Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Engineering

Four Equivalent Relations between MCP and CP and Its Implication in Quantum and Information Theory

Wan Xiao-long

Published May 25, 2022• 2 citations
Publisher

Abstract

(Norraw) modal propositional logic (MCP) tries to be a strict model for quantum and information theory, but it has serious difficulties in syntax, semantics, and metaphysics. According to the guiding definition of minimal hidden variables, this study presents four equivalences between MCP and classical propositional logic (CP). It concludes that MCP is CP containing minimal syntactic hidden variables, modal axiom represents the classification of CP formulas, and possible world is CP formula as assignment background, which establishes a simpler and unified basis for quantum information theory. The black modal of classical propositional logic (BCP), as an assisted discovery method, reveals that modal nature is actually to express the interdependence between things and their environment in a mathematical way, which can express the holism, dialectics, and uncertainty of metaphysics, and use mark hiding (e.g., f (x) is a function of x, f () is the functional expression, functional-e for short, and “□” is, in fact, a cluster of functional-es in which superscript and subscript are hidden) that does not affect the effectiveness of reasoning to express a more simplified, efficient, and consistent artificial intelligence.

DOI
10.1155/2022/1586534

Related papers

Quantum logic

Alireza Moazzen

This article employs a descriptive-analytical method with an algebraic and axiomatic approach to address quantum logic. Through several thought experiments, it is demonstrated that the distributive law for propositions does not hold in quantum mechanics. Therefore, the logic governing this domain of science is not classical logic. In fact, the algebra corresponding to the logic of quantum propositions is an orthomodular lattice, which is why the distributive law is not necessarily valid for such propositions. In quantum logic, various definitions are offered for the implication operator. Since none of these operators satisfy the deduction theorem, a suitable implication operator with this property must be defined. This objective is achieved by defining a unified quantum logic. By constructing a Lindenbaum-Tarski algebra for this unified quantum logic, it can be shown that it is indeed a quantum logic. Dishkant presented an embedding of quantum logic into the Br-modal logic system. This provides an appropriate semantics for quantum logic. A suitable model for this logic is the closed subspaces of a Hilbert space.

•Nov 1, 2025•0 citations

On the Philosophical Foundation of Mesoscopic Information Theory

Gang Liu

This paper lays out the philosophical foundation of mesoscopic information theory. Taking S4 modal logic and the M₃ diamond modular lattice as its mathematical skeleton, this work puts forward four modal‑probability bridging axioms which form a minimally‑sufficient axiom set: the modal necessity axiom, the modal‑probability consistency axiom, the modal‑probability regularity axiom, and the distributive‑law breaking axiom. This constraint system realizes a controlled transition from the Boolean base of S4 to the non‑Boolean M₃ lattice, embedding modal operators and probability measures within one unified formal framework. Global completeness is not pursued; only internal self‑consistency and traceable reasoning are required. This axiomatic apparatus yields a unified philosophical account of mesoscopic‑level features including modal tension, quasi‑quantum‑like behaviors and informational phase transitions. Empirical adequacy and real‑world case studies are reserved for future versions.

•Aug 25, 2026•0 citations

Strict implication in quantum modal logic

Kenji Tokuo

Abstract Quantum logic has no canonical implication connective, and polynomial candidates differ in the inferential principles they satisfy. We study a relational strict implication in quantum modal logic and interpret it through the modal accessibility relation. On reflexive modal frames, a formula formed with this connective is valid exactly when its antecedent semantically entails its consequent. The connective therefore internalizes semantic consequence. The forcing condition links modal accessibility to non-orthogonality. Consequently, any formula whose principal connective is this strict implication has the same truth value throughout each component determined by non-orthogonality. The truth set of each such formula therefore belongs to the Boolean algebra of unions of these components. We also establish further semantic laws and analyze quantum disjunction, contraposition, and nested implications.

Logic Journal of IGPL•Sep 4, 2026•0 citations

On the Philosophical Foundation of Mesoscopic Information Theory (v.2.0)

