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arxiv: 2306.13903 · v3 · pith:3OIRUZJ5new · submitted 2023-06-24 · 🧮 math.LO · cs.CC

On the local consequence of modal Product logic: standard completeness and decidability

Amanda Vidal This is my paper

Pith reviewed 2026-05-24 08:35 UTC · model grok-4.3

classification 🧮 math.LO cs.CC
keywords modal product logiclocal consequencestandard completenessdecidabilityKripke modelsvalued accessibilitycrisp accessibilityproduct algebra
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The pith

Modal product logics reduce constructively to propositional product logic, making them decidable and standard complete for both valued and crisp accessibility.

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

The paper shows that local consequence relations for modal extensions of product logic, defined over Kripke models with either valued or crisp accessibility, reduce directly to consequence in ordinary propositional product logic. This reduction is constructive and applies whether semantics are taken over the full class of product algebras or restricted to the standard product algebra on the unit interval. The result immediately yields decidability for the modal systems and shows that the local consequence relation does not change when moving from all product algebras to the standard one alone. A reader would care because the same reduction settles both the valued-accessibility case, where it strengthens earlier theoremhood results, and the crisp-accessibility case, where no prior decidability or standard-completeness proofs existed.

Core claim

The central claim is that a constructive reduction exists from the modal product logics to propositional product logic; as direct consequences, the modal systems are decidable and the local consequence relation induced by the class of all product algebras coincides with the relation induced by the standard product algebra on [0,1]. The reduction and its corollaries hold uniformly for both the valued-accessibility semantics and the crisp-accessibility semantics.

What carries the argument

The constructive reduction of modal product logic to propositional product logic, which preserves local consequence relations in both accessibility settings.

If this is right

  • All the modal product logics under study are decidable.
  • Standard completeness holds: local consequence over all product algebras equals local consequence over the standard [0,1] product algebra.
  • In the valued-accessibility setting the results extend previous decidability theorems from theoremhood alone to arbitrary local consequence relations.
  • In the crisp-accessibility setting the results supply the first decidability and standard-completeness theorems for local modal product logics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction technique might be adapted to obtain decidability results for modal extensions of other continuous t-norm logics.
  • Because the reduction is constructive, it supplies an explicit method for turning modal consequence problems into propositional ones that can be fed to existing product-logic decision procedures.
  • The coincidence of full-class and standard-algebra consequence suggests that finite-model or finite-chain arguments developed for the standard algebra may transfer back to the modal setting without loss.

Load-bearing premise

The reduction from the modal language to the propositional language preserves local consequence over both the full class of product algebras and the standard [0,1] algebra.

What would settle it

A concrete modal formula (or finite set of premises and conclusion) such that the local consequence holds in the modal logic over all product-algebra Kripke models but fails after the reduction in propositional product logic would show the reduction does not preserve consequence.

read the original abstract

We study local consequence relations in modal extensions of product logic over Kripke models with either valued (fuzzy) or crisp accessibility relations. In both settings, we consider semantics over the full class of product algebras as well as over the standard product algebra on $[0,1]$. Our main result is a constructive reduction of these modal logics to propositional product logic. As consequences, we prove that all the resulting systems are decidable and standard complete, i.e., the local consequence relation over all product algebras coincides with the one induced by the standard product algebra. In the valued-accessibility case, our methods strengthen previous results on decidability by extending them from theoremhood to arbitrary local consequence relations, and covering standard completeness. In the crisp case, the techniques are substantially different and yield, to the best of our knowledge, the first decidability and standard completeness results for local modal product logics with crisp accessibility relations.

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

0 major / 2 minor

Summary. The paper studies local consequence relations for modal extensions of product logic over Kripke frames with either valued (fuzzy) or crisp accessibility relations. Semantics are given both over the full class of product algebras and over the standard product algebra on [0,1]. The central claim is a constructive reduction of these modal systems to propositional product logic; the reduction is asserted to preserve local consequence in both the full and standard semantics, from which decidability and standard completeness are derived. The valued-accessibility case extends prior decidability results from theoremhood to arbitrary local consequences and adds standard completeness; the crisp-accessibility case yields what the authors claim are the first such results.

Significance. If the reduction is faithful, the work supplies decidability for local consequence (rather than merely theoremhood) in the valued case and the first decidability and standard-completeness theorems for the crisp case. The explicitly constructive character of the reduction is a clear strength: it directly yields an effective decision procedure and supports the standard-completeness claim without additional non-constructive arguments. These results sit squarely inside the program of algebraic and many-valued modal logic and would be of interest to researchers working on completeness and decidability for fuzzy modal systems.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'modal extensions of product logic' is used without naming the precise modal axioms or the signature of the modalities; a single sentence listing the operators and the base axioms would improve immediate readability.
  2. The manuscript would benefit from an explicit, self-contained statement of the translation function (including how modal formulas are mapped to propositional product formulas) before the proof that it preserves consequence; this would make the constructive character easier to verify at a glance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment, including the recommendation to accept. No major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity: constructive reduction to propositional product logic is independent

full rationale

The paper's central result is a constructive reduction of the modal product logics (valued and crisp accessibility) to propositional product logic, preserving local consequence over product algebras and the standard [0,1] algebra. This yields decidability and standard completeness as direct consequences. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain; the reduction is presented as a translation that is faithful by construction to the Kripke semantics and algebra semantics, without reducing the target theorems to the inputs by definition. The argument is self-contained against external benchmarks in the base propositional product logic.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; the paper relies on standard axioms of product algebras and modal logic without introducing new free parameters or entities.

axioms (1)
  • domain assumption Standard semantics of product logic and modal Kripke models
    The paper builds on established definitions in fuzzy modal logic.

pith-pipeline@v0.9.0 · 5679 in / 1195 out tokens · 26561 ms · 2026-05-24T08:35:15.847857+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    math.LO 2026-05 unverdicted novelty 7.0

    Resolves open problem by proving natural extensions of modal Gödel logics are incomplete for finite models and supplies new complete axiomatizations.

Reference graph

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