Logic Realism Theory: Physical Foundations from Logical Constraints
Abstract
Logic Realism Theory (LRT) treats the three classical logical laws (Determinate Identity, Non-Contradiction, and Excluded Middle) as ontological constraints on physical instantiation, not merely rules of inference. The framework distinguishes I∞ (all representable configurations) from AΩ (the L₃-admissible subset that can be physically instantiated as stable records). This boundary condition generates quantum structure: vehicle-invariance under mathematically equivalent decompositions forces the Born rule via Gleason's theorem; local tomography requirements select complex Hilbert space over real alternatives. The framework is falsifiable (L₃ violations in stable records would refute it) and derives (not merely accommodates) the complex structure of quantum mechanics confirmed by Renou et al. (2021). Extensions to quantum field theory, general relativity, and cosmology are outlined. Full technical derivations appear in companion papers.
1. The Discovery
No stable experimental record, across the history of experimental science, has ever been documented as instantiating a direct violation of the classical laws of Identity, Non-Contradiction, or Excluded Middle. No detector record is both triggered and not triggered in the same respect. No logged outcome is both P and not-P in the same respect and at the same time. No measurement apparatus has yielded a contradiction at the level of actualized, stable records.
This pattern is not plausibly treated as a local artifact of one instrument class or one domain. It recurs across quantum experiments, particle physics, condensed matter, and astrophysics wherever results are registered as public records. Whatever we say about pre-measurement descriptions or quantum superposition, the level of actualized outcomes exhibits determinate, non-contradictory form.
Tahko (2009) defends the Law of Non-Contradiction as a metaphysical principle governing reality’s structure rather than a semantic convention or psychological limitation. The present work extends this logic-realist position in two ways: first, it generalizes the stance to the full $L_3$ package (Identity, Non-Contradiction, and Excluded Middle jointly); second, it develops the structural consequences that follow when the $L_3$ is treated as an ontological constraint on physical instantiation rather than merely on abstract possibility.
The track record matters. $L_3$ is not an unexamined assumption. Philosophers have actively sought violations: Priest defends true contradictions, paraconsistent logics formalize contradiction-tolerance, quantum logic challenges distributivity, vagueness theorists probe borderline cases. These are not marginal efforts; they represent sustained, rigorous attempts to show that $L_3$ can fail. Human minds explore these violations fluently. Yet across a century of precision physics, no stable experimental record has ever instantiated one. The asymmetry is stark: representation permits what instantiation forbids. This asymmetry is the empirical foundation of Logic Realism Theory.
Positioning. LRT is “physics-first” in the sense that it starts from what any workable physical theory must deliver: stable, public records with determinate content. The claim is not that logic is an interpretive overlay on physics, but that $L_3$ admissibility is a boundary condition on physical instantiation. That boundary condition is operationally load-bearing: it restricts the space of viable state spaces, measures, and correlation structures, and it yields empirical discriminators rather than mere reinterpretations.
The discovery claim is modest but sharp: physics only proceeds because its public outputs are $L_3$ shaped. Logic Realism Theory treats this not as a human convention but as evidence that instantiation itself is constrained. The methodology is empirical (observe universal conformity of records) → structural (identify the constraint pattern) → derivational (determine what that pattern requires). The task is to develop what physics looks like under that constraint.
It from Bit, Bit from Fit. Wheeler’s famous slogan “It from Bit” proposed that physical reality (It) emerges from information (Bit). LRT accepts this but goes one layer deeper: information itself requires a grounding. Bits exist because $L_3$ permits distinction; without Determinate Identity, there is no fact of the matter about whether something is 0 or 1. The complete slogan is therefore: It from Bit, Bit from Fit: physical structure (It) emerges from informational structure (Bit), which emerges from logical admissibility (Fit = $L_3$). This three-level picture distinguishes LRT from purely operational reconstruction programs: we derive not just “what mathematical structure fits the data” but “why that structure is the only one logic permits.”
1.1 The Core Distinction
The framework distinguishes two domains (defined formally in §2.1-2.3):
$I_\infty$: The space of all representable configurations: everything that can be specified, described, conceived, or formally expressed, without restriction to coherence or consistency. This includes contradictions, impossibilities, and violations of every logical principle. $I_\infty$ carries no ontological commitment; it is simply the totality of what representation permits.
$A_\Omega$: The constraint-class of configurations that satisfy the Three Fundamental Laws of Logic ($L_3$: Identity, Non-Contradiction, and Excluded Middle). These are the configurations that can be physically instantiated as stable records.
The relationship between them is the foundation of Logic Realism Theory.
1.2 The Research Program
If the $L_3$ constrains which configurations can be instantiated, what follows for the structure of physics? What constraints does the $L_3$ place on admissible physical theories, measurable quantities, and dynamical evolution? What can we derive from the $L_3$ rather than postulate empirically?
The answers turn out to be substantial. Key structural features of quantum mechanics (most centrally the Born rule and the admissible form of nonlocal correlations) follow from applying the $L_3$ rigorously to physical systems, with full derivations developed in companion work. These are not independent empirical facts about quantum mechanics but consequences of the requirement that physical records satisfy Identity (Id).
