Rule of Idempotence

Definition
The rule of idempotence is two-fold:


 * Conjunction is idempotent:
 * $$p \dashv \vdash p \land p$$


 * Disjunction is idempotent:
 * $$p \dashv \vdash p \lor p$$

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

Alternative rendition
These can alternatively be rendered as:


 * $$\vdash p \iff \left({p \land p}\right)$$
 * $$\vdash p \iff \left({p \lor p}\right)$$

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

Proof by Natural deduction
These are proved by the Tableau method.

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

As can be seen by inspection, the truth values under the main connectives match for each model.

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