|Substructural logics vs classical logic
Department of Education
In this talk, we argue that the fine-grained analysis of propositional connexion which characterises substructural logics, as opposed to classical logic, is a reason for considering such logics as plausible candidates for the formal investigation of vast fragments of ordinary discourse, not just as smooth logics (Aberdein and Read, 200+) of merely technical interest.
1) Are substructural logics alternative to classical logic? A criterion of genuine rivalry between logics presented by means of sequent calculi - is suggested according to which Quine’s meaning variance attack can be defused at least as regards the substructural logic LL (roughly corresponding to linear logic without exponentials and without additive constants). In a nutshell, genuine rivalry between propositional logics having the same similarity type is possible whenever corresponding connectives share the same operational rules in the respective calculi, yet the presence of different structural rules yields different sets of provable sequents and thus permits a disagreement across logics (Paoli, 2003).
2) Are substructural logics philosophically useful? The distinction between lattice and group connectives, typical of substructural logics, suggests plausible solutions to well-known philosophical puzzles such as McGee’s paradox or the lottery paradox (Paoli, 2005).
3) Can we use substructural logics to model implication? An argument can be advanced to the effect that the substructural logic LL is the most promising logic of relevant implication. In particular, we contend that a plausible logic of relevant implication should lack both contraction and lattice distribution (Paoli, 200+).
4) Can we use substructural logics to model defeasible conditionals? We introduce and motivate a conditional logic based on the logic LL. Its hallmark is the presence of three logical levels (each one of which contains its own conditional connective), linked to one another by means of appropriate distribution principles. Such a theory affords a solution to a long-standing open problem in conditional logic: in fact, we retain suitable versions of both Substitution of Provable Equivalents and Simplification of Disjunctive Antecedents, while still keeping out such debatable principles as Transitivity, Monotonicity, and Contraposition.
Aberdein A., Read S. (200+), “The philosophy of alternative logics”, to appear in L. Haaparanta (Ed.), The Development of Modern Logic; Oxford University Press,
Paoli F. (2003), "Quine and Slater on paraconsistency and deviance", Journal of Philosophical Logic, 32, pp. 531-548.
Paoli F. (2005), “The ambiguity of quantifiers”, Philosophical Studies, 124, 3, pp. 313-330.
Paoli F. (200+), “Implicational paradoxes and the meaning of logical constants”, Australasian Journal of Philosophy, forthcoming.