Material nonimplication

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 not P implies Q. This is more naturally stated as that the material nonimplication from P to Q is true is only true 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)

References

Logical connectives

Tautology $\top$

NAND $\uparrow$ Converse implication $\leftarrow$ Implication $\rightarrow$ OR $\or$

Negation $\neg$ XOR $\oplus$ Biconditional $\leftrightarrow$ Statement

NOR $\downarrow$ Nonimplication $\nrightarrow$ Converse nonimplication $\nleftarrow$ AND $\and$

Contradiction $\bot$

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