Definition:Logical Not

Definition
The logical not or negation operator is a unary connective whose action is to reverse the truth value of the statement on which it operates.


 * $$\neg \, p$$ is defined as: $$p$$ is not true, or It is not the case that $$p$$ is true.

Thus the statement $$\neg \, p$$ is called the negation of $$p$$.

$$\neg \, p$$ is voiced not $$p$$.

In the statement $$\neg \, p$$, the scope of $$\neg$$ is $$p$$.

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


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

T & : \mathbf A_{\mathcal M} = F \\ F & : \mathbf A_{\mathcal M} = T \end{cases}$$

Truth Function
The logical not connective defines the truth function $$f^\neg$$ as follows:

$$ $$

Truth Table
The truth table of $$\neg \, p$$ is as follows:


 * $$\begin{array}{|c||c|} \hline

p & \neg \, p \\ \hline F & T \\ T & F \\ \hline \end{array}$$

Notational Variants
Various symbols are encountered that denote the concept of the logical not: