellian, and we reached a contradiction. The supposition thatjy is not russellian leads us to a contradiction in a similar way. Frege immediately acknowledged the importance of the paradox but he wrongly saw the culprit in the extension of a concept being an object to which the concept may apply. That is, in his solution to the paradox that he described in the Appendix to the second volume of Grundgesetze (1903), Frege simply proscribed the applicability of concepts to their extensions. It is well known that this solution is not enough to ban the paradoxes. 8 The reactions to the discovery of the inconsistency in Frege's system have been of different kinds. Some logicians, for instance Russell, gave up the attempt to base arithmetic on a theory of extensions, as Frege did. Godel was right to point out in his review of Russell and Whitehead's Principia Mathematica (1910-13) that the discovery of the paradoxes produced in Russell 'a pronounced tendency to build up logic as far as possible without the assumption of the objective existence of such entities as classes and concepts', visible in his no class theory interpretation (cf. below). Other logicians, more extensionally minded, continued to pursue the extensionalist programme under the influence of Cantor, but realized that in order to do that in a consistent way, the Fregean notion of extension has to be disentangled from such intentionally flavoured notions as concepts, properties, etc. For instance, Zermelo wrote: ... it follows further as already appears in a much simpler way from Russell's antinomy, to be sure that it is not permissible to treat the extension of every arbitrary notion as a set and therefore that the customary definition of a set is too wide.9 Ramsey adhered to the second programme. In his Foundations, he criticized Russell and Whitehead for not taking into account the extensionalist attitude of Cantorian set theory. Frege's logicist programme has been revived some twenty years ago under the name of neo-logicism. A group of philosophers and logicians, including Charles Parsons, George Boolos, Crisipin Wright, John Burgess, Richard Heck Jr, Harold Hodes and others have shown that although the Axiom V leads to inconsistencies, and in this respect arithmetic cannot be obtained from a theory of extensions, it can nevertheless be obtained from a theory of logical objects. That is, with each concept F, an object x is associated construed as 'the number of F'. The association of the concept to its number must obey the so called 'number principle' (or 'Hume's principle') which asserts that
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The number of F's = the number of G's if and only if the F's and G's are in one—one correspondence. Frege showed in the Grundgesetze that the number principle can be derived from his Axiom V. But as Axiom V has been shown by Russell to lead to inconsistency, the crux of the neo-logicism programme has been to show that by giving up the Axiom V and retaining only the number principle, Frege's logicist programme can be salvaged: the addition of this principle to the secondorder system to be found in the Begriffsschrift or Grundgesetze yields a derivation of (Frege's) arithmetic.10 Boolos regrets that too much attention has been paid to Russell's paradox with the effect of obscuring the real merits of Frege's positive achievements, that is, the derivation of arithmetic from the number principle (and the second-order system of the Begriffsschrift]ft).. The inconsistency of Frege's theory of extensions was generally acknowledged to be due to Frege's gluing too closely together the notions of extension, concepts, properties and other 'arbitrary notions', as Zermelo put it. Cantor perceived this problematic aspect of Frege's theory of extensions much earlier than Russell's discovery of the paradox. In his review of Frege's Grundlagen, he pointed out that sets cannot be identified with the extensions of concepts because the latter are quantitatively too indeterminate.11 In an earlier paper with Jaakko Hintikka,12 I argued (reviving some arguments in Burge (1884)) that Frege's attitude towards extensions was shaped by the influence of a long-standing tradition in logic which emphasized the logical primacy of intensions over extensions . . . In fact I do hold that the concept is logically prior to its extension and I regard as futile the attempt to base the extension of a concept or a class not in the concept but on individual things.l3 This attitude stood in deep contrast to that of Cantor and Dedekind who thought of classes and numbers not as constituted by appeal to properties and concepts but formed by 'abstraction' from the corresponding elements. Frege continuously criticized the extensionalists' idea that a class is formed by grouping together elements, insisting instead that concepts or properties are needed to 'hold' together the elements of the class: Of course, one must not then regard a class as constituted by the objects (individual, entities), that belong to it; for removing the objects, one would then also be removing the class constituted by them. Instead, one must regard the class as constituted by the charactestic marks, i.e. the properties which an object must have if it is to belong to it.14 True enough, unlike many representatives of the intensionalist tradition, Frege had an extensional view of concepts, in the sense that he construed
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them as functions from objects into truth-values which are to be found at the referential (Bedeutung] level of language. In addition to this, he also had an extensional view of the extensions of concepts, given through his Axiom V. But this should not obscure the fact that his theory of extensions is subordinated to his theory of concepts and that extensions existed for Frege only as extensions of concepts. In this respect, the view we defended in the article mentioned above is drastically different from that held by philosophers like Dummett, Stanley, Heck Jr, Demopoulos, and others who credit Frege with the view that once a set of individuals is given, then the totality of arbitrary functions and hence of concepts based on them is completely determined.15 I think Carnap is much closer to the truth when he perceived the attitude of his teacher in foundational issues to be completely different from the extensionalist attitude shared by Ramsey: I think we should not let ourselves be seduced by it into accepting Ramsey's basic premise, viz. that the totalities of properties already exist before their characterization by definition ... I think we ought to hold fast to Frege's dictum that in mathematics only that may be taken to exist whose existence has been proved (and he meant proved in finitely many ways) ,16 There is no guarantee, from the way Frege conceived the operation of abstraction, that there is a concept for every extension. The process of abstraction, as it is traditionally conceived, allows one to compare objects a, b, and c in order to detect the properties they have in common and the ones that separate them. In the end, by ignoring the latter, one gets to a common concept under which each of a, b, and c fall. The concept arrived at does not include the properties which were 'abstracted' neither the properties held in common by a, b, and c. As Frege put it, the concept of a book is not constituted by the property of having printed pages, as the concept of female is not constituted by having small children fed through mammary glands for the simple reason that the concept in question does not have such glands (Frege 1979, p. 71). The conclusion we draw from such examples was that, if abstraction functions for Frege in this way, there is no guarantee that the totality of concepts applicable to a, b, and c is not applicable to other objects too. In that conclusion we were not alone. Hao Wang noticed that Frege's extensions are too closed to concepts and too far away from extensions, a fact which places Frege closer to Russell, in that an extension subordinated to a concept is more likely to suggest a type hierarchy of extensions than the iterative conception of a set which emerged from Cantor's work. The generated principles operating behind Cantor's theory of transfmite ordinals go a long way towards accepting the existence of arbitrary subdomains and thus rely on a way of thinking from which Frege has estranged himself:
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Cantor asserts that the two principles of generation [he had formulated] give us the ability to break through every barrier in the formulation of real, whole numbers. This certainly suggests a belief that these methods take the ordinal number 1o sequence arbitrarily far. In other words, Frege's complete subordination of extensions to concepts placed himself out of two long-range developments which influenced the mathematicians' thinking about the foundations of their field: The development of the idea of an arbitrary function in extension The development of the idea of arbitrary set.
