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Conditionals and Modalities in Constructive Quantum Logics

This paper introduces and axiomatizes iEx-logic, a constructive quantum logic that combines intuitionistic and orthomodular logics via the Sasaki hook, characterizing its extensions as a product of intermediate and orthomodular logic lattices while exploring modal extensions.

Original authors: Juan P. Aguilera (TU Wien), Guillaume Massas (Chapman University)

Published 2026-07-01

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CC BY 4.0

Original authors: Juan P. Aguilera (TU Wien), Guillaume Massas (Chapman University)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to build a universal rulebook for how things "connect" and "change." In our everyday world, we have two very different rulebooks that seem to contradict each other:

The "Common Sense" Rulebook (Intuitionistic Logic): This is how we usually think. If I say "It is raining," and I know "If it is raining, the grass is wet," then I can conclude "The grass is wet." This logic is strict about cause and effect. It's like a well-organized library where every book has a clear place, and you can't have a book that is both "on the shelf" and "not on the shelf" at the same time.
The "Quantum" Rulebook (Orthomodular Logic): This is how the subatomic world works. In this world, things can be in a superposition (both here and there). The rules are looser; you can't always distribute ideas the way you do in common sense (e.g., "A and (B or C)" doesn't always equal "(A and B) or (A and C)"). It's like a magic trick where the rules of the game change depending on how you look at them.

For a long time, logicians (the people who study these rulebooks) struggled to find a single system that could handle both of these worlds, especially when you add a tricky ingredient: Implication (the "If... then..." part).

The Problem: The Missing Link

The authors of this paper, Juan Aguilera and Guillaume Massas, noticed that while they had a "Common Sense" rulebook and a "Quantum" rulebook, they didn't have a good way to say "If A, then B" in the Quantum world that made sense to the Common Sense world.

In the Quantum world, the standard "If... then..." rule is called the Sasaki Hook. It's a bit weird: it says "If A, then B" is true if either A is false, or A and B are both true. It's like a safety valve. But when you try to mix this with the strict "If... then..." of Common Sense, things break.

The Solution: The "Ex-Logic" Bridge

The authors decided to build a bridge. They started with a foundation they had already built called Fundamental Logic (which is a basic, minimal set of rules that both worlds agree on). Then, they added a new, super-strong rule called (iEx).

Think of (iEx) as a "Universal Translator." It's a complex, multi-part rule that ensures that no matter which world you are in (Common Sense or Quantum), the way you combine ideas (AND, OR, NOT) and the way you say "If... then..." doesn't lead to a contradiction.

They proved that if you take this new system, which they call iEx-logic, you get something amazing:

It is the exact intersection of the two worlds.
It contains only the rules that are true in both the Common Sense world and the Quantum world.
It is the "safest" possible logic you can have that respects both realities.

The Big Discovery: The Product of Two Worlds

The most exciting part of their discovery is how they described the structure of this new logic. They found that the entire landscape of these new rules is just a product of two simpler landscapes.

Imagine you have two Lego sets:

Set A: All the possible variations of Common Sense logic (Intermediate Logics).
Set B: All the possible variations of Quantum logic (Orthomodular Logics).

The authors proved that if you want to build a new logic that sits in the middle, you don't need to invent new bricks. You just need to take one brick from Set A and one brick from Set B and snap them together. The resulting structure is a valid "Constructive Quantum Logic."

This means the universe of these mixed logics is perfectly organized. It's not a messy pile; it's a neat grid where every possible mixed logic corresponds to a specific pair of a Common Sense rule and a Quantum rule.

Adding "Maybe" and "Must" (Modalities)

In the final section, the authors tried to add Modalities to their bridge. Modalities are words like "Must" (Necessity) and "Maybe" (Possibility).

In Common Sense: "It must be raining" (based on evidence).
In Quantum: "It might be raining" (based on probability).

They found that adding these words makes the bridge much harder to build. The rules for "Must" and "Maybe" behave differently in the two worlds, and they don't always play nice together. However, they managed to build a special, restricted version of the bridge (called miEx-algebras) where these words do work, provided you follow some very specific, strict instructions. They showed that even with these extra words, the "Product" rule still holds: the mixed logic is still just a combination of a "Must/Maybe" Common Sense logic and a "Must/Maybe" Quantum logic.

Summary

In simple terms, this paper says:

We found a way to perfectly merge the logic of everyday reasoning with the logic of quantum physics.
We did this by creating a new "If... then..." rule that works in both worlds.
We proved that any logic that tries to mix these two worlds is just a simple combination of a specific "Common Sense" rule and a specific "Quantum" rule.
We also showed how to add "Must" and "Maybe" to this mix, though it requires being very careful with the rules.

The paper doesn't claim this will cure diseases or build faster computers right now. It is a pure mathematical map, showing us exactly how these two different ways of thinking can coexist in a single, consistent system.

Technical Summary: Conditionals and Modalities in Constructive Quantum Logics

Problem Statement
The paper addresses the challenge of unifying constructive reasoning (intuitionistic logic) and quantum reasoning (orthomodular logic) within a single propositional framework. While "fundamental logic," introduced by Holliday, generalizes both by extending the signature $\{\land, \lor, \neg\}$ , it lacks a robust notion of implication. Orthologic (minimal quantum logic) typically lacks a conditional satisfying modus ponens, whereas intuitionistic logic relies heavily on the implication connective.

