Material nonimplication

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Venn diagram of [math]\displaystyle{ P \nrightarrow Q }[/math]

Material nonimplication or abjunction (Latin ab = "from", junctio =–"joining") is the negation of material implication. That is to say that for any two propositions [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math], the material nonimplication from [math]\displaystyle{ P }[/math] to [math]\displaystyle{ Q }[/math] is true if and only if the negation of the material implication from [math]\displaystyle{ P }[/math] to [math]\displaystyle{ Q }[/math] is true. This is more naturally stated as that the material nonimplication from [math]\displaystyle{ P }[/math] to [math]\displaystyle{ Q }[/math] is true only if [math]\displaystyle{ P }[/math] is true and [math]\displaystyle{ Q }[/math] is false.

It may be written using logical notation as [math]\displaystyle{ P \nrightarrow Q }[/math], [math]\displaystyle{ P \not \supset Q }[/math], or "Lpq" (in Bocheński notation), and is logically equivalent to [math]\displaystyle{ \neg (P \rightarrow Q) }[/math], and [math]\displaystyle{ P \land \neg Q }[/math].


Truth table

[math]\displaystyle{ P }[/math] [math]\displaystyle{ Q }[/math] [math]\displaystyle{ P \nrightarrow Q }[/math]
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.

[math]\displaystyle{ P \nrightarrow Q }[/math]   [math]\displaystyle{ \Leftrightarrow }[/math]   [math]\displaystyle{ \neg (P \rightarrow Q) }[/math]
Venn0100.svg   [math]\displaystyle{ \Leftrightarrow }[/math]   [math]\displaystyle{ \neg }[/math] Venn1011.svg

In classical logic, it is also equivalent to the negation of the disjunction of [math]\displaystyle{ \neg P }[/math] and [math]\displaystyle{ Q }[/math], and also the conjunction of [math]\displaystyle{ P }[/math] and [math]\displaystyle{ \neg Q }[/math]

[math]\displaystyle{ P \nrightarrow Q }[/math]   [math]\displaystyle{ \Leftrightarrow }[/math]   [math]\displaystyle{ \neg( }[/math] [math]\displaystyle{ \neg P }[/math] [math]\displaystyle{ \lor }[/math] [math]\displaystyle{ Q) }[/math]   [math]\displaystyle{ \Leftrightarrow }[/math]   [math]\displaystyle{ P }[/math] [math]\displaystyle{ \land }[/math] [math]\displaystyle{ \neg Q }[/math]
Venn0100.svg   [math]\displaystyle{ \Leftrightarrow }[/math]   [math]\displaystyle{ \neg( }[/math] Venn1010.svg [math]\displaystyle{ \lor }[/math] Venn0011.svg[math]\displaystyle{ ) }[/math]   [math]\displaystyle{ \Leftrightarrow }[/math]   Venn0101.svg [math]\displaystyle{ \land }[/math] Venn1100.svg


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.


The symbol for material nonimplication is simply a crossed-out material implication symbol. Its Unicode symbol is 219B16 (8603 decimal): ↛.

Natural language


"p minus q."

"p without q."


"p but not q." "q is false, in spite of p."

Computer science

Bitwise operation: A&(~B)

Logical operation: A&&(!B)

See also


External links