Russell's work in foundations At the stage when Russell was working on his book The Principles of Mathematics (1903), he came to learn indirectly of the work done by Frege from Peano with whom Frege had been corresponding from 1894 to 1896. Greatly impressed by Frege's work, Russell added an Appendix (Appendix A) to his Principles., where he presented some of Frege's ideas. Russell was already aware at this stage of the problem raised by the Burali-Forti paradox of the greatest ordinal number and of Cantor's paradox of the largest cardinal number. In the analysis of these paradoxes he came upon his own paradox (1902) which we discussed above. In the Appendix B of the Principles, Russell sketched for the first time a solution to the paradoxes under the name 'the doctrine of types'. Essentially the 'doctrine' requires of each variable occurring in a prepositional function to be assigned a type and for the type of the class determined by the prepositional function (p(x) to be higher than the type of any of its members. This move, which results in the proscription of all expressions of the form x € x, x 0 x is immediately seen to block the paradoxes. In the same Appendix, Russell introduced the term predicative to distinguish those properties (p(x) which determine classes from those, called impredicative, which do not, but he did not devise a general criterion for distinguishing between them. Between 1903 and 1908, Russell worked on the details of'the doctrine of types'. He came more and more under the influence of Poincare's criticism of the logicist programme. The resulting system is described in his article 'Mathematical Logic as Based on the Theory of Types' (1908) which is based on his own interpretation of Poincare's vicious circle principle and on his earlier 'doctrine of types'. Poincare had identified the source of the paradoxes in the presence of a vicious circle. He noticed that in each paradoxical case, there is a definition of an object in terms involving implicitly the object itself.19 For instance, in the case of Richard paradox, a certain real number is defined in terms of
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the supposed totality of all real numbers, and in the case of the paradox of Burali-Forti, the largest ordinal is defined in terms of the supposed totality of all ordinals. Poincare adopted Russell's terminology and called predicative all definitions which do not involve the vicious circle and impredicative those which do (implicitly or otherwise) and are to be proscribed. Russell adopted Poincare's vicious circle principle but gave it a more technical interpretation as 'Whatever contains an apparent variable must not be a possible value of that variable' which finally led to his Ramified Theory of Types (RTT) to be sketched below. The Simple Theory of Types (STT) is a more detailed presentation of Russell's earlier 'doctrine of types'. Each variable is assigned a given type. If the type of the variable V is n, then the property term 'xxtp(xy formed from the prepositional function (y}) whenjy is of the same type as x where (f>(x)) is a formula of STT. We may think that the individuals in the universe of discourse are taken to be the objects of type 0 and the properties of the objects of type 0 are taken to be the objects of type 1. In general, the objects of type ra+ 1 are conceived as properties of objects of type n. Ramsey noticed later on in his Foundations that already the STTprevents Russell's paradox by containing no vicious circle principle with respect to property formation. The class term x ~> (x G x) is not a legitimate one in this language no matter what the type of 'x' is. Russell, however, thought that this solution does not suffice and that STT has to be enriched with the technical implementation of his reformulation of the vicious circle principle given above. The outcome was the Ramified Theory of Types (RTT) in which every type greater than 0 is divided into levels. In a more modern formulation (confusingly, Russell sometimes used 'type' to cover both types and levels), each (predicate) variable of type n is assigned a natural number k as its level. Then the level of any term of the form x
Ramsey and the Notion of Arbitrary Function
245
proposition one of the names of an individual by a variable. But now notice that functions of individuals (all of which have type 1) may have different levels. The elementary functions of individuals have level 0. From the elementary function of two individuals, (pxy, we obtain by generalization a function of one individual Vxtpxy which is not an elementary function of individuals any longer (except in the case where there is a finite number of individuals in the universe of discourse). In agreement with the vicious circle principle, this function is assigned one level higher than the level of x, that is, the level 1. But in this function we can equally well treat '
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is most likely due, as Godel remarked, by the discovery of the paradoxes which had produced in Russell 'a pronounced tendency to build up logic as far as possible without the assumption of the objective existence of such entities as classes and concepts'. This attitude is most visible in the no class theory interpretation of the Principia, which is a contextual elimination of classes analoguous to the contextual elimination of descriptions. Thus where in the Principles the expression lx(p(x}' was a class expression (the class of those things which satisfy the prepositional function <£>(#)), in section *20 of the Principia the authors still cling to this view while immediately qualifying it by pointing out that talk of classes should be taken with a grain of salt: such talk is to be contextually eliminated in the same way as talk of descriptions. Even more, in the Introduction to the second edition of Principia, Russell remarks that he does not make any difference between, on one side, the meaning of the symbol l x(p(x)' and that of the symbol
Ramsey's theory The exposition in this section is based entirely on Ramsey's Foundations of Mathematics (1925) which I see as describing Ramsey's development towards reaching the notion of arbitrary function in three stages: (a) (b) (c)
The criticism of Principia for defending a notion of set and function which do not do justice to the extensionalist attitude of Cantorian set theory; Ramsey's definition of predicative functions; Ramsey's definition of an impredicative function which led to a complete extensionalization of this notion.