This introduction presents the Logic Realism Theory framework and demonstrates its scope. Section 2 develops the formal structure: $I_\infty$, $A_\Omega$, and the $L_3$ as admissibility condition. Section 3 establishes the vehicle/content distinction and its role in quantum mechanics. Section 4 sketches the derivation of quantum structure from the $L_3$, with full technical details in companion papers. Section 5 addresses objections. Section 6 outlines the research program extending LRT to quantum field theory, general relativity, and cosmology. Section 7 concludes.
The goal is not to argue for Logic Realism Theory as one interpretation among others but to show that it generates a productive research program with testable consequences. If the $L_3$ constrains instantiation, physics has logical foundations. The aim here is to show what follows if that constraint is treated as real and load-bearing rather than as a mere feature of language.
1.3 Scientific Status: Popper and Lakatos
Popperian falsifiability. Logic Realism Theory satisfies Popper’s demarcation criterion: it makes risky predictions that could be empirically refuted. The framework predicts that (1) all stable records are $L_3$ admissible, (2) probability measures are vehicle-invariant, (3) complex QM is required for local tomography, and (4) objective collapse parameters (if they exist) are derivable from fundamental constants. Each prediction specifies conditions under which LRT would be falsified, not merely adjusted. A single stable record instantiating a direct $L_3$ violation (say, a detector simultaneously registering “triggered” and “not triggered” in the same respect) would refute the core thesis.
The framework is not unfalsifiable. It takes empirical risk.
Lakatosian research program. Lakatos distinguished progressive from degenerating research programs by their ability to predict novel facts. LRT qualifies as progressive on three counts:
Hard core: The $L_3$ (Identity, Non-Contradiction, Excluded Middle) as constraints on physical instantiation. This is non-negotiable; abandoning it would abandon the program.
Protective belt: Specific implementations (vehicle-invariance formulation, Hilbert space representation, interface criterion) are negotiable. These can be refined, replaced, or extended without abandoning the hard core.
Progressive predictions: Reconstruction theorems (Hardy, 2001; Masanes & Müller, 2011) establish local tomography as a foundational axiom constraining field choice. Building on this, LRT predicted complex QM over real alternatives before Renou et al. (2021) designed the experimental test. The framework predicts derivability constraints on collapse parameters before such experiments are performed. It predicts correlation bounds (Tsirelson not PR-box) and information preservation (unitary black hole evaporation) in advance of decisive experimental confirmation. These are novel predictions, not post-hoc accommodations.
1.4 Companion Papers
This Position Paper establishes the LRT framework. Detailed technical derivations appear in companion papers:
- Hilbert Space Paper: Derives complex Hilbert space from Determinate Identity via local tomography
- Born Rule Paper: Derives the Born rule from vehicle-weight invariance
- QFT Statistics Paper: Derives the symmetrization postulate (bosons/fermions) from Determinate Identity
- GR Extension: Explores spacetime implications; derives identity continuity constraints and quadratic kinematic structure (programmatic)
The papers can be read independently, but together form a unified derivation from foundational logic to quantum mechanics and beyond.
One-world realism. LRT shares structural features with Many-Worlds (Everett): quantum states evolve unitarily, branching structure is real, decoherence explains apparent classicality. But LRT differs fundamentally in ontology. In Many-Worlds, all branches are equally instantiated: the universe continually splits into coexisting worlds. In LRT, branching structure exists in $I_\infty$ (the representational space), but only one $L_3$-admissible outcome history is ever instantiated in $A_\Omega$. The Born rule weights don’t describe “how much” of reality each branch gets; they describe the objective disposition of a single world toward its possible futures. This is one-world realism with Everettian structure: the mathematics of branching without the ontology of multiplication.
Commitments, Non-Commitments, and Falsifiers
What LRT commits to:
- Stable public records are $L_3$ admissible (satisfy Identity, Non-Contradiction, and Excluded Middle jointly).
- This constraint is ontological (about what can be instantiated), not merely semantic (about how we describe things).
- Outcome-measure must be invariant under mathematically equivalent decompositions (different ways of describing the same event cannot assign different total weights).
- The representational vehicle (quantum state) encodes outcome-possibility structure determinately, even when outcomes themselves are not yet sharp.
What LRT does not commit to:
- No commitment to collapse as a physical mechanism. “Actualization” is category-transition (representable → instantiated), not a dynamical process requiring new physics.
- No commitment that state vectors are ontologically fundamental entities “out there” as Platonic objects. They are representational vehicles within the theory.
- No denial that alternative ontologies (Bohmian, Many-Worlds, etc.) exist. The claim is that any empirically adequate theory must respect the constraint that stable records belong to $A_\Omega$.
- No claim that $L_3$ constraints derive all of physics. Specific field content, coupling constants, and initial conditions remain empirical inputs.
Falsification criteria (examples):
- A stable, reproducible public record instantiating a direct $L_3$ violation (e.g., detector output simultaneously registering “triggered” and “not triggered” in the same respect).
- A consistent, repeatable measurement scenario where mathematically equivalent decompositions of the same event yield systematically different total probability weights (violating vehicle-invariance).
