finite_boolean_logic_embeds_into_positive_ratios
plain-language theorem explainer
Finite Boolean events with positive weight embed their likelihood ratios into the positive reals. Researchers working on discrete-to-continuous bridges in Recognition Science would cite this when moving from Boolean logic to ratio-based comparisons. The proof is a one-line wrapper that packages the event ratio with its positivity witness.
Claim. Let $R$ be a finite weighted Boolean reality over $Fin n$. For nonempty events $A, B : Fin n → Bool$, there exists $r > 0$ such that $r$ equals the likelihood ratio of $A$ to $B$ under $R$.
background
FiniteBooleanReality is a structure that assigns a strictly positive real weight to each element of Fin n. EventNonempty asserts that a Boolean event on Fin n evaluates to true at least once. The module supplies a discrete-to-continuous bridge in which likelihood ratios of such events remain positive whenever both events are nonempty.
proof idea
One-line wrapper that applies eventRatio_pos to the nonempty hypotheses to obtain positivity and uses rfl to equate the ratio.
why it matters
This theorem supplies the embedding step that lets finite Boolean comparisons enter the positive-ratio domain, supporting the LogicAsFunctionalEquation development. It furnishes the discrete foundation required for ratio comparisons that later align with the Recognition Composition Law. No downstream uses are recorded yet.
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