Ramsey identified three major defects in Principia. We discussed the first one in connection with Frege and Russell: Principia proposed a definition of a class which applies only to definable ones and which is therefore inadequate to capture all the functions in extensions dealt with in Gantorian set theory. The second defect concerns the way paradoxes are solved. He points out that to block Russell's paradox, it is not necessary to use the full force of the vicious-circle principle and the Ramified Theory of Types, and that the distinction according to types serves this purpose already. In connection with this critique, Ramsey blamed the authors of Principia for not noticing an
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important distinction among the paradoxes, i.e. that some of them are purely logical (like Russell's paradox or the Burali-Forti's paradox), while others are more semantical (linguistic) or epistemological, like the Liar paradox, Richard's paradox, the hererological paradox, and so on. He observed, and this is a point endorsed later on by Tarski (1935), and by Godel (1944), that the Simple Theory of types is seen right away to solve the logical paradoxes, and the same theory solves the epistemological paradoxes too, after a closer analysis of those. The third major defect ofPrincipiaa concerns the treatment of identity, something which I am not going to discuss in this chapter. Ramsey's notion of predicative function
Ramsey's notion of predicative function and proposition has a different meaning to Russell's or Poincare's. Against Russell, he makes a first radical step towards disentangling the notion of elementary proposition from the symbols used to express it by noticing that an elementary proposition symbol and a non-elementary prepositional symbol could be instances of the same proposition (e.g. '(pa. (pb . c' expresses the same proposition as \/x(px if a, b, c were all the individuals). By 'being instances of the same propositions' he means agreement and disagreement with the same truth-values (in Wittgenstein's sense), making it thus obvious that he had in mind a purely extensional view of this notion.23 Moreover, against the definabilitist style ofPrincipia, he noticed that there are propositions which are not definable in language, as for instance the proposition \/x(px which is the logical product of all the propositions even of those which involve individuals for which we do not have a name. The crux of Ramsey's extensionalist view of functions comes into play when he deals with functions of functions. To illustrate, consider one such function f((px], which has as its only argument the function of individuals (px. When quantifying over the function of individuals so that we get the logical product V(pf((px} and the logical sum 3x), we have to specify the range of the universal and existential quantifiers, that is, the domain of the functions (px. Ramsey notices that we cannot accomplish this task without defining what a function of individuals is. One would think that the same problem arises when we deal with an elementary function of individuals (px (here (p is constant), and quantify over the individuals x obtaining the logical sum and product 3x(px, V#. Aren't we, in this case too, compelled to specify what an individual is? The situation is not analogical, however, for according to Ramsey the range of individuals is objective, while the range of functions is not, because the range of functions is a range of symbols 'actual or possible [which] is not objectively fixed, but depends on our methods of constructing them and requires more precise definition'. 24 We see from this answer that Ramsey's road
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towards reaching a fully extensional notion of a function is still full of traps. He is not yet able at this stage to make a clear distinction between a function and the symbol which refers to it, and he hesitates to recognize that a function has an ontological status which has nothing to do with our methods of constructing it. Ramsey's answer is nevertheless clear enough to delimit his position from the view attributed to Frege by Dummett, Heck and Stanley which was mentioned earlier, namely the view according to which once we are given a set of individuals, we are also given the set of all arbitrary functions ranging over them. In clarifying what he means by an objective notion of a function, Ramsey contrasts his view with that developed in Principia, where the domain of the functions is fixed in a subjective way, that is, the notion of a function is specified in terms of the expressions of the language. That he regards as the source of the difficulties that led to the impasse of the Axiom of Reducibility. Instead, he adopts quite a different method: I, on the other hand, shall adopt the entire original objective method which will lead to a satisfactory theory in which no such axiom is required. The method is to treat functions of functions as far as possible in the same way as functions of individuals. . . . I propose similarly to determine the symbols which can be determined as arguments in 'f((x-J)
V p,\/y(p(x,y]
where (f is an atomic function of individuals, are all examples of predicative functions of individuals. The second thing which deserves attention is that in his discussion concerning the functions of (predicative) functions of individuals, Ramsey argued that Russell's vicious circle is false. To illustrate his train of thought, consider one of
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the above functions of individuals, f(x,y] in which we replace (one of) the free variables, sayjy by the individual constant b, and treat (p as a varible. The result is an atomic function of (predicative) functions i^bM p, and \/x(px are such functions. Consider now the atomic function of (predicative) functions of individuals, (p(x, b}. We may quantify over the variable/and get the function Vipip(x, b} which is the logical product
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section 'Prepositional Functions in Extension', Ramsey addresses the question of how these two projects may be reconciled. His answer is unambiguous. He came to realize that if the logicist programme is going to succeed, one has to have in one's logical system the same notion of a function as that acknowledged as existing in mathematics. Since the latter is purely extensional and arbitrary, so must be the former. Ramsey thus came to realize he has to extend the notion of a function to cover also non-predicative propositional functions: The only practicable way is to do it as radically and drastically as possible; to drop altogether the notion of that (pa says about a what (pb says about b\ to treat propositional functions like mathematical functions, that is, extensionalize them completely. Indeed it is clear that, mathematical functions being derived from propositional, we shall get an adequately extensional account of the former only by taking a completely extensional view of the latter.28 He is well aware of the fact that he cannot give an explicit definition of a function in extension and accordingly contents himself with explaining this notion rather than defining it. His explanation is given in terms of the notion of correlation, that is, a relation in extension between propositions and individuals, which to every individual associates a unique proposition, the individual being the argument to the function, and the proposition its value. In specifying the nature of this correlation, he explicitly observes that it may be 'practicable or impracticable' which is just another way of saying that the correlation is simply an arbitrary association between individuals and propositions. Most importantly, he remarks on page 53 that his notion of arbitrary function in extension is free from contradictions, in the same way as the notion of predicative function is. It is obvious that these remarks have to be understood in the same spirit as his criticisms of the vicious circle principle: a foundational programme which starts from the notion of extension as ontologically given is not liable to end up in the inconsistencies which arise in logical systems anchored in intensional notions. At the same time, this notion of arbitrary function ensures that there are enough classes needed for the fulfilment of the logicism programme, given the fact that now a class can be defined from each such function in extension (i.e. the corresponding characteristic function).30 or\
