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.
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.
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.
This chapter provides a probabilistic framework for formulating classical probability theory, quantum probability, thermodynamics, diffusion, and the Wiener integral using a set of four axioms or principles. It explains everything that conventional quantum information theory and classical probability theory achieve. We want to emphasize that this framework is not an interpretation of quantum mechanics such as “Many-Worlds,” “Bohm’s Theory,” or the “Copenhagen interpretation.” It is much more general and can be viewed as a probability algorithm that calculates probabilities of future events. As a result, previously perplexing paradoxes find resolution. In particular, the superposition principle takes on a new meaning. Our probabilistic framework stands apart from the Hilbert space formalism. It relies solely on elementary set theory, classical logic, and complex numbers. Consequently, this theory is accessible for instruction in educational settings. This framework can be regarded as an axiomatic approach to probability in the sense of Hilbert. In his sixth of the twenty-three open problems presented at the International Congress of Mathematicians in Paris in 1900, Hilbert called for an axiomatic probability theory.
This paper, based on two fundamental physical axioms—Information Conservation (A1) and Computability (A2)—systematically argues for the intrinsic logical structure necessary for describing physical reality. Research shows that in aself-referential universe model constrained by local observations, the truth-valuejudgment logic of physical propositions necessarily deviates from classical Booleanlogic. The core root lies in the “local-global inequivalence” of physical systems: theinformation contained in the global physical state cannot be completely determinedby any finite observational procedure within a finite spacetime region. This leads tothe invalidity of the law of excluded middle within the system, and consequently,the logical structure mathematically manifests as a non-distributive orthomodularlattice, i.e., Birkhoff–von Neumann quantum logic. This structure is proven tocorrespond exactly to the subobject lattice in the ∞-topos TA constructed previously. This theoretical framework finally leads to a testable cosmological prediction:the central value of the non-Gaussianity parameter fNL of the cosmic microwavebackground (CMB) temperature fluctuations is about −3.2. Next-generation CMBexperiments (such as CMB-S4) will be able to test this prediction precisely, andthe results can be used to evaluate whether the quantum logical structure emergingfrom the basic axioms is suitable for describing the real universe.
(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.