Disjunction with Tautology/Proof 1

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Theorem

$p \lor \top \dashv \vdash \top$


Proof

By the tableau method of natural deduction:

$p \lor \top \vdash \top$
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: $\lor \mathcal I_1$ 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

$\Box$



By the tableau method of natural deduction:

$\top \vdash p \lor \top$
Line Pool Formula Rule Depends upon Notes
1 1 $\top$ Premise (None)
2 1 $p \lor \top$ Rule of Addition: $\lor \mathcal I_2$ 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.