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SUFFICIENT CONDITIONS FOR LOCAL TABULARITY OF A POLYMODAL LOGIC

Part of: General logic

Published online by Cambridge University Press:  27 January 2025

ILYA B. SHAPIROVSKY Open the ORCID record for ILYA B. SHAPIROVSKY [Opens in a new window]
ILYA B. SHAPIROVSKY*
Affiliation:
NEW MEXICO STATE UNIVERSITY LAS CRUCES NM, USA
*
E-mail: [email protected]
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Abstract

On relational structures and on polymodal logics, we describe operations which preserve local tabularity. This provides new sufficient semantic and axiomatic conditions for local tabularity of a modal logic. The main results are the following.

We show that local tabularity does not depend on reflexivity. Namely, given a class $\mathcal {F}$ of frames, consider the class $\mathcal {F}^{\mathrm {r}}$ of frames, where the reflexive closure operation was applied to each relation in every frame in $\mathcal {F}$. We show that if the logic of $\mathcal {F}^{\mathrm {r}}$ is locally tabular, then the logic of $\mathcal {F}$ is locally tabular as well.

Then we consider the operation of sum on Kripke frames, where a family of frames-summands is indexed by elements of another frame. We show that if both the logic of indices and the logic of summands are locally tabular, then the logic of corresponding sums is also locally tabular.

Finally, using the previous theorem, we describe an operation on logics that preserves local tabularity: we provide a set of formulas such that the extension of the fusion of two canonical locally tabular logics with these formulas is locally tabular.

Keywords

modal logiclocally tabular logiclocally finite algebrafinite heightpretransitivitylexicographic sum of Kripke frames

MSC classification

Primary: 03B45: Modal logic (including the logic of norms) 03B62: Combined logics 06E25: Boolean algebras with additional operations (diagonalizable algebras, etc.)
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 26
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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