In Boolean logic, logical NOR, non-disjunction, or joint denial Peirce used <math>\overline{\curlywedge}</math> for non-conjunction and <math>\curlywedge</math> for non-disjunction (in fact, what Peirce himself used is <math>\curlywedge</math> and he didn't introduce <math>\overline{\curlywedge}</math> while Peirce's editors made such disambiguated use). Note that most uses in logical notation of <math>\sim</math> use this for negation.
In 1913, Sheffer described non-disjunction and showed its functional completeness. Sheffer used <math>\mid</math> for non-conjunction, and <math>\wedge</math> for non-disjunction.
In 1935, Donald L. Webb described non-disjunction for <math>n</math>-valued logic, and use <math>\mid</math> for the operator. So some people call it Webb operator, So some people call the operator Peirce arrow or Quine dagger.
In 1944, Church also described non-disjunction and use <math>\overline{\vee}</math> for the operator.
In 1954, Bocheński used <math>X</math> in <math>Xpq</math> for non-disjunction in Polish notation.
APL uses a glyph that combines a with a .
Properties
NOR is commutative but not associative, which means that <math>P \downarrow Q \leftrightarrow Q \downarrow P</math> but <math>(P \downarrow Q) \downarrow R \not\leftrightarrow P \downarrow (Q \downarrow R)</math>.
Functional completeness
The logical NOR, taken by itself, is a functionally complete set of connectives. This can be proved by first showing, with a truth table, that <math>\neg A</math> is truth-functionally equivalent to <math>A \downarrow A</math>. Then, since <math>A \downarrow B</math> is truth-functionally equivalent to <math>\neg (A \lor B)</math>,
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