Negation, Relevance and the Theory of Meaning

The paper assumes a largely Dummettian framework for the justification of logical laws and a certain priority of proof over truth. The paper dissolves certain arguments by Dummett against classical logic, on the basis that negation is not amenable to proof-theoretic justification. However, classical logic cannot, given Dummett’s principles for self-justifying logical laws, be the whole of logic; in particular its implication connective is problematic. I argue that from a Dummettian position, somewhat modified in the light of the forgoing considerations, relevance logic can be viewed as employing only connectives governed by self-justifying rules of inference. But arguably, the same is true for intuitionistic logic. So surprisingly, the upshot is a certain pluralism, contrary to what one might expect from Dummett’s writings. Whether the laws governing negation are (quasi-)classical or (quasi)intuitionistic cannot be decided on the basis of a proof-theoretic justification of logical laws. Other meaning-theoretical resources need to be appealed to in order to settle this question. The paper concludes with some reflections on whether *ex contradictione quodlibet*, common to both, intuitionistic and classical logic, can be quite in accordance with Dummett’s arguments against holism.

Dummett’s proof-theoretic argument against classical logic hinges on the fact that we cannot formulate harmonious introduction and elimination rules for classical negation that satisfy Dummett’s constraints on self-justifying rules of inference. Adding classical negation to the positive fragment of a natural deduction formulation of logic does not result in a conservative extension thereof. Classical negation violates the principle of compositionality, i.e. that to understand a sentence I only need to understand its constituents, and the complexity condition, i.e. that the verification of a sentence proceeds only via less complex sentences: there are sentences A such that A can only be proven by showing that ~A leads to absurdity. Intuitionistic negation, on the contrary, is purportedly shown by Dummett to be capable of a proof theoretic justification. However, I shall give an that also intuitionistic negation cannot be so justified, as in order to justify a connective as a negation, in the introduction and elimination rules there already is implicitly assumed a notion of negation. The meaning of negation hence cannot be given by introduction and elimination rules for the symbol ~. The argument naturally leads to an adoption of a point by Peter Geach, that Fa and ~Fa are of the same complexity and can only be understood together. The complexity condition can then of course be trivially met. But meaning is compositional: ~Fa is composed out of Fa and ~, as negation makes a systematic contribution to the meanings of sentences in which it occurs. Understanding ~Fa and Fa is one thing, rejecting compositionality quite another thing. I shall argue that there still is a problem for the classical logician, as he cannot meet the compositionality constraint. This is due to theorems like Av(A>B), ((A>B)>A)>A and (A>B)v(B>A), which cannot be proved, in a system of natural deduction, without appeal to negation rules. If a logical law is true only in virtue of the logical constants occurring in it, then the conclusion must be that either v and > are not genuine logical constants or that we cannot construe the meanings of those theorems as composed out of the meanings of v and >. Now Geach’s point is one about predicates and the classical logician may certainly extend it to all kinds of sentences and may claim that the meanings of > and v also cannot be given by merely appealing to introduction and elimination rules for the constants, but in addition rules for ~(AvB) and ~(A>B) are needed. Once more, I shall argue that this is just to reject compositionality. The upshot of the discussion of classical logic is that at least its > connective is not an independent operation on sentences, contrary to what is suggested by rules governing it in the usual natural deduction systems, and that classical logic is best understood as talking only about ~ and &. If implication tells us something that is independent of conjunction and negation, then > is not a proper implication connective.

This way relevance logic lies. Of course it is no surprise that my conclusion on classical logic follows from Dummett’s considerations. Classical logic, formulated using only ~ and &, is a subsystem of intuitionistic logic. The question then is rather: is classical logic all of logic? Accepting Dummett’s criteria, the answer must be: no. The rules governing the implication connective of a relevance logic like *R* can easily be seen to be justified by Dummettian criteria and hence determine the meaning of → entirely. The problem for classical > iis one concerning the non-conservativeness of classical negation. Concerning *R*, the complexity and compositionality conditions can be met, given the conservativeness of negation over the positive fragment of *R* and the Geachean view on negation. *R* is, from this perspective, probably the most natural relevance logic, as it uses no restrictions on reiteration. Geach’s point does not in itself establish (quasi-)classical negation rules. I argue that both, the intuitionistic negation rules and the quasi-classical negation rules of *R* are reasonable from different perspectives. The result is that classical, intuitionist and relevant logic are all perfectly in order, from the proof-theoretic point of view, although reflecting on Dummett’s rejection of holism puts some doubt on *ex contradictione quodlibet* as a rule fundamentally connected to the meaning of negation. Whether there is a concept of negation which is primary and settles certain issues in the realism/antirealism debate cannot be established by appeal to proof-theoretical considerations only.