# Tense Logics over Lattices

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Part of the Lecture Notes in Computer Science book series (LNCS,volume 13468)

## Abstract

Lattice theory has close connections with modal logic via algebraic semantics and lattices of modal logics. However, one less explored direction is to view lattices as relational structures based on partial orders, and study the modal logic over them. In this paper, following the earlier steps of Burgess and van Benthem in the 1980s, we use the basic tense logic and its nominal extensions with binary modalities of infimum and supremum to talk about lattices via standard Kripke semantics. As the main results, we obtain a series of complete axiomatizations of lattices, (un)bounded lattices over partial orders or strict preorders. In particular, we solve an axiomatization problem left open by Burgess (1984).

### Keywords

• tense logic
• lattice
• completeness
• step-by-step
• hybrid logic
• modal logic

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1. 1.

Burgess calls the strict order satisfying $$\texttt{FOASym}$$ and $$\texttt{FOSTrans}$$ partial order.

2. 2.

Here “unbounded” means there are no greatest and least elements.

3. 3.

Note that the canonicity is based on the canonical model of normal tense logics.

4. 4.

E.g., by Proposition 1 (2), a frame validating $$\mathtt {CR_\textsf {G}}$$ also has the property $$\forall x\forall y (xRy \rightarrow \exists t(yRt))$$ (obtained by taking $$y=z$$ in the original corresponding first-order property).

5. 5.

Two formulas $$\textsf {F}p \wedge \textsf {F}q \rightarrow \textsf {F}(\textsf {P}p \wedge \textsf {P}q)$$, $$\textsf {P}p \wedge \textsf {P}q \rightarrow \textsf {P}(\textsf {F}p \wedge \textsf {F}q)$$ are mentioned in  as alternatives for $$\mathtt {CR_\textsf {G}}$$ and $$\mathtt {CR_\textsf {H}}$$, but they are also not valid over $$\mathfrak {L}_{<}$$.

6. 6.

This condition is to break the potential symmetry and reflexivity in $$\mathcal {M}^c_\mathbb{S}\mathbb{L}$$.

7. 7.

Note that this can indeed happen, e.g., when $$\textsf {G}\bot \wedge \textsf {H}\bot \in \varSigma _0$$ thus no $$\textsf {F}\varphi$$ or $$\textsf {P}\varphi$$ is in $$\varSigma _0$$.

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## Acknowledgement

The authors thank John Burgess, Johan van Benthem, and the two anonymous reviewers of WoLLIC2022 for relevant pointers and suggestions. The research is supported by NSSF grant 19BZX135.

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Correspondence to Yanjing Wang .

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