DUMMETT ON IMPREDICA TIVITY

Alan WEIR The Queen's University of Belfast.

[T]he vicious circle principle in its first form applies only if the entities involved are con­ structed by ourselves. If, however, it is a question of objects that exist independently of our constructions, there is nothing in the least absurd in the existence oftotalities con­ taining members, which can be described ... only by reference to this totality. Kurt Godel 'Russell's Mathematical ' p. 136. 1

Godel' s view ofimpredicative specification and ofthe vicious circle principle which was wielded against it by Russell and Poincare is a very widespread one, perhaps widespread enough to count as ortho­ doxy. But it has been forcefully challenged by Michael Dummett who has argued that impredicative characterisations, though not to be rejected out of hand are, roughly speaking, guilty until proved in­ nocent. Moreover, according to Dummett, the question of the legiti­ macy of impredicative or characterisation arises whether or not the domain in question is conceived of as wholly independent of human thought and activity. In this paper I attempt to defend the Godelian position and rebut Dummett's criticisms: impredicative definition, I will argue, is un­ objectionable in mind-independent domains (whatever exactly they are); it is, moreover, innocent until proved guilty where applied to entities thought of as in some way the product of human activity so

l. In P. Schilpp (ed.): The Philosophy ofBertrand Russell (New York: Tudor, 1944), pp. 123-152. See also F.P. Ramsey: 'The Foundations of ', p. 41 in The Foundations ofMathematics (London, 193 l ); also p. 192 of F ounda­ tions (London: 1978). 66 long as they are not intentional constructs but rather, in Adam Fergu­ son's words, are 'the result ofhuman action but not ofany human de­ sign'. 2 In the first section I address the question of what exactly im­ predicativity is; in the second I look at Dummett's objections. The third section criticises in tum Dummett's objections whilst the fourth concluding section summarises the argument against Dummett.

I: What is lmpredicativity?

The notion of impredicativity is often introduced as applicable to mathematical objects, - numbers, sets, and so forth - and linked to some sort ofreflexivity which is viewed as viciously circular. Thus among Russell's formulations one finds: No totality can contain members defined in terms of itself. ' as Based on the Theory of Types', p. 753 Whatever involves all ofa collection must not be one ofthe collection. ibid., p. 63 Given any of objects such that, if we suppose the set to have a total, it will contain members which presuppose this total, then such a set cannot have a total. Principia Mathematica, p. 37 These characterisations have justly drawn criticism on account of the obscurity ofthe notions of'definability', 'involving' and 'presu­ pposing', cf. Godel op. cit. p. 135f. One way round this is to apply the concept of impredicativity, at least in the first instance, to expressions and formulae in interpreted languages. Thus definite and indefinite descriptions, in one com­ mon usage, are classic cases of impredicative expressions. If one takes a standard formal regimentation of a definite description such as, to adapt Ramsey's example, 'the tallest person in the room' ( con­ strued, as against Russell, as a genuine singular term), we would get

2. Adam Ferguson: An Essay on the History ofCivil Society (London: 1767), p. 187. 3. Reprinted in Logic and Knowledge, ed. R.C. Marsh (London: George Allen and Unwin, 1956.