- An experimental realization of the Renou et al. (2021) network scenario that violates the real-quantum-theory bound, which would support complex QM over real-Hilbert-space QM and confirm LRT’s local tomography requirement.
- Empirical confirmation of objective collapse with free parameters that cannot be derived from fundamental constants (violating the parsimony requirement).
2. The Framework: Formal Development
2.1 The Space of Representable Configurations
$I_\infty$ is not a domain of entities. It carries no ontological commitment. It is the totality of what can be specified, described, or conceived, including configurations that violate every logical principle.
Examples of elements in $I_\infty$:
- Consistent configurations: “electron with spin up”
- Inconsistent configurations: “electron with spin both up and down in the same basis”
- Impossible configurations: “round square,” “married bachelor”
- Formally specified contradictions: paraconsistent theorems, dialetheia
- Vague specifications: “somewhat round square”
- Indeterminate specifications: “object without determinate identity”
The last example is crucial. $I_\infty$ includes configurations that violate Determinate Identity itself, configurations specified as lacking determinate identity. These are representable (we just represented one) but not instantiable.
$I_\infty$ is closed under representation operations:
- Negation: If configuration $c \in I_\infty$, then “not-$c$” $\in I_\infty$
- Conjunction: If $c_1, c_2 \in I_\infty$, then “$c_1$ and $c_2$” $\in I_\infty$ (even if contradictory)
- Disjunction: If $c_1, c_2 \in I_\infty$, then “$c_1$ or $c_2$” $\in I_\infty$
- Quantification: Universal and existential quantification over $I_\infty$ elements
This closure property ensures $I_\infty$ contains every representational construction, including those that build contradictions from consistent elements.
2.2 The Three Fundamental Laws as Ontological Constraints
The $L_3$ comprises three classical logical principles, understood here as constraints on instantiation rather than constraints on inference or description:
Determinate Identity (Id): Every instantiated configuration is determinately what it is, independent of description or decomposition. For any admissible ways of describing or decomposing configuration $i \in A_\Omega$, they all pick out the same $i$. Formally, this is the requirement that identity is invariant under admissible redescription:
\[\text{For any descriptions } d, d' \text{ of the same configuration: } d(i) = d'(i) = i\]This is not the trivial claim that identity is reflexive ($i = i$). It is the substantive claim that instantiated configurations have determinate identity: they are self-identical in a way that does not depend on how they are described, measured, or decomposed. A configuration without determinate identity is not vague or indeterminate; it fails to be a configuration at all.
Non-Contradiction (NC): No instantiated configuration simultaneously possesses and lacks a property in the same respect. For any well-defined property $P$ and configuration $i \in A_\Omega$:
\[\neg(P(i) \land \neg P(i))\]The “in the same respect” qualification is essential. An electron can have spin-up relative to the z-axis and spin-down relative to a rotated axis. But it cannot have spin-up and spin-down relative to the same measurement basis.
Excluded Middle (EM): For any well-defined property $P$ applicable to an instantiated configuration $i \in A_\Omega$, either $i$ possesses $P$ or $i$ lacks $P$:
\[P(i) \lor \neg P(i)\]Here EM is a constraint on instantiation for sharply specified properties, not a claim about what an agent can know or prove. The law applies to instantiated configurations with respect to well-defined properties, properties that have determinate applicability conditions. It does not require that we know which disjunct holds, only that exactly one does hold.
These three laws are not independent. They are three aspects of determinacy: a configuration is determinate when it is self-identical (Id), possesses no contradictory properties (NC), and has definite status with respect to all applicable properties (EM). Together they define what it means for a configuration to be instantiable.
2.3 Qualitative and Quantitative Identity
Determinate Identity admits a natural distinction between two aspects:
Qualitative identity concerns what kind of thing a configuration is: an electron rather than a photon, a spin-up state rather than spin-down, configuration A rather than configuration B. Qualitative identity is categorical: a configuration either is or is not of a given type.
Quantitative identity concerns how much or where precisely: the magnitude of a field, the position along a continuum, the degree of some continuous parameter. Quantitative identity admits continuous variation while preserving qualitative type.
This distinction has structural consequences:
Lemma (Bounded Distinguishability). For any two configurations $c_1, c_2 \in A_\Omega$ with the same qualitative identity but different quantitative identity, there exists a continuous path in $A_\Omega$ connecting them.
Proof sketch. Qualitative identity is preserved under continuous variation of quantitative parameters. Since both endpoints are $L_3$-admissible and differ only quantitatively, and since $A_\Omega$ is closed under $L_3$-preserving limits, the intermediate configurations along any monotonic interpolation remain admissible. $\square$
This lemma grounds the continuity structure of physical configuration space. Configurations of the same type form connected components; discontinuities in physical evolution require qualitative identity changes (which §2.5 links to temporal ordering).
The distinction also illuminates permutation symmetry (see QFT Statistics Paper, §4): intrinsically identical particles share qualitative identity, and any state attributing quantitative differences that would distinguish them violates Id.
2.4 The Admissibility Condition
Definition. Let $A_\Omega$ be the constraint-class of $L_3$ admissible configurations in $I_\infty$. Define:
\[L_3(i) := \text{Id}(i) \land \text{NC}(i) \land \text{EM}(i)\]Then:
\[A_\Omega := \{ i \in I_\infty : L_3(i) \}\]where Id is Determinate Identity, NC is Non-Contradiction, and EM is Excluded Middle. A configuration here is a candidate state-description in $I_\infty$. An instantiated configuration is one that satisfies the $L_3$ constraints (belongs to the constraint-class $A_\Omega$).
Empirical claim. All stable physical records appear to fall within $A_\Omega$. No experimental record has been documented as belonging to $I_\infty$ \ $A_\Omega$. For avoidance of doubt: this is a claim about stable, public records, not about all internal theoretical descriptions or counterfactual propositions.
Substantive thesis. Defining $A_\Omega$ does not explain why the physical world is confined to it. The definition specifies the constraint-class. The substantive claim is that this confinement is not contingent (a lucky regularity) and not merely methodological (an artifact of how we measure), but rooted in what it is to be physically instantiated. Logic Realism Theory is the claim that instantiation itself is governed by the $L_3$. The research program asks what physics looks like under that constraint.
2.5 Joint Inadmissibility and Temporal Ordering
Two individually admissible configurations may be jointly inadmissible. Let $c_1, c_2 \in A_\Omega$ be configurations such that:
\[c_1 \land c_2 \notin A_\Omega\]This occurs when the conjunction violates NC or EM. For example, “particle at position $x_1$” and “particle at position $x_2 \neq x_1$” are individually admissible but jointly inadmissible for the same particle at the same time.
Joint inadmissibility generates temporal ordering. If $c_1$ and $c_2$ cannot be co-instantiated but can be sequentially instantiated, time emerges as the ordering structure that separates jointly inadmissible configurations.
Definition (Temporal Ordering): Configurations $c_1, c_2 \in A_\Omega$ stand in temporal relation “$c_1$ before $c_2$” if:
- $c_1 \land c_2 \notin A_\Omega$ (jointly inadmissible)
- Both $c_1$ and $c_2$ are instantiated
- $c_1$ instantiates, then ceases, then $c_2$ instantiates
Time is not geometry. Time is the logical sequencing necessitated by joint inadmissibility of individually admissible configurations.
3. Vehicle/Content and Quantum Mechanics
3.1 Quantum States as Representational Vehicles
Having distinguished representable content from instantiated records (§2), we now apply the same vehicle/content split to quantum theory: the state vector functions as a representational vehicle that encodes a structured space of outcome-possibilities, while measurement records are the instantiated outputs constrained by the $L_3$.
The vehicle/content distinction explains why minds can represent what reality cannot instantiate: the representing is always $L_3$ admissible, even when the represented is not. The same distinction explains quantum superposition without paradox.
3.2 The Asymmetry as Evidence
This vehicle/content distinction generates a testable asymmetry. If the $L_3$ constrained only representation (as psychologism claims), then:
- Representing violations should be difficult or impossible (since vehicles must satisfy the $L_3$)
- Instantiating violations should be possible (since reality would be unconstrained)
If the $L_3$ constrains only instantiation (as logic realism claims), then:
- Representing violations should be possible (vehicles satisfy the $L_3$ while contents need not)
- Instantiating violations should be impossible ($A_\Omega$ is defined by $L_3$ satisfaction)
The empirical record supports the second pattern. Conceiving contradictions is cognitively trivial; instantiating them has never been observed. This asymmetry (representation permits what instantiation forbids) is the empirical foundation introduced in §1.
3.3 Superposition as Representational Vehicle
| The vehicle/content distinction becomes essential when analyzing quantum mechanics. A quantum state $ | \psi\rangle$ is not a direct description of an instantiated configuration but a representational vehicle encoding outcome-possibilities. |
Consider the paradigm case: a qubit in superposition
\[|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\]This state is a well-defined mathematical object within the theory. The state vector is a precise representational vehicle that can be manipulated without contradiction. Whatever physical substrate realizes the situation, the representation is determinate as a vehicle: it is the specific superposition with coefficients $(1/\sqrt{2}, 1/\sqrt{2})$, with definite inner products with other states, with determinate evolution under unitary operators.
| The content (what the vehicle represents about outcome-possibilities) concerns what will happen when measurement in the {$ | 0\rangle$, $ | 1\rangle$} basis occurs. The state does not describe a configuration that both has and lacks a definite value (NC violation), nor does it describe a configuration outside the true/false dichotomy (EM violation). It describes a physical situation such that when measurement occurs, exactly one outcome will be recorded, and that record will be $L_3$ admissible. |
3.4 What Superposition Represents
The critical question: what does superposition represent in the LRT framework?
| Not a contradiction. The state $\alpha | 0\rangle + \beta | 1\rangle$ does not represent a configuration that is both “definitely 0” and “definitely 1.” It represents a physical situation that is not yet determinate with respect to the sharp property “has definite value 0 or 1 in this basis.” |
Not vagueness. The state is not a vague or imprecise specification of “either 0 or 1 but we don’t know which.” The superposition has definite mathematical structure: specific coefficients, definite phase relations, determinate inner products with other states. It is precisely what it is as a representational vehicle.
