Algebraizable Weak Logics

TL;DR

This paper extends algebraic logic to weak logics not closed under uniform substitution, introducing algebraizability concepts and analyzing their properties, with applications to team semantics and dependence logic.

Contribution

It develops a new framework for algebraizability of weak logics, including loose and strict versions, and proves a version of the Isomorphism Theorem within this setting.

Findings

Classical inquisitive and dependence logics are strictly algebraizable.

Intuitionistic versions of these logics are only loosely algebraizable.

The framework links algebraizability of weak logics to their schematic fragments.

Abstract

We extend the framework of abstract algebraic logic to weak logics, namely logical systems which are not necessarily closed under uniform substitution. We interpret weak logics by algebras expanded with an additional predicate and we introduce a loose and strict version of algebraizability for weak logics. We study this framework by investigating the connection between the algebraizability of a weak logic and the algebraizability of its schematic fragment, and we then prove a version of Blok and Pigozzi's Isomorphism Theorem in our setting. We apply this framework to logics in team semantics and show that the classical versions of inquisitive and dependence logic are strictly algebraizable, while their intuitionistic versions are only loosely so.