Material nonimplication

Venn diagram of $P\nrightarrow Q$

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 $P$ and $Q$, the material nonimplication from $P$ to $Q$ is true if and only if the negation of the material implication from $P$ to $Q$ is true. This is more naturally stated as that the material nonimplication from $P$ to $Q$ is true only if $P$ is true and $Q$ is false.

It may be written using logical notation as $P\nrightarrow Q$, $P\supset ̸Q$, or "Lpq" (in Bocheński notation), and is logically equivalent to $\mathrm{\neg }(P\to Q)$, and $P\wedge \mathrm{\neg }Q$.

Definition

Truth table

$P$	$Q$	$P\nrightarrow Q$
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.

$P\nrightarrow Q$	$\iff$	$\mathrm{\neg }(P\to Q)$
	$\iff$	$\mathrm{\neg }$

In classical logic, it is also equivalent to the negation of the disjunction of $\mathrm{\neg }P$ and $Q$, and also the conjunction of $P$ and $\mathrm{\neg }Q$

$P\nrightarrow Q$	$\iff$	$\mathrm{\neg }($	$\mathrm{\neg }P$	$\vee$	$Q)$	$\iff$	$P$	$\wedge$	$\mathrm{\neg }Q$
	$\iff$	$\mathrm{\neg }($		$\vee$	$)$	$\iff$		$\wedge$

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