Many-Valued Coalgebraic Modal Logic: One-step Completeness and Finite Model Property

TL;DR

This paper extends coalgebraic modal logic to many-valued settings, establishing one-step completeness and finite model property for various algebraic structures using canonical models and filtration techniques.

Contribution

It generalizes one-step completeness to finitely many-valued coalgebraic modal logic and proves the finite model property for these logics.

Findings

Proves one-step completeness for many-valued coalgebraic modal logic.

Establishes the finite model property using filtration techniques.

Applies results to multiple algebraic structures including Łukasiewicz and FL$_{ew}$-algebras.

Abstract

In this paper, we investigate the many-valued version of coalgebraic modal logic through predicate lifting approach. Coalgebras, understood as generic transition systems, can serve as semantic structures for various kinds of modal logics. A well-known result in coalgebraic modal logic is that its completeness can be determined at the one-step level. We generalize the result to the finitely many-valued case by using the canonical model construction method. We prove the result for coalgebraic modal logics based on three different many-valued algebraic structures, including the finitely-valued {\L}ukasiewicz algebra, the commutative integral Full-Lambek algebra (FL$_{ew}$-algebra) expanded with canonical constants and Baaz Delta, and the FL$_{ew}$-algebra expanded with valuation operations. In addition, we also prove the finite model property of the many-valued coalgebraic modal logic by using the filtration technique.