View source for Material nonimplication

{{Short description|Logical connective}}
{{Use dmy dates|date=April 2024}}
[[File:Venn0100.svg|thumb|240px|[[Venn diagram]] of <math>P \nrightarrow Q</math>]]
'''Material nonimplication''' or '''abjunction''' ({{Etymology|la|ab|away||junctio|to join}}) is a term referring to a logic operation used in generic circuits and [[Boolean algebra]].<ref>{{cite journal|last=Berco |first=Dan|last2=Ang|first2=Diing Shenp|last3=Kalaga| first3=Pranav Sairam|title=Programmable Photoelectric Memristor Gates for In Situ Image Compression|journal=Advanced Intelligent Systems| volume=2|issue=9 |year=2020|doi=10.1002/aisy.202000079|page=5|doi-access=free}}</ref> It is the [[negation]] of [[Material conditional|material implication]]. That is to say that for any two [[proposition]]s <math>P</math> and <math>Q</math>, the material nonimplication from <math>P</math> to <math>Q</math> is true [[if and only if]] the negation of the material implication from <math>P</math> to <math>Q</math> is true.  This is more naturally stated as that the material nonimplication from <math>P</math> to <math>Q</math> is true only if <math>P</math> is true and <math>Q</math> is false.

It may be written using logical notation as <math>P \nrightarrow Q</math>, <math>P \not \supset Q</math>, or "L''pq''" (in [[Bocheński notation]]), and is logically equivalent to <math>\neg (P \rightarrow Q)</math>, and <math>P \land \neg Q</math>.

==Definition==

===Truth table===

{{2-ary truth table|0|0|1|0|<math>A \nrightarrow B</math>}}

===Logical equivalences===

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

{| style="text-align: center; border: 1px solid darkgray;"
|-
| <math>P \nrightarrow Q</math>
| &emsp;&emsp;<math>\Leftrightarrow</math>&emsp;&emsp;
| <math>\neg (P \rightarrow Q)</math>
|-
| [[File:Venn0100.svg|50px]]
| &emsp;&emsp;<math>\Leftrightarrow</math>&emsp;&emsp;
| <math>\neg</math> [[File:Venn1011.svg|50px]]
|}

In [[classical logic]], it is also equivalent to the negation of the [[Logical_disjunction|disjunction]] of <math>\neg P</math> and <math>Q</math>, and also the [[Logical_conjunction|conjunction]] of <math>P</math> and <math>\neg Q</math>

{| style="text-align: center; border: 1px solid darkgray;"
|-
| <math>P \nrightarrow Q</math>
| &emsp;&emsp;<math>\Leftrightarrow</math>&emsp;&emsp;
| <math>\neg(</math>
| <math>\neg P</math>
| <math>\lor</math>
| <math>Q)</math>
| &emsp;&emsp;<math>\Leftrightarrow</math>&emsp;&emsp;
| <math>P</math>
| <math>\land</math>
| <math>\neg Q</math>
|-
| [[File:Venn0100.svg|50px]]
| &emsp;&emsp;<math>\Leftrightarrow</math>&emsp;&emsp;
| <math>\neg(</math>
| [[File:Venn1010.svg|50px]]
| <math>\lor</math>
| [[File:Venn0011.svg|50px]]<math>)</math>
| &emsp;&emsp;<math>\Leftrightarrow</math>&emsp;&emsp;
| [[File:Venn0101.svg|50px]]
| <math>\land</math>
| [[File:Venn1100.svg|50px]]
|}

==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 [[Arrow (symbol)#Arrows_in_Unicode|219B<sub>16</sub>]] (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==
* [[Material conditional|Implication]]
* [[Complement (set theory)#Relative_complement|Set difference]]

==References==
{{Reflist}}

==External links==
*{{Commonscatinline}}

{{Logical connectives}}

[[Category:Logical connectives]]


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