Philosophical Logic: Philosophical Papers 9781501717314

Table of contents :
Contents
Introduction
Acknowledgements
The Heterological Paradox
The Paradox of the Liar
The Paradoxes of Confirmation
Confirmation Theory and the Concept of Evidence
The Epistemology of Subjective Probability
The Epistemology of Subjective Probability
"And Next"
"And Then"
Time, Change, and Contradiction
A Modal Logic of Place
Subject Index
Index of Persons

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Philosophical Logic

Philosophical Papers of Georg Henrik von Wright I Practical Reason II Philosophical Logic III Truth, Knowledge and Modality

Philosophical Logic GEORG HENRIK VON WRIGHT Philosophical Papers Volume II

Cornell University Press Ithaca, New York

© G. H. Von Wright 1983 All rights reserved. Except for brief quotations in a review, this book, or parts thereof, must not be reproduced in any form without permission in writing from the publisher. For information address Cornell University Press, 124 Roberts Place, Ithaca, New York 14850. First published 1984 by Cornell University Press International Standard Book Number 0-8014-1674-4 Library of Congress Catalog Card Number 83-71774

Printed in Great Britain

Contents

Introduction

vii

Acknowledgements

xiii

The Heterological Paradox

1

The Paradox of the Liar

25

The Paradoxes of Confirmation

34

Confirmation Theory and the Concept of Evidence

44

The Epistemology of Subjective Probability

56

The Logic of Preference Reconsidered

67

"And Next"

92

"And Then"—with a Note on Quantification in Discrete Time

103

Time, Change, and Contradiction

115

A Modal Logic of Place

132

Subject Index

141

Index of Persons

143

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Introduction In 1957 appeared (published by Routledge & Kegan Paul, London), a collection of my essays entitled Logical Studies. It contained eight papers, five of which had been published in the preceding ten years. The present volume may be regarded as a continuation to Logical Studies. It was, at the time, my plan to search all the important branches of logic and present my findings and reflections in a number of successive papers. In the course of time the itinerary assumed unexpected new directions and the author's ambitions became more restricted. My first target after the publication of Logical Studies was the concept of negation. In 1959 I published a paper "On the Logic of Negation".1 It elaborates a distinction between two concepts of negation—a weak and a strong. Weak negation signifies (mere) absence or denial, "nothingness". Its logic is "classical"; in particular the Law of Excluded Middle is valid for it. Strong negation is an affirmation, usually on the basis of something which is present in or true of something. For it the Law of Excluded Middle does not hold unrestrictedly. Its logic therefore is "non-classical". I think that the distinction between the two negations is of vital importance, and its further elaboration has continued a main theme of my endeavours in philosophical logic. It may be used for clarifying the ideas of predication and truth, the notion of omission in action theory, and the difference between "dialectical" and "traditional" logic. The paper on negation is not included here—partly because I am no longer satisfied with its formal aspects, and partly because I hope in future publications to be able to present its main ideas better and more fully. My next target were the antinomies (paradoxes) of logic. In my treatment of them some reflexes also of the distinction between the two kinds of negation can be discerned. The paradoxes traditionally belong to the most lively debated matters in logic. Attempts to "solve" them have contributed decisively to the development of logic after Frege. To me the fascination of the antinomies has been that they challenge reflection about the most basic ideas of logical thinking: property and proposition, truth and demonstration, the meaning of "contradiction". These ideas are intertwined in their roots. The antinomies make us aware of this. There is no unique way of untwisting the connections—and therefore no one way of "solving" the paradoxes either. It is hardly possible any longer to treat the antinomies in an entirely 1

In Commentationes physico-mathematicae Societatis Sclentiarum Fennicae, XXII (4), 1959; also available as reprint by University Microfilms International.

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Introduction

novel manner. Every serious effort to deal with them is likely to resemble one or several which have been made before. This certainly holds true of my contributions—and I have tried to give an account of their relations to some other views of the matter, in particular to those of Wittgenstein in his later writings. But I hope there is enough originality in what I have to say to make my own way through the mazes interesting to follow by others. The paper on the heterological paradox appeared in the Commentationes physico-mathematicae, XXIV (5), 1960 of the Finnish Society of Sciences (Societas Scientiarum Fennica). "The Paradox of the Liar" is based on a lecture which I gave in the University of Lund in 1957. In preparing the essay for publication I was relying on criticisms and suggestions made by Professor Peter Geach. I am particularly indebted to him for observations, in the concluding section of the paper, on sense and reference and man's "semantic omnipotence". A paper by W.H. Baumer, "Von Wright's Paradoxes", Philosophy of Science, 30, 1963, stirred a return to an early interest of mine, viz. probabilistic inductive logic. Baumer had found an error in my formal treatment of the so-called Paradoxes of Confirmation and suggested a new treatment of them which, however, I do not think successful. Trying to put my error right, I was led to rethink Confirmation Theory after my own fashion. This resulted, not only in the paper "The Paradoxes of Confirmation"—but also in a more general statement of my position "A Note on Confirmation Theory and on the Concept of Evidence" in Scientia, 105, 1970. Both papers are included in the present volume. Probability had been my earliest research topic in logic and philosophy. It had occupied me continuously from my first published paper "Der Wahrscheinlichkeitsbegriff in der modernen Erkenntnisphilosophie" (Theoria, 4, 1938) to the completion ten years later of my first book published abroad A Treatise on Induction and Probability (Routledge & Kegan Paul, London, 1951). It was not, however, until the mid-1950s, when I was lecturing at Cornell and some other universities in the United States that my views on the "meaning" of this notoriously controversial notion reached their mature state. For many years I was working on a new monograph on the subject—to be called "An Essay in Probability"—but other occupations in philosophical logic distracted me from ever giving a final touch to the planned work. Those later views of mine of probability are by no means entirely original —and this may have been a contributory reason why I never worked them out in full detail. I had met them in the "philosophical hints" which mathematicians—beginning with James Bernoulli—sometimes drop in their writings on probability, and later I thought I found them anticipated in some manuscripts of Wittgenstein from the early 1930s.2 To put this 2 Published in the posthumous works of his called Philosophische Bemerkungen (1964) and Philosophische Grammatik (1969).

Introduction

ix

view in a nutshell: a numerical probability is a measure which we hypothetically assign to the occurrence of a (generic) event in a sequence of repeatable experiments or observations. We use this measure for calculating probabilities of relative frequencies of the event in question and we adjust it, if needed, on the basis of statistical experience. A probability hypothesis is not objectively true or false as the frequency theorists would have it; nor is it primarily the expression of a subjective belief in the occurrence of individual happenings. It is "beyond" truth and falsehood. It is not a "hypothesis" about an unknown reality—but its use is in a characteristic way subject to experimental control. Of Wittgenstein's position I have given a relatively full account in an essay.3 My own views are reflected in some minor publications from the 1950s and 1960s, among them the "Note on Confirmation Theory" mentioned above. Another short paper representing the same standpoint was originally a contribution to a symposium, with I.J. Good and L. Savage, on "current views of subjective probability" in Stanford in 1960. This paper is reprinted here as a "supplement" to the two papers on confirmation theory, but also as an "introduction" to the subsequent essay on preference logic. It was not, as might have been expected, my work on the logic of norms, but rather my study of subjective probability, that paved the way for me to a Logic of Preference or, as I also proposed to call it, Prohairetic Logic (from the Greek word for preferential choice npoaipeais).4 F.P. Ramsey had thought that relations of preference and indifference in (conditioned) options could be used for defining (subjective) probabilities. As I tried to argue in my Stanford paper, this idea involves a circularity (cf. below, pp. 60ff). Ramsey himself recognized that his procedure rested on certain "laws of preference", but he made no effort to work out these laws or principles in a systematic way. The task, however, must be considered basic in relation to the "subjectivists'" attempts to metricize a utility- or value-function. Shortly after writing the Stanford paper I had some ideas about how to accomplish the task. After having presented my thoughts in lectures in the University of Edinburgh, I publicized them in a small book called The Logic of Preference (Edinburgh University Press, 1963). Ideas akin to those of mine had been expounded by Soren Hallden in his pioneering essay On the Logic of 'Better' (Uppsala, 1957). 3

"Wittgenstein's Views on Probability", Revue Internationale de Philosophic, 23, 1969. Reprinted with revisions in the collection of papers Wittgenstein, Basil Black well, Oxford, 1982. 4 In an earlier paper "On the Logic of Some Axiological and Epistemological Concepts" (Ajatus, 17, 1952) I had made a first attempt to approach the notion of value from a logical point of view. This attempt was inspired by my then recent discovery of "deontic logic". I soon realized, however, that I had entered a blind alley and that a completely fresh attempt in a new direction Was required.

x

Introduction

Preference logic has turned out to be much more tricky than I had initially expected. It has also remained to this day relatively little developed (cf. below, pp. 67f). One can hardly say that the subject has acquired for itself the same "established" status as is undoubtedly now the case with its "companion branch" deontic logic. Dissatisfaction with the prevailing state of affairs was an urge for me to return to the subject once again in order to clarify and improve upon my earlier position and also to take the treatment of the "absolute" value-concepts (good, bad, and neutral) a step further. This I did in a paper "The Logic of Preference Reconsidered" published in Theory and Decision, 3, 1972. The essay has, as far as I can see, remained relatively little noticed. I have included it in the present collection. It is likely to remain my last word on the subject—and it is my hope that it will inspire others to do more research into this obscure corner of philosophical logic. I came to tense-logic (temporal logic, the logic of time) in a somewhat roundabout way. The discovery of deontic logic in the early 1950s had made me interested in norms. Norms are, primarily, rules of (human) action. And actions consist, normally, in the bringing about or prevention of changes in the world. I therefore came to the opinion that a (more) fully developed Logic of Norms (deontic logic) will have to be based on a Logic of Action and this, in turn, on a Logic of Change. In Norm and Action (Routledge & Kegan Paul, London, 1963) a first attempt was made to work out this hierarchy of "logics" in some detail. A change is, in its simplest form, a transition in time from one stage in the history of a state of affairs to the next. In the description of such processes is presupposed that the time medium has a discrete structure—or at least that time is "chopped up" in discrete bits such as successive days, hours, seconds, etc. For purposes of the description a connective "and next" may be used. It is a kind of non-associative and non-symmetrical conjunction. Some rudiments of its logic have been presented already in Norm and Action, and a fuller systematization given in a paper "'And Next'" in Acta Philosophica Fennica, XVIII, 1965. An attractive and, from a philosophical point of view, interesting property of this logic is that the notion of tautology (in its original Wittgensteinian sense) provides a semantic criterion of logical truth for its formulas. This observation links my work on the logic of the connective "and next" with earlier efforts of mine to extend the notion of tautology beyond the confines of "classical" propositional logic.5 At the time when I conceived of a change logic I knew nothing about tense-logic and had no idea that there also existed a hidden connection between change logic and modal logic. I was eager to study yet another 5

Cf. my papers "On the Idea of Logical Truth" (1948) and "Form and Content in Logic" (1949) reprinted in Logical Studies.

