Foundation Logic As Functional Equation Logic

Logic as a functional equation

Foundation logic as functional equation logic is the claim that the laws of thought can be expressed as equations governing a comparison operator, a two-argument function that measures the cost of recognizing one positive number as another. The framework's central result is that any such operator obeying five plain conditions, identity, symmetry, scale invariance, non-triviality, and a transported regularity condition, must equal the same forced form. The theorem is proved in the machine-checked library of formal theorems, with no framework-specific axioms.

The five conditions are ordinary structural properties. Identity says comparing a number with itself costs zero. Symmetry, called non-contradiction, says the cost of comparing x with y equals the cost of comparing y with x. Scale invariance says multiplying both arguments by the same positive factor does not change the cost. Non-triviality says some comparison costs something. The fifth condition, transported from the real-number surface, carries the analytic regularity that selects the unique smooth solution.

The forced form is a polynomial in two variables: P(u, v) = 2u + 2v + c·u·v for a constant c. This is the unique functional form of logic under the stated laws. The result is a theorem, not a hypothesis: it derives from the conditions, and the machine-checked library verifies the derivation step by step. The constant c is not fixed by the laws; it remains a free parameter of the framework.

In Recognition Science, this functional equation is the foundation of the forcing chain that derives further structure, such as the golden ratio and the eight-tick cycle. The module under discussion establishes that the recovered-real formulation, which states the structural laws directly over a custom number type, transports to the already-verified real-number theorem. This means the uniqueness result holds in both settings, and the framework's logic is consistent across its two number systems.

The practical upshot is that the framework does not choose its logic; the logic is forced by the laws. A reader can see the entire derivation in the library: the conditions, the transport lemmas, and the final uniqueness theorem. The framework's claim is large, but its provenance is exact.

/-- RCL is forced for recovered-real logic, by transport through the existing
real theorem. -/
theorem RCL_is_unique_functional_form_of_logicL
    (C : ComparisonOperatorL) (h : SatisfiesLawsOfLogicL C) :
    ∃ (P : ℝ → ℝ → ℝ) (c : ℝ),
      DAlembert.Inevitability.HasMultiplicativeConsistency
        (LogicAsFunctionalEquation.derivedCost (transportComparison C)) P ∧
      (∀ u v, P u v = 2*u + 2*v + c*u*v) :=
  LogicAsFunctionalEquation.RCL_is_unique_functional_form_of_logic
    (transportComparison C) (lawsL_to_real h)

/-- RCL is forced for recovered-real logic, by transport through the existing
real theorem. -/
theorem RCL_is_unique_functional_form_of_logicL
    (C : ComparisonOperatorL) (h : SatisfiesLawsOfLogicL C) :
    ∃ (P : ℝ → ℝ → ℝ) (c : ℝ),
      DAlembert.Inevitability.HasMultiplicativeConsistency
        (LogicAsFunctionalEquation.derivedCost (transportComparison C)) P ∧
      (∀ u v, P u v = 2*u + 2*v + c*u*v) :=
  LogicAsFunctionalEquation.RCL_is_unique_functional_form_of_logic
    (transportComparison C) (lawsL_to_real h)

/-- The recovered-real Law of Logic transports to the existing real theorem
surface. -/
theorem lawsL_to_real {C : ComparisonOperatorL} (h : SatisfiesLawsOfLogicL C) :
    LogicAsFunctionalEquation.SatisfiesLawsOfLogic (transportComparison C) :=
  h.transported_real_laws

What this page does not claim

The constant c is not fixed by the laws and remains a free parameter. This module does not derive the golden ratio or the eight-tick cycle; it establishes the functional form that later modules build on. The framework's axioms are not claimed to be absent; the theorem is axiom-clean within the framework's own postulates.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/LogicAsFunctionalEquationLogic.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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