Kripke Completeness of Strictly Positive Modal Logics over Meet-semilattices with Operators

TL;DR

This paper investigates the completeness of strictly positive modal logics over meet-semilattices with operators, establishing foundational links between algebraic and relational semantics.

Contribution

It develops a foundational completeness theory for spi-logics, connecting algebraic meet-semilattice semantics with Kripke frame semantics.

Findings

Established conditions for completeness over meet-semilattices with operators

Linked algebraic and relational semantics for spi-logics

Provided a framework for analyzing consequence relations in spi-logics

Abstract

Our concern is the completeness problem for spi-logics, that is, sets of implications between strictly positive formulas built from propositional variables, conjunction and modal diamond operators. Originated in logic, algebra and computer science, spi-logics have two natural semantics: meet-semilattices with monotone operators providing Birkhoff-style calculi, and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations for a completeness theory that aims to answer the question whether the two semantics define the same consequence relations for a given spi-logic.