Definition:Logical NOR

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 Quine.

Boolean Interpretation

From the above, we see that the boolean interpretations for $\mathbf A \downarrow \mathbf B$ under the model $\mathcal M$ are:

$\left({\mathbf A \downarrow \mathbf B}\right)_{\mathcal M} = \begin{cases} T & : \mathbf A_{\mathcal M} = F \text{ and } \mathbf B_{\mathcal M} = F \\ F & : \text {otherwise} \end{cases}$

Complement

The complement of $\downarrow$ is the disjunction operator.

Truth Function

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

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

Truth Table

The truth table of $p \downarrow q$ and its complement is as follows:

$\begin{array}{|cc||c|c|} \hline p & q & p \downarrow q & p \lor q \\ \hline F & F & T & F \\ F & T & F & T \\ T & F & F & T \\ T & T & F & T \\ \hline \end{array}$

Notational Variants

Various symbols are encountered that denote the concept of NOR:

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