Material nonimplication

Material nonimplication or abjunction (from Latin ab 'away' and junctio 'to join') is a term referring to a logic operation used in generic circuits and Boolean algebra.^[1] It is the negation of material implication. That is to say that for any two propositions ${\displaystyle P}${\displaystyle P} and ${\displaystyle Q}${\displaystyle Q}, the material nonimplication from ${\displaystyle P}${\displaystyle P} to ${\displaystyle Q}${\displaystyle Q} is true if and only if the negation of the material implication from ${\displaystyle P}${\displaystyle P} to ${\displaystyle Q}${\displaystyle Q} is true. This is more naturally stated as that the material nonimplication from ${\displaystyle P}${\displaystyle P} to ${\displaystyle Q}${\displaystyle Q} is true only if ${\displaystyle P}${\displaystyle P} is true and ${\displaystyle Q}${\displaystyle Q} is false.

It may be written using logical notation as ${\displaystyle P\nrightarrow Q}${\displaystyle P\nrightarrow Q}, ${\displaystyle P\not \supset Q}${\displaystyle P\not \supset Q}, or "Lpq" (in Bocheński notation), and is logically equivalent to ${\displaystyle \neg (P\rightarrow Q)}${\displaystyle \neg (P\rightarrow Q)}, and ${\displaystyle P\land \neg Q}${\displaystyle P\land \neg Q}.

Material nonimplication may be defined as the negation of material implication.

In classical logic, it is also equivalent to the negation of the disjunction of ${\displaystyle \neg P}${\displaystyle \neg P} and ${\displaystyle Q}${\displaystyle Q}, and also the conjunction of ${\displaystyle P}${\displaystyle P} and ${\displaystyle \neg Q}${\displaystyle \neg Q}

falsehood-preserving: The interpretation under which all variables are assigned a truth value of "false" produces a truth value of "false" as a result of material nonimplication.

The symbol for material nonimplication is simply a crossed-out material implication symbol. Its Unicode symbol is 219B₁₆ (8603 decimal): ↛.

"p minus q."

"p without q."

"p but not q."

"q is false, in spite of p."

Bitwise operation: A & ~B. This is usually called "bit clear" (BIC) or "and not" (ANDN).

Logical operation: A && !B.

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