On Intermediate Inquisitive and Dependence Logics: An Algebraic Study

TL;DR

This paper develops an algebraic semantics for intermediate inquisitive and dependence logics, proving completeness and establishing representation theorems, offering an alternative to team semantics.

Contribution

It introduces inquisitive and dependence algebras, proving their completeness and establishing algebraic and model-theoretic properties for these logics.

Findings

Algebraic semantics are complete for all intermediate inquisitive and dependence logics.

Finite, core-generated, well-connected algebras witness validity of formulas.

Representation theorems connect team and algebraic semantics.

Abstract

This article provides an algebraic study of intermediate inquisitive and dependence logics. While these logics are usually investigated using team semantics, here we introduce an alternative algebraic semantics and we prove it is complete for all intermediate inquisitive and dependence logics. To this end, we define inquisitive and dependence algebras and we investigate their model-theoretic properties. We show they are elementary structures axiomatised by Horn sentences and we use our completeness result to state a version of algebraisability for inquisitive and dependence logics. We then focus on finite, core-generated, well-connected inquisitive and dependence algebras: we show they witness the validity of formulas true in inquisitive algebras, and of formulas true in well-connected dependence algebras. Finally, we obtain Representation Theorems for finite, core-generated, well-connected, inquisitive and dependence algebras and we prove some results connecting team and algebraic semantics.