A boolean-algebraic approach to completeness for normal modal predicate logics

Abstract

This paper introduces an innovative methodology for demonstrating completeness for normal modal predicate logics. Traditional proofs typically involve constructing canonical models, wherein possible worlds are defined as maximal consistent sets possessing specific properties, with a heavy reliance on the Barcan Formula to affirm the existence of these worlds. Our approach deviates from the classical method by utilizing Boolean algebras and ultrafilters to construct models. Unlike conventional methods, our construction of possible worlds does not depend on the Barcan Formula; rather, these properties are ensured by Tarski’s Lemma. Furthermore, our proof distinguishes itself from other Boolean-algebraic completeness proofs in two key respects: it employs Kripke semantics instead of algebraic semantics and exclusively relies on ultrafilters, thereby offering a more concise approach. This methodology facilitates a natural extension from modal propositional logic to modal predicate logics and circumvents the added complexity of Q-filters. In our model, the equivalence class of each theorem of a normal modal predicate logic is a member of all worlds, while the equivalence class of each non-theorem is a member of some worlds. Consequently, the structure of these worlds ensures that non-theorems are false in the model.