Tense Logic via Truth Degrees: An Algebraic Completeness Result for Kashima's Calculus

We study the minimal tense logic $K_t$ from an algebraic and proof-theoretic perspective. We introduce the degree-of-truth-preserving logic associated with the class of tense Boolean algebras. We then introduce a sequent calculus for this logic and establish its soundness and completeness with respect to tense Boolean algebras by means of an adaptation of the Lindenbaum–Tarski construction. Consequently, this calculus provides an additional syntactic presentation of the minimal tense logic $K_t$. We also provide a second, purely syntactic proof of completeness. Furthermore, we establish an algebraic soundness and completeness theorem for Kashima’s Gentzen-style calculus. To this end, we develop an adaptation of the Lindenbaum–Tarski construction to Kashima’s nested sequent framework, which allows us to construct the algebraic semantics directly from the proof-theoretic system. This yields a direct algebraic completeness proof for Kashima’s calculus and connects its nested-sequent formulation with the algebraic semantics of tense Boolean algebras.