Introduction

xi

non-associative and non-symmetrical temporal connective "and then" which did not presuppose that the successive moments in time constitute a discrete manifold. In the logic for "and then" also temporal quantifiers, the notions of "always" and "sometimes", could be defined. The calculus was evidently going to be much more complex than I had anticipated. I was lecturing about it a trimester at Pittsburgh in 1966 when I met Arthur Prior and came to understand that I was really moving in parallel, although narrower, grooves to his. I was, however, anxious to go my own route to the end, and this resulted in the essay "'And Then'" published later in the same year in the Commentationesphysico-mathematicae, XXX, 7 of the Finnish Society of Sciences. The notion of tautology could no longer be used for defining logical truth in the calculus of "and then". But in view of what was known about logical truth in the monadic predicate calculus and the calculus of "and next", it was tempting to think that the notion of a (history-)tautology would provide a criterion of truth in a logic which combined the connective "and next" with a temporal quantifier. I wrote a paper on the problem, called "'Always'".6 In all evidence I had succeeded in axiomatizing the combined calculus, but my idea of truth as tautology of histories of arbitrary length will not work, for reasons which I came to see later. This essay is not included here—but the axiomatization of the combined system is appended as a "Note on Quantification in Discrete Time" to the paper "'And Then'". In a paper "Tense Logic and the Logic of Change", in Logique et Analyse, No. 34, 1966, J.E. Clifford proved that my Logic of Change described in the paper '"And Next'" was deductively equivalent with a modification of a system of tense-logic described by Prior in Time and Modality (1957). Krister Segerberg gave this system an elegant and much simplified reformulation in a paper "On the Logic of 'Tomorrow'" in Theoria, 33, 1967; and Prior discussed at length the essays '"And Next'" and '"And Then'" in his classic work Past, Present and Future (1967). In the meantime I had myself summed up my contributions to systems of tense-logic and modal logic in a paper "Quelques remarques sur la logique du temps et les systemes modales", in Scientia, 102,1967. It is not included here. In Autumn 1968 I gave the Eddington Memorial Lecture at Cambridge and chose as a topic for it "Time, Change, and Contradiction". The lecture referred back to the work of Prior on time and my own work on change, but its main question was how the three notions mentioned in the title are mutually related. What I think of as a new avenue in tense-logic was also opened up, leading to a study of the logic of the envision of time 6

Theoria, 34, 1968.

xii

Introduction

into "bits" of ever shorter and shorter duration. This avenue is still awaiting exploration.7 I was in Cambridge to give the Eddington Lecture when I got news of the unexpected death of Arthur Prior on a lecturing tour in Norway. A translation of the lecture into Norwegian appeared about a year later and was dedicated to his memory. Of all those who were breaking new ground in philosophical logic in the 1950s and 1960s, Prior was the one whose approach and tastes were most similar to my own. Modal logic, which I started doing in the early 1950s, had turned out to be an enormously fertile discipline with offshoots in many directions. Deontic Logic, Epistemic and Doxastic Logic, Temporal Logic may all be regarded legitimate daughters of modal logic; and she may still bear further offspring. Since there is a modal logic of time (tense-logic) why not also one of space? The possibility continued to tease my thinking for many years until in the mid-1970s I hit upon the two systems for the notions of "nearby" and "somewhere else". They are described in a paper "A Modal Logic of Place", first published in a Festschrift8 for Nicholas Rescher, and reprinted as the last essay in the present volume. A completeness proof for the calculus of "somewhere else" had by that time already been published by Krister Segerberg.9 An essential simplification of the system was suggested by Professor Robert Stalnaker in the course of lectures which I gave at Cornell in 1976. I have tried to standardize my use of symbols and technical expressions as far as it has seemed to me expedient. Some inconsistencies may nevertheless strike the reader. Thus, for example, in some of the earlier papers I use lower case letters as schematic representations for proposition (as in the phrase "the proposition /?"). Later, however, I have come to prefer the use of such letters as schemas for which sentences may be substituted. When referring to propositions I use that-clauses (as in the phrase "the proposition that /?"). In all essays revisions have been made and occasionally errors have been corrected. The revisions are, on the whole, minor only; in some cases they have been added in footnotes marked "(1983)". Georg Henrik von Wright

7

The only contributions to it, known to me, are two papers by Maria Louisa Dalla Chiara, "Istanti e individui nelle logice temporali", Rivista difilosofia, 64, 1973 and "Von Wright on Time, Change and Contradiction" (forthcoming). 8 The Philosophy of Nicholas Rescher, ed. by Ernest Sosa, D. Reidel Publishing Co., Dordrecht, Holland, 1979. 9 '"Somewhere else' and 'at some other time'", Wright and Wrong, Mini-Essays in honor of Georg Henrik von Wright, Abo, 1976; also "A note on the logic of elsewhere", Theoria, 46, 1980.

Acknowledgements The papers in this volume originally appeared as follows: "The Heterological Paradox", Soc. Sci. Fenn., Comm. Phys.-Math., XXIV (5) (1960), 3-26. "The Paradox of the Liar", Philosophical Essays dedicated to Gunnar Aspelin\ Gleerups, Lund, 1963, pp. 295-306. "The Paradoxes of Confirmation", Aspects of Inductive Logic, ed. by J. Hintikka and P. Suppes; North-Holland Publishing Co., Amsterdam, 1966, pp. 208-18. "Confirmation Theory and the Concept of Evidence", Scientia, 105 (1970), 595-606. "The Epistemology of Subjective Probability", Proceedings of the I960 International Congress: Logic, Methodology and Philosophy of Science, ed. by E. Nagel, P. Suppes, and A. Tar ski; Stanford University Press, 1962, pp. 330-9. "The Logic of Preference Reconsidered", Theory and Decision, 3 (1972), 140-67. "And Next", Acta Philosophica Fennica, XVIII (1965), 293-304. "And Then", Soc. Sci. Fenn., Comm. Phys.-Math., XXX (7) (1966), 1-11.. "Time, Change, and Contradiction", Time, Change, and Contradiction', Cambridge University Press, 1969, pp. 1-32. "A Modal Logic of Place", The Philosophy of Nicholas Rescher, ed. by Ernest Sosa (© D. Reidel Publishing Co., Dordrecht, Holland), pp. 65-73. The papers are here republished with the kind permission of the copyrightholders.

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The Heterological Paradox Few things have had a more stimulating effect on the development of modern logic than the discovery of the antinomies of set theory round the turn of the century and the subsequent discussion of these and other paradoxes. The idea that there is a solution to the problem of the antinomies such that, the solution having been found, the paradoxes would cease to puzzle logicians, may have been natural to entertain in the days when Russell put logic and Zermelo set theory on a new basis. Today it appears wiser to think that the antinomies will remain a permanent topic of discussion to receive fresh attention over and over again by logicians. Rather than speaking of the efforts of the logicians as of so many proposed solutions to a problem, we should think of them as different, but not necessarily mutually exclusive or even mutually competing, ways of treating a subjectmatter. In this paper one of the antinomies, viz. the one known as the Heterological or Grelling Paradox, is selected for treatment along lines which, as far as the author knows, are somewhat novel. The treatment to which this one Paradox is subjected is meant to be an illustration of a more general method which can be applied to other antinomies as well. Some remarks about the general significance of the method are made in the five Appendices (pp. 17-24) of the present essay. I

The Grelling or Heterological Paradox is one of the so-called semantic antinomies. It is called "semantic", because it concerns the relation of linguistic expressions to their meanings. The linguistic expressions concerned in this particular antinomy are words, and the meanings are properties. It may be asked, whether a property should be called the meaning or rather the reference (denotation, "bearer") of the (word which is a) name of this property. I do not believe, however, that the distinction between meaning and reference, important as it is in itself, is relevant to our discussion of the paradox. The concept of a "word" gives rise to interesting problems. One may distinguish between the word and the word-sign ("icon", "picture"), and say that a word is a word-sign in use (or associated with a meaning). One also distinguishes between the word(-sign) as a type and as a token (of a type). I do not think that these distinctions are relevant to our discussion of

2

The Heterological Paradox

the paradox. I shall here regard the notion of a word as being unproblematic. (See, however, remarks in Appendix I.) Words which are names of properties are, for example, "red" or "new" or "animal". It is clear that words may not only name but also have properties. (I assume that a word can be regarded as a kind of thing. For comments on this assumption see Appendix I.) A property of a given word can be, for example, that it contains so and so many letters or syllables, that it is long or short, that it occurs so and so many times in a certain context, that it is a proper name or a noun or a verb, etc. Since words have properties and some words are names of properties, it may happen that a word has exactly that property of which it is a name. Examples are easily found. Thus the word "pentasyllabic" is pentasyllable (fivesyllabic). The word "short" is a short word, the word "old" is an old word in the English language, the word "used" is used, "noun" is a noun, etc. A word which has a property of which it is itself a name is called autological. A word which is not autological is called heterological. It is clear that a vast majority of words, even of those which are names of properties, are heterological. (Be it observed in passing that one and the same word may be a name of several properties; we then call it autological, if it has at least one of the properties which it names.) II

Before we proceed to constructing the antinomy, we ought to make the two notions "autological" and "heterological" more precise. We propose the following definition: x is autological if, and only if, x has a property of which x is a name. The definition contains a variable, "x". What does this variable "stand for", what is its range (range of significance)? The range of the variable might also be called the Universe of Discourse of the definition. A most general answer to our question is that "jt" stands for any thing (i.e. for anything to which some property may be significantly attributed). The definition then says, that a thing is autological, if – – –. I think that this is a quite legitimate answer to our question, and that it is not impaired by the generality and vagueness of the concept of a thing, nor by the difficulties which may be caused by the idea of a totality of all things. One can, however, also narrow the Universe of Discourse of the definition. One can limit it to words, or even to names of properties only. Then the definition says, on which conditions a word is autological, or on which conditions a name of a property is autological.