Ramsey's attitude towards foundations Let us return now to the distinction introduced by Carnap between two theses of logicism discussed in the Introduction of this chapter. The concept of arbitrary function in extension allowed Ramsey to take a different stand
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from Russell and Whitehead on the logical status of some of the axioms used in Principia. Unlike the Axiom of Reducibility which Ramsey finds to be neither a tautology nor a contradiction, he believes the Multiplicative Axiom to be a tautology when classes are reduced to functions in extension. He notices, on the other side, that the axiom is not a tautology if, as in Principia, the only existing classes are the definable ones. Garnap pointed out that both Grundlagen and Principia constitute a fulfilment of the logicist programme in style (a). But he also observed that Principia does not constitute a reduction of arithmetic in style (b), given the fact that the proofs of some theorems (e.g. for every natural number there is a greater one) requires the Axiom of Infinity which is not a logical truth, as Russell himself was quick to recognize. Carnap does not voice any opinion as to whether he considers Grundlagen to be a reduction of arithmetic to logic in sense (b). The reason for his silence may have to do with the inconsistency in Frege's system discovered by Russell. We have indicated earlier in connection with the discussion around the neologicism programme, that Boolos (among other logicians) has carried out a reduction of arithmetic to the second-order system of Begriffschrift plus ththee number principle. He came to the conclusion, though, that although the number principle is expressed in logical vocabulary, it is not a principle of pure logic,31 and therefore the reduction carried out within the neo-logicist programme is not one in sense (b). These considerations motivated Boolos to go even further than Garnap in restating Frege's and Russell's aims in the foundations of mathematics. According to him, Frege's aim in Grundgesetze was not a reduction of arithmetic to logic in sense (b) but only the derivation of arithmetic from some pure logical principles together with the number principle. And he argued that the same attitude was shared by Russell too: his aim in Principia was not a reduction in style (b) but the derivation of a sufficiently large portion of mathematics from some primitive logical propositions and other mathematical (extra-logical) propositions like the Axiom of Infinity. In view of all this, Boolos proposed to rebaptize Frege's programme as The Logical Analysis of Arithmetic and Russell's and Whitehead's programme in Principia as the Logical Analysis of Mathematics.^ We are not in the position to evaluate whether Boolos' assessment of Russell's programme is the correct one. There are passages in Ramsey which suggest, against Boolos, that he perceived the task of Principia to be a reduction of mathematics to logic in the sense (b) ,33 Independently of that, however, there is no doubt that Ramsey's own aim in the Foundations is clearly different from the one credited by Boolos to Frege and Russell. Like Wittgenstein, under whose sharp criticisms Russell gave up the Axiom of Reducibility in the second edition of Principia (1925), Ramsey criticized the use of the very same Axiom in Principia on the grounds that it is not a truth of logic. In fact he
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stated explicitly that all the primitive propositions ofPrincipia, with the exception of the Axiom of Reducibility, are tautologies (in Wittgenstein's sense), and the other propositions follow from the primitive ones by inference principles which preserve truth.34 Ramsey thus believed he had carried out a reduction of mathematics to logic in the sense (b) above. We know today that he was wrong, for neither the Multiplicative Axiom nor the Axiom of Infinity which he believed to be tautologies, are not so. It should be said to his credit though, that he was well aware of the fact that neither of them can be shown in a straightforward way to be a primitive tautology in Wittgenstein's sense. For instance, he believed the Multiplicative Axiom to follow from primitive tautologies through adequate inference rules: It will probably be objected that, if it [the Multiplicative Axiom] is a tautology, it ought to be able to be proved, i.e. deduced from the simpler primitive propositions which suffice for the deduction of the rest of mathematics.35 He acknowledged, however, right away that he was unable to offer any formal proof of this fact, which may turn out to be too complicated for him: But it does not seem to me in the least unlikely that there should be a tautology, which could be stated in finite terms, whose proof was, nevertheless, infinitely complicated and therefore impossible for us.36 We know today that he could not find such a proof for the simple reason that there is none. In modern jargon, the Axiom of Choice has been shown to be independent from the other axioms of set theory, a result which lay, however, many years ahead of Ramsey's work. This means, among other things, that there are models of set theory in which the Axiom of Choice is false, and it cannot, therefore, be a tautology. Ramsey faces similar difficulties with the Axiom of Infinity, which he believed to be a tautology despite his failure to find a proof of it in his logical system.37 The explanation he put forward for his failure is an intricate combination of Wittgensteinian considerations on saying and showing. He observes that any statement asserting that there are at least n individuals can be seen to be either a tautology or a contradiction. First the statement 'There is an individual' (i.e., '(Bx)x = x)' reduces to a = aV b = bV ...
which is a disjunction of tautologies if there is at least one individual, and a sheer nonsense if there is no such individual. Thus the statement 'There is an individual' is either a tautology or a contradiction.
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A similar reasoning applies to the statement 'there are at least two individuals', i.e., '(3x,j>). x =y which reduces to
a = b V a = c V b = c V ... This statement is a tautology, if there are at least two individuals, and a contradiction otherwise. Ramsey concludes that 'A little reflection will make it clear that this will hold not merely of 2, but of any other number, finite or infinite'.38 He ends up, however, in thinking that there is no genuine assertion which says that there are n individuals, where n is the number of individuals of the 'whole world': It is only when we take, not a limited universe of discourse, but the whole world, that nothing can be said about the number of individuals in it.39 Ramsey concludes that, since the Axiom cannot be proved, it must be taken as a primitive: Similarly the Axiom of Infinity in the logic of the whole world, if it is a tautology, cannot be proved, but must be taken as a primitive proposition. And this is the course which we must adopt, unless we prefer the view that all analysis is self-contradictory and meaningless.40 I do not want to go here into the intricate details of the above argument, which I find rather obscure. The Axiom of Infinity is certainly not a tautology but a rather highly non-trivial set-theoretical principle. Boolos has pointed out that in a Principia-styled logical system where numbers are construed as classes (we disregard the 'no class theory' interpretation), one cannot show that there are at least two individuals without assuming the Axiom of Infinity. Accordingly, if this Axiom is taken as primitive, Ramsey has to face the same fate as Russell: Whether extensions, subextensions, or numbers are logical objects or not, it may seem, from a Fregean point of view, that Russell's definition of numbers as certain sort of class fails in two respects: invoking the axiom of infinity invalidates a claim to have shown numbers to be logical objects; defining them as certain classes (of sets of individuals) forbids him from thinking he has shown them to be logical objects.42 Endnotes 1.
