# Double Negation/Formulation 2

## Theorem

- $\vdash p \iff \neg \neg p$

## Proof 1

By the tableau method of natural deduction:

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

1 | $p \implies \neg \neg p$ | Theorem Introduction | (None) | Double Negation Introduction: Formulation 2 | ||

2 | $\neg \neg p \implies p$ | Theorem Introduction | (None) | Double Negation Elimination: Formulation 2 | ||

3 | $p \iff \neg \neg p$ | Biconditional Introduction: $\iff \mathcal I$ | 1, 2 |

$\blacksquare$

## Proof 2

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, the truth values under the main connective are true for all boolean interpretations.

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

$\blacksquare$

## 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

- 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 3.6$: Reference Formulae: $RF \, 7$ - 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 4.7$: The Derivation of Formulae: $D \, 17$ - 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Theorem $\text{T110}$ - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 6$: Using logical equivalences: $14$ - 1988: Alan G. Hamilton:
*Logic for Mathematicians*(2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Example $1.6 \ \text{(c)}$ - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.14$: Exercise $12 \ (1)$