Many-valued coalgebraic modal logic: One-step completeness and finite model property

Abstract

In this paper, we investigate the many-valued version of coalgebraic modal logic through the 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. We prove the result for coalgebraic modal logics based on three different many-valued algebraic structures, namely the finitely-valued Łukasiewicz algebra, the commutative integral Full-Lambek algebra (FLew-algebra) expanded with canonical constants and Baaz Delta, and the FLew-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.