# Double Negation/Formulation 1

## Contents

## Theorem

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

## Proof 1

### Double Negation Introduction

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $p$ | Premise | (None) | ||

2 | 2 | $\neg p$ | Assumption | (None) | ||

3 | 1, 2 | $\bot$ | Principle of Non-Contradiction: $\neg \mathcal E$ | 1, 2 | ||

4 | 1 | $\neg \neg p$ | Proof by Contradiction: $\neg \mathcal I$ | 2 – 3 | Assumption 2 has been discharged |

$\blacksquare$

### Double Negation Elimination

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $\neg \neg p$ | Premise | (None) | ||

2 | $p \lor \neg p$ | Law of Excluded Middle | (None) | |||

3 | 3 | $p$ | Assumption | (None) | ||

4 | 4 | $\neg p$ | Assumption | (None) | ||

5 | 1, 4 | $\bot$ | Principle of Non-Contradiction: $\neg \mathcal E$ | 4, 1 | ||

6 | 1, 4 | $p$ | Rule of Explosion: $\bot \mathcal E$ | 5 | ||

7 | 1 | $p$ | Proof by Cases: $\text{PBC}$ | 2, 3 – 3, 4 – 6 | Assumptions 3 and 4 have been discharged |

$\blacksquare$

## Proof 2

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:

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

## Sources

- 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): $\S 2.3.3$