Frank Plumpton Ramsey, 'The Foundations of Mathematics', Proceedings of the London Mathematical Society, ser. 2, Vol. 25, Part 5, pp. 338-84, 1925, reprinted in
254
2.
3. 4. 5. 6. 7. 8. 9. 10.
11.
12. 13.
14. 15.
16. 17. 18. 19. 20. 21.
F. P. Ramsey: Critical Reassessments Frank Plumpton Ramsey, The Foundations of Mathematics, New York, The Humanities Press, 1950, pp. 1—61, to which references are given. R. Carnap, 'The Logicist Foundations of Mathematics', published originally in German in 1931, and translated in Putnam and Benaceraf (eds), Philosophy of Mathematics, pp. 31-41. The same distinction is taken up in G. Boolos, 'The Advantages of Honest Toil Over Theft', in Logic, Logic, and Logic, Harvard University Press, 1998, p. 271. Ramsey, The Foundations, p. 3. The quotation is from Russell's The Principles of Mathematics, 1903, p. 3. Idem, p. 4. Frege, Foundations of Arithmetic, section 107. Idem, section 107. George Boolos, 'Gotlob Frege and the Foundations of Arithmetic' in Logic, Logic, and Logic, Harvard University Press, 1988, pp. 143-54. Quine, 1955, and Geach, 1956. Ernst Zermelo, 'Neuer Beweis fur die Moglichkeit einer Wohlordnung', Mathematische Annalen, LXV (1908): 107-28; quotation from page 107. A proof of the relative consistency of the resulting system may be found in Boolos' paper 'The Consistency of Frege's Foundations of Arithmetic' published in 1987 and reprinted in Logic, Logic, and Logic, Harvard University Press, 1998. For a proof of how arithmetic can be developed from the Number Principle together with the second-order system of Grundgesetze, see Richard Heck jnr, 'The Development of Arithmetic in Frege's Grundgesetze der Arithmetik', Journal of Symbolic Logic, 58, pp. 579-601. G. Cantor, 'Rezension der Schrift von G. Frege, Die Grundlagen der Arithmetik', in Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, E. Zermelo (ed.), Berlin, Springer, 1932, pp. 440-1. Hintikka and Sandu, 1991. G. Frege, A critical elucidation of some points in E. Schroder's 'Algebra der Logik', in P. Geach and M. Black (eds), Translationsfrom the Philosophical Writings ofGottlob Frege, 1952, pp. 86-106; quotation from p. 106. G. Frege, 'Letter to Peano', undated, in Correspondence, p. 109. M. Dummett, Frege's Philosophy of Language, 1973, p. 177; Heck Jr, R. G. and J. Stanley, 'Reply to Hintikka and Sandu, Frege and Second-Order Logic', Journal of Philosophy, 90, 1993, 424, n. 1. R. Carnap, 'Die logizistiche Grundlegung der Mathematik', Erkenntnis, II (1931): 91-105; quotation from p. 102. Hao Wang, From Mathematics to Philosophy, New York, Routledge, 1974; p. 210. Hallet, Cantorian Set Theory and Limitation of Size, 1990, p. 58. H. Poincare, 'Les mathematiques et la logique', Revue de Metaphysique et de Morale, Mai, 1906. Russell, Introduction to Mathematical Philosophy, London: Allen and Unwin, 1919, p. 141. K. Godel, 'Russell's Mathematical Logic', 1944, reprinted in K. Godel, Collected Works; p. 137. Similar remarks are to be found in S. Feferman, 'The Development
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22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.
255
of Programs for the Foundations of Mathematics in the First Third of the 20th Century', p. 14. Principia Mathematica, section *20, p. 186, Cambridge: Cambridge University Press, 1936. Ramsey, Foundations, pp. 33-4. Idem, p. 37. Idem, p. 38. Idem,pAl. Idem, p. 47. Idem, p. 52. Idem, p. 52 or 54. Idem, p. 54. This point is still debated, C. Wright, 'Is Hume's Principle Analytic?', in Bob Hale and Crispin Wright, The Reason's Proper Study, forthcoming. Boolos, 1998, p. 154, and p. 271. Foundations. Idem. Idem. Idem. Idem. Idem. Idem. Idem. Boolos, 1998, p. 257. Boolos, p. 261.
References Boolos, G. (1998) 'The Advantages of Honest Toil over Theft' in Boolos, Logic, Logic, and Logic, Cambridge, MA: Harvard University Press, pp. 255—74. (1998) 'The Consistency of Frege's Foundations of Arithmetic' in George Boolos (1998), pp. 182-201. (1998) 'Gotlob Frege and the Foundations of Arithmetic' in Boolos (1998), pp. 143-54. Burge, T., 'Frege on Extensions of Concepts, from 1884 to 1903', Philosophical Review, 63, pp. 1-34. Cantor, C. (1932) 'Rezension der Schrift von G. Frege, Die Grundlagen der Arithmetik', in Gesammelte Abhandlungen mathematischen undphilosophischen Inhalts, E. Zermelo (ed.), Berlin: Springer, pp. 440-1. Carnap, R., 'The Logicist Foundations of Mathematics', in Putnam and Benaceraf (eds), Philosophy of Mathematics, pp. 440-1. (1931) 'Die logizistiche Grundlegung der Mathematik', Erkenntnis, II: 91-105.