Admissible structure poised toward outcomes. The superposition represents a physical situation whose public records, when produced in that measurement context, are always $L_3$ admissible and mutually exclusive. The state encodes how the physical situation is disposed toward those outcomes via the Born rule probabilities.
3.5 Measurement Contexts and Sharp Properties
| The measurement basis determines which properties are sharp. For state $ | \psi\rangle$ = $\alpha | 0\rangle + \beta | 1\rangle$: |
-
Property $P_1$: “has value 0 in computational basis,” not sharp for $ \psi\rangle$ prior to measurement -
Property $P_2$: “corresponds to state vector $ \psi\rangle$ in Hilbert space,” sharp for the representational vehicle -
Property $P_3$: “would yield outcome 0 with probability $ \alpha ^2$ upon computational basis measurement,” sharp for the representational vehicle
| The third property is crucial. The quantum state determinately possesses the property “encodes probability $ | \alpha | ^2$ for outcome 0.” This probability is not epistemological uncertainty about a pre-existing value; it is a determinate feature of how the representational vehicle characterizes the physical situation. |
When we say measurement “actualizes” an outcome, we mean: the physical situation transitions to a configuration that produces a stable record displaying a sharp property assignment. The record is $L_3$ admissible. The transition is from a situation represented by a superposition (vehicle encoding outcome-possibilities) to a situation represented by a definite eigenstate (vehicle encoding sharp property assignment).
3.6 Entanglement and Non-Local Correlations
Entangled states present a case where the vehicle/content distinction clarifies non-locality.
Consider the Bell state:
\[|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\]| This is a single representational vehicle encoding a two-particle configuration. The vehicle is well-defined in the theory: it is determinately the state $ | \Phi^+\rangle$, with definite coefficients and inner products. What it represents is a physical situation where neither particle’s outcome is sharp relative to measurement basis prior to measurement, yet the outcomes will be perfectly correlated. |
The key insight: the representational vehicle is non-local. It cannot be factorized into separate vehicles for particles A and B. The physical situation is represented as a unified configuration spanning both locations. When measurement occurs at one location, the global vehicle transitions to a new global vehicle representing a configuration where both particles now produce correlated sharp records.
This is not action at a distance within the physical situation itself. The vehicle was already global; measurement reveals which of the globally-correlated outcomes actualizes. The correlation is encoded in the vehicle structure prior to measurement.
Bell inequality violations demonstrate that no local hidden variable account can reproduce these correlations. Within LRT, any hidden-variable completion compatible with Bell must either be nonlocal or contextual. LRT predicts that outcome-structure is globally constrained at the level the vehicle encodes.
3.7 Summary: Layered Representation
The picture that emerges has three layers:
Layer 1: Representable configurations ($I_\infty$) All specifications, including contradictions and impossibilities. No constraint.
Layer 2: Quantum states as representational vehicles in the theory Well-defined Hilbert space vectors. These are $L_3$-consistent descriptions within the mathematical formalism. They encode outcome-possibilities via probability distributions, but they are not themselves “instantiated records” in the sense of experimental outputs.
Layer 3: Measurement outcomes as instantiated records Sharp property assignments in stable, public records. These are the $L_3$-admissible configurations that experimental records display. Determinately 0 or 1, spin-up or spin-down, etc. Only this layer is directly instantiated as public record.
Layer 2 is the theory’s vehicle for representing how Layer 3 distributions arise from physical situations. The vehicle/content distinction operates at each level:
- Mind states (vehicles satisfying the $L_3$) represent impossible configurations (content in $I_\infty$)
- Quantum states (vehicles in mathematical formalism) represent outcome-possibility structures (content pointing toward Layer 3 records)
- Measurement outcomes (instantiated records) represent physical properties (content as property assignments)
The framework is realist about all three layers in different senses: $I_\infty$ exists as the space of representational possibility, quantum states exist as well-defined vehicles within our best theory, and measurement outcomes exist as actual instantiated records in $A_\Omega$.
4. From Vehicle-Invariance to the Born Rule
4.1 The Derivation Strategy
The Born rule states that the probability of measuring outcome $\phi$ given state $\psi$ is:
\[P(\phi|\psi) = |\langle\phi|\psi\rangle|^2\]This is not postulated in LRT but derived from the requirement that representational vehicles characterize physical situations determinately. The derivation proceeds in four steps:
- Vehicle-invariance requirement: Id (Determinate Identity) applied to representational vehicles
- Measure structure: Vehicle-invariance generates additivity and non-contextuality constraints
- Gleason’s theorem: These constraints uniquely determine the measure form
- Born rule: The $\lvert\psi\rvert^2$ form emerges as the unique solution
4.2 Vehicle-Invariance: The Core Requirement
| A quantum state $ | \psi\rangle$ is a representational vehicle encoding outcome-possibilities for measurement contexts. For any measurement context M with possible outcomes {$\phi_i$}, the state assigns probabilities: |
The question: what constrains this probability assignment?
Key insight: A single physical situation can be described using different bases, different decompositions, different mathematical representations, all equivalent in the sense that they describe the same measurement event. Vehicle-invariance is the requirement that these equivalent descriptions assign the same total probability weight to the event.
