🤖 AI Summary
This work establishes, for the first time, a Lindström-style maximality theorem for Fitting’s modal logic based on finite Heyting algebras and crisp Kripke frames, without assuming linearity or co-atomicity. By introducing precise truth-value tests, Booleanization techniques, and derived existential modalities, and by leveraging compactness, the Tarski union property, and strong bisimulation invariance, the paper develops an abstract model-theoretic framework tailored to non-classical settings. The main contribution is the proof that any abstract logic extending Fitting’s system and satisfying these three properties is 1-expressively equivalent to Maruyama’s version of Fitting logic, thereby establishing its maximal expressive power and yielding a definability characterization for each precise truth-value fiber.
📝 Abstract
We establish a Lindström-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.