# Material nonimplication

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Material nonimplication
Venn diagram of $A \nrightarrow B$

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."

(A'+B)'

C: (A&(~B))