In Recognition Science, logic emerges as the structure of cost-minimizing configurations rather than being imposed externally.
A proposition is formalized as a PropConfig with positive ratio r; its cost is defect(r). consistent_minimum_cost proves that any consistent configuration satisfies consistent_cost c ≥ 0 with equality if and only if c.ratio = 1.
A contradiction is formalized as a ContradictionConfig requiring complementary ratios whose product is 1. contradiction_positive_cost proves that the total cost is strictly positive unless both ratios equal 1, at which point logical_contradiction_impossible yields classical falsehood.
logic_from_cost assembles these facts: consistent configurations achieve zero cost while contradictions cannot, so logical consistency is exactly the minimum-cost structure.
why_logic_is_real strengthens the conclusion: only ratio-1 states stabilize (defect = 0), contradictions are forbidden, and existence at ratio 1 is the unique zero-cost state, making reality logical by cost necessity.
The same emergence appears in functional form: laws_of_logic_imply_dalembert_hypotheses shows that the four Aristotelian constraints on a comparison operator translate directly into the hypotheses of the d'Alembert inevitability theorem, forcing the unique reciprocal cost J(x) = (x + x⁻¹)/2 − 1.