# 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 1

By the tableau method of natural deduction:

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

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

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

3 | 2 | $p \lor \neg p$ | Rule of Addition: $\lor \II_1$ | 2 | ||

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

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

6 | 1 | $p \lor \neg p$ | Rule of Addition: $\lor \II_2$ | 5 | ||

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

8 | $\neg \neg \paren {p \lor \neg p}$ | Proof by Contradiction: $\neg \II$ | 1 – 7 | Assumption 1 has been discharged | ||

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

$\blacksquare$

## Proof 2

By the tableau method of natural deduction:

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

1 | $\neg\neg (p \lor \neg p)$ | Theorem Introduction | (None) | Negation of Excluded Middle is False/Form 2 | ||

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

$\blacksquare$

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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:

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