Kripke Semantics for Predicate BI

Abstract

We extend the Kripke semantics of propositional BI to predicates and quantifiers. Let M = (M,e, ∙, ⊑) be a preordered commutative monoid, viewed as a preordered monoidal category. Recall, from Chapter 4, that a Kripke model of propositional BI and hence of α λ, is a triple

$$\left\langle {\left[ {{M^{op}},Set} \right],| = ,\left[\kern-0.15em\left[ - \right]\kern-0.15em\right]} \right\rangle ,$$

in which $\left[\kern-0.15em\left[ - \right]\kern-0.15em\right]$ is a partial function from P(L),the collection of BI propositions, or α λ-types, over a language L of propositional letters (or atomic types) to obj([M ^op, Set]) and in which $| = \subseteq M \times { \cup _{\phi \in P(L)}}$ is a satisfaction relation. Recall also that the definition of a Kripke model of propositional BI given in Chapter 4, makes essential use of Day’s tensor product construction and its right adjoint [Day, 1970].