# Double Negation/Formulation 1/Proof 2

## Theorem

- $p \dashv \vdash \neg \neg p$

## Proof

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, appropriate truth values match for both boolean interpretations.

$\begin{array}{|c||ccc|} \hline
p & \neg & \neg & p \\
\hline
F & F & T & F \\
T & T & F & T \\
\hline
\end{array}$

Hence $p \dashv \vdash \neg \neg p$.

$\blacksquare$

## Double Negation from Intuitionistic Perspective

The intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom.

This in turn invalidates the Law of Double Negation Elimination from the system of intuitionistic propositional logic.

Hence a difference is perceived between Double Negation Elimination and Double Negation Introduction, whereby it can be seen from the Principle of Non-Contradiction that if a statement is true, then it is not the case that it is false.

However, if all we know is that a statement is not false, we can not be certain that it *is* actually true without accepting that there are only two possible truth values.

Such distinctions may be important when considering, for example, multi-value logic.

However, when analysing logic from a purely classical standpoint, it is common and acceptable to make the simplification of taking just one Double Negation rule:

- $p \dashv \vdash \neg \neg p$