Beyond the Limits of Thought

Beyond the Limits of Thought GRAHAM PRIEST

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 I RP 40 West 20th Street, New York, NY 1 00 1 1 -42 1 1 , USA 1 0 Stamford Road, Oakleigh, Melbourne 3 1 66, Australia © Cambridge University Press 1 995 First published 1 995 Printed in Great Britain at the University Press, Cambridge A catalogue record for this book is available from the British Library Library of Congress cataloguing in publication data

Priest, Graham. Beyond the limits of thought / Graham Priest. p. cm. Includes bibliographical references. ISBN 0 521 45420 4 (hardback) I. Limit (Logic) 2. Concepts. I. Title. BC 1 99 .L54P75 1 995 1 60 - dc20 94- 1 9 1 64 CIP ISBN 0 521 45420 4 hardback

Woodcut of a fantastic depiction

of the

solar system, German School, 17th century.

Photo: Bridgeman Art Library/Private collection

Contents

Preface

page xiv

Introduction

1

Beyond the limit

3

I The limits of thought 2 Dialetheism 3 The history of limits 4 The structure of the book

5 . . . and the role of logic in it

Part I The limits of tbougbt in pre-Kantian philosopby The limits of expression Introduction

I 2 3 4 5 6 7 8 9

The flux

Plato against Cratylus

The stability of meaning Aristotle on substance and change Prime matter The problem with prime matter Subject and form Cusanus on God Objects and categories Conclusion

2 The limits of iteration Introduction

I Generated infinities 2 . . . in Aristotle 3 Time 4 Motion

3 4 6 7 8

9 II 11 12 13 IS 17 18 20 21 23 24 2S

26 26 27 27 28

30

x

Contents 5 6 7 8 9

The continuum Infinite Qarts Aguinas' cosmological argument Leibniz' repair The QrinciQle of sufficient reason

CQP! g , and presumably it is not the case that g* f g* .

64

The limits of thought in pre-Kantian philosophy

thought. The argument shows that God is not in the set of things that can be conceived (Transcendence) . Yet Anselm obviously is conceiving God in putting forward the argument (Closure). Anselm is, in general, aware of the contradiction involved in saying anything about the inconceivable. In chapter 65 of the Monologion he tries to defuse the problem by insisting that our concepts do not apply to God literally, but only by analogy (per aliud ). This solution cannot be sustained. For in the claim that God is inconceivable 'inconceivable' must have its literal sense, or the whole force of the claim is lost. God must therefore be literally conceivable, if only as the inconceivable. In any case, the move will not remove this particular problem. As I have just observed, the logic of the argument itself requires that 'God is conceiv­ able' is true (in exactly the same sense that conceivability is used in the conclusion), so the argument itself locks Anselm into this claim.

4.3

The characterisation principle

As we have seen, Anselm ends in a contradiction at the limit of thought, but he does so by assuming the CP: that an object characterised as an, or the, object satisfying certain properties does indeed have those properties. This certainly looks like a logical truth; and, if it is, we can prove the existence of God very simply. The conditions taken to characterise God are normally called the perfections (omniscience, omnipotence, etc.). Let Px be a predicate expressing the conjunction of these. Let Ex be the predicate 'x exists', if it is not one of the conjuncts already. (It is usually taken to be so.) Now consider the condition 'Px A Ex' . By the CP, a (the) thing satisfying these properties satisfies these properties. Something, therefore, satisfies these properties. Hence there is some existing thing satisfying P, i.e., God exists. 6 The trouble with this argument is that, as far as its logic goes, the property P is absolutely arbitrary. We could therefore prove the exis­ tence of every thing, possible and impossible (after all, Px does not have to be consistent) - which is slightly too much. The problem is, actually, even worse than this. If the CP were correct, we could establish an arbitrary sentence, 'IjJ, by applying the CP to the condition x = x A 'IjJ. The CP, then, is not a logical truth. I think that it appears so plausible because the claim that a (the) thing that is P is P is easily confused with the claim that everything that is P is P, which is a logical truth. 6

This is essentially Descartes' version of the Ontological Argument in the 5th Meditation.

