Algebraizable Weak Logics

Abstract

We extend the standard framework of abstract algebraic logic to the setting of logics which are not closed under uniform substitution. We introduce the notion of weak logics as consequence relations closed under limited forms of substitutions and we give a modified definition of algebraizability that preserves the uniqueness of the equivalent algebraic semantics of algebraizable logics. We provide several results for this novel framework, in particular a connection between the algebraizability of a weak logic and the standard algebraizability of its schematic fragment. We apply this framework to the context of logics defined over team semantics and we show that the classical version of inquisitive and dependence logic is algebraizable, while their intuitionistic versions are not. * The authors would like to thank Tommaso Moraschini for the very useful discussions and pointers to the literature, and Fan Yang and Fredrik Nordvall Forsberg for their valuable comments and remarks to an early draft of this article.