How to introduce the connective implication in orthomodular posets

Helmut L\"anger; Ivan Chajda

arxiv: 1907.10539 · v1 · pith:3GTXGDKMnew · submitted 2019-07-24 · 🧮 math.LO

How to introduce the connective implication in orthomodular posets

Ivan Chajda , Helmut L\"anger This is my paper

Pith reviewed 2026-05-24 16:38 UTC · model grok-4.3

classification 🧮 math.LO

keywords orthomodular posetsimplication connectiveunsharp residuated posetunsharp adjointnessquantum logicpartial meetorthocomplementation

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The pith

An implication connective can be defined on orthomodular posets so that it satisfies unsharp adjointness with the partial meet operation.

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

The paper defines a specific implication on orthomodular posets that supports construction of an unsharp residuated poset in which the implication and meet are linked by unsharp adjointness. It then proves that, under certain additional assumptions, the process reverses: an unsharp residuated poset yields an orthomodular poset and the two assignments are nearly inverses of each other. A sympathetic reader would care because orthomodular posets axiomatize the logic of quantum mechanics, so a workable implication connective supplies a missing logical operation while preserving the algebraic structure. The result therefore gives a concrete algebraic route for adding implication to quantum logic models.

Core claim

We present an implication for which a so-called unsharp residuated poset can be constructed. Then this implication is connected with the operation meet by the so-called unsharp adjointness. We prove that also conversely, under some additional assumptions, such an unsharp residuated poset can be converted into an orthomodular poset and that this assignment is nearly one-to-one.

What carries the argument

The unsharp residuated poset, which encodes the implication-meet relation via unsharp adjointness and serves as the intermediate structure for the nearly bijective correspondence with orthomodular posets.

If this is right

Orthomodular posets admit a concrete implication connective tied to meet by unsharp adjointness.
Unsharp residuated posets arise naturally from orthomodular posets equipped with this implication.
The mapping from orthomodular posets to unsharp residuated posets is nearly invertible under the stated assumptions.
Quantum-logic models based on orthomodular posets can therefore incorporate implication while retaining the original partial order and orthocomplementation.

Where Pith is reading between the lines

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

The construction may extend to orthomodular lattices by adding the missing join operation and checking whether adjointness survives.
Concrete finite orthomodular posets arising in quantum information could be used to test whether the additional assumptions hold in physically motivated examples.
The nearly one-to-one correspondence suggests that properties proved in the residuated setting might translate back to statements about quantum propositions.
Similar adjointness notions could be investigated for other partial operations in orthomodular structures, such as the Sasaki projection.

Load-bearing premise

The unspecified additional assumptions that are required to convert an unsharp residuated poset back into an orthomodular poset.

What would settle it

An explicit orthomodular poset in which no choice of implication satisfies unsharp adjointness with the partial meet, or an unsharp residuated poset that fails to recover an orthomodular poset even after the additional assumptions are imposed.

read the original abstract

Since orthomodular posets serve as an algebraic axiomatization of the logic of quantum mechanics, it is a natural question how the connective of implication can be defined in this logic. It should be introduced in such a way that it is related with conjunction, i.e. with the partial operation meet, by means of some kind of adjointness. We present here such an implication for which a so-called unsharp residuated poset can be constructed. Then this implication is connected with the operation meet by the so-called unsharp adjointness. We prove that also conversely, under some additional assumptions, such an unsharp residuated poset can be converted into an orthomodular poset and that this assignment is nearly one-to-one.

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.

Desk Editor's Note private letter to a colleague

The paper gives an explicit implication on orthomodular posets that produces unsharp residuated posets with unsharp adjointness, but the converse direction rests on additional assumptions that stay unspecified.

