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:

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

Notational Variants
Various symbols are encountered that denote the concept of NOR: