Rule of Commutation

Definition
This rule is two-fold:

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

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

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

Proof by Natural Deduction
By the tableau method:


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

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


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

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

Proof by Truth Table
Let $$v: \left\{{p, q}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a logical formula $$\phi$$ of two variables $$p, q$$.

We see that $$v \left({p \and q}\right) = v \left({q \and p}\right)$$ for all interpretations $$v$$.

Hence the result by the definition of interderivable.

We see that $$v \left({p \or q}\right) = v \left({q \or p}\right)$$ for all interpretations $$v$$.

Hence the result by the definition of interderivable.