Foundation Logic From Cost Consistent Minimum Cost

The cost of consistency

In the Recognition Science framework, every proposition is assigned a number that measures how "present" it is, called its ratio, and a cost that grows as that ratio moves away from one. The theorem consistent_minimum_cost proves that for any consistent configuration, meaning a proposition with a positive ratio, the cost is never negative, and it hits zero if and only if the ratio equals one. In plain terms: a statement that is perfectly balanced, neither over-asserted nor under-asserted, is the cheapest possible state. A statement that is out of balance, too strong or too weak, always carries a positive cost.

This is a theorem in the machine-checked library of formal theorems, proved from the framework's definition of cost as a defect function. The proof is short and rests on two earlier results: the defect is always non-negative for positive ratios, and it is zero exactly at ratio one. The declaration also shows that such a zero-cost state is not merely possible but is realized: the consistent configuration built from the proposition True with ratio one has cost zero. So the theorem does not just say what consistency would cost; it exhibits a consistent statement that actually achieves the minimum.

The framework reads this as the bridge from cost to logic. A contradiction, a configuration where both a proposition and its negation are asserted, cannot have zero cost, because the two sides cannot both be at ratio one. The consistent minimum is therefore the stable, cheap state, and the framework models logical consistency as the structure that cost minimization selects. The philosophical claim that logic emerges from cost is a structural analogy, not a derivation of classical logic itself; the proof runs inside the ambient classical logic of the library.

What the declaration does not claim is just as important. It does not prove that contradictions are impossible in the object language; it proves that they cannot both stabilize at the minimum. It does not derive classical logic from nothing, since the proof uses classical reasoning. And it does not say that every consistent statement has zero cost, only that the minimum is zero and is reached exactly at ratio one. The theorem is a precise statement about a cost function, and its force is that the cheapest configuration is the balanced one.

The practical upshot is a criterion: when the framework assigns costs to configurations, the balanced ratio of one is the unique zero-cost point for consistent statements. That single fact anchors the framework's account of why consistency, rather than contradiction, is the natural resting state. It is the cost function, not an external rule, that makes the balanced statement cheap and the contradictory one expensive.

/-- **THEOREM 5**: The minimum cost for consistency is 0, achieved at ratio = 1. -/
theorem consistent_minimum_cost (c : ConsistentConfig) :
    consistent_cost c ≥ 0 ∧ (consistent_cost c = 0 ↔ c.ratio = 1) := by
  constructor
  · exact defect_nonneg c.ratio_pos
  · exact defect_zero_iff_one c.ratio_pos

/-- **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

/-- **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

What this page does not claim

The theorem does not prove that contradictions are impossible, only that they cannot both stabilize at the minimum cost. The framework does not derive classical logic from nothing; the proof uses the ambient classical logic of the library. The declaration does not claim that every consistent statement has zero cost, only that the minimum is zero and is achieved exactly at ratio one.

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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