Material nonimplication

Venn diagram of $A \nrightarrow B$

$\and$ $\Leftrightarrow$ $\Leftrightarrow \neg$

Material nonimplication or abjunction (latin ab = "from", junctio =–"joining") 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 P does not imply Q.

It may be written using logical notation as:

p⊅q
Lpq
p↛q

Definition

Truth table

p q $~\nrightarrow$
T T F
T F T
F T F
F F F

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 8603 (decimal).

Natural language

Rhetorical

"It's not the case that p implies q."
"p but not q."

Colloquial

"Just because p, doesn't mean q."

Boolean algebra

Further information: Boolean algebra

(A'+B)'

Computer science

Bitwise operation: A&(!B)

Logical operation: A&&(!B)