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Lattices of Finitely Alternative Normal Tense Logics

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Abstract

A finitely alternative normal tense logic \(T_{n,m}\) is a normal tense logic characterized by frames in which every point has at most n future alternatives and m past alternatives. The structure of the lattice \(\Lambda (T_{1,1})\) is described. There are \(\aleph _0\) logics in \(\Lambda (T_{1,1})\) without the finite model property (FMP), and only one pretabular logic in \(\Lambda (T_{1,1})\). There are \(2^{\aleph _0}\) logics in \(\Lambda (T_{1,1})\) which are not finitely axiomatizable. For \(nm\ge 2\), there are \(2^{\aleph _0}\) logics in \(\Lambda (T_{n,m})\) without the FMP, and infinitely many pretabular extensions of \(T_{n,m}\).

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Acknowledgements

Thanks are given to the referee for a very insightful comment on the description of \(\Lambda (T_{1,1})\) and related problems which leads to a revision of the manuscript of this paper. This work was supported by Chinese National Funding of Social Sciences (Grant No. 18ZDA033).

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Correspondence to Qian Chen.

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Ma, M., Chen, Q. Lattices of Finitely Alternative Normal Tense Logics. Stud Logica 109, 1093–1118 (2021). https://doi.org/10.1007/s11225-021-09942-5

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