The Heterological Paradox

3

If the Universe of Discourse of the definition is the totality of things, then in order to decide whether a given thing is autological or not, we ought to make up our minds on the following three questions: (1) Is x a name of a property? If the answer is no, we decide that x is not autological. If the answer is yes, we proceed to question (2) Of which property (properties) is A: a name? When this has been settled, we have still to answer question (3) Has x this property (at least one of these properties)? If and only if the answer to this last question is yes, is x autological. It may be thought that question (1), whether x is the name of a property, can be settled in the affirmative only by mentioning some property of which x is a name. I shall accept this. Thus an affirmative answer to question (1) automatically answers, at least partially, question (2) as well. If the Universe of Discourse of the definition is the totality of words, then, when the definition is being applied to a given x, it is already assumed that this A: is a word. And if the Universe of Discourse is the totality of names of properties, then, when the definition is being applied to a given x, it is already assumed that question (1) above can be answered in the affirmative for this x. Since, in discussing the paradox, anything which has been assumed in constructing it may afterwards be questioned, it is practical to conceive of the Universe of Discourse in a manner which makes a minimum of assumptions and which does not beg any of the questions (l)-(3) above. We proceed to the concept "heterological". Its definition is as follows: x is heterological if, and only if, it is not the case that x is autological. Substituting, in this definition, for "autological" its defining expression, we get: x is heterological if, and only if, it is not the case that x has a property, of which x is a name. Since "autological" and "heterological" are interdefinable, we could dispense with one of the terms (and yet have the paradox). We need not introduce the word "heterological" at all, but could simply say "not autological" in its place. But we could equally well dispense with the word "autological" in favour of the two-word phrase "not heterological". It might be observed in passing that the extension of the term "autological" is not affected by the choice between the alternative conceptions, which we mentioned, of the range of the variable. But the extension of the term "heterological" is affected by the choice. If the Universe of Discourse of the definition is restricted to names of properties, then, for example, the word "red" is heterological, but the word "Napoleon" is neither autological nor heterological—since it is not the name of any property at all, but of a man. If the universe is restricted

4

The Heterological Paradox

to words, the word "Napoleon" is heterological, but the emperor Napoleon is neither autological nor heterological. If, however, the universe comprises the totality of things, we shall have to say of the emperor too that he is heterological—since he is not a word, but a man. It may be suggested that the most natural way of using the term "heterological" is for names of properties which have not got the property they name. This would speak in favour of restricting the Universe of Discourse of our definitions of "autological" and "heterological" to names of properties. This question of naturalness, however, is of no importance to the discussion of the paradox.

Ill In the course of our considerations so far, we have introduced two new words into our language, viz. the words "autological" and "heterological". The two new words, it would seem, are names of properties. The word "autological" names the property which a thing has if, and only if, it (a) is a name of a property and (b) has a property of which it is a name. The word "heterological" again names the property which a thing has if, and only if, it (a) is not a name of any property or (b) is a name of some property but has not got any property of which it is a name. We could also say that the word "autological" names the property of autologicality, and the word "heterological" the property of heterologicality. To say that "autological" and "heterological" are names of properties is, however, to make a very consequential commitment. In view of what will happen later, we may have to question our willingness to make this commitment.

IV I now proceed to the construction of the antinomy. The paradox originates from an application of the definition either of the concept "autological" or of the concept "heterological" to the word "heterological". (No paradox, however, springs from an application of either concept to the word "autological".) We ask: Is the word "heterological" autological or heterological? Let the thesis that the word is heterological be presented for consideration. Is this thesis true or false? When a definition containing a free variable is being applied to an individual thing in the range of the variable, we substitute for the variable in the definition a name of this thing in the range. What is substituted for the variable is thus a name of a thing—not the thing itself. This is obvious enough, if the things under consideration are some extra-lingual entities

The Heterological Paradox

5

such as, say, men or plants or numbers. It is perhaps less obvious, if the things are linguistic entities, such as names or words or sentences. If the thing, to which the definition is to be applied, is a name, then what has to be substituted for the variable is the name of a name. Thus when the definition of the concept "heterological" is applied to the word "heterological", we substitute for "x" in the definition a name for this word. The customary way of naming words is by the use of quotes. I shall use this customary way here too. And although the quotation-mark method of naming words is not free from problems, I shall not for a moment doubt that its use in the reasoning which takes us to the paradox is perfectly sound and uncontroversial. The application of the definition of "heterological" to the word "heterological" gives us in the first place: "heterological" is heterological if, and only if, it is not the case that "heterological" is auto logical. Substituting for the phrase "heterological' is autological" its definiens, we get: "heterological" is heterological if, and only if, it is not the case that "heterological" has a property, of which "heterological" is a name. Now in order to decide, whether the thesis that "heterological" is heterological is true or not, we must answer the above questions (l)-(3), when for "x" we substitute "'heterological"'. The questions are: (1') Is "heterological" a name of some property? (2') Of which property (properties) is "heterological" a name? (3') Has "heterological" any property of which it is a name? As regards questions (1') and (2') we tentatively gave the answer that "heterological" names the property of heterologicality, viz. the property which a thing has on condition that it is not autological. Accepting this, the truth of the thesis that "heterological" is heterological hinges on question (3'), whether "heterological" has or has not this property which it is thought to name. To say that "heterological" is heterological is tantamount to saying that it is not the case that "heterological" has the property which it names. But to say that "heterological" has the property which it names is tantamount to saying that "heterological" is heterological. The truth-condition of our thesis therefore becomes: "heterological" is heterological if, and only if, it is not the case that "heterological" is heterological. This truth-condition is of the general form: p if, and only if, it is not the case that p. (In the symbolism of the prepositional calculus: p «-» ~ p.)

6

The Heterological Paradox

According to "ordinary" logic this is a contradiction. For, if/? is or is not true, then to say that p is true if, and only if p is not true is tantamount to saying that/? is both true and not true. I shall not for a moment doubt the contradictory character of the form "p if, and only if, it is not the case that/?". By the Grelling or the Heterological Paradox I shall understand the finding that the truth-condition of the thesis that "heterological" is heterological is a contradiction. V

In derivations and discussions of the Grelling paradox it is assumed as a matter of course that "heterological" is not the name of any other property beside (possibly) the property of heterologicality. It is, however, trivially the case that the word "heterological" could be the name of just any property. The word could, for example, also mean " pentasyllable ". Let us therefore assume, for the sake of argument, that the word "heterological" already occurred in our language prior to its introduction by the definition which we have given to it here. Then it will either be the case that it names some property (other than heterologicality) which it also possesses, or it will be the case that it does not name any such property. In the first case we shall have to say that the word "heterological" is autological. (The word "heterological" is autological if, for example, it also means "hexasyllabic".) And in this case there will be no paradox at all. For, the truth-condition of the thesis that "heterological" is heterological cannot now be stated in the form of an "if and only if" proposition. We should have to weaken the bi-conditional to a (simple) conditional and say that // "heterological" is heterological, then it is not the case that "heterological" is heterological. This is of the general form "/? -» ~ p" which is equivalent to ~ p alone. " ~ p" here stands for "it is not the case that 'heterological' is heterological", and this is the same as "'heterological' is autological". Only in the second case is there a paradox. For then the only ground on which the word "heterological" could be declared autological is that it happened to possess the property of heterologicality. The truth-condition of the thesis that the word "heterological" is heterological could be stated in the form of a bi-conditional—and thus become a contradiction. As is shown by these considerations, it is not essential to the Grelling paradox that the words "autological" and "heterological" (or any other words) should be univocal. What is essential is only that the truth-condition of the thesis that "heterological" is heterological can be stated in the form

The Heterological Paradox

1

of a bi-conditional. The assumption of univocality warrants this possibility and thus the occurrence of the paradox. VI

We must now stop to consider exactly what we have done, when we produced the antinomy. When an antinomy is described in the usual, carefree manner, which one may find even in respectable works on logic, it easily looks as though we in an antinomy, quasi, prove a thesis and its contradictory, i.e. prove a contradiction. One says, for example: The word "heterological" must surely be either autological or heterological (i.e. not autological). Now assume that it is autological, i.e. that it has the property which it names. Then the word "heterological" is heterological. But if it is heterological it has the property which it names, viz. the property of heterologicality. And if it has the property which it names, it is by definition autological. Thus if "heterological" is autological, it is heterological—and if it is heterological, it is autological. But this is a superficial and gravely misleading way of presenting the paradox. It takes for granted the very things which a true presentation of the antinomy will force us to question. Under no circumstances must we let ourselves be induced into saying that we have proved a contradiction. For a contradiction is something which by its very nature cannot be proved. The meaning of "proof", one might also say, excludes this. If, instead of saying that we have proved a contradiction we say that we have derived one, this would be a legitimate way of speaking. But then the meaning of "derivation" stands in need of clarification. I here propose to clarify it in the following manner: When we say that in the Grelling antinomy a contradiction is being derived what we mean is that we have proved that from certain premisses a contradiction logically follows. Thus in producing the antinomy, something has been proved. But that which has been proved is not a contradiction, but some true proposition. This true proposition has the form of a conditional, and can be stated as follows: If" heterological" names a property which a thing has if, and only if, it is not autological, then "heterological" is heterological if, and only if, it is not heterological. The conditional proposition, which we have proved, is of the general form "If/?, then (q if and only if not q)". Or, in the symbolism of the prepositional calculus:p -> (q «-+ ~ q). The consequent of the proved conditional proposition is, as was already noted, a contradiction. And a contradiction is a logically false proposition.

