# Double Negation/Double Negation Elimination/Sequent Form/Formulation 2

## Theorem

- $\vdash \neg \neg p \implies p$

## Proof

By the tableau method of natural deduction:

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

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

2 | 1 | $p$ | Double Negation Elimination: $\neg \neg \EE$ | 1 | ||

3 | $\neg \neg p \implies p$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged |

$\blacksquare$

## Also see

## 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): $\S 4.7$: The Derivation of Formulae: $D \, 16$ - 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 5$: Theorem $\text{T11}$ - 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules: Theorem $40$