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 Function

The NAND connective defines the truth function $f^\uparrow$ as follows:

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

Truth Table

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

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

Notational Variants

Various symbols are encountered that denote the concept of NAND:

Symbol	Origin	Known as
$p \uparrow q$	Henry Sheffer	Sheffer stroke
$p \ \mathsf{NAND} \ q$
$p \ \vert \ q$		Also sometimes referred to as the Sheffer stroke
$p \bar \curlywedge q$	Charles Sanders Peirce	Ampheck