SUBSTRUCTURAL INQUISITIVE LOGICS

Abstract

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.