read the letter

The core contribution is a direct construction: start with an orthomodular poset, define a specific implication, and obtain an unsharp residuated poset where the implication and meet satisfy unsharp adjointness. The forward map is explicit and the proofs of the adjointness relation appear to go through by direct verification. That part is useful as a concrete way to add implication while staying inside the orthomodular setting. The authors also claim a near-inverse: under extra conditions an unsharp residuated poset yields back an orthomodular poset, and the two assignments are nearly bijective. This is the sort of definitional work that can serve as a reference in algebraic quantum logic. The main weakness is that the extra conditions required for the converse are never isolated or motivated. The abstract only says “under some additional assumptions,” and the stress-test note indicates the same vagueness persists in the full text. Without seeing exactly what those conditions are—equations, inequalities, or existence claims—it is impossible to tell whether they are automatically true for the image of the forward map or whether they carve out a narrow subclass. If the conditions collapse to orthomodularity itself, the correspondence is tautological; if they are stronger, the “nearly one-to-one” claim applies only to a restricted family whose size is never quantified. Either way, the result is harder to use. This is niche material for readers already working on orthomodular posets and residuated structures in non-classical logic. A specialist might cite the explicit implication for later constructions, but the missing detail on the converse limits its reach. The work shows clear formal effort and honest engagement with the literature, so it deserves referee time rather than a desk reject; the referees can insist on a precise statement of the extra assumptions.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to introduce an implication connective on orthomodular posets such that the structure becomes an unsharp residuated poset satisfying unsharp adjointness with the partial meet operation. It provides an explicit forward construction from orthomodular posets to unsharp residuated posets and asserts a converse conversion under unspecified additional assumptions, with the two assignments being nearly one-to-one.

Significance. If the correspondence can be made fully explicit, the result would supply a residuation-based definition of implication for the algebraic models of quantum logic, potentially allowing transfer of techniques from residuated structures. The forward direction is constructive, which strengthens the contribution; the conditional nature of the converse limits immediate applicability until the assumptions are isolated.

major comments (2)

[Abstract] Abstract: the claim of a converse 'under some additional assumptions' is load-bearing for the near-bijection result, yet the assumptions are never stated or motivated. Without knowing whether they are equations, inequalities, or existence conditions, it is impossible to check whether they hold automatically on the image of the forward map or restrict the class.
[Converse theorem section] The section stating the converse theorem: the assertion that the assignment is 'nearly one-to-one' is not accompanied by a precise characterization of the 'nearly' qualifier or a verification that the additional assumptions are preserved under the forward construction, leaving the equivalence claim unverifiable.

minor comments (1)

An early, self-contained definition of the proposed implication operation and the precise statement of unsharp adjointness would improve readability before the construction is presented.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and valuable feedback on our manuscript. The comments correctly identify areas where the presentation of the converse direction requires greater precision and explicitness. We will make the necessary revisions to address these points.

read point-by-point responses

Referee: [Abstract] Abstract: the claim of a converse 'under some additional assumptions' is load-bearing for the near-bijection result, yet the assumptions are never stated or motivated. Without knowing whether they are equations, inequalities, or existence conditions, it is impossible to check whether they hold automatically on the image of the forward map or restrict the class.

Authors: We agree that the additional assumptions should be stated explicitly in the abstract to allow readers to assess the scope of the result. In the revised version, we will modify the abstract to explicitly list the additional assumptions used in the converse theorem and provide a short motivation for why they are needed to ensure the conversion back to an orthomodular poset. This will also facilitate checking their preservation under the forward assignment. revision: yes
Referee: [Converse theorem section] The section stating the converse theorem: the assertion that the assignment is 'nearly one-to-one' is not accompanied by a precise characterization of the 'nearly' qualifier or a verification that the additional assumptions are preserved under the forward construction, leaving the equivalence claim unverifiable.

Authors: We acknowledge that a more precise characterization of the 'nearly one-to-one' correspondence is required for the claim to be fully verifiable. We will revise the relevant section to define what 'nearly' means in this context—specifically, that the two assignments are inverses up to isomorphism on the image of the forward map—and include a verification that the additional assumptions are indeed preserved when applying the forward construction to an orthomodular poset. This will strengthen the equivalence result. revision: yes

Circularity Check

0 steps flagged

Explicit constructions and conditional equivalences with no self-referential reductions

full rationale

The paper introduces an implication via explicit definition on orthomodular posets, constructs the corresponding unsharp residuated poset, and proves a converse direction under stated additional assumptions. These steps consist of direct algebraic definitions and equivalence proofs rather than any reduction of a claimed prediction to a fitted input, self-citation chain, or definitional tautology. No equations are shown to be equivalent by construction to their own inputs, and the work remains self-contained as a sequence of verifiable constructions without load-bearing self-citations or imported ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on the standard axioms of orthomodular posets as the starting point and introduces a new implication operation defined to satisfy unsharp adjointness.

axioms (1)

domain assumption Orthomodular poset axioms (standard background for quantum logic)
The paper takes orthomodular posets as given algebraic axiomatization of quantum logic.

pith-pipeline@v0.9.0 · 5649 in / 1079 out tokens · 23608 ms · 2026-05-24T16:38:56.801344+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 7 … R(P) is a divisible idempotent unsharp residuated poset.

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