Definition:Logical NOR

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NOR (that is, not or), is a binary connective, written symbolically as $p \downarrow q$, whose behaviour is as follows:

$p \downarrow q$

is defined as:

neither $p$ nor $q$ is true.

$p \downarrow q$ is voiced:

$p$ nor $q$

The symbol $\downarrow$ is known as the Quine arrow, named after Willard Van Orman Quine.

Truth Function

The NOR connective defines the truth function $f^\downarrow$ as follows:

\(\ds \map {f^\downarrow} {\F, \F}\) \(=\) \(\ds \T\)
\(\ds \map {f^\downarrow} {\F, \T}\) \(=\) \(\ds \F\)
\(\ds \map {f^\downarrow} {\T, \F}\) \(=\) \(\ds \F\)
\(\ds \map {f^\downarrow} {\T, \T}\) \(=\) \(\ds \F\)

Truth Table

The characteristic truth table of the logical NOR operator $p \downarrow q$ is as follows:

$\begin {array} {|cc||c|} \hline

p & q & p \downarrow q \\ \hline \F & \F & \T \\ \F & \T & \F \\ \T & \F & \F \\ \T & \T & \F \\ \hline \end {array}$

Boolean Interpretation

The truth value of $\mathbf A \downarrow \mathbf B$ under a boolean interpretation $v$ is given by:

$\map v {\mathbf A \downarrow \mathbf B} = \begin {cases}

\T & : \map v {\mathbf A} = \map v {\mathbf B} = \F \\ \F & : \text {otherwise} \end {cases}$

Notational Variants

Various symbols are encountered that denote the concept of logical NOR:

Symbol Origin Known as
$p \downarrow q$ Willard Quine Quine arrow
$p \mathop {\mathsf {NOR} } q$
$p \mathop \bot q$
$p \curlywedge q$ Charles Sanders Peirce Ampheck

The all-uppercase rendition NOR originates from the digital electronics industry, where, because NOR is Functionally Complete, this operator has a high importance.

Also see

  • Results about logical NOR can be found here.