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. It is the negation of material implication. That is to say that for any two propositions P {\displaystyle P} and Q {\displaystyle Q} , the material nonimplication from P {\displaystyle P} to Q {\displaystyle Q} is true if and only if 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 Bocheński notation), and is logically equivalent to ¬ ( P → Q ) {\displaystyle \neg (P\rightarrow Q)} , and P ∧ ¬ Q {\displaystyle P\land \neg Q} .

Definition

Truth table

Logical equivalences 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 ¬ P {\displaystyle \neg P} and Q {\displaystyle Q} , and also the conjunction of P {\displaystyle P} and ¬ Q {\displaystyle \neg Q}

Properties 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.

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

Natural language

Grammatical "p minus q." "p without q."

Rhetorical "p but not q." "q is false, in spite of p."

Computer science Bitwise operation: P & ~Q. This is usually called "bit clear" (BIC) or "and not" (ANDN). Logical operation: P && !Q.

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