Material nonimplication

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.^[1] It is the negation of material implication. That is to say that for any two propositions ${\displaystyle P}$ and ${\displaystyle Q}$, the material nonimplication from ${\displaystyle P}$ to ${\displaystyle Q}$ is true if and only if the negation of the material implication from ${\displaystyle P}$ to ${\displaystyle Q}$ is true. This is more naturally stated as that the material nonimplication from ${\displaystyle P}$ to ${\displaystyle Q}$ is true only if ${\displaystyle P}$ is true and ${\displaystyle Q}$ is false.

Venn diagram of ${\displaystyle P\nrightarrow Q}$

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

Definition

Truth table

${\displaystyle A}$	${\displaystyle B}$	${\displaystyle A\nrightarrow B}$
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.

${\displaystyle P\nrightarrow Q}$	${\displaystyle \Leftrightarrow }$	${\displaystyle \neg (P\rightarrow Q)}$
	${\displaystyle \Leftrightarrow }$	${\displaystyle \neg }$

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

${\displaystyle P\nrightarrow Q}$	${\displaystyle \Leftrightarrow }$	${\displaystyle \neg (}$	${\displaystyle \neg P}$	${\displaystyle \lor }$	${\displaystyle Q)}$	${\displaystyle \Leftrightarrow }$	${\displaystyle P}$	${\displaystyle \land }$	${\displaystyle \neg Q}$
	${\displaystyle \Leftrightarrow }$	${\displaystyle \neg (}$		${\displaystyle \lor }$	${\displaystyle )}$	${\displaystyle \Leftrightarrow }$		${\displaystyle \land }$

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. This is usually called "bit clear" (BIC) or "and not" (ANDN).

Logical operation: A && !B.

See also

• Implication
• Set difference