Material nonimplication explained

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.^[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 \nrightarrow Q , P \not \supset Q , or "Lpq" (in Bocheński notation), and is logically equivalent to \neg (P → Q) , and P \land \neg Q .

Definition

Logical Equivalences

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

P \nrightarrow Q	\Leftrightarrow	\neg (P → 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

Notes and References

1. Berco . Dan. Ang. Diing Shenp. Kalaga. Pranav Sairam. Programmable Photoelectric Memristor Gates for In Situ Image Compression. Advanced Intelligent Systems. 2. 9 . 2020. 10.1002/aisy.202000079. 5. free.