Formal statement: Let E be a measurement event (identified with projector P_E). If descriptions $d$ and $d’$ both represent E using different orthonormal bases or decompositions, then:
\[P_d(E|\psi) = P_{d'}(E|\psi)\]The probability assigned to E cannot depend on which mathematically equivalent description we use.
This is not a physical assumption about measurement apparatus. It is a logical requirement from Id: if the representational vehicle determinately characterizes the physical situation, then probability assignments that vary with mere representational choice would violate that determinacy. The vehicle would fail to pick out a determinate probability structure.
4.3 From Vehicle-Invariance to Additivity
Vehicle-invariance immediately generates the additivity constraint.
Consider a measurement with outcomes {$\phi_1$, $\phi_2$, $\phi_3$}. We can describe the event “outcome is $\phi_1$ or $\phi_2$” in two equivalent ways:
Description 1: Treat the event as a single composite outcome E = {$\phi_1$, $\phi_2$}.
Description 2: Treat the event as the disjunction of two separate outcomes $\phi_1$ and $\phi_2$.
Vehicle-invariance requires these assign the same probability:
\[P(E|\psi) = P(\phi_1|\psi) + P(\phi_2|\psi)\]This is additivity: the probability of a disjoint union of outcomes equals the sum of individual probabilities.
Key point: Additivity is not assumed as a probability axiom. It is derived from vehicle-invariance. The representational vehicle must assign the same weight to an event regardless of whether we describe it as “composite outcome E” or “$\phi_1$ or $\phi_2$.”
4.4 From Vehicle-Invariance to Non-Contextuality
Vehicle-invariance also generates non-contextuality of the probability measure.
| Consider a two-dimensional system (qubit). We can measure in the computational basis {$ | 0\rangle$, $ | 1\rangle$} or in the Hadamard basis {$ | +\rangle$, $ | -\rangle$}. Both are valid measurement contexts for the same system. |
| Now consider a specific outcome, the projector $P_0 = | 0\rangle\langle 0 | $. This projector can be expressed in multiple bases. Both expressions represent the same measurement event. |
Vehicle-invariance requires:
\[P(P_0|\psi) = \text{same value regardless of which basis decomposition we use}\]This is non-contextuality: the probability assigned to a projector cannot depend on which other projectors it is measured alongside (which basis embeds it).
Crucial distinction: We are not claiming that measurement outcomes are non-contextual in the sense that values pre-exist. We are claiming that the probability measure over outcomes must be non-contextual in the sense that projector weights don’t vary with basis choice. The Kochen-Specker theorem concerns value assignment; this concerns measure assignment. These are compatible.
4.5 Gleason’s Theorem
We now have two constraints derived from vehicle-invariance:
- Additivity: For disjoint outcomes, $P(E_1 \cup E_2) = P(E_1) + P(E_2)$
- Non-contextuality: Projector weights independent of basis embedding
Gleason’s theorem (1957) proves that in Hilbert spaces of dimension $\geq 3$, any probability measure satisfying these two constraints must have the form:
\[P(P_\phi|\psi) = \text{Tr}(\rho P_\phi)\]where $\rho$ is a density operator and $P_\phi$ is the projector onto outcome $\phi$.
| For pure states, $\rho = | \psi\rangle\langle\psi | $, giving: |
This is the Born rule. The $\lvert\psi\rvert^2$ form is the unique probability measure satisfying the constraints that vehicle-invariance imposes.
What Gleason’s theorem does: It proves uniqueness. Given additivity and non-contextuality, the Born rule is the only possibility (in dimension $\geq 3$).
What LRT contributes: The justification for additivity and non-contextuality. These are not independent probability axioms but consequences of Id (Determinate Identity) applied to representational vehicles. The Born rule is derived, not postulated.
4.6 Physical Interpretation
The Born rule derivation reveals its physical meaning. The rule is not an independent postulate about measurement but a consequence of requiring that:
- Quantum states are determinate vehicles (Id applied to representation)
- Vehicles encode probability structure (outcomes are $L_3$ admissible but which actualizes is probabilistic)
- Probability assignments are vehicle-invariant (don’t vary with representational choice)
| The $ | \langle\phi | \psi\rangle | ^2$ form emerges as the unique way to assign probabilities that respects these requirements. |
Why the squared magnitude? Three reasons:
| First, from inner product structure. The Born rule uses the inner product $\langle\phi | \psi\rangle$, which encodes distinguishability structure. The squared magnitude $ | \langle\phi | \psi\rangle | ^2$ converts distinguishability (which can be negative/complex-valued) into probability (which must be real and non-negative). |
Second, from normalization. For state $\psi$ expanded in basis ${\phi_i}$: \(|\psi\rangle = \sum_i c_i |\phi_i\rangle\) Normalization requires $\langle\psi|\psi\rangle = 1$, giving $\sum_i |c_i|^2 = 1$. This matches probability normalization $\sum_i P(\phi_i|\psi) = 1$ precisely when $P(\phi_i|\psi) = |c_i|^2$.
Third, from Gleason’s uniqueness. Given additivity and non-contextuality, the exponent 2 is not chosen but forced. Any other exponent would violate one of these constraints.
