Material nonimplication

Material nonimplication or abjunction (Latin ab = "away", 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.

Venn diagram of $P\nrightarrow Q$

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 $P\land \neg Q$.

Definition

Truth table

$A$	$B$	$A\nrightarrow B$
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 $\neg Q$

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

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
• Set difference

External links

• Media related to Material nonimplication at Wikimedia Commons