Hypergendered Logic Source

Hypergendered Logic — Page 34

HGL Source Framework · Page 34 of 293 · Source: Hypergendered Logic

HGL has no general monotonicity axiom 
 
𝑃𝐴𝐴∧𝐵
(
) ⇒𝑃𝐴𝐴
( ).
A countermodel is any HGL interpretation containing the irreducibly joint developmental mode 
 
𝑃𝐴𝐴∧𝐵
(
)
while lacking the separately designated 
 mode. Such a model is permitted by PA irreducibility. Hence 
𝑃𝐴𝐴
( )
the lifting is invalid in general. 
Proposition 4 — De Morgan remains valid inside PA conditions 
Classically, 
 
¬ 𝐴∧𝐵
(
) ≡¬𝐴∨¬𝐵.
If HGL treats logically equivalent inner conditions as equivalent descriptions of the same designated 
condition, then 
 
𝑃𝐴¬ 𝐴∧𝐵
(
)
(
) ≡𝑃𝐴¬𝐴∨¬𝐵
(
).
This does not imply 
. 
𝑃𝐴¬𝐴
(
) ∨𝑃𝐴¬𝐵
(
)
Proposition 5 — De Morgan remains valid outside PA, IH, and PH atoms 
For any HGL states 
, 
𝑆, 𝑇
 
¬ 𝑆∧𝑇
(
) ≡¬𝑆∨¬𝑇
and 
 
¬ 𝑆∨𝑇
(
) ≡¬𝑆∧¬𝑇.
The transformation occurs at the outer Boolean level and does not migrate a NOT through either PA or IH. 
Proposition 6 — IH projection 
By definition of inheritance, 
 
𝐼𝐻𝑆
( ) ⇒𝑆.
This is a core HGL axiom. It formalizes the minimum content of “inherits the properties of.” 
Proposition 7 — Possession does not prove inheritance 
The converse 
 
𝑆⇒𝐼𝐻𝑆
( )
is invalid. Consider a model in which is independently instantiated rather than inherited. is true and 
 
𝑆
𝑆
𝐼𝐻𝑆
( )
is false. Therefore 
 
𝑆 ⊭ 𝐼𝐻𝑆
( ).
Proposition 8 — Inheritance of negation is not negation of inheritance 
From projection, 
 
𝐼𝐻¬𝑆
(
) ⇒¬𝑆.
But 
 only says the inheritance relation from is absent. A model can satisfy independently while 
¬𝐼𝐻𝑆
( )
𝑆
𝑆
failing to inherit , so 
 does not entail 
. Therefore 
𝑆
¬𝐼𝐻𝑆
( )
¬𝑆
 
𝐼𝐻¬𝑆
(
)≢¬𝐼𝐻𝑆
( ).

Source Canon: Verbatim mathematical and modal logic. Interactive navigation and operator lookups available at /#hgl-part:34.