On Modal Logics of Full Products of Neighborhood Frames

2026-06-30

AI Summary

This study addresses the axiomatization of multimodal logics induced by three natural neighborhood functions within the framework of full product neighborhoods, focusing on the frame classes validating the modal logics T and D. By combining three monomodal logics and introducing an interaction axiom (mix), the authors construct the trimodal logics Tx+T and Dx+D. The work innovatively extends known product logic results for S4 and D4 to the weaker systems T and D, revealing that the role of (mix) varies across logical contexts: in S4, (mix) is equivalent to (sub), thereby enabling an axiomatization of the full product logic over topological spaces. The main contributions establish that Tx+T equals the product logic T×T×T augmented with (mix), and similarly, Dx+D equals D×D×D plus (mix).

Abstract

On the product of two neighborhood frames, three natural neighborhood functions can be defined: the horizontal one assigning to a point (x, y) the set of all supersets of the Cartesian product of U and y, where U is a neighborhood of x; the vertical analog; and the product neighborhood function assigning as neighborhoods all supersets of sets Cartesian products of U and V, for neighborhoods U of x and V of y. We define the tri-modal logics Tx+T and Dx+D of classes of full products equipped with all three neighborhood functions of neighborhood frames validating the logic T or D; thereby extending known product results for S4 and D4 to weaker systems. Two interaction principles arise: (sub) = []p -> [1]p & [2]p and (mix) = []p -> [1][2]p & [2][1]p, where the modality [] stands for the product neighborhood function and [1], [2] the horizontal and vertical ones. Namely, we show that Tx+T = T*T*T + (mix) and Dx+D = D*D*D + (mix), where * denotes fusion. Notably, (sub) and (mix) are equivalent over S4*S4*S4 and thus S4*S4*S4 + (mix) axiomatizes the logic of full products of topological spaces.

Problem

• modal logic
• neighborhood frames
• product
• full products
• interaction principles

Innovation

• neighborhood frames
• product logics
• tri-modal logic
• interaction axioms
• fusion