8

The Heterological Paradox

"q if, and only if, not q" is false for any value of "£'£"' which is a contradiction. We could call the contradiction The Heterological Paradox. Since we do not wish to have a contradiction in our calculus, we ought to take steps to remove it. In order to see, how the contradiction could be removed, we examine, how the contradiction came about. The last step in producing the contradiction was a substitution. Was this substitution permitted? By the rules, as we had formulated them, it was not permitted. For the rules only said that for a P-symbol in a proved formula, another P-symbol may be substituted throughout. But we had not said that "£ " could be handled as a P-symbol with regard to substitutability. That the contradiction occurs could now be taken as a ground for saying that "£ " must not be regarded as a symbol of the same kind (type, category) as the P-symbols, although it resembles P-symbols by virtue of the way in which it is combined with TV-symbols to form expressions. This way of avoiding the contradiction would correspond to the "way out", adopted in our previous informal discussion of the Heterological Paradox. Although the word "heterological" resembles property-words as far as its use to form sentences is concerned, it cannot be treated as a property-word in the sense that, if something is true for the meaning of any property-word, then it is also true for the meaning of "heterological". We can of course decide to call "£" too a P-symbol (by virtue of its resemblance to the other P-symbols)—but then we must distinguish between two senses of being a P-symbol: the sense in which anything is a P-symbol which may occur to the left of a T- or a TV-symbol in a well-formed expression, and

16

The Heterological Paradox

the sense in which anything is a P-symbol which may be substituted for the symbol standing to the left of a T- or a N-symbol in a theorem. It might be suggested that another "way out" would be to declare, not the substitution which we made, but the definition of "£" which we gave, "illegitimate". To say that the definition is illegitimate merely because of its somewhat unusual form seems to me to be, not only arbitrary, but plainly false. Since we can use it to define "£ " without contradiction. Actually, the definition as introduced by us into our calculus was flawless. (It was only the unwarranted assumption that the symbol defined was a P-symbol which led to the contradiction.) The definition teaches us the use of "{" for purposes of forming expressions and proving theorems in the calculus. It is true that it does not serve any purpose which a definition may serve. It has not the same powers of eliminating the defining terms from contexts, where they occur, as has an "ordinary", explicit definition. This is a peculiarity, not a defect, of it. Definitions with this peculiarity are sometimes called non-pascalian. But even to say that a non-pascalian definition is illegitimate for the purpose of defining P-symbols seems to me to be false. For if we defined a symbol "0" by means of the definition "0'A" = dfX'X9" and said that the symbol so defined was to be treated as a P-symbol, we should not get any contradiction in our calculus. (This would correspond to a definition of "autological".) The non-pascalian definition would now be of the kind called impredicative. Thus a definition cannot be declared illegitimate because it is nonpascalian or because it is impredicative. I can see no ground for declaring the definition of "£" illegitimate at all. It is true that it cannot be used to define a P-symbol, because this would lead to contradiction. But this only amounts to saying that the symbol "£", which it is used to define, is not a P-symbol. The only thing which may be called "illegitimate" here is the assumption, if we happen to make it, that "£" is a P-symbol. XII

Actually, in setting up the calculus, it is not necessary to have two distinct sets of symbols, the T- and the P-symbols. We can have only one set. Let us call them K-symbols and denote them by lower case letters "a", "ft", . . . . An atomic expression would now be defined as a succession of two Ksymbols. For example: "#£" would be an atomic expression, and so would "for" be. The rule of substitution would now be that for any K-symbol, which occurs in an axiom or theorem of the calculus, another K-symbol may be substituted throughout. For example: from "aft -» (ab v erf)", which is a theorem, we derive by substitution "aa -> (aa v aa)", which is therefore a theorem too.

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17

We could, however, even within this simplified calculus mark a distinction which would correspond to our previous distinction between T- and P-symbols. For, we could say that, in a given expression "xy ", the K-symbol to the left occurs as P-symbol, and the K-symbol to the right as ^-symbol. (One and the same K-symbol may then occur both as P-symbol and as r-symbol in the same atomic formula, or as P-symbols in one and as Tsymbols in another atomic formula.) Of "£", when added to the calculus, we should then have to say either that it is not one of the F-symbols, or that it is a K-symbol but not subject to the rule of substitution when it occurs as P-symbol. I hope that these considerations suffice to show that it was not the informal character of our presentation which was responsible for the peculiar nature of our proposed "solution" of the paradox, and that also a formalized presentation would have taken us to essentially the same end. This being so, it may be asked, why we should have given an informal presentation at all of the paradox. The answer is that it is only in the informal presentation that the derivation of the paradox can be stated as an argument from certain premisses to a certain conclusion (the contradiction), and the "way out" as an argument modo tollente from the falsehood of the conclusion to the falsehood of some of the premisses. In the formalized presentation the paradox originates thanks to a manipulation of symbols which is not permitted by the rules of the calculus. This notpermitted move in the game may be tempting, because of its strong resemblance to certain permitted moves. If the move was not expressly forbidden by the rules, then making it shows that it must be forbidden because of leading to contradiction. Therefore, if making the move can be described as acting on the assumption (premiss) that it could be permitted, then the paradox shows that this assumption (premiss) was false. But this argument from the assumption to the contradiction and back from the rejection of the contradiction to the rejection of the assumption, cannot be formalized in the calculus. APPENDIX I A characteristic of the treatment of the Heterological Paradox, which we have given in the paper, is that it tries to show that in the derivation of the antinomy use is being made of an entity, viz. the property of heterologicality, which the paradox proves to be non-existing. The derivation of the antinomy may thus be regarded as a kind of reductio ad absurdum argument. The observation that in the derivation of some of the best known antinomies—the Russell Paradox, the Heterological Paradox, and the

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The Heterological Paradox

Liar—use is made of a provably non-existing entity, is not new. The first to make the observation was, as far as I know, Bochvar.3 The formula of the predicate calculus, which may be said to show that there can be no such property as that of heterologicality is, ultimately, the formula Px v - Px or its equivalent according to the laws of the prepositional calculus ~ (Px& ~ Px). The formulae mentioned say that, given an arbitrary property P and an arbitrary thing x, then it is always the case that x either is P or is not P, and never the case that x both is P and is not P. To express that these formulae hold quite generally for any thing we may "quantify" them in the variable x, thus obtaining the new formulae (x)(Px v ~ Px) and (x) - (Px & ~ Px), of which the second can also be written in the form ~ (Ex)(Px & ~ Px). And in order to express that these last formulae hold quite generally for any property as well, we may quantify them in the variable P too, thus obtaining the new formulae (P)(x)(Pxv - Px)and(P) ~ (Ex)(Px& - Px), of which the second can also be written in the form ~ (EP)(Ex)(Px& - Px). These formulae can be used as a ground for saying that there is no such property as heterologicality, only if we take for granted that the fault in the paradox is with improperly and not with the thing with which we are manipulating in constructing the antinomy. This thing is the word "heterological". An alternative way out of the paradox would therefore be to say that there is no such thing as this word. This second way out may seem more "artificial" than the one adopted here. If we deny that the entity named by "'heterological'" is a thing but regard words as a kind of thing, then we would have to dispute that the named entity is a word. The conclusion would be that there is no such word as "heterological".4 An alternative conclusion would be that not all words are things (since the word "heterological" is not one). I do not think that either of these conclusions can be brushed aside as being absurd. To do this would be to take a much too simpleminded view of the notions of "thing" and "word". What I have done in my discussion of the paradox is therefore to show what follows for the concept of heterologicality, //we take the thing- and word-character of "heterological" for granted. If, instead, I had taken for granted the property-character of heterologicality, I might have used the paradox for drawing conclusions about this property's name, viz. that it is not a thing or not a word.

3 See the review in The Journal of Symbolic Logic, 11 (1946), 129. See also Quine, Mathematical Logic (1940), p. 128 and, for a more detailed exploration of the idea, Valpola, "Elementare Untersuchungen der Antinomien von Russell, Grelling-Nelson und Eubulides", Theoria, 19(1953), 183-8. 4 Cf. Valpola, op. c//., p. 186.