The Born rule is not a brute fact about quantum mechanics but the unique probability measure compatible with vehicle-invariance in Hilbert space.
4.7 Comparison with Standard Approaches
| Approach | Born Rule Status | Justification |
|---|---|---|
| Copenhagen | Postulated | “It works” |
| Many-Worlds | Derived? | Decision-theoretic (contested) |
| Bohmian | Emergent | Quantum equilibrium (conditional) |
| GRW | Modified | Stochastic collapse postulate |
| Relational QM | Relational | Perspective-dependent |
| LRT | Derived | Vehicle-invariance (Id) |
The LRT derivation is rigorous (using Gleason’s theorem), unconditional (not dependent on equilibrium assumptions), and grounded in logical constraints (not decision theory or perspective).
5. Objections and Replies
5.1 “This Is Just Semantic Relabeling”
Objection. You define $I_\infty$ and $A_\Omega$, observe that physics uses $L_3$ language, and declare victory. But all you have done is relabel “measurement outcomes” as “$A_\Omega$ instantiations” and “quantum states” as “vehicles in $I_\infty$.” This is circular.
Reply. The objection confuses two questions: (1) whether $A_\Omega$ exists as a mathematical definition, and (2) whether that definition does explanatory work.
Yes, $A_\Omega$ is defined to be the $L_3$ admissible subset. That is not in dispute. The substantive claim is that this definitional boundary is operationally load-bearing: it restricts what kinds of state spaces, probability measures, and correlation structures can interface with stable records. The framework then derives specific mathematical consequences (Hilbert space, Born rule, Bell-bound violations but not PR-box violations) from that constraint.
A semantic relabeling would leave all structure unexplained. LRT derives structure: vehicle-invariance forces the Born rule, $L_3$ boundaries exclude super-quantum correlations, local tomography selects complex over real Hilbert space (Renou et al. 2021). These are not renamings but restrictions with empirical bite.
5.2 “The L-Triad Constraint Is Trivial”
Objection. Identity, Non-Contradiction, and Excluded Middle are tautologies. Of course physical records satisfy them: records are defined to be determinate, non-contradictory outcomes. You have not discovered a constraint on physics; you have stipulated what counts as a “record” and then noted that records meet your stipulation.
Reply. The constraint is not trivial because its violation is conceivable and its enforcement has mathematical consequences.
Conceivability: Paraconsistent logics (da Costa, Priest) provide rigorous formal systems where contradictions are tolerable. Impossible-worlds semantics models scenarios where logical laws fail. These frameworks are internally consistent and mathematically well-developed. The fact that we can formally specify $L_3$ violations shows they are not inconceivable.
Non-observation: Despite conceivability, physical reality never produces $L_3$ violations at the record level. No detector has ever simultaneously fired and not-fired. No measurement has yielded both A and ¬A. This universal conformity requires explanation.
Mathematical consequences: Treating the $L_3$ as a boundary condition on instantiation restricts mathematical structure. Vehicle-invariance (from Id) forces Born-rule probability. Non-Boolean representational structure (from $L_3$ on vehicles vs. $L_3$ on records) requires complex Hilbert space. Excluded Middle on outcomes constrains correlation bounds (Tsirelson not PR-box). These are derivations, not tautologies.
5.3 “You’ve Smuggled in Quantum Mechanics”
Objection. The vehicle/content framework presupposes Hilbert space, superposition, and measurement: essentially the full apparatus of quantum mechanics. You claim to derive quantum structure, but you have built it into your starting definitions. This is circular reasoning disguised as derivation.
Reply. The framework does not presuppose quantum mechanics. It presupposes only: (1) representational vehicles can be mathematically specified, (2) those vehicles encode outcome-possibility structure, (3) outcome records are $L_3$ admissible, and (4) vehicle-invariance holds for probability assignments.
These are minimal structural requirements, not quantum-specific assumptions. Classical probability theory satisfies (1)–(4) using real-valued distribution functions as vehicles. The question is: what additional structure emerges when vehicles must encode quantum phenomena (interference, entanglement, contextuality)?
The derivation proceeds as follows:
- Hilbert space: Emerges from requiring that vehicles encode interference patterns while respecting vehicle-invariance. Classical probability (real functions) cannot represent interference without violating Born-rule additivity.
- Superposition: Falls out of requiring that vehicles can represent situations where outcome-likelihoods are determinate but outcomes themselves are indeterminate.
- Born rule: Derived in §4 from vehicle-invariance plus non-contextuality, not presupposed.
The appearance of circularity arises because quantum mechanics is the output of the constraint structure. Once derived, quantum formalism becomes the natural language for discussing vehicles, but that does not mean it was assumed at the start.
5.4 “This Doesn’t Solve the Measurement Problem”
Objection. The measurement problem asks: how does unitary evolution (Schrödinger equation) produce definite outcomes (wavefunction collapse)? You have renamed “collapse” as “vehicle-to-record transition” and declared the problem solved. But you have not explained the mechanism, the timing, or why one outcome occurs rather than another.
Reply. Correct, and that is intentional. LRT does not claim to solve the measurement problem by providing dynamical collapse mechanisms or hidden-variable completions. Instead, it reframes the problem by distinguishing what requires mechanistic explanation from what requires structural explanation.