65

The limits of conception

It is usually reckoned that the Ontological Argument was destroyed by Kant (Critique of Pure Reason, A598 B626 ff.), who argued that exis­ tence is not a predicate. There is, as a matter of fact, no problem about a syntactic existence predicate. In orthodox logic the formula 3yy x is such a thing. What Kant actually says is that existence is not a determin­ ing predicate. On a reasonably generous reading of the Critique this can be interpreted as the view that existence is not a predicate that can be used in a valid instance of the CP, as versions of the Ontological Argument require. Leaving Kant aside, there certainly are instances of the CP that are true: the highest mountain in the world, for example, is, undoubtedly, the highest mountain in the world. However, the essential point remains: the CP cannot be assumed in general. And until the instances of it that are used in Anselm's arguments are legitimised (which no one has ever succeeded in doing), this is sufficient to sink the arguments of Proslogion 2 and 1 5. Anselm, then, did not succeed in establishing the contradiction at the limit of conception. The matter is different when we turn to Berkeley. =

=

4.4

Berkeley's master argument for idealism

The context of Berkeley's concerns is idealism. Idealisms come in many kinds. Berkeley'S idealism was of a very strong kind, to the effect that nothing which is not itself a mind exists, unless it is before a mind in some sense: perceived, conceived, let us just say thought. As he puts it in the Principles of Human Knowledge, section 6: all the choir of heaven and the furniture of the earth, i n a word, all those bodies which compose the mighty frame of the world, have not any substance without a mind . . . their being is to be perceived or known.

As is clear from the last sentence of this quotation, Berkeley held a much stronger view: that all things (except, possibly, minds themselves) are essentially thought. It is not just that there is nothing that is not thought; there could be nothing. Important as it is in Berkeley's philoso­ phy, this stronger view need not concern us here. Hence, slightly inaccu­ rately, I will call the claim that everything is thought of, 'Berkeley's Thesis' . Berkeley advanced a number of arguments for his Thesis. Most of these are direct arguments depending on various empiricist assump­ tions, and have been well hammered in the literature. 7 He has one 7

See, for example, Tipton ( 1 9 74), Pitcher ( 1 977), Dancy ( 1 987).

66

The limits of thought in pre-Kantian philosophy

argument of a different kind, however; one on which, he says, he is prepared to rest everything. This will be the only one that will concern us here. It is given about three quarters of the way through the first of the Three Dialogues Between Bylas and Philonous. 8 Philonous is Berkeley's mouthpiece, Hylas the unlucky stooge. I enumerate for future reference: P H I LO N OUS .

i.

11.

iii. iv. v. VI.

vii . viii. ix. x. xi. xii .

I am content to put the whole upon this issue . If you can conceive it possible for any mixture or combination of qualities , or any sensible object whatever to exist without the mind , then I will grant that it actually be so. HYLAS . If it come to that, the point will soon be decided. What more easy than to conceive of a tree or house existing by itself, independent of, and unperceived by any mind whatever . I do at this present time conceive them existing after this manner. P H I LO N OUS . How say you, Hylas, can you see a thing that is at the same time unseen? HYLAS . No , that were a contradiction. P H I LO N OUS . Is it not as great a contradiction to talk of conceiving a thing which is unconceived? HYLAS. It is. P H I LON 0 u s . The tree or house therefore which you think of, is conceived by you? HYLAS . How should it be otherwise? P H I LO N OUS . And what is conceived is surely in the mind? HYLAS . Without question, that which is conceived is in the mind. P H I LO N OUS . How then came you to say, you conceived a house or a tree existing independent and out of all minds whatever? HYLAS . That was I own an oversight .

The most difficult problem concerning this argument is to know what, exactly, it is. The fact that it is given informally, and in dialogue form at that, makes this a very sensitive issue. Because of this and the fact that several commentators have taken the argument to be little more than a sophism, I will formalise it. This will help to see what, if anything, is wrong with it; and just as importantly, what is not.

