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

Proof by Natural deduction

These are proved by the Tableau method.

$p \vdash p \land p$

Line		Pool	Formula	Rule	Depends upon
1		1	$p$	P	(None)
2		1	$p \land p$	$\land \mathcal I$	1, 1

$\blacksquare$

$p \land p \vdash p$

Line		Pool	Formula	Rule	Depends upon
1		1	$p \land p$	P	(None)
2		1	$p$	$\land \mathcal E_1$	1

$\blacksquare$

$p \vdash p \lor p$

Line		Pool	Formula	Rule	Depends upon
1		1	$p$	P	(None)
2		1	$p \lor p$	$\lor \mathcal I_1$	1

$\blacksquare$

$p \lor p \vdash p$

Line		Pool	Formula	Rule	Depends upon
1		1	$p \lor p$	P	(None)
2		2	$p$	A	(None)
3		1	$p$	$\lor \mathcal E$	1, 2-2, 2-2

$\blacksquare$

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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.

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

$\blacksquare$

Sources

• Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{II}: \S 3$: Theorem $\text{T41}$
• Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{II}: \S 5$: Theorem $\text{T47}$
• E.J. Lemmon: Beginning Logic (1965): $\S 1.5$: Theorem $33$, Exercise $1 \ \text{(b)}$