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Which road we wish to go, which moral we wish to draw from the paradox is, I believe, very much a matter of indifference. We can let the paradox, i.e. the contradiction which we reach as the result of a certain argument, "recoil" modo tollente either on the notion of a property, as we preferred to do in the paper, or on the notion of a thing or on the notion of a word. There is, however, a further way out which we have not so far mentioned at all. This way consists in pronouncing neither upon the property-character of heterologicality nor on the thing-character of "heterological" but in saying simply that "heterological" and heterologicality do not match one another as thing and property, according to the standards of the predicate calculus. This would be the most "neutral" conclusion of all. It might be termed a "Janus face conclusion", from which one could then proceed in one of two directions: The road in the one direction terminates in the conclusion that heterologicality is no property, since, if it were, it would both belong and not belong to one and the same thing, viz. the word "heterological". This was the direction we choose and the end we reached in our discussion in the paper. But we need not proceed as far as this end. We could also say that heterologicality is a property ("behaves like a property") up to a singular point. This point is, when we predicate the property of its own name. The road in the second direction terminates in the conclusion that "heterological" is no thing, since, if it were, it would both possess and lack one and the same property, viz. the property of heterologicality. But we need not proceed as far as to this conclusion. We might also say that the word "heterological" is a thing ("behaves like a thing") up to a singular point, namely to the point when we attribute to the word the property which it names.5

APPENDIX II It is an interesting observation that we get an antinomy only by trying to answer the question, whether "heterological" is autological or heterological —and not by raising the same question about "autological". Similarly, the Russell Paradox originates, when we ask whether the extension of the 5 Wittgenstein, in Remarks on the Foundations of Mathematics (rev. ed., IV, 59), compares an antinomy (he is then speaking of the Russell Paradox) to "something that towers above the propositions and looks in both directions like a Janus head". I do not understand quite clearly the meaning of the simile. It does not seem to me to be related to the idea, propounded in this paper, of the Janus face character of the conclusion which, in the most non-committing way, takes us out of the paradox.

20

The Heterological Paradox

property "extension which is not a member of itself" is a member of itself or not—whereas the question, whether the extension of the property "extension which is a member of itself" is a member of itself or not, does not lead to paradox. The Liar again originates from the question, whether the proposition "this proposition is not true" is true or not—but no paradox springs from the question whether the proposition "this proposition is true" is true or not. Remembering that a thing is heterological, if it has not got any property, which the thing itself names, we may take these observations as an indication that the concept of negation holds a crucial position in the construction of these three antinomies.6 The "negative" nature of heterologicality must not be misunderstood. It does not lie in the fact that we defined "heterological" as meaning "not autological", thus introducing "autological" first and "heterological" second. We could equally well have started with a definition of "heterological" (not mentioning "autological") and subsequently defined "autological" as meaning "not heterological" (see above, p. 3). By calling "heterological" negative I do not mean that the word has been introduced as an abbreviation for a phrase ("not autological") containing the word "not" or some other word signifying negation. The negativity of "heterological" is a peculiarity of the concept and not of the word, and consists in the fact that a thing is said to be heterological, not on the ground that it has such and such characteristic features, but on the ground that it has not those features, the presence of which in a thing (word) makes this thing autological. Similar remarks could be made about the negative nature of the concept "extension which is not a member of itself" and the concept "not true". The "way out" of the Grelling and Russell antinomies, which is here suggested, is the conclusion that the negative property involved is not a property in the sense of the predicate calculus. The corresponding "way out" of the Liar is the conclusion that the negative proposition involved is not a proposition in the sense of the prepositional calculus. The negative concepts, which give rise to antinomies, are thus of a different logical category or level or type from those concepts, of which they are the negatives. This can also be expressed by saying that, if "P" names a property, it does not always follow that "not P" names a property in the same sense of "property"; and if "/?" expresses a proposition, it does not always follow that "not p" expresses a proposition in the same sense of "proposition". When the conclusion does not follow, I shall call the entity named by the phrase containing the negation-word essentially negative. The antinomies of Grelling and Russell and the Liar can be said to 6

Cf. Valpola, op. ci/., p. 187. The author, however, seems to overlook the existence of "negation-free" antinomies, such as e.g. Curry's Paradox.

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21

establish or demonstrate the "essential negativity" of certain concepts. But this is not to say that the notion of essential negativity could be used to "explain" or "solve" the antinomies. This would be the case only if the notion of essential negativity had independent criteria which would make it possible to show—independently of the antinomy—that e.g. heterologicality is not a property in the same sense as those properties, the absence of which in their names makes us call the names heterological. This condition, for all I can see, is not fulfilled. It is not the essential negativity of the entities, which shows that they must not be treated as entities of the same category as those, of which they are the negatives. It is the fact that treating them thus leads to antinomies, which shows that the entities are what I have here called "essentially negative".

APPENDIX III The way of treating the Heterological Paradox which I have adopted in this paper seems to me related to Wittgenstein's comments on the paradox in Remarks on the Foundations of Mathematics (rev. ed., Ill, 79 and VII, 28) and to some other of his remarks on paradoxes and contradictions in general. "Why shouldn't it be said", Wittgenstein asks (op. cit.9 p. 395), "that such a contradiction as: 'heterological' € heterological = ~ ('heterological' € heterological), shews a logical property of the concept'heterological'." This, in my view, is exactly what the antinomy, when rightly understood, does. It reveals to us a logical feature of the concept "heterological". This feature is that the concept, although it may function as a predicate in true propositions of the subject-predicate form, yet does not, unlike most concepts with this function, in all cases obey the law of contradiction. The exception is the paradox. Therefore we do not notice the exception, until we become aware of the paradox. Wittgenstein then goes on to saying (p. 396) that '"h' € h = ~ ('h' € h) might be called 'a true contradiction'". That the contradiction is true means, he says, that "it is proved; derived from the rules for the word 'h'". To say that the contradiction is "true" would seem to me to be very misleading—and therefore also to say that it is "proved". For the lesson which the contradiction teaches us, I have tried to argue, is a lesson modo tollente, i.e. because we insist that a contradiction is not true and therefore cannot be proved, we are forced to question the validity of an assumption which we made in the derivation of the contradiction. (To say, as Wittgenstein does, that the contradiction is derived from the rules for the word "heterological" is, of course, quite in order.) That this assumption is not

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The Heterological Paradox

valid for the concept "heterological" is the truth which the paradox reveals. Wittgenstein also says that the truth of the contradiction means: this really is a contradiction, and so you cannot use the word "*/z'" as an argument in "£ € /z". To this we can agree. For it amounts to saying that the truth is not of the contradiction, but of the proposition that, on such and such assumptions, we get a contradiction. But, as we have seen (p. 11), there is a sense in which we can say that "'/*' € h = - ('h' € h)" is true—but not a true contradiction. To call "*/*' € A s= ~ ('/z' £ h)" true entails saying that the pair of predications "P" and "not P" are not always mutually exclusive, and this again amounts to saying that there is a use of "not" for predication which is different from that use of "not" which conforms to the Law of Contradiction. u < /i' € h = ~ ('A' € /*)", w/ie/i /rae, is therefore not a contradiction (unless we wish to give a new sense to the term "contradiction").

APPENDIX IV The analogy between an antinomy and division by 0 in arithmetic seems to me a good analogy and worth further exploration.7 Let us assume that we formulated a rule to the effect that for any real numbers m, k, and /, if mk equals ra/, then k equals /. Then someone uses the rule to "prove" that, say, since 0 times 5 equals 0 times 7, therefore 5 equals 7. What shall we then say? Since we insist upon the falsity of the conclusion, one thing to do would be to let it "recoil" modo tollente on some of the premisses used in the proof. One such premiss (perhaps better "presupposition") is that 0 is a (real) number. This we might now reject and say that the "paradox" shows that 0 is not to be regarded as one of the (real) numbers. Historically, the idea that 0 is one of the numbers was a hard one to acquire. At an earlier stage in the history of mathematics, the above conclusion from the "paradox" might therefore have seemed plausible. Not so today. The reasons which we have for regarding 0 as one of the (real) numbers are strong enough to withstand the "paradox". The obvious thing to say after the discovery of the "paradox" would therefore be, not that 0 is not a number, but that we have discovered an 7

The analogy is referred to in several places in Wittgenstein's Remarks on the Foundations of Mathematics.

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exception to the rule of division, which we gave. // ought to be reformulated. The rule, when reformulated, would say that for any numbers ra, /:, and /, if m is different from 0 and mk equals ml, then k equals /. This rule holds good for all numbers (including 0), but it enables us to derive the result that k equals / only for such values of m which are different from 0. Let us now compare this with our proposed "way out" of the Grelling Paradox. The conclusion modo tollente from the contradiction was that heterologicality is not a property. This corresponds to the suggested conclusion from 5 = 7 that 0 is not a number, //"the conclusion in question from the Grelling Paradox seems plausible, it must be because we do not feel that the grounds for calling heterologicality a property outweigh this ground (viz. the contradiction) against calling it thus. We could, I think, imagine circumstances under which we would insist upon calling heterologicality a property. (These circumstances would have to be somewhat similar to those which make us insist upon calling 0 a number.) Then the conclusion which we drew from the paradox would appear highly "artificial". But we could now draw a different conclusion, corresponding to the one which we actually draw from the paradox that 5 equals 7. We could say that, for any word (or thing) x, ifxis not a name of the property of heterologicality itself, then x is heterological if, and only if, it is not the case that x has got a property, of which x is a name. This holds good for all words (including the word "heterological"), but it enables us to derive a truth-condition for the thesis that x is heterological only for such values of x which do not name heterologicality.

