Hypergendered Logic Source
Hypergendered Logic — Page 179
T.9 Stage-Modal Strengthening is not modal necessitation When a trait is possible at one Archaddress and necessary at another, call that a Stage-Modal Strengthening or address-relative modal-status change. Do not call it the rule of necessitation. This terminodistinction prevents proof-system necessitation from being confused with biological evolution of the accessible phenotype-space. T.10 No cross-address monotonicity axiom HGL does not adopt W_a subseteq W_b merely because b is a later address, nor the reverse inclusion. A later Gamma may open possibilities, close possibilities, make a possibility necessary, make a former necessity merely possible, or restructure trait compatibility. Rule-structural enrichment is therefore not identical to set-theoretic expansion of accessible phenotype worlds. Appendix U — Biological Metaexpressability proof catalogue U.1 Proof: global governance does not imply uniform output Let Γ act on two regions r1 and r2 with different biological inputs B1 and B2. Let the stage-valid outputs be P1=Γ(B1) and P2=Γ(B2), with P1≠P2. Γ is global because it governs both regions. Therefore global governance is consistent with nonuniform local phenotype. U.2 Proof: static Γ does not imply static phenotype Let Γ be constant and let θ1≠θ2. Choose a nonconstant phenotype-generation function F such that F(θ1;Γ)≠F(θ2;Γ). Then Γ constancy and phenotype change are jointly satisfiable. Therefore Γ(t1)=Γ(t2) does not logically entail P(t1)=P(t2). U.3 Proof: actual entails possible By validity, the actual phenotype realization belongs to W_n. If p is true at that actual realization, there exists at least one W_n realization satisfying p. Hence ◇_n p. U.4 Proof: necessity entails actuality If □_n p, p holds at every W_n realization. The actual valid realization is a member of W_n. Therefore p holds actually. U.5 Countermodel: possibility does not entail actuality Let W_n contain two realizations v1 and v2. Let p be true only at v2 while v1 is actual. Then ◇_n p is true but p is not actual. U.6 Countermodel: same stage does not imply same phenotype Let two subjects share Γ_n but have different maturity vectors θA and θB. If F is sensitive to maturity, their outputs differ while stage remains the same. U.7 Countermodel: same phenotype does not imply same stage Let Γ_3 and Γ_20 both permit current output p while differing on a counterfactual phenotype q. The two subjects can currently express p identically even though ◇_20 q and NOT ◇_3 q distinguish their stages.
Source Canon: Verbatim mathematical and modal logic. Interactive navigation and operator lookups available at /#hgl-part:179.