Connexive Modal Logic

Connexive logic is a neglected direction in non-classical logic. In the present paper, first an axiomatic system of connexive propositional logic is presented. This logic, C, is shown to be sound and complete with respect to a class of relational models. It seems that this semantics is, in fact, the first known intuitively plausible interpretation of a system of connexive logic. The presentation of C suggests that connexive logic is constructive. It is a variant of David Nelson’s constructive logics with strong negation. In Nelson’s logics the verification conditions of implications are dynamic, whereas all falsification conditions are static conditions of falsification on the spot. In C, both the verification and the falsification conditions of implications are dynamic. This is enough to ensure that C is connexive and can be given a comprehensible and clear interpretation in terms of information states. In a second step, the language of the system C is extended by the modal operators and ♦ to obtain a connexive analogue of the smallest normal modal propositional logic K. Aiming at a connexive analogue of K that can be faithfully embedded by a modal translation into QC, quantified C, we arrive at a system that will be called CK, connexive K. The system CK is a connexive version of the constructive modal logic FSK characterized in [12]. CK is shown to be sound and complete with respect to relational models and to be decidable. We shall also critically discuss the evaluation clauses for the modal operators that are induced by the standard translation from modal propositional logic into first-order logic. In the context of the connexive base logic, the falsification clauses of formulas A induced by the standard translation appear to be intuitively implausible. In any case, both syntactic duality axioms ∼ A ↔ ♦ ∼ A and ∼ ♦ ↔ ∼ A fail to hold. It seems that CK is the first system of connexive modal logic considered in the modal logic literature. This paper may therefore be seen as a contribution to establishing connexive modal logic as a respectable branch of modal logic. Advances in Modal Logic, Volume 5. c © 2005, Heinrich Wansing. 368 Heinrich Wansing 1 Aristotle’s Theses and Boethius’ Theses The following principle is well-known as “Aristotle’s Thesis”: