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arxiv: 2606.31862 · v1 · pith:O57YQJLXnew · submitted 2026-06-30 · 🧮 math.LO · cs.LO

Modal Measurable Logics via a Modal Loomis-Sikorski Representation Theorem

Nick Bezhanishvili , Jim de Groot , Lawrence S. Moss This is my paper

Pith reviewed 2026-07-01 02:15 UTC · model grok-4.3

classification 🧮 math.LO cs.LO
keywords modal logicmeasurable spacesLoomis-Sikorski theoremJonsson-Tarski dualitycompletenessinfinitary logicsigma-idealdynamical systems
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The pith

A modal extension of the Loomis-Sikorski theorem establishes completeness for modal measurable logics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops modal measurable logics as a modal extension of infinitary classical logic that includes countable meets and joins. It introduces a semantics based on measurable spaces equipped with a designated modal sigma-ideal. Using a restriction of Jonsson-Tarski duality together with this modal Loomis-Sikorski extension, the authors prove that these logics are complete for the new semantics. This matters for providing a logical framework suited to measure-theoretic investigations in dynamical systems and point-free ergodic theory. Sympathetic readers would value the bridge it creates between modal logic and measurable structures.

Core claim

The central discovery is a completeness theorem for modal measurable logics with respect to Kripke-like semantics on measurable spaces that incorporate a modal sigma-ideal. The proof proceeds by restricting Jonsson-Tarski duality and extending the Loomis-Sikorski theorem to the modal setting, thereby showing that every consistent theory has a model in this semantics.

What carries the argument

The modal extension of the Loomis-Sikorski representation theorem, which represents the modal measurable algebras in terms of measurable spaces with modal sigma-ideals.

Load-bearing premise

The newly formulated modal extension of the Loomis-Sikorski theorem is valid and integrates with the restricted Jonsson-Tarski duality to prove the completeness result.

What would settle it

Discovery of a modal measurable logic formula that holds in every measurable space with a modal sigma-ideal but is not provable from the axioms of the logic, or vice versa.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper defines modal measurable logics as a modal extension of infinitary propositional logic allowing countable meets and joins. It introduces a Kripke-style semantics on measurable spaces equipped with a designated modal sigma-ideal. The central result is a completeness theorem obtained by combining a restriction of Jónsson-Tarski duality with a modal extension of the Loomis-Sikorski representation theorem.

Significance. If the modal extension of the Loomis-Sikorski theorem is valid and composes correctly with the restricted duality, the result supplies a representation theorem linking modal logic to measurable spaces. This could support applications in dynamical systems and point-free ergodic theory by providing a semantics that respects measure-theoretic structure. The approach follows standard representation-theoretic methods without introducing free parameters or circular definitions.

major comments (2)
  1. [The modal Loomis-Sikorski theorem] The modal extension of the Loomis-Sikorski theorem (whose statement appears to be the key technical step) must be verified to preserve the countable completeness properties under the modal operators; without an explicit check that the modal sigma-ideal is closed under the relevant operations, the completeness claim for the full language remains open.
  2. [Completeness proof] The restriction of Jónsson-Tarski duality is applied to the algebra of measurable sets modulo the modal sigma-ideal; it is necessary to confirm that this restriction still yields a modal algebra whose ultrafilters correspond exactly to the points in the measurable space, otherwise the completeness direction may fail for formulas involving countable joins.
minor comments (2)
  1. Clarify the precise definition of a 'modal sigma-ideal' early in the paper, including its interaction with the modal operators, to make the semantics reproducible.
  2. The abstract claims a completeness proof but supplies no outline of the derivation; adding a high-level proof sketch in the introduction would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below. Both concerns can be met by adding explicit lemmas and remarks that isolate the required closure and correspondence properties; these additions clarify the existing arguments without changing the results.

read point-by-point responses

  1. Referee: The modal extension of the Loomis-Sikorski theorem (whose statement appears to be the key technical step) must be verified to preserve the countable completeness properties under the modal operators; without an explicit check that the modal sigma-ideal is closed under the relevant operations, the completeness claim for the full language remains open.

    Authors: We agree that an isolated verification improves readability. The proof of the modal Loomis-Sikorski theorem (Theorem 4.5) already establishes that the modal sigma-ideal is closed under the modal operators and under countable meets and joins, but the argument is distributed across several steps. We will insert a new standalone lemma (Lemma 4.3) stating the closure properties before the main representation argument. This makes the preservation of countable completeness under the modal operators fully explicit. revision: yes

  2. Referee: The restriction of Jónsson-Tarski duality is applied to the algebra of measurable sets modulo the modal sigma-ideal; it is necessary to confirm that this restriction still yields a modal algebra whose ultrafilters correspond exactly to the points in the measurable space, otherwise the completeness direction may fail for formulas involving countable joins.

    Authors: The restricted duality is defined so that the modal operators descend to the quotient algebra and preserve the required Boolean operations, yielding a modal algebra by construction. The ultrafilter correspondence with points of the measurable space follows from the underlying Loomis-Sikorski representation. To address the concern directly, we will add a short paragraph immediately after the statement of the restricted duality (in the proof of Theorem 5.2) that records the preservation of countable joins and the exact recovery of points via ultrafilters. This confirms completeness for formulas with countable joins. revision: yes

Circularity Check

0 steps flagged

No significant circularity; completeness derived from external representation theorems

full rationale

The paper establishes completeness for modal measurable logics by combining a restriction of Jonsson-Tarski duality with a modal extension of the Loomis-Sikorski theorem on measurable spaces equipped with a modal sigma-ideal. These steps invoke established results from modal algebra and measure theory rather than defining the target semantics or completeness in terms of themselves, fitting parameters to subsets of data, or relying on load-bearing self-citations whose content reduces to the present work. The derivation chain remains independent of the paper's own inputs and is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim depends on the validity of the modal extension of Loomis-Sikorski and the applicability of restricted Jonsson-Tarski duality; these are treated as background results rather than derived inside the paper.

axioms (2)
  • standard math Jonsson-Tarski duality holds when restricted to the relevant class of algebras or frames
    Invoked to obtain the representation needed for completeness.
  • domain assumption A modal extension of the Loomis-Sikorski theorem exists and applies to the measurable setting with sigma-ideals
    This is the key representation tool used to prove completeness.
invented entities (1)
  • modal sigma-ideal no independent evidence
    purpose: Designates the modal component in the Kripke-like semantics on measurable spaces
    New construct introduced to define the semantics for the modal measurable logics.

pith-pipeline@v0.9.1-grok · 5623 in / 1376 out tokens · 38276 ms · 2026-07-01T02:15:27.964928+00:00 · methodology

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