Material nonimplication or abjunction (Latin ab = "away", 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 and Q, the material nonimplication from P to Q is true if and only if the negation of the material implication from P to Q is true. This is more naturally stated as that the material nonimplication from P to Q is true only if P is true and Q is false. It may be written using logical notation as, , or "Lpq" (in Bocheński notation), and is logically equivalent to , and.

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 \neg P and Q, and also the conjunction of P and \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: A&(~B) Logical operation: A&&(!B)