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:

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: