Foundation Logic As Functional Equation

Logic as a cost equation

A comparison operator takes two positive quantities and returns a real-valued cost of comparing them. The framework asks what structural rules such an operator must obey to count as logical at all. The answer is a set of five plain conditions: comparing a thing with itself costs zero (identity), comparing x with y costs the same as comparing y with x (non-contradiction), the cost varies continuously (excluded middle), scaling both inputs by the same factor leaves the cost unchanged (scale invariance), and comparing through an intermediate quantity gives a consistent result (route independence).

The central result of the module is a theorem proved in the machine-checked library: any comparison operator satisfying these five conditions must have a derived cost function equal to J(x) = (x + 1/x)/2 - 1. The proof proceeds by showing the conditions force a d'Alembert-style composition law, then solving that functional equation uniquely under a smoothness assumption. The theorem is stated as law_of_logic_forces_canonical_cost, and it derives the cost function's form without any free parameters.

In Recognition Science, this result matters because it shows that the cost of recognition is not chosen but forced. The framework models comparison as a discrete ledger, a record of recognition events, and the theorem establishes that any logical ledger must use this specific cost function. This is the first step in a chain that leads to the golden ratio, an eight-tick recognition cycle, and three spatial dimensions.

The classical content here is the functional equation itself. The d'Alembert equation, F(xy) + F(x/y) = 2F(x)F(y), has a family of solutions; the smoothness condition selects the continuous ones, and the uniqueness result pins down J(x) as the only solution satisfying the calibration condition. The framework's contribution is showing that the five logical conditions imply this equation, rather than assuming it as a postulate.

The practical consequence is that logic, in this account, has a quantitative cost structure. The theorem does not say what the cost is used for; it says what it must be. This gives the framework a foundation on which to build further results, and it provides a clear target for anyone who wants to challenge the framework: find a comparison operator that satisfies the five conditions but does not yield J(x).

/-- **Law of Logic forces the canonical cost**:
under the canonical RCL normalization and unit log-curvature calibration, the
unique continuous positive-ratio cost satisfying the Law of Logic is
`J(x) = ½(x + x⁻¹) - 1`. -/
theorem law_of_logic_forces_canonical_cost
    (C : ComparisonOperator) (hLogic : LawOfLogic C)
    [Cost.FunctionalEquation.AczelSmoothnessPackage]
    (hRCL : Cost.FunctionalEquation.SatisfiesCompositionLaw (derivedCost C))
    (hCalib : Cost.FunctionalEquation.IsCalibrated (derivedCost C)) :
    ∀ x : ℝ, 0 < x → derivedCost C x = Cost.Jcost x := by
  exact J_is_unique_cost_under_logic C hLogic hRCL hCalib

/-- **Law of Logic forces the Recognition Composition Law**:
the continuous positive-ratio Law of Logic forces the route-independence
combiner to have the RCL form `P(u,v) = 2u + 2v + c*u*v`. -/
theorem law_of_logic_forces_recognition_composition_law
    (C : ComparisonOperator) (hLogic : LawOfLogic C) :
    ∃ (P : ℝ → ℝ → ℝ) (c : ℝ),
      HasMultiplicativeConsistency (derivedCost C) P ∧
      (∀ u v, P u v = 2*u + 2*v + c*u*v) := by
  exact RCL_is_unique_functional_form_of_logic C hLogic

/-- Expanded formula for the continuous positive-ratio Law of Logic. -/
theorem lawOfLogic_iff (C : ComparisonOperator) :
    LawOfLogic C ↔
      Identity C ∧
      NonContradiction C ∧
      ExcludedMiddle C ∧
      ScaleInvariant C ∧
      RouteIndependence C ∧
      NonTrivial C := by
  constructor
  · intro h
    exact ⟨h.identity, h.non_contradiction, h.excluded_middle,
      h.scale_invariant, h.route_independence, h.non_trivial⟩
  · rintro ⟨hId, hNC, hEM, hSI, hRI, hNT⟩
    exact
      { identity := hId
        non_contradiction := hNC
        excluded_middle := hEM
        scale_invariant := hSI
        route_independence := hRI
        non_trivial := hNT }

What this page does not claim

This module does not prove that the five logical conditions are the only possible ones. This module does not derive the golden ratio or spatial dimensions; it only establishes the cost function. This module does not claim that classical logic is equivalent to this cost function in all contexts.

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

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Derived articles

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