Structural explanation provided: Why are outcomes always determinate? Because instantiated records are $L_3$ admissible (Excluded Middle + Identity). Why do probabilities follow $\lvert\psi\rvert^2$? Because vehicle-invariance forces Born-rule structure. Why does “collapse” appear discontinuous? Because it represents a category transition (vehicle → record), not dynamics within a single layer.
Mechanistic questions left open: When exactly does the transition occur? What physical criterion marks the boundary? Why this particular outcome rather than that one? These are legitimate questions, but they are empirical/technical questions, not foundational puzzles.
Why this is progress: If the structural features follow necessarily from logical constraints, then measurement is not mysterious in principle: it is the $L_3$ enforcing itself at the interface. The remaining questions are “how does nature implement this enforcement?” not “why does enforcement occur at all?”
6. Research Program and Open Questions
6.1 Extension to Quantum Field Theory
The current development focuses on non-relativistic quantum mechanics. Extension to quantum field theory requires addressing:
Lorentz covariance. How do $L_3$ constraints interact with Lorentz transformations? The vehicle-invariance principle must generalize to include frame-independence.
Vacuum structure. QFT’s vacuum is not the “empty state” but a complex ground state with quantum fluctuations. How does this fit within the $I_\infty$/$A_\Omega$ framework?
Gauge structure. Local gauge symmetry ($U(1)$, $SU(2)$, $SU(3)$) is central to the Standard Model. Does gauge structure emerge from vehicle-invariance requirements for field configurations, or is it an additional physical input?
6.2 Gravitational Extension
General relativity presents unique challenges:
Spacetime as emergent. If the $L_3$ constrains instantiation, and spacetime is the arena of instantiation, does spacetime structure itself emerge from $L_3$ constraints?
Black hole information. LRT’s requirement that vehicle-layer information is never destroyed (only transformed) predicts that black hole evaporation must be unitary. This aligns with recent results from AdS/CFT.
Quantum gravity unification. The measurement problem in quantum gravity asks: what constitutes a measurement when spacetime itself is quantized? LRT’s answer: measurements occur when geometric configurations instantiate as $L_3$ admissible records.
6.3 Testable Predictions
Several predictions distinguish LRT from alternative frameworks:
Complex vs. real QM. LRT derives that complex Hilbert space is required for local tomography. The Renou et al. (2021) network scenario confirms complex QM over real alternatives, consistent with this LRT requirement.
Collapse parameters. If objective collapse mechanisms (GRW, Penrose-Diósi) are confirmed, LRT predicts their parameters must be derivable from fundamental constants rather than being free.
Correlation bounds. LRT predicts Tsirelson bound for quantum correlations ($S \leq 2\sqrt{2}$ for CHSH) but excludes PR-box correlations that would allow super-quantum correlations. Any observation of correlations exceeding the Tsirelson bound would falsify the $L_3$ constraint framework.
Information preservation. Fundamental dynamics must preserve vehicle-layer information. Tests: precision tests of unitarity, black hole information recovery via Hawking radiation correlations.
7. Conclusion
Logic Realism Theory offers a novel approach to quantum foundations: treating the Three Fundamental Laws of Logic as constraints on physical instantiation rather than merely on rational discourse. The framework derives quantum structure (complex Hilbert space, Born rule, measurement postulates) from logical constraints plus minimal physical inputs, transforming interpretive puzzles into structural necessities.
The core insight: It from Bit, Bit from Fit. Physical structure emerges from informational structure, which emerges from logical admissibility. Physics proceeds because its outputs are $L_3$ shaped, not by convention but by constraint. Vehicle-invariance forces the Born rule; local tomography forces complex Hilbert space; anti-holism forces parts to ground wholes. These are not separate postulates but consequences of a single requirement: Determinate Identity.
What this achieves: LRT reduces quantum postulates (Born rule becomes theorem), explains quantum structure (why $\lvert\psi\rvert^2$ not $\lvert\psi\rvert$), and generates testable constraints (complex QM requirement—confirmed by Renou et al. 2021—plus derivable collapse parameters and correlation bounds). It reframes the measurement problem by distinguishing structural explanation (why definite at all) from mechanistic explanation (what process produces definiteness), addressing the first while leaving the second to empirical investigation.
The wager: If the Three Fundamental Laws of Logic are constraints on physical instantiation, then quantum mechanics is not contingent but necessary: the unique stable interface between non-Boolean possibility and Boolean actuality. The evidence so far: universal $L_3$ conformity at the record level, Born rule as forced consequence, complex QM experimentally distinguished from real alternatives. The research program invites testing this wager against nature.
Acknowledgments
This research was conducted independently. I thank the related online communities for critical feedback on early drafts.
AI Assistance Disclosure: This work was developed with assistance from AI language models including Claude (Anthropic), ChatGPT (OpenAI), Gemini (Google), Grok (xAI), and Perplexity. These tools were used for drafting, editing, literature review, and exploring mathematical formulations. All substantive claims, arguments, and errors remain the author’s responsibility. Human-Curated, AI-Enabled (HCAE).
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