Definition:Logical NAND

Definition
NAND (that is, "not and"), is a binary connective, written symbolically as $$p \uparrow q$$, whose behaviour is as follows:


 * $$p \uparrow q$$ is defined as: "$$p$$ and $$q$$ are not both true."

"$$p \uparrow q$$" is voiced "$$p$$ nand $$q$$".

The symbol $$\uparrow$$ is known as the "Sheffer stroke", named after Henry Sheffer, who proved an important result about this operation.

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


 * $$\left({\mathbf A \uparrow \mathbf B}\right)_{\mathcal M} = \begin{cases}

F & : \mathbf A_{\mathcal M} = T \text{ and } \mathbf B_{\mathcal M} = T \\ T & : \text {otherwise} \end{cases}$$

Complement
The complement of $$\uparrow$$ is the conjunction operator.

Truth Table
The truth table of $$p \uparrow q$$ and its complement is as follows:

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

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


 * $$p \ \texttt{NAND} \ q$$;
 * $$p \ | \ q$$, also sometimes referred to as the Sheffer stroke;
 * $$p \bar \curlywedge q$$, this notation originating with Charles Sanders Peirce.