Connexive implications in Substructural Logics

TL;DR

This paper explores connexive implications within substructural logics and residuated lattices, identifying conditions for connexive principles and their relation to Boolean algebras, with philosophical implications for plausible inference.

Contribution

It provides a detailed analysis of connexive implications in substructural logics, establishing their equivalence to the Glivenko property and connecting them to philosophical logic.

Findings

Connexive principles often equivalent to the Glivenko property.

Characterization of weak and strong connexivity.

Discussion on relevance to Polya's logic of plausible inference.

Abstract

This paper is devoted to the investigation of term-definable connexive implications in substructural logics with exchange and, on the semantical perspective, in sub-varieties of commutative residuated lattices (FLe-algebras). In particular, we inquire into sufficient and necessary conditions under which generalizations of the connexive implication-like operation defined in [6] for Heyting algebras still satisfy connexive theses. It will turn out that, in most cases, connexive principles are equivalent to the equational Glivenko property with respect to Boolean algebras. Furthermore, we provide some philosophical upshots like e.g., a discussion on the relevance of the above operation in relationship with G. Polya's logic of plausible inference, and some characterization results on weak and strong connexivity.