Disjunction elimination

Disjunction elimination

Type	Rule of inference
Field	Propositional calculus
Statement	If a statement $P$ implies a statement $Q$ and a statement $R$ also implies $Q$, then if either $P$ or $R$ is true, then $Q$ has to be true.
Symbolic statement	${\frac {P\to Q,R\to Q,P\lor R}{\therefore Q}}$

In propositional logic, disjunction elimination (sometimes named proof by cases, case analysis, or or elimination), is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof. It is the inference that if a statement $P$ implies a statement $Q$ and a statement $R$ also implies $Q$, then if either $P$ or $R$ is true, then $Q$ has to be true. The reasoning is simple: since at least one of the statements P and R is true, and since either of them would be sufficient to entail Q, Q is certainly true.

An example in English:

If I'm inside, I have my wallet on me.

If I'm outside, I have my wallet on me.

It is true that either I'm inside or I'm outside.

Therefore, I have my wallet on me.

It is the rule can be stated as:

${\frac {P\to Q,R\to Q,P\lor R}{\therefore Q}}$

where the rule is that whenever instances of "$P\to Q$", and "$R\to Q$" and "$P\lor R$" appear on lines of a proof, "$Q$" can be placed on a subsequent line.

Formal notation

The disjunction elimination rule may be written in sequent notation:

$(P\to Q),(R\to Q),(P\lor R)\vdash Q$

where $\vdash$ is a metalogical symbol meaning that $Q$ is a syntactic consequence of $P\to Q$, and $R\to Q$ and $P\lor R$ in some logical system;

and expressed as a truth-functional tautology or theorem of propositional logic:

$(((P\to Q)\land (R\to Q))\land (P\lor R))\to Q$

where $P$, $Q$, and $R$ are propositions expressed in some formal system.

Proof [1]

This rule can be derived from constructive dilemma.

Step	Proposition	Derivation
1	$P\to Q$	Given
2	$R\to Q$	Given
3	$P\lor R$	Given
4	$(P\to Q)\land (R\to Q)$	Conjunction introduction (1,2)
5	$Q\lor Q$	Constructive dilemma (4,3)
6	$Q$	Idempotent law (5)

See also

• Disjunction
• Argument in the alternative
• Disjunct normal form