Material nonimplication
Material nonimplication or abjunction (Latin ab = "away", junctio= "to join") is the negation of material implication. That is to say that for any two propositions  and , the material nonimplication from  to  is true if and only if the negation of the material implication from  to  is true.  This is more naturally stated as that the material nonimplication from  to  is true only if  is true and  is false.

It may be written using logical notation as , , or "Lpq" (in Bocheński notation), and is logically equivalent to , and .

Definition

Truth table

Logical Equivalences

Material nonimplication may be defined as the negation of material implication.

In classical logic, it is also equivalent to the negation of the disjunction of  and , and also the conjunction of  and

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 219B16 (8603 decimal): ↛.

Natural language

Grammatical
"p minus q."

"p without q."

Rhetorical
"p but not q."

"q is false, in spite of p."

Computer science
Bitwise operation: A&(~B) 

Logical operation: A&&(!B)

See also
 Implication
 Boolean algebra
 Set difference

References

External links

Logical connectives