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 implication. That is to say that for any two propositions P and Q, if P does not imply Q, then P is the material nonimplication of 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

(A'+B)'

Computer Science

C: (A&(~B))