Conceptions of truth in intuitionism

Abstract

Intuitionism's disagreement with classical logic is standardly based on its specific understanding of truth. But different intuitionists have actually explicated the notion of truth in fundamentally different ways. These are considered systematically and separately, and evaluated critically. It is argued that each account faces difficult problems. They all either have implausible consequences or are viciously circular.

Notes

Often what the intuitionists attack under the name of LEM is not actually LEM but the principle of bivalence; strictly speaking, these two should be distinguished (see e.g. Dummett 1978 , p. xix). However, under certain default assumptions, accepted by intuitionists, these two are equivalent (see e.g. Pagin 1998 ). Consequently, in what follows I shall not be overly pedantic on this issue.

This is not the only possibility; some, most notably Heyting, rebut the whole notion of truth, and proceed directly to give a meaning-explanation of the logical constants, an explanation under which LEM etc. fail. But the basic question of this paper (see below), whether one should adopt the actualist or the possibilist approach, remains. I shall argue that Heyting is still deeply committed to the actualist picture, with all its problems; see below.

Also some recent constructivists such as Bishop and Bridges avoid talk of truth, but their explanation of constructivism closely leans on the possibilist picture (see below). Thus Bishop explains ‘the basic constructivist goal’ as being ‘that mathematics concern itself with the precise description of finitely performable abstract operations … Thus by “constructive” I shall mean a mathematics that describes or predicts the results of certain finitely performable, albeit hypothetical, computations within the set of integers’ (Bishop 1970, p. 53 my emphasis). Bridges, in turn, writes: ‘Constructive mathematics is distinguished from its traditional counterpart, classical mathematics, by the strict interpretation of the phrase “there exists” as “we can construct” ’ (Bridges 1997 , my emphasis).

Some (e.g. Heyting; see below) rather want to eliminate the concept of truth and replace it e.g. by the possession of proof or by the possibility of proof. For my purposes here, however, this makes little difference.

Possibly just because of the problem just mentioned, namely, that Brouwer realized that very few would accept his austere actualist notion of truth and falsity as being actually proved and being actually refuted, respectively (van Stigt 1990 , p. 248).

One might think that he was destined to fail, because it is arguable that the idea of an absolutely unsolvable problem makes no sense intuitionistically; see e.g. Heyting 1958b , Dummett ( 1977 , p. 17), Pagin 1998 ; see also below.

I have put this together from three different, very similar sources: Brouwer 1951 , 1951 –52 and 1955 .

I shall return to views like this (which I shall call ‘liberalized actualism’) later in the paper.

See van Stigt (1990, p. 212); cf. also the quote from Heyting 1959 above. Brouwer appeals especially often to the meaninglessness of classical mathematics in Brouwer 1912.

See, however, the comments in Heyting 1930 , quoted above.

Dummett 1963 (not to be confused with Dummett 1982 with the same title); in his 1978 introduction to the collection in which it was published Dummett regarded this paper ‘as very crude and as mistaken in many respects’ (Dummett 1978 ).

Very recently, Dummett has become less cautious; he now states that ‘[t]he canons of correct reasoning in intuitionistic mathematics debar us from restricting true statements to those we have actually proved’ (Dummett 2003 , p. 19).

See Prawitz ( 1987 , p. 137). Prawitz originally presented it in a somewhat different form in a different context; I have slightly modified it.

Dummett 1982 , 1994a .

In his early classic, 1959 paper ‘Truth’, Dummett had already recommended accepting the redundancy theory of truth.

See below. Note how similar Dummett's view here is to that of Heyting's.

Indeed, Dummett himself has often expressed the desirability of a non-temporal notion of truth.

To make the examples more concrete: take H to be e.g. the Paris-Harrington sentence (finitary Ramsey's theorem), and assume that the methods of proof available in Bob's time are limited to (a conservative extension of) Heyting Arithmetic HA, which does not prove H. It is, however, provable with the help of (intuitionistically acceptable) transfinite induction, which we may imagine to become accepted only much later.

The situation is even worse; the Gödelian argument goes through if one merely assumes that these methods are, even if not captured by a formalized system (i.e. not recursively enumerable, or ), at least somewhere in the arithmetical hierarchy (i.e. for some n); cf. Appendix.

Cf. note 5.

This is because ‘A is not provable’ is true (according to intuitionism) if and only if it is provable; a proof that A is not provable entails that A → B holds for any B, and thus in particular that A → ⊥, that is, that ¬A, cf. Dummett ( 1977 , p. 17).

Not as a reaction to Prawitz but already before their debate begun.

The set of natural numbers is effectively generated, and the standard model of arithmetic is recursive (it is the only recursive model of PA), whereas the totality of proofs (if the idea is assumed to make sense) cannot be even arithmetical; see Appendix. Hence the former notion is much more simple and accessible than the latter. Therefore, it would be quite preposterous to be realist only with respect to the latter but not with respect to the former.

Quoted in Brouwer 1912 !

Although, as I have argued, it is a widespread mistake to count Dummett as an adherent of this view.

While later surveying the literature, I have found out that the same question (‘What kind of possibility?’) has been raised, and sometimes a related argument hinted, by others: Parsons 1983 (passim), 1986 , 1997 , George 1993 , and Moore 1998 . For me, however, the argument occurred independently of these; it was rather inspired by Sellars' criticism of phenomenalism (see Sellars 1963 ). Interestingly, also Parsons ( 1986 ) notes the affinity between intuitionism and phenomenalism.

Cf. e.g. Brouwer 1933 , Heyting 1956b , Troelstra ( 1969 , pp. 3–4), Dummett ( 1977 , p. 19), Prawitz 1987 , 1994 , Martin-Löf 1995 .

Timothy Williamson once suggested in conversation that there is one more option, namely metaphysical possibility. I am personally inclined to agree, but the resulting view would certainly be quite alien for the anti-metaphysical spirit of intuitionism, and I doubt that no intuitionist is prepared to base intuitionism on this notion.

See e.g. Sundholm 1983 and the many references therein.