APPENDIX V Is an antinomy a "danger" to logical thinking? Was reasoning with the concepts, which are involved in the antinomy, not "safe" or "in order" before we had discovered the antinomy? (These questions are very unprecise.) Questions of this kind are raised and discussed over and over again by Wittgenstein in the Remarks on the Foundations of Mathematics. They are raised, not for the antinomies only, but for all contradictions which may occur in a context of logical or mathematical proof. Wittgenstein says (III, 82) that his aim is "to alter the attitude to contradiction and to consistency proofs". He seems, in places, to say that antinomies are harmless (I, Appendix III, 12) and that contradictions might be accepted (IV, 55-60), and he speaks of "the superstitious dread and veneration by mathematicians in face of contradiction" (I, Appendix III,

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The Heterological Paradox

17). It seems to me that Wittgenstein never succeeded in saying quite clearly what he wanted to say on the subject of antinomies, contradictions, and consistency-proofs. I have tried to advocate a view in this paper which, if correct, would show in what sense antinomies may be regarded as "harmless". What I mean by this can, for the Heterological Paradox, be stated as follows: The only way, in which the false premiss or illegitimate move in the calculus which is responsible for the heterological paradox shows its falsehood or illegitimacy, is in the production of the antinomy. To assume that the word "heterological" names a property (or that "£" in our formalism is a P-symbol) and to rely on this assumption in reasoning works without contradiction up to a singular point. This point is when we assume that the property of heterologicality in relation to its name, the word "heterological", is subject to the same rule which holds for all other properties in relation to their names, v/z. that the name is heterological if, and only if, it does not possess the property. To assume this of heterologicality is, by definition, self-contradictory. If I am right, the antinomies of logic do not require any "general theory" for their solution—be it a doctrine of distinction of logical types, a Vicious Circle Principle or some other general restriction on the definition of concepts. The antinomies do not indicate any disease or insufficiency in the "laws of thought" as we know them at present. The antinomies are not the result of false reasoning. They are the result of right reasoning from a false premiss. And their common characteristic seems to be that it is only the result, viz. the paradox, which makes us aware of the falsehood. Without the discovery of the paradox, the falsehood would therefore have remained for ever unknown—just as people might never have realized that fractions cannot be divided by zero, unless they had actually tried to do it and reached contradictory results.

The Paradox of the Liar The antinomies of logic have puzzled people ever since they were discovered —and will probably continue to puzzle us for ever. We should, I think, not regard them as much as problems awaiting a solution, but rather as providing a perpetual raw-material for thinking. They are important, because thinking about them challenges the ultimate questions of all logic —and therefore of all thinking. I shall here discuss one of the oldest of all known antinomies, the Liar. The discussion will bring us in touch with many of the most important topics in logic and semantics: the definition of truth, the distinctions between sense and reference, use and mention, type and token, sentence and proposition, the meaning of contradiction and of proof. I. A "Superficial" Presentation of the Paradox I shall first give what I propose to call a "superficial" presentation of the antinomy. I write down the sentence: This sentence is not true, and raise the question: Is that which I have written true or not? It must be either one or the other.1 If it is true, then—by its own claim—it is not true. But if it is not true, then—by "the law of double negation"—it is true. Since it must be either true or not true, it is both true and not true. The above may sound convincing, but it is not very clear. Is the argument meant to be a demonstration of something? If so, is it successful? If successful, what does it demonstrate? II. A "Sophisticated" Presentation of the Paradox We cannot answer these questions unless we first re-state the argument. I shall now give what could be termed a "sophisticated" presentation of the antinomy. We need a general statement of the conditions for calling a sentence 1

This, I think, is true, in spite of all criticism which may be levelled against the argument as here presented. The alternative "true-not true" is more comprehensive than the alternative "true-false". Not everything is either true or false, since not everything is capable of having truth-value. But of everything which is neither true nor false it can be truly said that it is not true. Cf. my paper "On the Logic of Negation" in Soc. Sci. Fenn., Comm. Phys.-Math., XXII, 4 (1959) and above, p. 9, fn. 1.

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The Paradox of the Liar

true. It might be given as follows: (1)

"p " is true if, and only if, p.

It is understood that the letter "/?" stands (in both places) for an arbitrary sentence. If we insert or substitute for this schematic letter (two tokens of) the sentence about which we conducted the paradoxical argument in (1)—I shall sometimes speak of it as "the top sentence"—we get: (2) "This sentence is not true " is true if, and only if, this sentence is not true. We use sentences to speak about various things, e.g. the weather. If we want to speak about a sentence, we must mention the sentence, i.e. use some sign to refer to it. A conventional way of referring to sentences in written language is to enclose (tokens of) them within quotation marks. According to this convention about the use of quotes, an expression(-token) in quotes refers to the (type-)expression quoted. As schematic representations of arbitrary sentences we here use single lower case letters of the English alphabet. Considering the use of quotes to form names of expressions, it is convenient to use single lower case letters within quotes as schematic representations of names of arbitrary sentences. It should be observed, however, that this is a different use of quotes from their use to form names of expressions. Under the latter use, a letter in quotes is a name of the quoted letter (and not a schematic representation of a name of the sentence which is schematically represented by that letter). There seems to be no objection to using quotes in both ways—as long as we do not confuse the two uses. Thus a name of the top sentence is: "this sentence is not true". But this is not the only sign which refers to the top sentence. Another sign is the two first words of the top sentence itself, i.e. the sign "This sentence". For, if we ask: What do these two words, as they occur in the top sentence refer to?, the answer is: To the top sentence. And a third sign which refers to the top sentence is, obviously, the words "the top sentence". The identity of reference of two expressions can be stated in the form of an equation, for example: (3) "This sentence is not true " = This sentence. (Here the expression within quotes, including the quotes, refers to the expression within the quotes, excluding those quotes.) Now it is often thought that signs which refer (uniquely) to the same thing—also called names with the same bearer or denotation—may be substituted for one another in a sentence salva veritate, as one says, meaning without altering the truth-value of (that which is said in) the sentence. Accepting this principle of substitution and the above condition for calling a sentence true, we can produce the antinomy of the Liar in a very neat

The Paradox of the Liar

27

manner. We need only substitute in (2) for the first expression within quotes the expression which, according to (3), has the same reference. We get: (4) This sentence is true if, and only if, this sentence is not true. This suffices. All further developments are superfluous "decorations" only. The argument which has taken us from (1) to (4) I shall call the Liar. By calling the Liar an antinomy, we shall mean that the conclusion of the argument, or (4), is a contradiction. III. Criticism of the Use of "Sentence" To this presentation may be objected that, although it is sophisticated in some ways, it is very crude in others. What is a sentence? An answer could be that a sentence is a pattern of words (sentence-type), or a token of a pattern of words (sentence-token).2 And it may be thought that a sentence —type or token—is not the kind of thing which can be true or false. Should I not therefore rather than having written "This sentence is not true" have written "This statement is not true"? If somebody then had asked me: Which statement?, I could have answered: The statement which I made in writing down the words "This statement is not true". These words, which I wrote down, form a sentence. An alternative suggestion is that I ought to have written "This proposition is not true". If asked: Which proposition?, I could have answered: The proposition expressed by the sentence "This proposition is not true". A statement is made by somebody. A sentence does not make a statement by itself. But a sentence may be used—by somebody—to make a statement. Can a sentence express a proposition "by itself"? Certainly not "quite by itself". The sentence must be identified as belonging to a certain language. It must also be understood in a certain way. "Proletarians in all countries unite." Does this, in English, express a proposition? The answer depends upon whether we understand the mood of the sentence to be the imperative or the indicative mood. I think it must be admitted that in a primary sense truth-value is an attribute of statements and propositions. Only in a secondary sense may a sentence be said to be true or false. Thus, if we have doubts about the correctness of the sentence "This sentence is not true", we can replace it by some other sentence which may 2

This, of course, cannot be the whole answer. Not every pattern of words is a sentence. The question which patterns shall count as sentences will not be discussed here. I shall assume, throughout the discussion, that the pattern of words "This sentence is not true" and the various alternatives which may take its place are sentences.

28

The Paradox of the Liar

appear less objectionable such as "This statement is not true" or "This proposition is not true" or "The proposition expressed by this sentence is not true" or—which would spare us the choice between "sentence", "statement", and "proposition"—simply "This is not true". All these replacements, it should be observed, are replacements of one sentence by another sentence. They are not replacements of a sentence by a statement or a proposition. The doubts about the original formulation of our top sentence concerned the appropriateness of the word "sentence" as a generic name of the individual entity or object, to which the demonstrative pronoun "this" refers. That the top sentence is a sentence we need not doubt—but whether the "this" in it refers to this sentence or to something else is not so clear. The objection we have just been discussing was that if the demonstrative pronoun refers to the statement made by the person who wrote this sentence, or to the proposition expressed by this sentence, then it is incorrect to append the generic name "sentence" to the "this" and say "this sentence". We should say instead "this statement" or "this proposition" or "the proposition expressed by this sentence". Or we should not append any generic name at all to the demonstrative, but simply say "This is not true". But irrespective of which formulation we choose, it is appropriate to ask: What is not true? and then we shall have to identify the statement made or the proposition expressed or the thing said in those words before we attempt to answer the question whether that, which was said, is true or not. I shall replace the original formulation of the top sentence by This is not true. And by the phrase "the top sentence" I shall henceforth understand this new formulation. Criticism similar to that which we have directed against the original formulation of the top sentence can also be made against the general condition of truth (1) which was applied to this sentence in the production of the antinomy. This condition does not mention sentences. But it is essential to the application of the condition that the schematic letter or variable "/?", which occurs in it, should be replaced by a sentence. This need be no cause of worry. What may give raise to doubts, however, is the fact that the condition uses the phrase "is true" as an attribute of sentences. If we think that sentences are not true or false, we must modify the condition. This can be done in several ways. One may suggest the following alternatives to the original formulation: (a) The proposition expressed by the sentence "/?" is true if, and only if, p. (b) The proposition that p is true if, and only if, p. (c) That p is true if, and only if, p. The question may be raised, whether these alternative formulations

