Material nonimplication

Material nonimplication
NIMPLY
Definition	xy‾
Truth table	(0100)
Logic gate
Normal forms
Disjunctive	xy‾
Conjunctive	x+y‾
Zhegalkin polynomial	x⊕xy
Post's lattices
0-preserving	yes
1-preserving	no
Monotone	no
Affine	no
Self-dual	no
	v; t; e;

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 $P$ 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 $P ↛ Q$ , $P \supset̸ Q$ , or "Lpq" (in Bocheński notation), and is logically equivalent to $\neg (P \to Q)$ , and $P \land \neg Q$ .

Definition

Truth table

$P$	$Q$	$P ↛ Q$
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.

$P ↛ Q$	$\Leftrightarrow$	$\neg (P \to Q)$
	$\Leftrightarrow$	$\neg$

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 $\neg Q$

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

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

Logical operation: P && !Q.

References

↑ Berco, Dan; Ang, Diing Shenp; Kalaga, Pranav Sairam (2020). "Programmable Photoelectric Memristor Gates for In Situ Image Compression". Advanced Intelligent Systems. 2 (9): 5. doi:10.1002/aisy.202000079.

External links

Media related to Material nonimplication at Wikimedia Commons

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[1] Berco, Dan; Ang, Diing Shenp; Kalaga, Pranav Sairam (2020). "Programmable Photoelectric Memristor Gates for In Situ Image Compression". Advanced Intelligent Systems. 2 (9): 5. doi:10.1002/aisy.202000079.

[1]

v t e Common logical connectives
Tautology/True $⊤$
Alternative denial (NAND gate) $\overline{\land}$ Converse implication $\Leftarrow$ Implication (IMPLY gate) $\Rightarrow$ Disjunction (OR gate) $\lor$
Negation (NOT gate) $\neg$ Exclusive or (XOR gate) $\oplus$ Biconditional (XNOR gate) $⊙$ Statement (Digital buffer)
Joint denial (NOR gate) $\overline{\lor}$ Nonimplication (NIMPLY gate) $⇏$ Converse nonimplication $⇍$ Conjunction (AND gate) $\land$
Contradiction/False $⊥$
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