Algebraizable Weak Logics

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.