Tense Logic via Truth Degrees: An Algebraic Completeness Result for Kashima's Calculus
Abstract
We study the minimal tense logic Kt 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 Kt. 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.
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BibTeX
@misc{figallo2026tense,
title = {{Tense Logic via Truth Degrees: An Algebraic Completeness Result for Kashima's Calculus}},
author = {Martín Figallo and Jonathan Sarmiento and Luis Pezzini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01547},
eprint = {2610.01547},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01547},
}
RIS
TY - UNPB
TI - Tense Logic via Truth Degrees: An Algebraic Completeness Result for Kashima's Calculus
AU - Martín Figallo
AU - Jonathan Sarmiento
AU - Luis Pezzini
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01547
UR - https://arxiv.org/abs/2610.01547
AB - 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.
ER -