FIRST-ORDER MODAL LOGIC VIA LOGICAL CATEGORIES

Abstract We extend the logical categories framework to first-order modal logic. In our modal categories , modal operators are applied directly to subobjects and interact with the background factorization system. We prove a Joyal-style representation theorem into relational structures formalizing a ‘counterpart’ notion. We investigate saturation conditions related to definability questions and we enrich our framework with quotients and disjoint sums, thus leading to the notion of a modal (quasi) pretopos . We finally show a way to build syntactic categories out of first-order modal theories.