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Demopoulos, W. and J. L. Bell (1993) 'Frege's theory of concept and objects and the interpretation of second-order logic', Philosophia Mathematica, 1, pp. 139—56. Dummett, M. (1973) Frege's Philosophy of Language, London: Duckworth. Feferman, S., 'The Development of Programs for the Foundations of Mathematics in the First Third of the 20th Century', to appear in Storia delXXSecolo: Logica, Institute dela Enciclopedia Italiana, Rome. Frege, G. (1950) The Foundations of Arithmetic,]. L. Austin (trans.), Oxford: Blackwell. (1967) The Basic Laws of Arithmetic: Exposition of the System, Berkeley, CA: California UP. (1952) 'A critical elucidation of some points in E. Schroder's Algebra derLogik', in P. Geach and M. Black (eds), Translations from the Philosophical Writings of Gottlob Frege, New York and Oxford, pp. 86-106. (1979) Posthumous Writings, Oxford: Blackwell. (1980) 'Letter to Peano', undated, in Gottfried Gabriel, et al. (eds), Philosophical and Mathematical Correspondence, Oxford: Blackwell. Geach, P. (1965) 'Frege's way out', Mind, 65, pp. 408-9. Godel, K. (1986) 'Russell's Mathematical Logic', 1944, repr. in S. Feferman, et al., Godel: Collected Works, vol. I, Oxford: Oxford University Press. Hallet, M. (1990) CantorianSet Theory and Limitation of Size, Oxford: Oxford University Press. Heck, R. Jr, 'The Development of Arithmetic in Frege's Grundgesetze der Arithmetik', Journal of Symbolic Logic, 58, pp. 579-601. Heck, Jr, R. G. and J. Stanley (1993) 'Reply to Hintikka and Sandu, Frege and Second-Order Logic', Journal of Philosophy, 90. Hintikka, J. and G. Sandu (1992) 'The Skeleton in Frege's Cupboard: The Standard versus Nonstandard Distinction', Journal of Philosophy, 89, pp. 290—315. Poincare, H. (1906) Les niathematiques etlalogique', Revue de Metaphysique et de Morale, May. Quine, W. V. (1955) 'Frege's way out', Mind, 64, pp. 145-59. Ramsey, F. P. (1925) 'The Foundations of Mathematics', Proceedings of the London Mathematical Society, ser. 2, Vol. 25, Part 5, pp. 338-84, repr. in Frank Plumpton Ramsey, The Foundations of Mathematics, New York: The Humanities Press, 1950, pp. 1—61. Russell, B. (1919) Introduction to Mathematical Philosophy, London: Allen and Unwin. Russell, B. and N. Whitehead (1910-13) Principia Mathematica, Cambridge: Cambridge University Press, 2nd edn, 1925. Wang, H. (1974) From Mathematics to Philosophy, New York: Routledge. Wright, C. (forthcoming) Ts Hume's Principle Analytic?', in Bob Hale and Crispin Wright, The Reason's Proper Study. Zermelo, E. (1908) 'Neuer Beweis fur die Moglichkeit einer Wohlordnung', Mathematische Annalen, LXV, 107-28.
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Reviews and critical notices 1922, 'Mr. Keynes on Probability'. The Cambridge Magazine, 11, n.l, January, pp. 3-5. Reprinted in The British Journal for the Philosophy of Science., 40, June 1989, ed. D. Mellor. 1922, 'Review of W. E. Johnson's Logic Part //', The New Statesman, vol. 19, July, pp. 468-70. 1923, 'Critical Notice of L. Wittgenstein's Tractatus Logico-Philosophicus', Mind, 32, October, pp. 465-78. Included in Braithwaite (1931), pp. 27-86. 1924, 'Review of C. K. Odgen and I. A. Richard's The Meaning of Meaning', Mind, 33, January, pp. 108-9. 1925, 'The New Principia', review of A. N. Whitehead and B. Russell's Principia Mathematica, vol. I, 2nd edn. Nature, vol. 116, n. 2908, July, pp. 127-8. 1925, Review of A. N. Whitehead and B. Russell's Principia Mathematica, vol. I, 2nd edn. Mind, vol. 34, n. 136, October, pp. 506-7.
Published papers 1925, 'Universals', Mind, 34, October, pp.401-17. Included in Braithwaite (1931), pp. 112-34, Mellor (1978), pp. 17-39 and Mellor (1990), pp. 8-30. 1925, 'The Foundations of Mathematics', Proceedings of the London Mathematical Society, 25, pp. 338-84. Included in Braithwaite (1931), pp. 1-61, Mellor (1978), pp. 151-212, Mellor (1990), pp. 164-224. 1926, 'Mathematics: Mathematical Logic', The Encyclopaedia Britannica, supplementary volumes constituting Thirteenth Edition, vol. 2. pp. 830-2. 1926, 'Universals and the "Method of Analysis"', Aristotelian Society Supplementary Volume VI, July, pp. 17-26. 1926, 'Mathematical Logic', The Mathematical Gazette, vol. 13, no. 184, October, pp. 185-94. Included in Braithwaite (1931), pp. 62-81, Mellor (1978), pp. 213-32, Mellor (1990), pp. 225-44. 1927, 'Facts and Propositions', The Aristotelian Society Supplementary Volume VII, July, pp. 153-70. Included in Braithwaite (1931), pp. 138-55, Mellor (1978), pp. 40-57, Mellor (1990), pp. 34-55. 1927, 'A Contribution to the Theory of Taxation', The Economic Journal, vol.37, no. 145, March, pp. 47-61. Included in Mellor (1990). 1928, 'A Mathematical Theory of Saving', The Economic Journal, vol. 38, no. 192, December, pp. 543-9. Included in Mellor (1990).
258
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1928, 'On a Problem of Formal Logic', Proceedings of the London Mathematical Society, ser. 2, vol. 30, part 4, pp. 338ff. 1929, 'Mathematics, Foundations of, The Encyclopaedia Britannica, 14th edn. 1929, 'Russell, Bertrand Arthur William', The Encyclopaedia Britannica, 14th edn.
