material nonimplication

{{Short description|Logical connective}}

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File:Venn0100.svg of $P\; \backslash nrightarrow\; Q$]]

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.{{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}} 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 $P\; \backslash nrightarrow\; Q$, $P\; \backslash not\; \backslash supset\; Q$, or "Lpq" (in Bocheński notation), and is logically equivalent to $\backslash neg\; (P\; \backslash rightarrow\; Q)$, and $P\; \backslash land\; \backslash neg\; Q$.