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 $$q$$ are 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 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
Alternative symbols that mean the same thing as $$p \downarrow q$$ are also encountered:


 * $$p \ \texttt{NOR} \ q$$;
 * $$p \bot q$$;
 * $$p \curlywedge q$$, this notation originating with Charles Sanders Peirce, and called the ampheck.