Material nonimplication

Material nonimplication or abjunction is a term referring to a logic operation used in generic circuits and <a href="/facts/Boolean_algebra/VkCToAmg">Boolean algebra</a>. It is the <a href="/facts/Negation/q5AQBvBj">negation</a> of <a href="/facts/Material_conditional/hYf4b7AE">material implication</a>. That is to say that for any two <a href="/facts/Proposition/0877MFMD">propositions</a> 
 
 
 
 P
 
 
 {\displaystyle P}
 
 and 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
, the material nonimplication from 
 
 
 
 P
 
 
 {\displaystyle P}
 
 to 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 is true <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the negation of the material implication from 
 
 
 
 P
 
 
 {\displaystyle P}
 
 to 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 is true. This is more naturally stated as that the material nonimplication from 
 
 
 
 P
 
 
 {\displaystyle P}
 
 to 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 is true only if 
 
 
 
 P
 
 
 {\displaystyle P}
 
 is true and 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 is false.
It may be written using logical notation as 
 
 
 
 P
 ↛
 Q
 
 
 {\displaystyle P\nrightarrow Q}
 
, 
 
 
 
 P
 ⊅
 Q
 
 
 {\displaystyle P\not \supset Q}
 
, or "Lpq" (in <a href="/facts/Boche%25C5%2584ski_notation/nWsm4IKA">Bocheński notation</a>), and is logically equivalent to 
 
 
 
 ¬
 (
 P
 →
 Q
 )
 
 
 {\displaystyle \neg (P\rightarrow Q)}
 
, and 
 
 
 
 P
 ∧
 ¬
 Q
 
 
 {\displaystyle P\land \neg Q}
 
.

Material nonimplication open-in-new

Material nonimplication