4.5

Analysis, stage I

There are a couple of preliminary issues that we can get sorted out straight away. Berkeley normally talks of conceiving, but sometimes talks of perceiving (ii). Although there is a world of difference between these two notions, Berkeley, because of his theory of perception, which is not relevant here, runs them together. As we shall see, nothing in the 8

And also in section 23 of the Principles of Human Knowledge.

The limits of conception

67

argument hangs on this and we shall do no injustice if we ignore this distinction. 9 More troublesome is the fact that Berkeley slides between a predicative use of 'conceives' - conceive y (v, ix, x), think of y (vii), y is in the mind (i, ix, x) - and a propositional use - conceive ofy as being F (ii, xi), conceive it possible for y to be F (i). Now, whatever connection there is between these two uses, we certainly cannot start by assuming one. I shall write the predicate as T and the propositional operator as T. These may be read canonically as 'is conceived' and 'it is conceived that', respectively. (I put both of these in the passive, since, although it is Hylas who is doing the conceiving, the particular agent in question is irrelevant to the argument.) In this notation, Berkeley's Thesis is the negation of: (0) Notice that in the propositional use Berkeley sometimes talks of conceiv­ ing x to be F (ii, xi), and sometimes of conceiving it possible for x to be F (i). But the 'possible' is really doing no real work here, as is witnessed by the fact that the modality occurs but once in the argument. Berkeley, like many people, thinks of 'conceive to be possible' as simply equivalent to 'conceive'. Clearly, conceiving a state of affairs to be possible entails conceiving that state of affairs. Berkeley thinks the converse also holds: note that Hylas tries to demonstrate that something can be conceived to be possible (i) by conceiving it (ii) . Philonous does not complain. Now, what is the argument supposed to prove? What is at issue is, as stated by Philonous (i), whether one can conceive that there is something that is not conceived. Hylas claims that he does conceive such a thing (ii): T3x-,Tx

(1)

showing that one can (

69

?

TC

TC 1\ -'TC 3X(TX 1\ -'TX)

Analysis, stage II

How a re the gaps in the argument to be filled in? ! ! The key is to go back and consider the object, c, which the reasoning is supposed to show to be inconsistent. As we noted, the exact nature of this is unimportant; all that is important is that it is some particular object which is not being con­ ceived. Now reasoning about an arbitrarily chosen object of a certain kind is very familiar to logicians. Its cleanest formalisation uses an indefinite description operator, E:. About how, exactly, this should behave we need not go into in detail. ! 2 All we need to assume is that if there are any objects satisfying , where 1 + (P)UnP) = D . (Note that I + (p)nnp) may be non-empty.) I will call the members of the pair the positive and negative extensions of P, respectively. For identity to have the right properties we also require that 1 + ( = ) = { (x,x) ; XED} . The language of dII is the language augmented by a set of individual constants, one for each member of D. For simplicity we take the set to be D itself, and specify that for all dED I(d) = d. Every formula in the language of dII , ep, is now assigned a truth value, v(ep) , in the set { { l } , {O} , { I ,O} } by the following recursive clauses. 1 E V(Pt l . . . tn) (I(t l ) . . . I(tn )) E I + (P)

° E V(Pt l . . . tn ) (I(t l ) . . . I(tn )) E I - (P)

1 E v(..., ep ) ° E (v) ° E v(""ep) 1 E v( ep) 1 E v(ep A 'IjJ) 1 E veep) and 1 E v('IjJ) ° E v('IjJ)

° E v(ep A 'IjJ) ° E veep) or

1 E v(Yxep) for all d E D I E v(ep(x/d))

° E v(Yxep) some d E D O E v(ep(x/d)) 2

3

This is more general than the Meyer collapse in one sense, since it applies to arbitrary models and equivalence relations. However, the Meyer collapse is stronger in another sense, since it handles not only extensional connectives, but also a non-extensional con­ ditional. The material in this chapter is part of Priest ( 1 992).

1 90

Limits and paradoxes of self-reference

Where