Foundation Logic From Cost Logical Contradiction Impossible

The contradiction theorem

Classical logic holds that a statement and its negation cannot both be true. The Recognition Science framework models propositions as configurations, each carrying a positive number called its ratio. A configuration is stable when its cost is zero. The theorem logical_contradiction_impossible states that no configuration can be both a proposition and its negation: if a configuration c satisfies P and also satisfies not P, the proof derives False. This is a direct formalization of the law of non-contradiction.

The proof is short. A ContradictionConfig structure holds a proposition P, a ratio for P, a ratio for not P, and a complementarity condition that the product of the two ratios equals 1. The theorem takes a contradiction configuration and hypotheses hP : c.P and hnotP : c.notP. The conclusion False follows immediately from hnotP applied to hP. The machine-checked library of formal theorems records this as a proved theorem, not an assumption.

The framework pairs this with two supporting results. contradiction_positive_cost shows that any contradiction configuration either has positive cost or sits at the singular point where both ratios equal 1. zero_cost_contradiction_forbidden shows that if a contradiction configuration somehow had zero total cost, it would still imply False. Together these state that a contradiction cannot be a stable, zero-cost configuration.

In Recognition Science, the cost function J(x) = (x + 1/x)/2 - 1 assigns zero cost only at ratio 1. A consistent configuration, one that is not contradictory, can achieve zero cost at ratio 1, as shown by consistent_zero_cost_possible. The framework's broader claim, stated in logic_from_cost, is that consistency is the minimum-cost structure: consistent configurations can have zero cost, all consistent configurations have non-negative cost, and zero cost occurs exactly at ratio 1.

What the theorem does not claim is equally important. It does not derive classical logic from cost. The proof runs inside Lean's ambient classical logic, using that logic to prove facts about cost-minimizing configurations. The philosophical thesis that logic emerges from cost is a structural analogy, not a derivation of logic itself. The framework does not claim that contradictions are physically impossible in the sense of being unrepresentable; it claims they cannot be stable configurations with zero cost.

/-- **THEOREM 2**: Logical contradictions are classically impossible.

    If both P and ¬P are true (ratio = 1, cost = 0), then P ∧ ¬P holds.
    But P ∧ ¬P = False.

    This shows: the cost framework respects classical logic.
    Contradictions can't stabilize because they can't exist. -/
theorem logical_contradiction_impossible (c : ContradictionConfig)
    (hP : c.P) (hnotP : ¬c.P) : False := hnotP hP

/-- **THEOREM 2**: Logical contradictions are classically impossible.

    If both P and ¬P are true (ratio = 1, cost = 0), then P ∧ ¬P holds.
    But P ∧ ¬P = False.

    This shows: the cost framework respects classical logic.
    Contradictions can't stabilize because they can't exist. -/
theorem logical_contradiction_impossible (c : ContradictionConfig)
    (hP : c.P) (hnotP : ¬c.P) : False := hnotP hP

/-- **THEOREM 4**: Consistent configurations can have zero cost.

    Unlike contradictions, a single proposition can stabilize at ratio = 1.
    This is the minimum-cost state for a proposition. -/
theorem consistent_zero_cost_possible :
    ∃ c : ConsistentConfig, consistent_cost c = 0 := by
  use ⟨True, 1, by norm_num⟩
  unfold consistent_cost
  exact defect_at_one

What this page does not claim

This answer does not claim that the framework derives classical logic from cost; the proof uses classical logic as its metalanguage. This answer does not claim that contradictions are physically unrepresentable, only that they cannot be stable zero-cost configurations. This answer does not claim that the theorem applies outside the framework's specific cost model.

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

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