Towards a uniform presentation of logical systems by indexed categories and adjoint situations

Logical Systems are paramount to almost every subject in computer science. This vast number of application areas had a deep influence on us and thus on how we perceive what a formal specification of a logical system should be. Lawvere s [29, 30] essential idea is that the fundamental relationship between syntax and semantics can be precisely formulated by adjoint functors. In this work, we show that Institutions from Goguen and Burstall [19] and Entailment Systems from Meseguer [36] are in its essence, a family of local adjoint situations between the syntactical and semantical aspects underlying these systems. These abstractions are named Indexed Frames. Also, from a categorical perspective, Tarski s consequence operator [40] can be formalized as Indexed Closure Operators, a construction that maps each language to the corresponding co-monad. Finally, in the framework of preorder categories, both concepts, Indexed Frames and Indexed Closure Operators are equivalent.