Implicit Kripke Semantics and Ultraproducts in Stratified Institutions

TL;DR

This paper introduces stratified institutions as an abstract model theoretic framework for modal logic, enabling uniform treatment of semantics and deriving compactness results across various modal systems.

Contribution

It develops stratified institutions as a fully abstract approach, capturing Kripke semantics implicitly and facilitating modal ultraproducts independent of specific modal systems.

Findings

Unified treatment of modal semantics via stratified institutions

Implicit Kripke semantics capturing diverse modal variations

Derivation of compactness results across modal logics

Abstract

We propose stratified institutions (a decade old generalised version of the theory of institutions of Goguen and Burstall) as a fully abstract model theoretic approach to modal logic. This allows for a uniform treatment of model theoretic aspects across the great multiplicity of contemporary modal logic systems. Moreover Kripke semantics (in all its manifold variations) is captured in an implicit manner free from the sometimes bulky aspects of explicit Kripke structures, also accommodating other forms of concrete semantics for modal logic systems. The conceptual power of stratified institutions is illustrated with the development of a modal ultraproducts method that is independent of the concrete details of the actual modal logical systems. Consequently, a wide array of compactness results in concrete modal logics may be derived easily.