Foundation Logic From Cost Zero Cost Contradiction Forbidden

The zero-cost ban on contradictions

Classical logic begins with a prohibition: a statement and its negation cannot both be true. This is the law of non-contradiction, and it is usually taken as a starting axiom, not something to be explained. Recognition Science (RS) asks a different question: could this prohibition be a consequence of something more basic, namely cost? The framework's answer is that a configuration with a proposition and its negation both fully present would have to pay a price that no stable state can bear.

In RS, a ledger (a discrete record of events) assigns a cost to each possible configuration. A proposition is modeled as a configuration with a positive ratio, a number measuring how present it is. A contradiction is a configuration where the ratio for P and the ratio for not-P both equal 1, meaning both are fully present. The theorem zero_cost_contradiction_forbidden, proved in the framework's machine-checked library of formal theorems, states that if such a contradiction were assigned zero total cost, it would still be impossible: the proof runs by deriving a contradiction from the assumption that both P and not-P hold. The cost assumption is not even needed for the impossibility; the logical contradiction itself suffices.

The surrounding theorems give the fuller picture. contradiction_positive_cost shows that a contradiction either has positive cost or sits at the singular point where both ratios equal 1. consistent_zero_cost_possible shows that a consistent configuration, one where P holds with ratio 1, can indeed achieve zero cost. The theorem logic_from_cost assembles these pieces: consistency can have zero cost, all consistent configurations have non-negative cost, and contradictions are either costly or at the singular point. The framework's claim is that logical consistency is the minimum-cost structure, not a structure imposed from outside.

The declaration does not claim to derive classical logic itself. The proof runs inside Lean's ambient classical logic, which supplies the law of non-contradiction from the start. The framework is explicit about this bootstrapping: it uses classical logic to prove that cost-minimization forbids contradictory object-level configurations. The philosophical thesis that logic emerges from cost is a structural analogy, not a derivation of logic from nothing. What the theorem establishes is narrower: within the framework's model, a contradiction cannot be a zero-cost stable state, while consistency can.

/-- **THEOREM 3**: Cost-zero contradictions imply classical impossibility.

    If a contradiction config has zero total cost, then:
    - ratio_P = 1 (so P "exists")
    - ratio_notP = 1 (so ¬P "exists")
    - But P ∧ ¬P is impossible

    Therefore: zero-cost contradictions are forbidden by logic itself. -/
theorem zero_cost_contradiction_forbidden (c : ContradictionConfig)
    (_h_zero : contradiction_cost c = 0)
    (hP : c.P) (hnotP : ¬c.P) : False := by
  exact 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

/-- **THE MAIN THEOREM**: Logic is the structure of cost minimization.

    1. Contradictions cannot have zero cost (they're unstable)
    2. Consistent propositions can have zero cost (they can stabilize)
    3. Therefore: the stable configurations are the logically consistent ones
    4. Therefore: logic = the structure of what can exist = what minimizes cost

    This proves: logical consistency is not imposed from outside.
    It emerges from the cost landscape. Logic is cheap. -/
theorem logic_from_cost :
    -- Consistency can achieve zero cost
    (∃ c : ConsistentConfig, consistent_cost c = 0) ∧
    -- Consistency cost is minimized at ratio = 1
    (∀ c : ConsistentConfig, consistent_cost c ≥ 0) ∧
    (∀ c : ConsistentConfig, consistent_cost c = 0 ↔ c.ratio = 1) ∧
    -- Contradictions have positive cost or are at the singular point
    (∀ c : ContradictionConfig,
      contradiction_cost c > 0 ∨ IsLogicalContradiction c) :=
  ⟨consistent_zero_cost_possible,
   fun c => defect_nonneg c.ratio_pos,
   fun c => defect_zero_iff_one c.ratio_pos,
   contradiction_positive_cost⟩

What this page does not claim

The theorem does not derive classical logic from cost; it uses classical logic as the ambient metalanguage. The theorem does not show that contradictions have infinite cost; it shows they cannot be zero-cost stable states. The philosophical thesis that logic emerges from cost is a structural analogy, not a proved derivation.

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)

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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