Hypergendered Logic Source

Hypergendered Logic — Page 14

HGL Source Framework · Page 14 of 293 · Source: Hypergendered Logic

𝑃𝐴¬ 𝑥∧¬𝑦
(
)
(
) ≡𝑃𝐴¬𝑥∨𝑦
(
).
10. Outer-PA De Morgan 
Now consider the user’s explicit outer example: 
 
¬𝑃𝐴𝑥
( ) ∧¬𝑃𝐴¬𝑦
(
).
By De Morgan at the outer state level, 
 
¬𝑃𝐴𝑥
( ) ∧¬𝑃𝐴¬𝑦
(
) ≡¬ 𝑃𝐴𝑥
( ) ∨𝑃𝐴¬𝑦
(
)
(
).
Likewise, 
 
¬𝑃𝐴𝑥
( ) ∨¬𝑃𝐴¬𝑦
(
) ≡¬ 𝑃𝐴𝑥
( ) ∧𝑃𝐴¬𝑦
(
)
(
).
Notice what De Morgan does not license: 
 
¬𝑃𝐴𝑥
( )≢𝑃𝐴¬𝑥
(
).
The outer negation stays outside unless HGL explicitly applies an antithetical operator that changes the 
content of the PA-state. 
11. Mixed-depth example 
Suppose 
 
𝑆= 𝑃𝐴𝑥∧¬𝑦
(
) ∧𝑃𝐴¬𝑦
(
).
Then 
 
¬𝑆≡¬𝑃𝐴𝑥∧¬𝑦
(
) ∨¬𝑃𝐴¬𝑦
(
).
If we also inspect the inner content of the first PA term, we may separately note 
 
¬ 𝑥∧¬𝑦
(
) ≡¬𝑥∨𝑦.
But we may not jump from 
 
¬𝑃𝐴𝑥∧¬𝑦
(
)
to 
 
𝑃𝐴¬𝑥∨𝑦
(
)
by De Morgan alone. That would move the negation through the PA operator, which is a different operation. 
This is one of the most important discipline rules in the whole field: 
De Morgan may rewrite what a negation scopes over. It does not give permission to move a 
negation across a non-Boolean operator. 
12. Eight worked De Morgan examples 
Example 1: inner conjunction 
 
𝑃𝐴¬ 𝑥∧𝑦
(
)
(
) ≡𝑃𝐴¬𝑥∨¬𝑦
(
).
Example 2: inner disjunction 
 
𝑃𝐴¬ 𝑥∨𝑦
(
)
(
) ≡𝑃𝐴¬𝑥∧¬𝑦
(
).
Example 3: outer conjunction of states 
 
¬ 𝑃𝐴𝑥
( ) ∧𝑃𝐴𝑦
( )
(
) ≡¬𝑃𝐴𝑥
( ) ∨¬𝑃𝐴𝑦
( ).

Source Canon: Verbatim mathematical and modal logic. Interactive navigation and operator lookups available at /#hgl-part:14.