Posthumous writings 1921, 'The Nature of Propositions', Rescher and Majer (1991), pp. 107-19. 1922, 'Paper for the Society', Rescher and Majer (1991), pp. 20-3. 1925, 'Epilogue', Braithwaite (1931), pp. 287-92, reprinted in Mellor (1990), pp. 245-50. 1926, 'Truth and Probability', Braithwaite (1931), pp. 156-98, reprinted in Mellor (1978), pp. 58-100 and Mellor (1990), pp. 52-94. 1927-29, 'The Nature of Truth'. Rescher and Majer (1991), pp. 6-24. 1927-29, 'The Coherence Theory of Truth', Rescher and Majer (1991), pp. 25-42. 1927-29, 'Judgement'. Rescher and Majer (1991), pp. 43-54. 1927-29, 'Knowledge and Opinion', Rescher and Majer (1991), pp. 55-66. 1927-29, 'Judgement and Time'. Rescher and Majer (1991), pp. 67-77. 1928, 'Reasonable Degree of Belief, Braithwaite (1931), pp. 199-203, reprinted in Mellor (1990), pp. 97-101. 1928, 'Statistics'. Braithwaite (1931), pp. 204-5, reprinted in Mellor (1990), pp. 102-3. 1928,' Chance'. Braithwaite (1931), pp. 206-11, reprinted in Mellor (1990), pp. 104-9. 1928, 'Universals of Law and of Fact', Mellor (1978), pp. 128-132, reprinted in Mellor (1990), pp. 140-44. 1929, 'Theories', Braithwaite (1931), pp. 212-36, reprinted in Mellor (1978), pp. 101-25 and Mellor (1990), pp. 112-36. 1929, 'General Propositions and Causality', Braithwaite (1931), pp. 235-55, reprinted in Mellor (1978), pp. 133-51 and Mellor (1990), pp. 145-63. 1929, 'Probability and Partial Belief, Braithwaite (1931), pp. 256-7, reprinted in Mellor (1990), pp. 95-6. 1929, 'Knowledge', Braithwaite (1931), pp. 258-9, reprinted in Mellor (1978), pp. 126-7 and Mellor (1990), pp. 110-11. 1929, 'Causal Qualities', Braithwaite (1931), pp. 260-2, reprinted in Mellor (1990), pp.137-9. 1929,'Philosophy', Braithwaite (1931), pp. 263-9, reprinted in Mellor (1990), pp. 1-7. 1987, 'The "Long" and "Short" of it or a Failure of Logic', American Philosophical Quarterly, 24, pp. 357-9, reprinted in Rescher and Majer (1991), pp. 124-7.
Collections of Ramsey's works 1931, R. B. Braithwaite (ed.), The Foundations of Mathematics and Other Logical Essays, London, Routledge and Kegan Paul. Later published by Littelfield, Adams and Co., Patterson, NJ, 1960.
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1978, D. Mellor (ed.), Foundations: Essays in Philosophy, Logic, Mathematics and Economic?,, London, Routledge and Kegan Paul. 1990, D. Mellor (ed.), Philosophical Papers, Cambridge, Cambridge University Press. 1991, N. Rescher and U. Majer (eds), On Truth. Original Manuscript Materials (1927-1929) from the Ramsey Collection at the University of Pittsburgh, Kluwer Academic Publishers.
Index
act 9,10,11,20,21,27,37,39,46 actino 2, 3, 7, 14, 16, 18, 26, 39, 42, 59, 61,62,67,87,101,115,140,146, 183, 184, 185, 186, 187, 192, 193, 197 Anscombe, E. 56, 68, 225, 226, 230, 235 Aristotle 116,118,136,137 Austin, J. 115, 116, 129, 136, 137,256 axiom 46, 47, 51, 71, 72, 73, 79, 85, 86, 87, 89, 91, 97, 98, 101, 139, 142, 148, 149,150,151,159,161,170,239, 242, 245, 251 axiom of choice 2, 245, 252 axiom of infinity 48, 252, 253 axiom of reducibility (Axiom der Reduzierbarkeit) 162, 164, 169, 171, 174, 175,238,245,251 bayesian 139, 141, 154, 157, 160 behaviourism 17,40 belief 2, 3, 8, 13, 18, 21, 25-33, 35, 38, 40,61,102,113,114,115,116, 145, 150, 153, 184, 186-92, 196, 199-208, 212, 216, 219, 226, 233 bet 151,152,153,158,159 betting 158, 159 Bradley, F. H. 55, 149, 159, 160, 220, 224, 227, 229, 235, 236 Braithwaite, R. B. 68, 70, 74, 102, 111, 135 Brentano,F. 7, 9, 10, 11, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 26, 27, 34, 40 Cantor, G. 170, 238, 240, 241, 242, 243, 246, 254, 255, 256 Carnap, R. 65, 66, 70, 75, 79, 80,81, 82, 83, 84, 97, 98, 99, 101, 102, 158, 160,
188, 237, 238, 242, 243, 250, 251, 254, 255 chance 15, 16, 187, 188, 189, 198, 205 concept 6, 10, 17, 18, 25, 36, 39, 55, 86, 93, 95, 100, 102, 226, 227, 234, 239, 240 correspondence 89, 91,92, 93, 111, 115, 118, 130, 132, 134, 135, 136, 140, 222,241,254,256 de Finetti, B. 139, 153, 154, 155, 160 Dedekind, R. 175,241 Dokic, J. 23, 40, 57, 62, 68, 137, 187, 193,197,216,218 Dretske, F. 202,218 Dutch Book 154 ethically neutral proposition 148, 149 evidence 50, 52, 53, 140, 188, 191, 195, 203,204,207,216,237 expectation 3, 25, 151, 156, 183, 186, 192 Frege, G. 4,49,55,65,68,161,163, 165, 166, 227, 232, 236, 237, 238, 239, 240, 241, 242, 243, 246, 248, 251,254,255,256 Gettier, E. 197, 198, 199, 200, 201, 202, 204 Goldman, A. 4, 202, 203, 206, 207, 208, 209,212,217,219 Grover, D. 118, 121, 126, 127, 135, 137 habit 5, 16, 17, 79, 110, 156, 183-7, 197 Hacking, I. 187, 193 Hilbert, D. 150, 160-81 Horwich,P. 122,136,138 Hume,D. 51,52,156,160,255
