Rule of Commutation

Definition
This rule is two-fold:


 * Conjunction is commutative:
 * $$p \and q \dashv \vdash q \and p$$


 * Disjunction is commutative:
 * $$p \or q \dashv \vdash q \or p$$

Its abbreviation in a tableau proof is $$\text{Comm}$$.

Alternative rendition
These can alternatively be rendered as:


 * $$\vdash \left({p \and q}\right) \iff \left({q \and p}\right)$$
 * $$\vdash \left({p \or q}\right) \iff \left({q \or p}\right)$$

They can be seen to be logically equivalent to the forms above.

Proof by Natural Deduction
By the tableau method:


 * $$p \and q \vdash q \and p$$:

By the same technique we can show $$q \and p \vdash p \and q$$.


 * $$p \or q \vdash q \or p$$:

By the same technique we can show $$q \or p \vdash p \or q$$.

Proof by Truth Table
We apply the Method of Truth Tables to the propositions in turn.

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

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

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

Also see

 * Rule of Association
 * Principle of Commutation