Hypergendered Logic Source
Hypergendered Logic — Page 148
𝑃𝐴φ ( ) ⇒𝑃𝐴ψ ( ). The semantic reason is that “developable due to ” and “developable due to ” answer different causal φ ψ attribution questions. A.4 IH as a source-sensitive relation A useful semantic representation is 𝐼𝐻𝑆 ( ) = ⟨𝑝𝑟𝑜𝑗𝑒𝑐𝑡𝑒𝑑 𝑐𝑜𝑛𝑡𝑒𝑛𝑡 𝑜𝑓 𝑆, 𝑠𝑜𝑢𝑟𝑐𝑒= 𝑆, 𝑑𝑒𝑝𝑡ℎ+ 1⟩. This notation is explanatory, not a literal set-theoretic definition. Its purpose is to make three facts obvious: 1. inherited content projects to source content; 2. source identity remains recorded; 3. an additional IH application increments depth. A.5 Projection function Define recursively: π 𝑃𝐴φ ( ) ( ) = 𝑃𝐴φ ( ), π ¬𝑆 ( ) = ¬π 𝑆 ( ), π 𝑆∧𝑇 ( ) = π 𝑆 ( ) ∧π 𝑇 ( ), π 𝑆∨𝑇 ( ) = π 𝑆 ( ) ∨π 𝑇 ( ), π 𝐼𝐻𝑆 ( ) ( ) = π 𝑆 ( ). Projection is useful for ordinary entailment. It is deliberately lossy: it throws away provenance. A.6 Why projection alone saturates but the full state need not If one repeatedly projects higher stages, many later non-opposite-stage statements can become logically weak or redundant. This is expected. The full hierarchy is not defined solely by . π For example, πIH¬SM2=¬SM2, while πIH¬SM20=¬SM20. The projected formulas may stand in implication relations. Their full inherited states still differ by source and depth. A.7 A provenance graph Represent each higher stage as a directed acyclic graph. For SF3: • root: SF3; • child 1: IHSF2; • child 2: IH¬SM2; • the SF2 source points to its four canonical components; • the SM2 source points to its four canonical components. The graph is not a mere display convenience. It is the data structure that preserves what Boolean flattening would erase.
Source Canon: Verbatim mathematical and modal logic. Interactive navigation and operator lookups available at /#hgl-part:148.