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:

$$ $$ $$ $$

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
Various symbols are encountered that denote the concept of NAND: