What Negation is not: Intuitionism and '0=1'

Abstract

It is evident that the differences between intuitionistic logic (and mathematics) and its classical counterpart(s) can be traced back at least in part to their differing treatments of negation. The intuitionistic failures of double negation elimination and classical reductio ad absurdum are the most obvious facts supporting this. As a result, the definition and/or rules formulated for negation are crucial in comparing these competing logics. One quite common method of handling intuitionistic negation, however, turns out to be untenable.The method in question is the definition of intuitionistic negation in terms of '0 = 1' and the conditional. In Elements of Intuitionism Michael Dummett writes that 'surprisingly, negation is definable in intuitionistic arithmetic ' (1977: 35), giving essentially 1 the following definition:This definition of negation is common among intuitionists, and its origin can be traced to Heyting's first formulations (1930) of intuitionistic reasoning.Before giving our objections, it should be pointed out that others have argued against this same definition. For example, Neil Tennant objects to defining negation in terms of '0 = 1' because this definition makes negation dependent on arithmetic, raising the question of 'how one would justify inferential passage from patent absurdities in a non-arithmetical discourse to the supposedly primal absurdity 0 = 1' (1999) in accounting for the behaviour of negation. On Tennant's account, if P and Q are two contrary sentences of some non-arithmetic discourse, say 'The ball is red' and 'The ball is green', then we would need to be able to justify (possibly non-logical) inferences from P and Q to '0 = 1', otherwise we cannot conclude ¬Q (i.e. Q → 0 = 1) from P. It is far from clear how the justification of this inference would proceed, and Tennant observes that for a relevance logician (such as himself) no justification seems possible. This is a subtle philosophical argument 2 for rejecting this manner of 1 Dummett actually writes this as a biconditional instead of using = df .2 A related philosophical criticism has to do with the use of '0 = 1' in the definition. One wonders why we should not just define ¬A as (A → 0 = 3) or (A → 0 = 247) or (A → 47 = 396). None of these seem any less arbitrary than using '0 = 1', yet one may worry that they impose different meanings on the negation operator. The objections raised in this paper, however, demonstrate that all definitions of this sort are inadequate, and thus that these quibbles are irrelevant.

Co-constructive Logics for Proofs and Refutations

“…In fact, this is clearer in Kolmogorov's [14] interpretation than even Brouwer's, which argues that constructive logic has to do "not with theoretical propositions but, on the contrary, problems". 3 In this setting, it is simple to see that what is central to constructivity is that it allows us to deal with undetermined formulas (or problems):…”

Tonking a theory of content: an inferentialist rejoinder

“…( [13], pp. [3][4] considered commitments because they are what one is committed to when one accepts as true a sentence with the dominant logical operator in question. Peacocke's insight is correct here, as the falsity of either Γ or of Φ would undermine the truth of (Γ ∧ Φ).…”

“…In linguistics proper, such theories are universally recognized as having the best syntax-semantics interface, and being the most rigorously testable, precisely because they recursively correlate natural language sentences with conditions under which those sentences are true in a model. Only the most near-sighted partisan would deny that the success of post-Montague semantics in empirical linguistics and computer science is a stunning confirmation of Davidson's order of explanation, 4 where one first recursively correlates sentences with truth conditions, and then tests these correlations via an account of logical consequence defined in terms of truth conditions.…”

“…Peacocke tries to craft a realistic theory of content that accommodates Michael Dummett's (late) Wittgensteinian belief 3 See, for example, Dummett [10] and [11], Tennant [29], and Wright [35]. 4 Given Montague's own uses of higher order logic, a categorial syntax framework devoid of transformations, and reliance upon truth-in-a-model as his basic notion, some Davidsonians might balk at my assertion that post-Montague grammar is the fulfillment of a Davidsonian style meaning theory. This would be wrong for three reasons: (1) From a linguistics perspective, Davidson's own remarks about the actual architecture of a theory of meaning are both sketchy and not consistent from article to article, (2) Montague is the Isaac Newton of empirical semantics and machine language translation precisely because he first showed how to recursively connect a robust fragment of natural language with an interpreted formal language, (something Davidson, participants in philosophical "theory of meaning" debates from the 60's until the present, as well as 1970's generative semanticists allied with Davidsonians, never accomplished), and (3) Davidson's own substantive philosophical views in no way rely upon anti-Montagovian positions.…”

“…For interesting details about this and other aspects of mod-one arithmetic, see Cook and Cogburn[4].…”