Encyclopedia

Material nonimplication

Logical connective

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Source updated September 13, 2026

Material nonimplication or abjunction (from Latin ab 'away' and junctio 'to join') is a term referring to a logic operation used in generic circuits and Boolean algebra. It 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 $P\nrightarrow Q$ , $P\not \supset Q$ , or "Lpq" (in Bocheński notation), and is logically equivalent to $\neg (P\rightarrow Q)$ , and $P\land \neg Q$ .

01Definition

Truth table

$P$	$Q$	$P\nrightarrow Q$
F	F	F
F	T	F
T	F	T
T	T	F

Logical equivalences

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

$P\nrightarrow Q$	$\Leftrightarrow$	$\neg (P\rightarrow Q)$
	$\Leftrightarrow$	$\neg$

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

$P\nrightarrow Q$	$\Leftrightarrow$	$\neg ($	$\neg P$	$\lor$	$Q)$	$\Leftrightarrow$	$P$	$\land$	$\neg Q$
	$\Leftrightarrow$	$\neg ($		$\lor$	$)$	$\Leftrightarrow$		$\land$

02Properties

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.

03Symbol

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

04Natural language

Grammatical

"p minus q."

"p without q."

Rhetorical

"p but not q."

"q is false, in spite of p."

05Computer science

Bitwise operation: P & ~Q. This is usually called "bit clear" (BIC) or "and not" (ANDN).

Logical operation: P && !Q.

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Sources and credits

This article is adapted from the Wikipedia article “Material nonimplication”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Venn0100.svg by Unknown author, Public domain

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