The Paradox of the Liar

29

should be regarded as identical in meaning. They all end with the phrase "is true if, and only if, p". The question therefore is, whether the phrases "The proposition expressed by the sentence '/?'" and "The proposition that/?" and "That/?" mean the same. The sentence "p" may express any proposition, or no proposition at all. In a code language, the sentence "it is raining" could be used to express the proposition that the sun is shining. The answer to the question: To which proposition does the phrase "The proposition expressed by the sentence '/?'" refer?, need not be: To the proposition that p. The phrase "The proposition expressed by the sentence '/?'" is not a uniquely descriptive phrase (a definite description). The phrase "The proposition that/?" is uniquely descriptive. Therefore the two phrases are not identical in meaning. As far as I can see, it is essential to the truth-condition which we are discussing that the question: Which proposition does the sentence "/?" express? can be answered: The proposition that p. For this reason it may be thought that a must be rejected as an attempted way of reformulating the condition of truth, or replaced by the following amplification of itself: (a') If the sentence "/?" expresses the proposition that p9 then the proposition expressed by the sentence "/?" is true if, and only if, p. Is the definite description "the proposition that p " identical in meaning with the phrase "that/?"? I believe that it is. I shall therefore regard the formulations b and c and the amended formulation a' as being identical in meaning. And since c is the briefest and handiest I shall henceforth use it as a formulation of the general condition of truth. The formulation thatp is true if, and only if, p seems to me to express a somewhat truistic but indubitable truth—it being understood that "/?" stands for an arbitrary sentence and "that/?" for the proposition expressed by this sentence. How this formulation is related to Tarski's definition of truth, I shall not discuss. Nor shall I discuss whether accepting this formulation and rejecting the formulation "p" is true if, and only if, p as being incorrect affects the so-called semantic theory of truth. IV. A Correct Presentation of the Antinomy We shall now produce the antinomy from the improved version of our top sentence and the improved version of the general condition of truth. By calling this presentation "correct" I mean that I can see no objection to the framing of the sentence or to the truth of the truth-condition. We start from (1)

That p is true if, and only if, /?.

Substituting for the variable "/?" the sentence "This is not true" we get

30

The Paradox of the Liar (2) That this is not true is true if, and only if, this is not true.

(No quotation-marks are needed, but if we wanted to make the thing clearer we could put brackets round the five first words in (2) or replace the whole phrase before the first "if" by the phrase It is true that this is not true. These alterations are purely notational, "stylistic".) What does the demonstrative pronoun in the top sentence refer to? We have rejected the possibility that it refers to the sentence itself. And we have accepted that it refers to the statement or proposition expressed by this sentence. Which is this proposition? Obviously the proposition that this is not true. Therefore "this" in the top sentence and "that this is not true" in (2) refer to the same thing. We thus have the identity: (3)

That this is not true = This.

By substitution in (2) according to (3) we get: (4)

This is true if, and only if, this is not true.

The result seems to me to be a very beautiful specimen of a self-contradiction. V. The Liar as a Demonstration We can generalize the general condition of truth and write it in the form t is true if, and only if, p. This is, of course, unintelligible unless some explanation is provided. The explanation we supply as follows: 'V", we say, shall refer to the proposition that p. The explanation can also be worked into the statement of the condition: (1) If"t" refers to the proposition that p, then t is true if, and only if, P. Substituting in (1) "'this'" for "'*'" and "this is not true" for V, we get: (2) If "this" refers to the proposition that this is not true, then this is true if, and only if, this is not true. If we may add

(3)

"This" refers to the proposition that this is not true, we can, by modus ponens, detach the consequent in (2) and obtain in conclusion:

(4) This is true if, and only if, this is not true. This, final, presentation of the antinomy has two advantages over the presentation given in the last section. It is more explicit. And it is independent of the, maybe questionable, principle about the substitutability of identities which we used in previous presentations.

The Paradox of the Liar

31

Since the step modoponente from (2) and (3) to (4) takes us to a contradiction, we can replace it by a step modo tollente from (2) and the negation of (4) to the negation of (3) which takes us to a necessary truth. Instead of (3) we now have (3') // is not the case that this is true if, and only if, this is not true. The conclusion to be detached is now: (4') It is not the case that "this " refers to the proposition that this is not true. If we take the view that the antinomy is a demonstration or proof and have to answer the question, what it proves, then we should, I think, answer that (4') is what it proves. Under no circumstances must we say that the Liar proves (4). For (4) is a contradiction. To say that one has proved a contradiction would be to affirm another contradiction and thus to speak the false. A contradiction is the very thing which cannot be proved. But a contradiction may occasionally become derived by logical argument from premisses. Such derivations are, in fact, an important proof-procedure in logic and mathematics, viz. for showing that something is wrong with the premisses. VI. Examination of the Thing Demonstrated Now let us examine the conclusion, which we reach, when the Liar is being viewed as a demonstration. Why is it that the word "this" does not refer to the proposition that this is not true? One would wish to frame the following objection against the truth of the conclusion: What expressions in a language mean is conventional. Any word or phrase can, by arbitrary convention, be made to refer to any thing. No "demonstration" can overrule this possibility and prevent the word "this" from referring to the proposition that this is not true. Therefore we cannot, after all, reject (3) as false and accept its contradictory (4') as true. It should be noted, in passing, that (4') does not say that the word "this" cannot refer to the proposition that this is not true. It says that the word in question does not (here) refer to the proposition in question. Here it is good to remember that the word "this" does not "by itself" refer to anything at all. But tokens of the type may be used for referring to various things. When we discuss, what the demonstrative "this" refers or does not refer to, it is important to be clear over which "this" we are speaking of. Therefore, let us ask: Which is the "this", of which (4') maintains that it does not refer to the proposition that this is not true? The answer is that it is the "this" which occurs in the top sentence. The top sentence expresses

32

The Paradox of the Liar

the proposition that this is not true. This proposition is the sense of the top sentence. What (4') says is thus that the reference of the word "this" in the top sentence is not the sense of that same sentence. It is then taken for granted that the sense of the top sentence is the proposition that this is not true. If we wish to build this presupposition into our formulation of the conclusion, we could re-state (4') as follows: //'the sense of the sentence "this is not true" is the proposition that this is not true, then the word "this" in that sentence does not refer to that proposition. By the idea of man's "semantic omnipotence" I here understand, approximately speaking, the idea that man has unlimited command over what expressions in a language shall mean, i.e. that he can, by arbitrary convention, make any word or phrase refer to just any thing and give to any expression just any sense. The conclusion (4') of the argument in the previous Section, be it observed, is not in conflict with our freedom, as such, to make any word refer to any thing—and thus in particular the demonstrative pronoun "this" to the proposition that this is not true. But our examination of the conclusion in this Section has shown that—speaking in general terms—if we associate a certain sense with a certain sentence, then we are not always free, after that, to make a certain word, which occurs in that sentence refer to just any thing. That man can thus come conditionally to restrict his "semantic omnipotence" is an observation of some interest. It is the lesson taught, for a particular case, by the antinomy of the Liar—and for other similar cases by other antinomies.3 We still have to answer one further objection to the conclusion which we reached in our presentation of the Liar as a demonstration. When we reject (4) and, modo tollente, conclude that (3) must be rejected too, we rely upon (a form of the) Law of Contradiction which says that no proposition is true if, and only if, not true. But could we not instead accept (4) and then, reflecting over the lesson taught by the antinomy, say something like this: Ordinarily, propositions are not true if, and only if, not true. But we have found an exception. This is the proposition that this is not true. For, if we make the word "this" in the sentence "this is not true" refer to 3 Mr P. Geach quotes an interesting example from Buridan (Sophismata, c. vi, sophisma v). The example, somewhat simplified, runs as follows: Let us stipulate (1) that in any true statement "A" shall stand for you, (2) that in any not-true statement "A11 shall stand for the donkey Brownie. Then either "A is a donkey" is true, or it is not true. In the first case, you are a donkey; in the second case, Brownie is not a donkey. But Brownie is a donkey. So the second alternative is excluded. It follows that you are a donkey. Buridan, when commenting on the sophisma, rightly concludes that such a stipulation as the above as to what "A " shall refer to is inadmissible. Each of (1) and (2) is a possible stipulation, but they are not combinable.

The Paradox of the Liar

33

the proposition that this is not true—and this we can do because of man's "semantic omnipotence"—then the proposition in question is true if, and only if, not true. We could say thus—and be right. But then (4), since we do not reject it as false but accept it as true, is no longer a contradiction. And if (4) is not a contradiction, the Liar is not an antinomy. Thus to accept (4') and with it the above conditional restriction on man's semantic omnipotence and to say that the Liar is an antinomy are correlatives. If we do the first, we must also do the second, and vice-versa. For to say that the Liar is an antinomy is to admit the self-refuting nature of the conclusion of a certain argument.

NOTE "A sentence cannot speak about itself." To suggest this could not possibly help us to solve the antinomy—and is besides false. For a sentence can perfectly well speak about itself. Sometimes it speaks the true, sometimes the false or not true about itself. For example: "This sentence contains five words", "This is a grammatically well-formed sentence", "This is a German sentence". In these examples the reference of the demonstrative pronoun "this" is a certain sentence, in which the demonstrative itself occurs. What is true or false is the proposition which the sentence in question is understood to express, when the demonstrative pronoun in it is understood to refer to the sentence itself. It is possible to understand the "this" in the sentence "This sentence is not true" or in "This is not true" as referring to the sentence—and not to the proposition expressed by the sentence. When the "this" is thus understood, we could say that the proposition which the sentence expresses is true. For, if we accept the view that sentences are neither true nor false, because they are not the kind of entity to which truth-value can be attributed, then it is true to say that the sentence "this sentence is not true"— or any other sentence for that matter—is not true. To say this is, of course, not to suggest that the sentence in question is false.4 Similarly, if "this" in the sentence "this is true" refers to the sentence itself, then the proposition that this is true is not true. Whether we should say that the proposition is, not only not true, but moreover false, is a question which I shall not discuss here.

4

Cf. above, fn. 1.