Gang Liu

Classical information theory and quantum information theory are respectively based on Boolean algebra and Hilbert space projection lattices, lacking a unified logical framework for cross-scale physical propositions covering classical, mesoscopic and quantum domains. Traditional physics attributes the classical-quantum transition to continuous spatial scale variation, which is essentially external parameter determinism and cannot explain the peculiar intermediate characteristics of mesoscopic systems. This paper constructs the philosophical and mathematical foundation of mesoscopic information theory based on the coupling of S4 modal logic and the non-distributive modular lattice M₃. A composite axiom system is proposed, and Axiom 3 (a–g) realizes the consistent embedding of S4 modal operators on non-distributive lattices. The cross-scale evolution of physical systems is reinterpreted as discrete phase transition of propositional lattice types, with an endogenous CHSH-type correlation threshold τ≈2.414 serving as the operational criterion for mesoscopic phase. A Möbius-weighted lattice entropy independent of Shannon and von Neumann entropy is established to characterize structural information of mesoscopic logical systems. Noise and decoherence are redefined as intrinsic signals of lattice structural transition rather than simple information dissipation. This study establishes the independent ontological status of mesoscopic propositional systems, breaks the scale-phenomenological paradigm in traditional cross-scale physics, and provides a new foundational framework for the unification of cross-scale information theory.

•Sep 19, 2026•0 citations

Philosophical Foundation of the Mesoscopic Order Parameter from the Point of View of Modal Information Theory(Abstract and Introduction)

Gang Liu

This paper reconstructs its philosophical foundation of Modal Information Theory (MIT). A complete deductive chain is constructed through Stone duality of Boolean lattices, Alexandroff topology and S4 modal logic correspondence, four modal-probability bridging axioms confining the structure to M3 diamond lattice, two-dimensional Hilbert space quantization, Gleason-Busch theorem and Tsirelson bound. The midpoint of the classical-quantum interval is defined as the mesoscopic order parameter η=1+√2≈2.4142, namely the silver critical value. Serving as the theoretical basis between classical and quantum regimes, it provides fundamental mathematical criterion for Liu Entropy and validates its dedicated mesoscopic information unit mobit. The paper demonstrates the minimality of the axiom system and extremal origin of the critical value, establishes partitioning criteria for classical, mesoscopic and quantum domains from both philosophy of science and quantum logic perspectives, and solidifies the axiomatic system of Modal Information Theory.

•Jul 30, 2026•0 citations

Classical Logic in the Quantum Context

Andréa Oldofredi

It is generally accepted that quantum mechanics entails a revision of the classical propositional calculus as a consequence of its physical content. However, the universal claim according to which a new quantum logic is indispensable in order to model the propositions of every quantum theory is challenged. In the present essay, we critically discuss this claim by showing that classical logic can be rehabilitated in a quantum context by taking into account Bohmian mechanics. It will be argued, indeed, that such a theoretical framework provides the necessary conceptual tools to reintroduce a classical logic of experimental propositions by virtue of its clear metaphysical picture and its theory of measurement. More precisely, it will be shown that the rehabilitation of a classical propositional calculus is a consequence of the primitive ontology of the theory, a fact that is not yet sufficiently recognized in the literature concerning Bohmian mechanics. This work aims to fill this gap.

