Замечания о релевантной коннексивной логике без сокращения

Abstract

This paper presents a metatheoretical study of CRW, a consistent connexive relevant logic without the contraction axiom. In CRW, Aristotle's and Boethius' theses are provable, and the antecedents and consequents of all its implicative laws share at least one propositional variable. What distinguishes CRW from other connexive relevant logics with a similar class of laws is the preservation of consistency: there is no formula in CRW such that both it and its negation are theorems. Moreover, owing to the absence of contraction, the standard derivation of Curry's paradox is blocked in CRW. The semantics of CRW is based on the notion of a commutative monoid — an algebra with a commutative associative operation and a neutral element. These properties allow the elements of the algebra's domain to be substantively interpreted as finite collections of empirical dossiers — concrete observations, experiments, and measurements. The paper proves the soundness of the axiomatization of CRW and its semantic completeness with respect to the calculus. It is shown that CRW is a conservative extension of the implicative fragment of the relevant logic RW. The completeness proof employs the canonical model method, building on the technique of A. Urquhart as adapted by Y. Weiss. Together, these results establish the adequacy of the axiomatics and semantics, i.e., the coincidence of the classes of provable and valid formulas of CRW.