Rule of Commutation/Disjunction/Formulation 1/Proof 1

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Theorem

$p \lor q \dashv \vdash q \lor p$


Proof

By the tableau method of natural deduction:

$p \lor q \vdash q \lor p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \lor q$ Premise (None)
2 2 $p$ Assumption (None)
3 2 $q \lor p$ Rule of Addition: $\lor \mathcal I_2$ 2
4 4 $p$ Assumption (None)
5 4 $q \lor p$ Rule of Addition: $\lor \mathcal I_1$ 4
6 1 $q \lor p$ Proof by Cases: $\text{PBC}$ 1, 2 – 3, 4 – 5 Assumptions 2 and 4 have been discharged


$\blacksquare$


By the tableau method of natural deduction:

$q \lor p \vdash p \lor q$
Line Pool Formula Rule Depends upon Notes
1 1 $q \lor p$ Premise (None)
2 2 $q$ Assumption (None)
3 2 $p \lor q$ Rule of Addition: $\lor \mathcal I_2$ 2
4 4 $p$ Assumption (None)
5 4 $p \lor q$ Rule of Addition: $\lor \mathcal I_1$ 4
6 1 $p \lor q$ Proof by Cases: $\text{PBC}$ 1, 2 – 3, 4 – 5 Assumptions 2 and 4 have been discharged


$\blacksquare$


Sources