Many-valued coalgebraic logic over semi-primal varieties
Alexander Kurz, Wolfgang Poiger, Bruno Teheux
TL;DR
This work develops a systematic method to lift classical coalgebraic logics from Boolean-algebra semantics to many-valued logics over semi-primal algebras, using Stone-type dualities and subalgebra adjunctions. It shows that one-step completeness and expressivity are preserved under the lifting, and that liftings of endofunctors on Set and BA yield endofunctors on Set_D and the semi-primal variety A with preserved dualities. The authors provide concrete lifting procedures and, in favorable cases, explicit axiomatizations of the lifted logics, applying the framework to classical modal logic, filter frames, and neighborhood frames. They also discuss limitations to finite truth-degrees and sketch extensions to broader dualities and potential infinite-valued settings, offering a foundation for further exploration of many-valued coalgebraic semantics. Overall, the paper bridges two rich traditions—coalgebraic modal logic and finitely-valued logics—by enabling principled, transfer-based reasoning and axiomatization across the semi-primal setting.
Abstract
We study many-valued coalgebraic logics with semi-primal algebras of truth-degrees. We provide a systematic way to lift endofunctors defined on the variety of Boolean algebras to endofunctors on the variety generated by a semi-primal algebra. We show that this can be extended to a technique to lift classical coalgebraic logics to many-valued ones, and that (one-step) completeness and expressivity are preserved under this lifting. For specific classes of endofunctors, we also describe how to obtain an axiomatization of the lifted many-valued logic directly from an axiomatization of the original classical one. In particular, we apply all of these techniques to classical modal logic.