Modal Measurable Logics via a Modal Loomis-Sikorski Representation Theorem

2026-06-30

AI Summary

This work introduces modal operators into classical logic of infinite order to support applications of measure theory in dynamical systems and point-free ergodic theory. To this end, the authors develop a formal system termed *modal measurable logic* and provide a Kripke-style semantics based on measurable spaces equipped with a designated modal σ-ideal. The central contribution lies in the novel integration of a restricted form of Jónsson–Tarski duality with a modal extension of the Loomis–Sikorski theorem, which together yield a completeness result for the proposed logic relative to the given semantics. This framework establishes a new logical foundation for reasoning about dynamical systems and ergodic-theoretic constructions in a point-free setting.

Abstract

We investigate a modal extension of the infinitary classical logic with countable meets and joins, formulated with an eye toward measure-theoretic work in dynamical systems and in point-free ergodic theory. We define a modal formalism in this language, which we call modal measurable logics. We also introduce a Kripke-like semantics for these logics in measurable spaces taking a designated modal sigma-ideal into consideration. Using a restriction of Jonsson-Tarski duality and a modal extension of the Loomis-Sikorski theorem, we prove completeness of modal measurable logics with respect to this new semantics.

Problem

• modal logic
• measurable spaces
• Loomis-Sikorski theorem
• sigma-ideal
• point-free ergodic theory

Innovation

• modal measurable logics
• Loomis-Sikorski theorem
• sigma-ideal
• Kripke-like semantics
• completeness