Substructural Inquisitive Logics

Abstract

This paper shows that any propositional logic that extends a basic substructural logic BSL (a weak, nondistributive, nonassociative, and noncommutative version of Full Lambek logic with a paraconsistent negation) can be enriched with questions in the style of inquisitive semantics and logic. We introduce a relational semantic framework for substructural logics that enables us to define the notion of an inquisitive extension of λ, denoted as ${\lambda ^?}$, for any logic λ that is at least as strong as BSL. A general theory of these “inquisitive extensions” is worked out. In particular, it is shown how to axiomatize ${\lambda ^?}$, given the axiomatization of λ. Furthermore, the general theory is applied to some prominent logical systems in the class: classical logic Cl, intuitionistic logic Int, and t-norm based fuzzy logics, including for example Łukasiewicz fuzzy logic Ł. For the inquisitive extensions of these logics, axiomatization is provided and a suitable semantics found.

Questions and Dependency in Intuitionistic Logic

“…Broadly speaking, Punčochář's aim is similar to the one we pursued, namely, to investigate how questions may be added, in the style of inquisitive semantics, to a propositional logic which is weaker than classical logic. In particular, in the two papers cited above Punčochář proposes two ways of generalizing inquisitive logic to a setting in which the underlying logic of statements is an intermediate logic, including intuitionistic logic (a further generalization to sub-structural logics is pursued in [21]).…”

Routley Star in Information-Based Semantics

“…We call our semantic framework information-based sematnics. It has been used in the context of inquisitive semantics (see [1,7,8]). Similar semantic frameworks were employed for example in [2,5,11].…”