Tho Modal Logic of Minimal Upper Bounds

TL;DR

This paper introduces the modal logic MIN, which interprets minimal upper bounds in information structures, and proves its equivalence in validity to the existing MIL logic, providing axiomatization and decidability results.

Contribution

The paper formalizes the modal logic MIN for minimal upper bounds and shows its equivalence to MIL, including axiomatization and decidability proofs.

Findings

MIN logic is equivalent to MIL in validity.

Axiomatization of MIN is achieved.

Decidability of MIN is established.

Abstract

To formalize patterns of information increase and decrease, Van Benthem (1996) proposed modal information logic (MIL), a modal logic over partial orders. In MIL, points are interpreted as information states and least upper bounds, when existent, as informational sums. A natural counterpart to this logic is the modal logic of minimal upper bounds (MIN), interpreting minimal, rather than least, upper bounds as informational sums. This paper presents the logic MIN, and in the main result, it is shown that the modal language cannot distinguish the two interpretations: a formula is valid in MIN if and only if it is valid in MIL. Leveraging the work of [11], as corollaries, an axiomatization of MIN and a proof of decidability are obtained.