The authors investigate the intersection of intuitionistic logic and orthomodular logic when the latter is equipped with the Sasaki implication, defined as $a \to b := \neg a \lor (a \land b)$ . The central problem is to axiomatize the set of formulas valid in both systems simultaneously in the full signature $\{\land, \lor, \to\}$ . Previous work established that the intersection of intuitionistic logic and orthologic (without implication) is strictly larger than fundamental logic and is axiomatized by the axiom (Ex). This paper seeks to extend that result to include implication.

Methodology
The authors employ an algebraic approach, utilizing the theory of fundamental algebras (structures generalizing both Heyting algebras and orthomodular lattices). The methodology proceeds in three main stages:

Algebraic Definition: The authors define Ex-algebras, a class of fundamental algebras that generalize both orthomodular algebras (equipped with Sasaki implication) and Heyting algebras. This involves introducing a new axiom, (iEx) ("implicative Ex"), which generalizes the earlier (Ex) axiom to the implication signature. They also define auxiliary axioms (nS, MP, M, W4) to capture the behavior of the Sasaki implication and modus ponens.
Embedding Theorem: The core technical contribution is the iEx-Embedding Theorem. The authors prove that any Ex-algebra $L$ $L$ can be subdirectly embedded into the product of an orthomodular algebra ( $O_L$ $O_{L}$ ) and a Heyting algebra ( $I_L$ $I_{L}$ ).
- The orthomodular component $O_L$ is constructed as a quotient of $L$ by the equivalence relation $a \sim b \iff \neg a = \neg b$ .
- The Heyting component $I_L$ is constructed via a representation using prime filters and the topology of open sets, similar to the Stone representation for distributive lattices.
- Crucially, the authors demonstrate that the implication connective $\to$ is preserved under these homomorphisms, specifically showing that the Sasaki implication in the quotient aligns with the definition in orthomodular algebras.
Modal Extension: The framework is extended to include modal operators ( $\Box, \Diamond$ ). The authors define miEx algebras (modal iEx algebras) and corresponding classes of modal algebras: IFS algebras (intuitionistic Fischer Servi) and DOM algebras (distributive orthomodular). They prove a modal version of the embedding theorem, showing that miEx algebras embed into the product of a DOM algebra and an IFS algebra.

Key Contributions and Results

Axiomatization of the Intersection: The paper provides a complete axiomatization for the joint validities of intuitionistic logic and orthomodular logic (with Sasaki implication). The logic, termed iEx-logic, is shown to be the intersection of the two systems.
- Theorem 1.1 & 4.2: The intersection is axiomatized over fundamental logic by the axiom (iEx) and four additional principles:
  1. $\neg(a \to b) = \neg(\neg a \lor (a \land b))$ (nS)
  2. $a \land (a \to b) = a \land b$ (MP)
  3. $(a \land b) \to b = 1$ (M)
  4. $a \to (b \land c) \le a \to b$ (W4)
- The authors also provide an alternative axiomatization decomposing (iEx) into three axioms: (Nu), (Vi), and (iCl), alongside the implication-specific axioms.
Characterization of the Lattice of Extensions: A major structural result is the characterization of the lattice of all logics extending iEx-logic (ICQ logics).
- Theorem 4.6: The lattice of ICQ logics is isomorphic to the product of the lattice of superintuitionistic logics and the lattice of superorthomodular logics ( $iSE \simeq iSO \times iSI$ ). This implies that any extension of iEx-logic is uniquely determined by its projection onto the intuitionistic and orthomodular components.
Modal Logic Intersection: The paper establishes that the intersection of a specific class of intuitionistic modal logics (IFS) and quantum modal logics (DOM) is exactly the logic of miEx algebras (Corollary 5.11). This is achieved by restricting the modal operators to satisfy specific distributivity and involutive properties (e.g., axiom I1 and M3) that ensure the quotient maps preserve the modal structure.

Significance and Claims
The paper claims to answer a specific question regarding the "jointly valid principles" of intuitionistic and orthomodular logics when the Sasaki implication is used. By introducing Ex-algebras and the iEx-Embedding Theorem, the authors demonstrate that the intersection of these two distinct logical traditions is not merely a weak combination but a structured system that decomposes cleanly into its constituent parts.

The significance lies in:

Unification: Providing a rigorous algebraic and axiomatic framework for "constructive quantum logics" that includes implication, a feature previously difficult to reconcile with quantum logic's non-distributive nature.
Structural Clarity: The product decomposition of the lattice of extensions suggests that reasoning in this combined logic can be reduced to reasoning in its intuitionistic and orthomodular components separately, provided the axioms of iEx-logic are satisfied.
Modal Robustness: The extension to modal operators shows that this algebraic approach is robust enough to handle modalities, provided specific constraints (like those in Fischer Servi algebras and DOM algebras) are met to ensure the preservation of modal operators under the embedding.

The authors conclude by listing open problems, including the computational complexity of iEx-logic, the independence of the proposed axioms, and the search for finite axiomatizations with limited variable counts. They explicitly state that the investigation of constructive quantum modal logics is left for future work, noting that the current modal results rely on specific algebraic restrictions to overcome difficulties in preserving modal operators under quotient maps.