Fredrik Dahlqvist

Completeness-via-canonicity for generalized positive modal logics

We show strong completeness with respect to a relational semantics of all positive modal logics with n-ary modalities which either (1) preserve joins or turn meets into joins in each of their arguments, or (2) preserve meets or turn joins into meets in each of their argument. This result lays the foundation of completeness-via-canonicity results for a large class of logics which includes standard positive modal logic, intuitionistic logic, the full distributive Lambek calculus and variations thereof.

The syntax and the semantics of these logics will be presented in the formalism of coalgebraic logics which will be introduced in a short tutorial assuming no prior knowledge on the topic. The coalgebraic framework greatly clarifies the connection between syntax and semantics, and in particular between canonical extensions (on the syntax side) and canonical models (on the semantics side). We will also show how it is modular and lends itself to further generalisations.