The Paradoxes of Confirmation i We consider generalizations of the form "All A are B". An example could be "All ravens are black". We divide the things, of which A (e.g. ravenhood) and B (e.g. blackness) can be significantly (meaningfully) predicated into four mutually exclusive and jointly exhaustive classes. The first consists of all things which are A and B. The second consists of all things which are A but not B. The third consists of all things which are B but not A. The fourth, finally, consists of all things which are neither A nor B. Things of the second category or class, and such things only, afford disconfirming (falsifying) instances of the generalization that all A are B. Since things of the first and third and fourth category do not afford disconfirming instances one may, on that ground alone, say that they afford confirming instances of the generalization. If we accept this definition of the notion of a confirming instance, it follows that any thing which is not A ipso facto affords a confirming instance of the generalization that all A are B. This would entail, for example, that a table, since notoriously it is not a raven, affords a confirmation of the generalization that all ravens are black. A consequence like this may strike one as highly "paradoxical". It may now be thought that a way of avoiding the paradox would be to give to the notion of a confirming instance a more restricted definition. One suggestion would be that only things of the first of the four categories, i.e. only things which are both A and B, afford confirmations of the generalization that all A are B. This definition of the notion of a confirming instance is sometimes referred to under the name "Nicod's Criterion". According to this criterion, only propositions to the effect that a certain thing is a raven and is black can rightly be said to confirm the generalization that all ravens are black. But if we adopt Nicod's Criterion as our definition of the notion of a confirming instance we at once run into a new difficulty. Consider the generalization that all not-B are not-A According to the proposed criterion we should have to say that only things which are not-B and not-^4 afford confirmations of this generalization. The things which are not-B and not-^4 are the things of the fourth of the four categories which we distinguished above. But, it is argued, the generalization that all A are B is the same as the generalization that all not-B are not-A. To say "all A are #" and to say "all not-B are not-^4 " appear to be but two ways of saying the same thing. It is highly reasonable, not to say absolutely necessary, to think that what constitutes a confirming or disconfirming instance of a generalization

The Paradoxes of Confirmation

35

should be independent of the way the generalization is formulated, expressed in words. Thus any thing which affords a confirmation and disconfirmation of the generalization g must also afford a confirmation and disconfirmation respectively of the generalization h, if "g" and "/z" are logically equivalent expressions. This requirement on the notion of a confirming instance is usually called "The Equivalence Condition". To accept Nicod's Criterion thus seems to lead to conflict with the Equivalence Condition. This conflict constitutes another Paradox of Confirmation. II

Before we proceed to a "treatment" of the paradoxes which we have mentioned, the following question must be asked and answered: Are confirmations of the generalization that all A are B through things which are not-A always and necessarily to be labelled "paradoxical", and never "genuine"? Simple considerations will show, I think, that the answer is negative. Let us imagine a box or urn which contains a huge number of balls (spheres) and of cubes, but no other objects. Let us further think that every object in the urn is either black or white (all over). We put our hand in the urn and draw an object "at random". We note whether the drawn object is a ball or a cube and whether it is black or white. We repeat this procedure—without replacing the drawn objects—a number of times. We find that some of the cubes which we have drawn are black and some white. But all the balls which we have drawn are, let us assume, black. We now frame the generalization or hypothesis that all spherical objects in the box are black. In order to confirm or refute it we continue our drawings. The drawn object would disconfirm (refute) the generalization if it turned out to be a white ball. If it is a black ball or a white cube or a black cube, it confirms the generalization. Is any of these types of confirming instance to be pronounced worthless? It seems to me "intuitively" clear that all the three types of confirming instance are of value here and that no type of confirmation is not a "genuine" but only a "paradoxical" confirmation. (Whether confirmations of all three types are of equal value for the purpose of confirming the generalization may, however, be debated.) I would support this opinion by the following ("primitive") argument: What we are anxious to establish in this case is that no object in the box is white and spherical. Not knowing whether there are or are not any white balls in the box, we run a risk each time when we draw an object from the box of drawing an object of the fatal sort, i.e. a white ball. Each time when the risk is successfully stood, we have been "lucky". We have been this, if the object which our hand happened to touch was a cube (and,

36

The Paradoxes of Confirmation

since we could feel it was a cube, need not be examined for colour at all); and we have been lucky, if the object was a ball which upon examination was found to be black. To touch a ball, one might say, is exciting, since our tension (fear of finding a white ball) is not removed until we have examined its colour. To touch a cube is not exciting at all, since it ipso facto removes the tension we might have felt. But to draw from the box is in any case exciting, since we do not know beforehand, whether we shall, to our relief, touch a cube, or touch a ball and, to our relief, find that it is black, or touch a ball and, to our disappointment, find that it is white. Let "S" be short for "spherical object in the box", "C" for "cubical object in the box", "£" for "black", and "W" for "white". All things in the box can be divided into the four mutually exclusive and jointly exhaustive categories of things which are S and B, S and W, C and B, and C and W. It is not connected with any air of paradoxality to regard things of all the four types as relevant (positively or negatively) to the generalization that all S are B. All things in the world can be divided into the four mutually exclusive and jointly exhaustive categories of things which are S and B, S but not B, Bbut not 5, and neither 5 nor B. Things of the first category obviously bear positively and things of the second category negatively on the generalization. But of the things of the third and fourth category some, we "intuitively" feel, do not bear at all on the generalization, have nothing to do with its content—and therefore "confirm" it only in a "paradoxical" sense. The categories of things C&B and S& W differ from the categories of things ~ S & B and - S & - B in this feature: All things of the first two categories are things in the box, but some things (in fact the overwhelming majority of things) of the last two categories are things outside the box. The things which we "intuitively" regard as affording "paradoxical" confirmations of the generalization that all 5 are B are those things of the 3rd and 4th category which are not things in the box. I shall here introduce the term range of relevance of a generalization. And I shall say that the range of relevance of our generalization above that all spherical things in the box are black is the class of things in the box. I now put forward the following thesis: All things in the range of relevance of a generalization may constitute genuine confirmations or disconfirmations of the generalization. The things outside the range are irrelevant to the generalization. They cannot confirm it genuinely. Since, however, they do not disconfirm it either, we may "by courtesy" say that they confirm it, though only "paradoxically". In order to vindicate my thesis I shall try to show, by means of a formal argument, that the irrelevance of the "paradoxical" confirmation consists in the fact that they are unable to affect the probability of the generalization. Showing this is one way, and a rather good one it seems to me, of dispelling the air of paradoxality attaching to these confirmations.

The Paradoxes of Confirmation

37

III

It is important to state explicitly the logico-mathematical frame of probability within which we are going to conduct our formal argument concerning the confirmation paradoxes. The probability concept of the confirmation theories of Carnap and Hintikka is a two-place functor which takes propositions (or, on an alternative conception, sentences) as its arguments. The probability concept used by us is a functor the arguments of which are characteristics (attributes, properties). Let "0" and *V" stand for arbitrary characteristics of the same logical type (order). The expression "P(0/«//)" may be read "the probability that a random individual is 0, given that it is y/". Instead of "is" we can also say "has the characteristic", and instead of "given" we can say "on the datum" or "relative to". We stipulate axiomatically that, for any pair of characteristics which are of the same logical type and such that the second member of the pair is not empty, the functor "P(/)n has a unique, non-negative numerical value. Furthermore, the functor obeys the following three axioms: Al. (Ex)y/x& (x)(y/x -> 0x) -> P(0/» = 1, A2. (Ex)vx -> P(0/» + P(~ 0/yO = 1, A3. (Ex)(xx & 0x) -+ P(/x) ' P(V/X & 0) = P(4> & V/X). It is a rule of inference of the calculus that logically equivalent (names of) characteristics are intersubstitutable in the functor "P(/)" ("Principle of Extensionality"). The application of probabilities, which are primarily associated with characteristics, to individuals is connected with notorious difficulties. The application is sometimes even said to be meaningless. This, however, is an unnecessarily restricted view of the matter. If x is an individual in the range of significance of 0 and i//9 and if it is true that P(0/y/) = p, then we may, in a secondary sense, say that, as a bearer of the characteristic y/, the individual x has a probability/? of being a bearer also of the characteristic 0. IV

If R is the range of relevance of the generalization that all A are B, and if this generalization holds true in that range, then it will also be true that (x)(Rx -» (Ax -> Bx)). This may be regarded as a "partial definition" of the notion of a range of relevance. For the sake of convenience, I shall introduce the abbreviation "Fx" for "Ax -> Bx". "F", we can also say, denotes the property which a thing has by virtue of the fact that it satisfies the prepositional function "Ax -» Bx". I define a second-order property *IIR by laying down the following truthcondition: The (first-order) property Xhas the (second-order) property

38

The Paradoxes of Confirmation

"IIR if, and only if, it is universally implied by the (first-order) property R. That Xis universally implied by R means that it is true that (x)(Rx -> Xx). The property 11 /?, in other words, is the property which a property has by virtue of belonging to all things in the range R. A property which belongs to all things in a range can also be said to be universal in that range. Assume we can order all things of which A, B and R can be significantly predicated into a sequence x\,X2, . . . , xn, . . . . Then we can define a sequence of second-order properties % i, % 2, . . . , 3 n, . . . as follows: The (first-order) property A" has the (second-order) property 9 „ if, and only if, it is true that Rxn -> Xxn. The property % „, in other words, is the property which a property has (solely) by virtue of belonging to a certain individual thing, //this thing is in the range R. ("//*" here means material implication.) For the sake of convenience, I introduce the abbreviation "#„" for the logical product of the first n properties in the sequence ^i, ^ 2 , . . . , ; ? * , . . . . "#„", we can also say, denotes the property which a property has by virtue of the fact that it is not missing from any of those of the first n things in the world which also are things in the range R. Finally, let "0" denote a tautological second-order property, i.e. a property which any first order property tautologically possesses—for example, the property of either having or not having the second-order property eUR (or 0 -> (P(VR/&n+l) > P(VR/&n) ~ P( 9n+l/#n) < !)• The first-order property R trivially has the second-order property 6&