Hypergendered Logic Source

Hypergendered Logic — Page 170

HGL Source Framework · Page 170 of 293 · Source: Hypergendered Logic

N.6 Entailment after projection 
Premises: 
 
𝐼𝐻𝑆
( ),   𝑆⇒𝑇.
Deduce , then . 
𝑆
𝑇
Do not deduce 
. 
𝐼𝐻𝑇
( )
N.7 SF2 ancestry 
SF2⇒IHSF1⇒SF1. 
N.8 SF2 opposite exclusion 
SF2⇒IH¬SM1⇒¬SM1. 
N.9 SF4 ancestry chain 
SF4⇒SF3⇒SF2⇒SF1. 
N.10 SF4 opposite exclusion 
SF4⇒¬SM3. 
Because SM3⇒SM2⇒SM1, one can also derive that any SM1 state would contradict the projected SF2 
exclusion already inherited in the female lineage. 
N.11 Depth calculation 
Stage-source IH depth is recursively defined, not inferred by visual counting alone. 
d_base(SF1)=d_base(SM1)=0. 
d(NOT S)=d(S). Negation changes truth-polarity/source content but does not itself add an IH layer. 
d(S AND T)=max(d(S),d(T)) for finite conjunctions; for a set-indexed limit closure use the supremum of 
constituent depths. 
d(IH(S))=d(S)+1. PH(S), UD(S), FD(S), and NOT FD(S) do not themselves add IH depth. 
Therefore d(SF8)=7 and d(IH(SF8))=8, so the maximum Stage-source IH depth of SF9 is 8. 
For a canonical limit lambda under the Limit-State Closure Constructor, 
d(SF^lambda)=sup_{beta<lambda}(d(SF^beta)+1), with the exact male mirror. Consequently 
d(SF^omega)=omega under the minimum canonical closure depth. A successor after a limit adds one further 
IH layer. This makes Stage-source depth ordinal-valued once transfinite stages are admitted. 
Hence d(IH(SF8))=8. 
Thus SF9 has maximum Stage-source IH depth 8. 
N.12 No IH distribution 
Premise 
. Projection yields 
, hence A and B. It does not yield 
 or 
. 
𝐼𝐻𝐴∧𝐵
(
)
𝐴∧𝐵
𝐼𝐻𝐴
( )
𝐼𝐻𝐵
( )
N.13 No PA decomposition under IH 
Premise 
. Projection yields 
. PA irreducibility blocks inference to 
 and 
, 
𝐼𝐻𝑃𝐴𝐴∧𝐵
(
)
(
)
𝑃𝐴𝐴∧𝐵
(
)
𝑃𝐴𝐴
( )
𝑃𝐴𝐵
( )
and hence certainly blocks inference to their IH versions.

Source Canon: Verbatim mathematical and modal logic. Interactive navigation and operator lookups available at /#hgl-part:170.