Disjunction with Tautology/Proof 1
Theorem
- $p \lor \top \dashv \vdash \top$
Proof
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \lor \top$ | Premise | (None) | ||
| 2 | 2 | $\top$ | Assumption | (None) | ||
| 3 | 3 | $p$ | Assumption | (None) | ||
| 4 | 3 | $p \lor \neg p$ | Rule of Addition: $\lor \II_1$ | 3 | ||
| 5 | 3 | $\top$ | Law of Excluded Middle | 4 | ||
| 6 | 1 | $\top$ | Proof by Cases: $\text{PBC}$ | 1, 2 – 2, 3 – 5 | Assumptions 2 and 3 have been discharged |
$\Box$
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By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\top$ | Premise | (None) | ||
| 2 | 1 | $p \lor \top$ | Rule of Addition: $\lor \II_2$ | 1 |
$\blacksquare$
Law of the Excluded Middle
This theorem depends on the Law of the Excluded Middle.
This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.
However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom. This in turn invalidates this theorem from an intuitionistic perspective.
The propositions:
- If it's not false, it must be true
and
- If it's not true, it must be false
are indeed valid only in the context where there are only two truth values.
From the intuitionistic perspective, these results do not hold.