Rule of Idempotence/Disjunction/Formulation 2

Theorem

The disjunction operation is idempotent:

$\vdash p \iff \paren {p \lor p}$

This can be expressed as two separate theorems:

Forward Implication

$\vdash p \implies \left({p \lor p}\right)$

Reverse Implication

$\vdash \left({p \lor p}\right) \implies p$

Proof

By the tableau method of natural deduction:

$p \iff \paren {p \lor p} $

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p$	Assumption	(None)
2		1	$p \lor p$	Rule of Addition: $\lor \II_1$	1
3			$p \implies \paren {p \lor p}$	Rule of Implication: $\implies \II$	1 – 2	Assumption 1 has been discharged
4		4	$p \lor p$	Assumption	(None)
5		5	$\neg p$	Assumption	(None)
6		4, 5	$p$	Modus Tollendo Ponens $\mathrm {MTP}_1$	4, 5
7		4, 5	$\bot$	Principle of Non-Contradiction: $\neg \EE$	6, 5
8		5	$p$	Proof by Contradiction: $\neg \II$	5 – 7	Assumption 5 has been discharged
9			$\paren {p \lor p} \implies p$	Rule of Implication: $\implies \II$	4 – 8	Assumption 4 has been discharged
10			$p \iff \paren {p \lor p}$	Biconditional Introduction: $\iff \II$	3, 9

$\blacksquare$

Sources

• 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.13$: Symbolism of sentential calculus
• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Theorem $\text{T47}$
• 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.2$: The Rule of Replacement: $19.$
• 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2)$: The remaining rules of inference: $18$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic: Exercise $(1) \ \text{(ii)}$