# Double Negation Elimination implies Law of Excluded Middle

## Theorem

Let the Law of Double Negation Elimination be supposed to hold:

- $\neg \neg p \vdash p$

Then the Law of Excluded Middle likewise holds:

- $\vdash p \lor \neg p$

## Proof

By the tableau method of natural deduction:

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

1 | 1 | $\neg \left({p \lor \neg p}\right)$ | Assumption | (None) | Assume the contrary | |

2 | 2 | $p$ | Assumption | (None) | Assume one disjunct | |

3 | 2 | $p \lor \neg p$ | Rule of Addition: $\lor \mathcal I_1$ | 2 | ||

4 | 1, 2 | $\bot$ | Principle of Non-Contradiction: $\neg \mathcal E$ | 3, 1 | ||

5 | 1 | $\neg p$ | Proof by Contradiction: $\neg \mathcal I$ | 2 – 4 | Assumption 2 has been discharged | |

6 | 1 | $p \lor \neg p$ | Rule of Addition: $\lor \mathcal I_2$ | 5 | ||

7 | 1 | $\bot$ | Principle of Non-Contradiction: $\neg \mathcal E$ | 6, 1 | also demonstrating a contradiction | |

8 | $\neg \neg \left({p \lor \neg p}\right)$ | Proof by Contradiction: $\neg \mathcal I$ | 1 – 7 | Assumption 1 has been discharged | ||

9 | $p \lor \neg p$ | Double Negation Elimination: $\neg \neg \mathcal E$ | 8 |

$\blacksquare$

## Comment

Thus the Law of Double Negation Elimination may be taken as an axiom instead of the Law of Excluded Middle.

## 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

- 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 2.2$: Theorems and Derived Rules: Theorem $44$ - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.2.2$: Derived rules