Encyclopedia Foundation Foundation Logic From Cost

ARTICLE 3 claims 3 theorems

Foundation Logic From Cost

Logical consistency is the minimum-cost structure of recognition configurations, and this module establishes the core theorems in a machine-checked way.

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Logic from Cost

In Recognition Science, recognition events carry a forced cost, and the central result is the unique cost function J(x) = (x + 1/x)/2 - 1. This module, LogicFromCost, applies that framework to propositions themselves. A proposition is modeled as a configuration with a positive ratio, and its cost is the defect of that ratio. A proposition is stable when its cost is zero, which happens exactly when its ratio equals 1.

The key theorems establish that consistency is the cheap state. A consistent configuration can achieve zero cost, and any consistent configuration has non-negative cost, with zero cost exactly when the ratio is 1. A contradiction, modeled as a configuration where both P and not-P have ratio 1, cannot have zero total cost; it either has positive cost or sits at the singular point where both ratios are 1. The theorem logic_from_cost_summary packages these facts: consistency can be free, contradictions are costly or singular, and nothing has infinite cost while ratio 1 has zero cost.

The module also shows that the stable states of a pre-logical cost structure form a Boolean algebra, with bitwise AND, OR, and NOT operations matching the classical truth tables. This is the structural analogy: the cost function's minima behave like logical consistency. The philosophical thesis that logic emerges from cost is precisely this, not a claim to derive classical logic itself, which would be circular. The proofs use the ambient classical logic of the formal system as the metalanguage.

THEOREM logic_from_cost_summary · IndisputableMonolith/Foundation/LogicFromCost.lean
logic_from_cost_summary · IndisputableMonolith/Foundation/LogicFromCost.lean:303 
/-- **LOGIC FROM COST: SUMMARY**

    The claim "consistent logic is a cost-minimizing state" is proven:

    | State | Cost | Can Exist? |
    |-------|------|------------|
    | Contradiction (P ∧ ¬P) | > 0 or singular | No (unstable) |
    | Consistency (just P) | ≥ 0, = 0 at ratio 1 | Yes (stable) |
    | Nothing (ratio → 0) | → ∞ | No (too expensive) |
    | Something (ratio = 1) | = 0 | Yes (free) |

    Therefore:
    - Logic is not imposed from outside
    - Logic emerges as the structure of cost minima
    - Reality is logical because logic is cheap
    - The cost landscape IS the logical landscape -/
theorem logic_from_cost_summary :
    -- Consistent configs can have zero cost
    (∃ c : ConsistentConfig, consistent_cost c = 0) ∧
    -- All consistent configs have non-negative cost
    (∀ c : ConsistentConfig, consistent_cost c ≥ 0) ∧
    -- Zero cost ↔ ratio = 1
    (∀ c : ConsistentConfig, consistent_cost c = 0 ↔ c.ratio = 1) ∧
    -- Contradictions are either costly or at the singular point
    (∀ c : ContradictionConfig,
      contradiction_cost c > 0 ∨ IsLogicalContradiction c) ∧
    -- Nothing has infinite cost
    (∀ C : ℝ, ∃ ε > 0, ∀ x, 0 < x → x < ε → C < defect x) ∧
    -- Something (1) has zero cost
    defect 1 = 0 :=
  ⟨consistent_zero_cost_possible,
   fun c => defect_nonneg c.ratio_pos,
   fun c => defect_zero_iff_one c.ratio_pos,
   contradiction_positive_cost,
   nothing_cannot_exist,
   defect_at_one⟩
THEOREM logic_from_cost_summary · IndisputableMonolith/Foundation/LogicFromCost.lean
logic_from_cost_summary · IndisputableMonolith/Foundation/LogicFromCost.lean:303 
/-- **LOGIC FROM COST: SUMMARY**

    The claim "consistent logic is a cost-minimizing state" is proven:

    | State | Cost | Can Exist? |
    |-------|------|------------|
    | Contradiction (P ∧ ¬P) | > 0 or singular | No (unstable) |
    | Consistency (just P) | ≥ 0, = 0 at ratio 1 | Yes (stable) |
    | Nothing (ratio → 0) | → ∞ | No (too expensive) |
    | Something (ratio = 1) | = 0 | Yes (free) |

    Therefore:
    - Logic is not imposed from outside
    - Logic emerges as the structure of cost minima
    - Reality is logical because logic is cheap
    - The cost landscape IS the logical landscape -/
theorem logic_from_cost_summary :
    -- Consistent configs can have zero cost
    (∃ c : ConsistentConfig, consistent_cost c = 0) ∧
    -- All consistent configs have non-negative cost
    (∀ c : ConsistentConfig, consistent_cost c ≥ 0) ∧
    -- Zero cost ↔ ratio = 1
    (∀ c : ConsistentConfig, consistent_cost c = 0 ↔ c.ratio = 1) ∧
    -- Contradictions are either costly or at the singular point
    (∀ c : ContradictionConfig,
      contradiction_cost c > 0 ∨ IsLogicalContradiction c) ∧
    -- Nothing has infinite cost
    (∀ C : ℝ, ∃ ε > 0, ∀ x, 0 < x → x < ε → C < defect x) ∧
    -- Something (1) has zero cost
    defect 1 = 0 :=
  ⟨consistent_zero_cost_possible,
   fun c => defect_nonneg c.ratio_pos,
   fun c => defect_zero_iff_one c.ratio_pos,
   contradiction_positive_cost,
   nothing_cannot_exist,
   defect_at_one⟩
THEOREM prelogical_boolean_fragment · IndisputableMonolith/Foundation/LogicFromCost.lean
prelogical_boolean_fragment · IndisputableMonolith/Foundation/LogicFromCost.lean:267 
/-- Pre-logical arithmetic cost minima induce Boolean-style stable operations. -/
theorem prelogical_boolean_fragment :
    (∀ a b : PreLogicalCost.StableState,
      (PreLogicalCost.band a b).bit = a.bit * b.bit) ∧
    (∀ a b : PreLogicalCost.StableState,
      (PreLogicalCost.bor a b).bit = a.bit + b.bit - a.bit * b.bit) ∧
    (∀ a : PreLogicalCost.StableState,
      (PreLogicalCost.bnot a).bit = 1 - a.bit) :=
  PreLogicalCost.stable_forms_boolean_algebra

What this page does not claim

This module does not derive classical logic itself; it uses the ambient classical logic of the formal system as the metalanguage. No claim is made that the philosophical thesis 'logic emerges from cost' is a proof of the nature of logic, only a structural analogy. The module does not connect these logical structures to the empirical predictions of particle masses or the fine-structure constant.

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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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