A Few Themes and Many Variations: The Semantics of Intuitionistic Propositional Logic

Intuitionistic propositional logic admits several closely related semantics. Among the

most familiar are algebraic semantics via Heyting algebras, Kripke semantics via partially

ordered sets of possible stages of information, and categorical semantics via Cartesian

closed or bi-Cartesian closed categories. These approaches are not unrelated alternatives.

Heyting algebras are the posetal form of categorical semantics; Kripke frames give rise to

Heyting algebras of upward-closed sets; and categorical semantics generalizes algebraic

semantics by replacing mere truth-values or entailment relations with objects and

morphisms capable of retaining proof structure. This report explains these relationships

and clarifies the sense in which some of these semantics are set-theoretic, algebraic, or

categorical.