Index identity 10, 11, 17, 28, 39, 47-9, 57, 85, 93, 98, 123, 125, 128, 135, 138, 143, 237,239,247 incompleteness 228,229 inconsistency 143, 153-5, 240, 241, 251 inconsistent 43, 142, 143, 154, 157 individual 17, 40, 47, 48, 65, 76, 95, 99, 104, 106, 120, 143-6, 158, 166, 173-9,220,223,230,236,238, 241-9, 250, 252, 253 induction 42, 50 2, 155, 156, 175, 183, 189,245 inquiry 64, 65, 104, 184, 190, 192, 193 intentionality 7,8, 12, 15, 16, 19, 23, 24, 29, 34, 38 intuitionism 50 James, W. 5,10,11,13,26,140,182, 183 judgement 9, 13, 73-5, 98, 113-5, 142, 146, 158, 166, 185, 186, 190, 206, 215 Kant, I. 68 KeynesJ. M. 1,2,44,50,51,104-12, 141-5, 155-7, 160 15-7,50,51,71-3,86-101,139, 141, 144, 145, 154, 155, 157, 159, 187, 191-3,239,250,252,256 Lewis, D. 70,79,81-6,91,95,97, 99-103 liar sentence 129-31 liar paradox 130,132,247 logic 2,3,140,157 logical positivism 5 logical atomism 232 logical constant (logical connectives) 4, 8,30-4, 135, 156, 181,221 logical form 33, 55, 65, 96, 118, 224-8, 231,233 logical relations 13, 43, 144, 145, 223 logicism 50, 251 logicist programme 44, 243, 250, 251 logical axioms 159 logical truth 77,81,84
261
mathematics 1, 3, 5, 6, 42-6, 50, 53, 64, 67, 68, 103, 130, 139, 162-5, 169, 170,171,179,180, 181,232,237, 238, 242, 243, 246, 250-2 meaning 14-16, 18, 22, 24, 28, 37, 42, 49, 55, 57-60, 65, 70, 72-85, 91, 97,98,100,101,116, 117,120, 122, 126, 135, 174, 183, 199,221, 222, 224, 226, 232, 237, 246-8, 253 Meinong, A. 11 mental content 12,13,21,39 mind 2,5,7,8,10,12,16,17,20,27, 29,30,33,34,39,45,73,74,105, 123, 165, 167, 170, 180, 197,214, 232 model 20, 65, 66, 88, 106, 108-12, 159, 185,233 Moore, G. 1,7,41,42,44,220,226, 227,229 object 9,10,203,240 objective reference 13,30 observational 77-9, 81, 86, 92, 100 occurrence 51, 128 ontology 226, 227, 233
law
paradox 142,143,245 (see also liar paradox and semantic paradox) partial belief 145 Peano,G. 130,230,243 Peirce, C. S. 155,182-93 perception 11,21, 144, 191, 196, 197, 203,206,209-14,217 perspectivism 212 postulate 12,79-100 pragmatism 3-5, 7-9, 14-17, 27-9, 32, 34, 38, 40, 41, 52, 59, 60, 118, 136, 138, 182-8, 191, 193 predicative 74, 74, 98, 171, 172, 174, 178,179,180,181,229,243,246, 247, 250 predicative function 161, 162, 176, 177, 248, 249
Th
Index
primary occurrence 223 Principia Mathematica 28, 31, 44-6, 49, 54, 161, 162, 166, 172, 174,175, 232, 238, 240, 245-8, 251-3, 255, 256 Prior, A. 116, 126, 129, 135, 136 probability 1-3, 5, 38, 39, 42, 51-3, 68, 111,114,139-45,147,149,151, 153-60, 183, 184, 187-9, 193, 203, 204 process 4, 9, 15, 72, 107, 178, 184, 187, 189, 194-8, 200-7, 242 property 12, 20, 26, 34, 47, 84, 94, 120, 122, 123, 135, 146, 167, 168, 187, 214, 215, 221, 230, 231, 234, 242, 244,245 prosentence 117, 121, 122, 125-7, 135, 136 Ramsey sentence 70, 75, 78, 80, 82-6, 88, 89, 95 realism 170, 178, 189, 193 redundancy 58, 124 reliabilism 202, 203, 207, 209, 212, 217, 219 Russell, B. 1-34, 38, 40, 41, 46, 48-55, 59, 61, 62, 64, 65, 69, 97, 98, 103, 114, 129, 138, 161-4, 166, 168-75, 177, 178, 180, 182, 187, 194-200, 202, 220, 222, 227-229, 231, 232, 236-40,243-9,251,253 Sahling,N-E. 1,2,6,115,138 semantic paradox 129 Sosa, E. 68, 205, 206, 208-13, 217, 219
Strawson, P. 52,66,69,116,136, 228-31, 236 Tarski, A. 57, 116, 129, 133, 135, 136, 138, 247 tautology 37, 46, 47, 49, 174, 190, 251, 253 theorem 1, 141, 149, 150, 154, 159, 161-81 theory of types 161, 162, 167, 168, 172, 173, 244, 245, 249 Tractatus 4, 27, 29, 31, 34-6, 40, 42-6, 48, 53, 54, 56-9, 64, 65, 67, 68, 129, 181,226,233,236 truth 2-4, 7-10, 13, 18-20, 28, 29, 34, 35, 37-9, 41, 43, 45, 47, 56, 57, 61, 64,67 truth bearer 57 truth function 4, 5, 31, 46, 48 truth table 51 universal 98, 162, 192, 220-36, 239, 247 utility 6,100, 105, 107-9, 139, 146,147, 151, 152, 154, 156, 183, 185 variable hypothetical
50,197
Weyl,H. 50,163-6,168,175 Wittgenstein, L. 50-4, 41-69, 116, 129, 171,181,182,226,227,236,238, 247,251,252 Zermelo, E. 164, 165, 167, 172, 240, 241,245,255,256
