Material nonimplication

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

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 [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].

Definition

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]
	[math]\displaystyle{ \Leftrightarrow }[/math]	[math]\displaystyle{ \neg }[/math]

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]
	[math]\displaystyle{ \Leftrightarrow }[/math]	[math]\displaystyle{ \neg( }[/math]		[math]\displaystyle{ \lor }[/math]	[math]\displaystyle{ ) }[/math]	[math]\displaystyle{ \Leftrightarrow }[/math]		[math]\displaystyle{ \land }[/math]

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