Foundation Logic As Functional Equation Logic Satisfies Laws Of Logic L

The recovered-real law

In classical logic, a comparison between two statements is a truth value: true or false. The Recognition Science framework studies a more general setting where a comparison between two quantities is itself a quantity, not a boolean. The declaration SatisfiesLawsOfLogicL states, in the framework's machine-checked library of formal theorems, that a comparison operator on a special kind of real number, called a recovered real, obeys four structural laws: identity (comparing a number with itself gives zero), non-contradiction (the comparison is symmetric), scale invariance (scaling both inputs leaves the comparison unchanged), and non-triviality (the comparison is not always zero).

The recovered reals are a framework-native construction: a number that is recovered from a recognition event, where a recognition event is a discrete record of a comparison being made. The operator is defined as a function taking two recovered reals and returning a recovered real. The four laws are stated directly over this type. The declaration also requires that when the operator is transported to ordinary real numbers, it satisfies the same laws in the already-verified real-number framework. This transport is the bridge that lets the framework reuse its existing theorems.

The key consequence is uniqueness. The theorem RCL_is_unique_functional_form_of_logicL proves that any comparison operator satisfying these laws must have a specific functional form: there exists a polynomial P such that P(u,v) = 2u + 2v + c·u·v for some constant c, where the derived cost (the comparison with 1) obeys a multiplicative consistency condition. This is not an assumption; it is forced by the laws. The proof works by transporting the recovered-real operator to the real-number surface and applying the already-proved real theorem there.

What this does not claim is important. It does not claim that classical logic is wrong; it generalizes it. It does not claim that the recovered reals are the only possible domain; it shows that if an operator satisfies these laws on this domain, it has that form. It does not claim that the constant c is determined; the theorem leaves it free. It does not claim that the structural laws are the only ones that matter; the analytic regularity conditions are transported from the real theorem, not re-proved here. The declaration is a structural statement about a functional equation, not a claim about physics or about what logic 'really is'.

The practical upshot is that the framework's logic is not arbitrary. If you accept the four structural laws as defining what a comparison operation should do, then the functional form is forced. This is a mathematical fact, checked by a machine, and it constrains what any model of recognition can look like. The reader can now see why the framework's cost function has the shape it does: it is not chosen, it is derived from a small set of structural requirements.

/-- 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)

/-- Recovered-real Law of Logic. The structural fields are native to
`LogicReal`; the analytic/polynomial regularity surface is explicitly
transported to the already-verified real theorem. -/
structure SatisfiesLawsOfLogicL (C : ComparisonOperatorL) : Prop where
  identity : IdentityL C
  non_contradiction : NonContradictionL C
  scale_invariant : ScaleInvariantL C
  non_trivial : NonTrivialL C
  transported_real_laws :
    LogicAsFunctionalEquation.SatisfiesLawsOfLogic (transportComparison C)

/-- 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 in the functional form is determined by the four structural laws. The recovered-real logic replaces or contradicts classical boolean logic. The structural laws are sufficient without the transported analytic regularity conditions.

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)

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