Abstract In this paper, the notion of structural completeness is explored in the context of a generalized class of superintuitionistic logics that also involve systems that are not closed under uniform substitution. We just require that each logic must be closed under $D$ -substitutions assigning to atomic formulas only $\vee$ -free formulas. For these systems, we introduce four different notions of structural completeness and study how they are related. We focus on superintuitionistic inquisitive logics that validate a schema called Split and have the disjunction property. In these logics, disjunction can be interpreted in the sense of inquisitive semantics as a question-forming operator. It is shown that a logic is structurally complete with respect to $D$ -substitutions if and only if it validates Split. Various consequences of this result are explored. For example, it is shown that every superintuitionistic inquisitive logic can be characterized by a Kripke model built from $D$ -substitutions. We also formulate an algebraic counterpart of this result that says that the Lindenbaum–Tarski algebra ${\mathscr{H}}$ of any inquisitive logic can be embedded into the Heyting algebra formed from left ideals of endomorphisms on ${\mathscr{H}}$ . Additionally, we resolve a conjecture concerning superintuitionistic inquisitive logics due to Miglioli et al. and show that a false conjecture about superintuitionistic logics due to Minari and Wroński becomes true in the broader space of regular generalized superintuitionistic logics.

In logic, the meaning of a sentence is usually reduced to its truth conditions. However, this makes sense only for declarative sentences. In order to model also the meaning of questions, inquisitive semantics replaces the truth-conditional approach relating sentences to possible worlds with a support-conditional approach that relates sentences to information states. The standard framework of inquisitive semantics is based on a crisp notion of an information state, defined as a set of possible worlds, and a crisp relation of informational support. This paper introduces and studies two refinements of the standard framework. The first refinement takes into account fuzzy information states (defined as fuzzy sets of possible worlds) and the second one introduces a notion of fuzzy informational support. The main result of this paper shows that in the resulting framework that fuzzifies inquisitive semantics in two different directions we obtain an abstract and very general version of a principle known from the basic inquisitive semantics as Truth-Support Bridge.

Abstract We study a version of public announcement logic with common knowledge based on the relevant logic $\textsf {R}$. Public announcements, as represented in our framework, are not necessarily truthful and accepted by all agents, nor is it assumed that beliefs are preserved under announcements. We establish a completeness result with respect to a relational semantics, and we show that an alternative semantics based on information states is dual to the relational one. We add a question-forming inquisitive disjunction operator to the language and prove a completeness result with respect to the information semantics.

Abstract In this paper, we study the epistemic extensions of distributive substructural inquisitive logics. Substructural inquisitive logics are logics of questions based on substructural logics of declarative sentences. They generalize basic inquisitive logic which is based on the classical logic of declaratives. We show that if the underlying substructural logic is distributive, the generalization can be extended to embrace also the epistemic modalities ‘knowing whether’ and ‘wondering whether’ that are applicable to questions. We construct a semantic framework for a language of propositional substructural logics enriched with a question-forming operator (inquisitive disjunction) and epistemic modalities. We show that within this framework, one can define a canonical model with suitable properties for any (syntactically defined) epistemic inquisitive logic. This leads to a general approach to completeness proofs for such logics. A deductive system for the weakest epistemic inquisitive logic is described and completeness proved for this special case using the general method.

In this article, a non-standard informational semantics for superintuitionistic modal logics is introduced and studied. It is based on algebraic structures that are interpreted as algebras of information states. The proposed semantics combines into one framework various features of standard relational and algebraic semantics. Especially, the connection to the algebraic semantics is explored in detail. The framework can be viewed as a generalization of inquisitive semantics that enables us to add inquisitive disjunction to any superintuitionistic modal propositional logic.