# Rule of Commutation/Disjunction/Formulation 1

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## Theorem

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

## Proof 1

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: $\lor \II_2$ 2
4 4 $p$ Assumption (None)
5 4 $q \lor p$ Rule of Addition: $\lor \II_1$ 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: $\lor \II_2$ 2
4 4 $p$ Assumption (None)
5 4 $p \lor q$ Rule of Addition: $\lor \II_1$ 4
6 1 $p \lor q$ Proof by Cases: $\text{PBC}$ 1, 2 – 3, 4 – 5 Assumptions 2 and 4 have been discharged

$\blacksquare$

## Proof 2

We apply the Method of Truth Tables.

As can be seen by inspection, in both cases, the truth values under the main connectives match for all boolean interpretations.

$\begin{array}{|ccc||ccc|} \hline p & \lor & q & q & \lor & p \\ \hline \F & \F & \F & \F & \F & \F \\ \F & \T & \T & \T & \T & \F \\ \T & \T & \F & \F & \T & \T \\ \T & \T & \T & \T & \T & \T \\ \hline \end{array}$

$\blacksquare$