Hypergendered Logic Source

Hypergendered Logic — Page 13

HGL Source Framework · Page 13 of 293 · Source: Hypergendered Logic

Thus revised antithetical productivity is not the claim that negation itself magically creates anatomy. It is the 
claim that HGL permits negative whole-state logical structure to become a source of inherited positive 
architecture in a later stage. 
The contrast is therefore: 
 
¬𝑆
means only that does not obtain, whereas 
𝑆
 
𝐼𝐻¬𝑆
(
)
means that a later state inherits the properties of that structured non- state. This distinction is the engine of 
𝑆
the higher zig-zag. 
Part III — De Morgan’s laws, carefully and at multiple levels 
8. De Morgan’s laws themselves 
For ordinary propositions and : 
𝐴
𝐵
 
¬ 𝐴∧𝐵
(
) ≡¬𝐴∨¬𝐵,
and 
 
¬ 𝐴∨𝐵
(
) ≡¬𝐴∧¬𝐵.
For any finite collection 
: 
𝐴1, ..., 𝐴𝑛
 
¬ ⋀
𝑛
𝑖=1𝐴𝑖
(
) ≡⋁
𝑛
𝑖=1¬𝐴𝑖,
and 
 
¬ ⋁
𝑛
𝑖=1𝐴𝑖
(
) ≡⋀
𝑛
𝑖=1¬𝐴𝑖.
These are ordinary classical equivalences. What becomes unusual in HGL is where the 
 live. They can live 
𝐴𝑖
inside a PA bracket, outside several PA brackets, or at several nested depths simultaneously. 
9. Inner-PA De Morgan 
Consider first-order Supramale: 
SM1 = PA(NOT x AND y) 
The inner antithetical developmental condition is 
 
𝑃𝐴¬ ¬𝑥∧𝑦
(
)
(
).
Applying De Morgan inside the PA bracket gives 
 
¬ ¬𝑥∧𝑦
(
) ≡𝑥∨¬𝑦.
Therefore 
 
𝑃𝐴¬ ¬𝑥∧𝑦
(
)
(
) ≡𝑃𝐴𝑥∨¬𝑦
(
)
provided PA respects substitution of logically equivalent inner conditions. This is not a distribution rule for PA; 
it is simply replacing one inner formula by a classically equivalent formula. 
Similarly,

Source Canon: Verbatim mathematical and modal logic. Interactive navigation and operator lookups available at /#hgl-part:13.