4.5.3. Model Theoretic versus Entailment Theoretic Connectives

Given an L-institution I, when L is a complete residuated lattice, we thus have two different definitions for each connective: one in terms of satisfaction by models and another one in terms of the semantic L-entailment system of I. It is important to establish the relationship between these two in order to be able to have an entailment-based calculus for the semantic consequence.

Consider the semantic L-entailment system of an L-institution such that L is a complete residuated lattice. Let *ρ* be a Σ-sentence and *ϕ* : Σ → Σ be a signature morphism. Then,


Let us further assume that L is a Heyting algebra. Then,


Let us further assume that L is a completely distributive Boolean algebra. Then

5. *ρ* is the entailment theoretic disjunction of *ρ*<sup>1</sup> and *ρ*<sup>2</sup> if it is the semantic disjunction of *ρ*<sup>1</sup> and *ρ*2.
