Venn diagram of 
  
    
      
        P
        ↛
        Q
      
    
    {\displaystyle P\nrightarrow Q}
Venn diagram of

Material nonimplication or abjunction (Latin ab = "from", junctio =–"joining") 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

True True False
True False True
False True False
False False False

Logical Equivalences

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

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

         
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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."

Computer science

Bitwise operation: A&(~B)

Logical operation: A&&(!B)

See also

References

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