A Lindstr\"om Theorem for Fitting's Modal Logic over a Finite Heyting Algebra

Abstract

We establish a Lindstr\"om-style maximality theorem for Maruyama's exact-truth-test presentation of Fitting's modal logic over a fixed finite Heyting algebra and crisp Kripke frames. Unlike the existing characterization over finite MTL-chains, no linearity or distinguished coatom is assumed. Exact truth tests yield Boolean tests for designated and non-designated values and a derived existential modality sufficient for the saturation argument. We prove that every abstract extension which is compact, has the Tarski Union Property, and is strongly invariant under bisimulation is $1$-expressively equivalent to Maruyama's version of Fitting's Heyting-valued modal logic. As a consequence, every exact-value fibre of an extension formula is definable in Maruyama's exact-truth-test modal language.