Hypergendered Logic Source

Hypergendered Logic — Page 151

HGL Source Framework · Page 151 of 293 · Source: Hypergendered Logic

1.​
 is the inner condition; 
𝑥∧¬𝑦
2.​
 forms the irreducible developmental mode; 
𝑃𝐴·( )
3.​
inner outer-state NOT forms the non-SF1 state; 
4.​
IH inherits that non-state; 
5.​
the outermost NOT denies that inheritance relation. 
No pair of adjacent operators may be silently swapped. 
B.13 User-style outer De Morgan 
 
¬𝑃𝐴𝑥
( ) ∧¬𝑃𝐴¬𝑦
(
) ≡¬ 𝑃𝐴𝑥
( ) ∨𝑃𝐴¬𝑦
(
)
(
).
This is valid and does not create any new PA mode. 
B.14 User-style OR counterpart 
 
¬𝑃𝐴𝑥
( ) ∨¬𝑃𝐴¬𝑦
(
) ≡¬ 𝑃𝐴𝑥
( ) ∧𝑃𝐴¬𝑦
(
)
(
).
B.15 IH of a De Morgan equivalent source 
If 
 
𝑆≡𝐵𝑇
and HGL adopts source-normalization for Boolean equivalents, then 
 
𝐼𝐻𝑆
( )
and 
 
𝐼𝐻𝑇
( )
may be treated as equivalent descriptions of one source. This is not a distribution rule; it is substitution of an 
equivalent source formula. 
Appendix C — Forty common invalid derivations and why they fail 
1.​
Invalid: 
. Why: NOT is outside PA in the premise and inside in the conclusion. 
¬𝑃𝐴𝑥
( ) ⇒𝑃𝐴¬𝑥
(
)
2.​
Invalid: 
 automatically. Why: requires an outer consistency law not supplied merely 
𝑃𝐴¬𝑥
(
) ⇒¬𝑃𝐴𝑥
( )
by syntax. 
3.​
Invalid: 
. Why: PA irreducibility. 
𝑃𝐴𝐴∧𝐵
(
) = 𝑃𝐴𝐴
( ) ∧𝑃𝐴𝐵
( )
4.​
Invalid: 
. Why: PA non-homomorphism. 
𝑃𝐴𝐴∨𝐵
(
) = 𝑃𝐴𝐴
( ) ∨𝑃𝐴𝐵
( )
5.​
Invalid: 
, therefore 
. Why: invalid causal lifting. 
𝐴∧𝐵⇒𝐴
𝑃𝐴𝐴∧𝐵
(
) ⇒𝑃𝐴𝐴
( )
6.​
Invalid: 
. Why: separate causes do not reconstruct a joint cause. 
𝑃𝐴𝐴
( ) ∧𝑃𝐴𝐵
( ) ⇒𝑃𝐴𝐴∧𝐵
(
)
7.​
Invalid: SF1 = PA(x) AND PA(NOT y). Why: corrected SF1 is exactly the single irreducible compound 
mode PA(x AND NOT y); it contains no constitutive IH/PH structure. 
8.​
Invalid: SM1 = PA(NOT x) AND PA(y). Why: corrected SM1 is the single irreducible compound mode 
PA(NOT x AND y). 
9.​
Invalid: SF2 is any old three-, four-, or six-term predecessor formula. Why: current canonical SF2 
has eight conjuncts including IH, PH, UD, standalone PA routes, and NOT FD(SF1). 
10.​ Invalid: SM2 is the old three-term PA conjunction. Why: same. 
11.​
Invalid: 
. Why: projection is one-way; inheritance carries source provenance. 
𝐼𝐻𝑆
( ) = 𝑆
12.​ Invalid: 
. Why: possession does not prove inheritance. 
𝑆⇒𝐼𝐻𝑆
( )
13.​ Invalid: 
. Why: inheritance of non-S differs from noninheritance of S. 
𝐼𝐻¬𝑆
(
) = ¬𝐼𝐻𝑆
( )
14.​ Invalid: 
. Why: S can be independently present. 
¬𝐼𝐻𝑆
( ) ⇒¬𝑆
15.​ Invalid: 
 and 
, therefore 
. Why: ordinary consequence does not invent a new 
𝐼𝐻𝑆
( )
𝑆⇒𝑇
𝐼𝐻𝑇
( )
inheritance source.

Source Canon: Verbatim mathematical and modal logic. Interactive navigation and operator lookups available at /#hgl-part:151.