Definition:Logical NOR

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Definition

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:

\(\displaystyle f^\downarrow \left({F, F}\right)\) \(=\) \(\displaystyle T\) $\quad$ $\quad$
\(\displaystyle f^\downarrow \left({F, T}\right)\) \(=\) \(\displaystyle F\) $\quad$ $\quad$
\(\displaystyle f^\downarrow \left({T, F}\right)\) \(=\) \(\displaystyle F\) $\quad$ $\quad$
\(\displaystyle f^\downarrow \left({T, T}\right)\) \(=\) \(\displaystyle F\) $\quad$ $\quad$


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:

$v \left({\mathbf A \downarrow \mathbf B}\right) = \begin{cases} T & : v \left({\mathbf A}\right) = v \left({\mathbf B}\right) = F\\ F & : \text{otherwise} \end{cases}$


Notational Variants

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

Symbol Origin Known as
$p \downarrow q$ Willard Quine Quine arrow
$p \ \mathsf{NOR} \ q$
$p \mathop \bot q$
$p \curlywedge q$ 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.


Sources