On pre-local tabularity above $\mathrm{S4}\times \mathrm{S4}$

TL;DR

This paper characterizes the structure of pre-local tabularity in normal extensions of the modal logic S4×S4, identifying exactly four such logics for finite height products and providing criteria for local tabularity.

Contribution

It precisely classifies pre-locally tabular logics above S4×S4 and offers an axiomatic criterion for local tabularity in this context.

Findings

Four pre-locally tabular logics in finite height extensions

Every non-locally tabular logic in this family is contained in these four

Criteria for local tabularity above product logics with Noetherian skeletons

Abstract

We investigate pre-local tabularity in normal extensions of the logic $\mathrm{S4}\times \mathrm{S4}$. We show that there are exactly four pre-locally tabular logics in normal extensions of products of finite height, and that every non-locally tabular logic in this family is contained in one of them. We also give an axiomatic criterion of local tabularity above the logic of products with Noetherian skeletons. Finally, we discuss examples of pre-locally tabular extensions of $\mathrm{S4}\times \mathrm{S4}$ outside this class, including logics with the converse and universal modalities.