Material nonimplication

Venn diagram of $A \nrightarrow B$

$\and$ 32px $\Leftrightarrow$ $\Leftrightarrow \neg$

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 not P implies Q. 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:

p⊅q

Lpq

p↛q

And is equivalent to:

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

"p but not q."

Boolean algebra

Further information: Boolean algebra

(A'+B)'

Computer science

Bitwise operation: A&(~B)

Logical operation: A&&(!B)

See also

• Implication