arXiv:2012.02040•Quantum Reports•Dec 5, 2020•5 citations
(Norraw) modal propositional logic (MCP) tries to be a strict model for quantum and information theory, but it has serious difficulties in syntax, semantics, and metaphysics. According to the guiding definition of minimal hidden variables, this study presents four equivalences between MCP and classical propositional logic (CP). It concludes that MCP is CP containing minimal syntactic hidden variables, modal axiom represents the classification of CP formulas, and possible world is CP formula as assignment background, which establishes a simpler and unified basis for quantum information theory. The black modal of classical propositional logic (BCP), as an assisted discovery method, reveals that modal nature is actually to express the interdependence between things and their environment in a mathematical way, which can express the holism, dialectics, and uncertainty of metaphysics, and use mark hiding (e.g., f (x) is a function of x, f () is the functional expression, functional-e for short, and “□” is, in fact, a cluster of functional-es in which superscript and subscript are hidden) that does not affect the effectiveness of reasoning to express a more simplified, efficient, and consistent artificial intelligence.
This article employs a descriptive-analytical method with an algebraic and axiomatic approach to address quantum logic. Through several thought experiments, it is demonstrated that the distributive law for propositions does not hold in quantum mechanics. Therefore, the logic governing this domain of science is not classical logic. In fact, the algebra corresponding to the logic of quantum propositions is an orthomodular lattice, which is why the distributive law is not necessarily valid for such propositions. In quantum logic, various definitions are offered for the implication operator. Since none of these operators satisfy the deduction theorem, a suitable implication operator with this property must be defined. This objective is achieved by defining a unified quantum logic. By constructing a Lindenbaum-Tarski algebra for this unified quantum logic, it can be shown that it is indeed a quantum logic. Dishkant presented an embedding of quantum logic into the Br-modal logic system. This provides an appropriate semantics for quantum logic. A suitable model for this logic is the closed subspaces of a Hilbert space.
This paper lays out the philosophical foundation of mesoscopic information theory. Taking S4 modal logic and the M₃ diamond modular lattice as its mathematical skeleton, this work puts forward four modal‑probability bridging axioms which form a minimally‑sufficient axiom set: the modal necessity axiom, the modal‑probability consistency axiom, the modal‑probability regularity axiom, and the distributive‑law breaking axiom. This constraint system realizes a controlled transition from the Boolean base of S4 to the non‑Boolean M₃ lattice, embedding modal operators and probability measures within one unified formal framework. Global completeness is not pursued; only internal self‑consistency and traceable reasoning are required. This axiomatic apparatus yields a unified philosophical account of mesoscopic‑level features including modal tension, quasi‑quantum‑like behaviors and informational phase transitions. Empirical adequacy and real‑world case studies are reserved for future versions.
Abstract Quantum logic has no canonical implication connective, and polynomial candidates differ in the inferential principles they satisfy. We study a relational strict implication in quantum modal logic and interpret it through the modal accessibility relation. On reflexive modal frames, a formula formed with this connective is valid exactly when its antecedent semantically entails its consequent. The connective therefore internalizes semantic consequence. The forcing condition links modal accessibility to non-orthogonality. Consequently, any formula whose principal connective is this strict implication has the same truth value throughout each component determined by non-orthogonality. The truth set of each such formula therefore belongs to the Boolean algebra of unions of these components. We also establish further semantic laws and analyze quantum disjunction, contraposition, and nested implications.
Classical information theory and quantum information theory are respectively based on Boolean algebra and Hilbert space projection lattices, lacking a unified logical framework for cross-scale physical propositions covering classical, mesoscopic and quantum domains. Traditional physics attributes the classical-quantum transition to continuous spatial scale variation, which is essentially external parameter determinism and cannot explain the peculiar intermediate characteristics of mesoscopic systems. This paper constructs the philosophical and mathematical foundation of mesoscopic information theory based on the coupling of S4 modal logic and the non-distributive modular lattice M₃. A composite axiom system is proposed, and Axiom 3 (a–g) realizes the consistent embedding of S4 modal operators on non-distributive lattices. The cross-scale evolution of physical systems is reinterpreted as discrete phase transition of propositional lattice types, with an endogenous CHSH-type correlation threshold τ≈2.414 serving as the operational criterion for mesoscopic phase. A Möbius-weighted lattice entropy independent of Shannon and von Neumann entropy is established to characterize structural information of mesoscopic logical systems. Noise and decoherence are redefined as intrinsic signals of lattice structural transition rather than simple information dissipation. This study establishes the independent ontological status of mesoscopic propositional systems, breaks the scale-phenomenological paradigm in traditional cross-scale physics, and provides a new foundational framework for the unification of cross-scale information theory.
This paper reconstructs its philosophical foundation of Modal Information Theory (MIT). A complete deductive chain is constructed through Stone duality of Boolean lattices, Alexandroff topology and S4 modal logic correspondence, four modal-probability bridging axioms confining the structure to M3 diamond lattice, two-dimensional Hilbert space quantization, Gleason-Busch theorem and Tsirelson bound. The midpoint of the classical-quantum interval is defined as the mesoscopic order parameter η=1+√2≈2.4142, namely the silver critical value. Serving as the theoretical basis between classical and quantum regimes, it provides fundamental mathematical criterion for Liu Entropy and validates its dedicated mesoscopic information unit mobit. The paper demonstrates the minimality of the axiom system and extremal origin of the critical value, establishes partitioning criteria for classical, mesoscopic and quantum domains from both philosophy of science and quantum logic perspectives, and solidifies the axiomatic system of Modal Information Theory.
It is generally accepted that quantum mechanics entails a revision of the classical propositional calculus as a consequence of its physical content. However, the universal claim according to which a new quantum logic is indispensable in order to model the propositions of every quantum theory is challenged. In the present essay, we critically discuss this claim by showing that classical logic can be rehabilitated in a quantum context by taking into account Bohmian mechanics. It will be argued, indeed, that such a theoretical framework provides the necessary conceptual tools to reintroduce a classical logic of experimental propositions by virtue of its clear metaphysical picture and its theory of measurement. More precisely, it will be shown that the rehabilitation of a classical propositional calculus is a consequence of the primitive ontology of the theory, a fact that is not yet sufficiently recognized in the literature concerning Bohmian mechanics. This work aims to fill this gap.