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arxiv: 2606.31853 · v1 · pith:TVAFD666new · submitted 2026-06-30 · 🧮 math.LO · cs.LO

Conditionals and Modalities in Constructive Quantum Logics

Juan P. Aguilera (TU Wien) , Guillaume Massas (Chapman University) This is my paper

Pith reviewed 2026-07-01 02:19 UTC · model grok-4.3

classification 🧮 math.LO cs.LO
keywords iEx-logicintuitionistic logicorthomodular logicSasaki hookquantum logicintermediate logicsmodal operatorsconstructive quantum logic
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The pith

iEx-logic is defined as the intersection of intuitionistic logic and orthomodular logic under the Sasaki hook, and its extensions form the product of the two lattices.

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

The paper axiomatizes iEx-logic as the common fragment of full intuitionistic logic and orthomodular logic when implication is taken to be the Sasaki hook. From this definition it derives a complete description of every logic that contains iEx-logic: each such logic is obtained by independently choosing one intermediate logic and one orthomodular logic. The resulting product structure separates the choice of constructive strength from the choice of quantum-logic structure. The same algebraic method is applied briefly to extensions that add modal operators.

Core claim

We obtain a characterization of the lattice of logics extending iEx-logic as the product of the lattice of intermediate logics and the lattice of orthomodular logics. iEx-logic itself is defined as the intersection of full intuitionistic logic and orthomodular logic with the implication connective interpreted as the Sasaki hook.

What carries the argument

iEx-logic, the intersection of intuitionistic logic and orthomodular logic under the Sasaki-hook interpretation of implication, whose extension lattice is the product of the two source lattices.

If this is right

  • Every pair consisting of an intermediate logic and an orthomodular logic determines a unique logic extending iEx-logic.
  • Properties that hold in all intermediate logics or in all orthomodular logics transfer separately to the combined systems.
  • The algebraic construction remains available when modal operators are added to iEx-logic.

Where Pith is reading between the lines

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

  • The product decomposition makes it possible to vary the degree of constructivity without altering the orthomodular axioms and vice versa.
  • Questions about decidability or interpolation that depend on only one factor can be settled by examining that factor alone.

Load-bearing premise

That iEx-logic is exactly the intersection of full intuitionistic logic and orthomodular logic when implication is interpreted as the Sasaki hook.

What would settle it

A concrete logic that properly extends iEx-logic yet cannot be expressed as the join of any intermediate logic with any orthomodular logic.

read the original abstract

We investigate logics that generalize both intuitionistic logic and quantum logic. In earlier work, we introduced Ex-logic, an extension of Holliday's fundamental logic that coincides with the intersection of orthologic and the implication-free fragment of intuitionistic logic. In this paper, we add an implication connective to Ex-logic and axiomatize iEx-logic, the intersection of full intuitionistic logic and orthomodular logic with the implication connective interpreted as the Sasaki hook. As a consequence, we obtain a characterization of the lattice of logics extending iEx-logic as the product of the lattice of intermediate logics and the lattice of orthomodular logics. We also explore the robustness of our algebraic approach by briefly discussing extensions of iEx-logic with modal operators.

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 iEx-logic as the intersection of full intuitionistic logic and orthomodular logic (with implication as the Sasaki hook), supplies an axiomatization extending prior Ex-logic work, and derives that the lattice of iEx-extensions is the product of the intermediate-logic lattice and the orthomodular-logic lattice. It also sketches modal extensions of iEx-logic.

Significance. If the axiomatization is sound and complete for the stated intersection semantics, the product-lattice result supplies a clean algebraic bridge between intuitionistic and quantum logics, allowing independent variation of the constructive and orthomodular components. This structural characterization is a substantive contribution to the study of combined logics.

major comments (2)
  1. [axiomatization section (likely §3)] The central claim in the abstract (and presumably §4) that the lattice of iEx-extensions is the product of the two lattices holds only if the supplied axiomatization of iEx-logic is both sound and complete for the intersection semantics. The manuscript must therefore contain explicit soundness and completeness theorems establishing that the axioms derive precisely the formulas valid under both the intuitionistic and orthomodular interpretations with the Sasaki hook; any gap would make the extension lattice differ from the intended product.
  2. [definition of iEx-logic and axiomatization] The definition of iEx-logic as the intersection (stated in the abstract) is load-bearing; the paper should verify that no formula valid in both semantics is underivable from the axioms and that no extra theorems are introduced by the axiomatization relative to the semantic intersection.
minor comments (2)
  1. [introduction] The abstract refers to 'earlier work' on Ex-logic; a brief self-contained recap of the implication-free fragment would improve readability.
  2. [preliminaries] Notation for the Sasaki hook should be introduced explicitly when first used, rather than assuming familiarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the importance of explicit soundness and completeness results. We agree that these are necessary to fully support the lattice characterization and will revise the manuscript to include them.

read point-by-point responses

  1. Referee: The central claim in the abstract (and presumably §4) that the lattice of iEx-extensions is the product of the two lattices holds only if the supplied axiomatization of iEx-logic is both sound and complete for the intersection semantics. The manuscript must therefore contain explicit soundness and completeness theorems establishing that the axioms derive precisely the formulas valid under both the intuitionistic and orthomodular interpretations with the Sasaki hook; any gap would make the extension lattice differ from the intended product.

    Authors: We agree that the central claim requires explicit soundness and completeness theorems. The manuscript currently presents the axiomatization as capturing the intersection and derives the lattice result from it, but we will add a dedicated subsection in §3 with full proofs: soundness showing that all theorems are valid in both semantics, and completeness showing that every formula valid under both interpretations is derivable from the axioms. revision: yes

  2. Referee: The definition of iEx-logic as the intersection (stated in the abstract) is load-bearing; the paper should verify that no formula valid in both semantics is underivable from the axioms and that no extra theorems are introduced by the axiomatization relative to the semantic intersection.

    Authors: This point is addressed by the same addition. The completeness theorem will establish that no semantically valid formula (in the intersection) is underivable, while soundness will confirm that the axiomatization introduces no extraneous theorems beyond those valid in both semantics. We will make these verifications explicit in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines iEx-logic explicitly as the intersection of IL and OML (Sasaki hook) and states that it supplies a separate axiomatization for this logic; the lattice-product characterization is presented as a consequence of that axiomatization rather than by redefinition or by renaming the input. The reference to prior work on Ex-logic concerns only the implication-free fragment and is not invoked as a uniqueness theorem or load-bearing premise for the new result. No equation or step reduces the target claim to a fitted parameter or self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard background in lattice theory, orthomodular lattices, and intuitionistic logic; no free parameters are introduced. Axioms are the defining axioms of the new logic plus background assumptions from prior literature on Ex-logic.

axioms (2)
  • domain assumption iEx-logic coincides with the intersection of intuitionistic logic and orthomodular logic under Sasaki hook interpretation
    This is the definitional premise stated in the abstract that enables the subsequent axiomatization and lattice result.
  • standard math Standard properties of orthomodular lattices and intermediate logics hold
    Invoked implicitly to form the product lattice structure.

pith-pipeline@v0.9.1-grok · 5660 in / 1369 out tokens · 24243 ms · 2026-07-01T02:19:05.827548+00:00 · methodology

discussion (0)

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Reference graph

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