prelogical_boolean_fragment
plain-language theorem explainer
Pre-logical cost minima on stable states induce arithmetic operations obeying Boolean algebra identities for conjunction, disjunction, and negation. Researchers deriving logic from recognition cost structures cite this when showing consistency minima behave like logical connectives. The proof reduces directly to the lemma that stable forms constitute a Boolean algebra.
Claim. Let $S$ be the set of stable states. For all $a,b$ in $S$, the bit value of the conjunction equals $a$.bit times $b$.bit, the disjunction equals $a$.bit plus $b$.bit minus their product, and the negation equals one minus $a$.bit.
background
In the Recognition Science framework, logical consistency emerges as the structure of cost-minimizing configurations. A proposition is a configuration $c$ that is true if it stabilizes (defect distance to zero) and false if it diverges (defect to infinity). The module proves that contradictions have infinite cost because they cannot both stabilize. PreLogicalCost defines StableState as configurations with zero defect, and band, bor, bnot as arithmetic operations on their bit values that preserve stability. This result relies on the upstream lemma that stable forms constitute a Boolean algebra, as well as cost definitions from ObserverForcing and MultiplicativeRecognizerL4.
proof idea
The proof is a one-line wrapper that applies the lemma stable_forms_boolean_algebra from PreLogicalCost.
why it matters
This declaration bridges cost minimization to logical structure, showing that the cost landscape coincides with the logical landscape. It supports the meta-principle that existence is the cost-minimizing state, as contradictions are forbidden by infinite cost. It fills the step where logic emerges from J-cost minimization, connecting to the unification of cost and logic derivations of the meta-principle.
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