Disjunction elimination

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 ${\displaystyle P}$ implies a statement ${\displaystyle Q}$ and a statement ${\displaystyle R}$ also implies ${\displaystyle Q}$, then if either ${\displaystyle P}$ or ${\displaystyle R}$ is true, then ${\displaystyle Q}$ 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.

Disjunction elimination

Type	Rule of inference
Field	Propositional calculus
Statement	If a statement ${\displaystyle P}$ implies a statement ${\displaystyle Q}$ and a statement ${\displaystyle R}$ also implies ${\displaystyle Q}$, then if either ${\displaystyle P}$ or ${\displaystyle R}$ is true, then ${\displaystyle Q}$ has to be true.
Symbolic statement	${\displaystyle {\begin{aligned}1.\quad &P\to Q\\2.\quad &R\to Q\\3.\quad &P\lor R\\\therefore \quad &Q\end{aligned}}}$

An example in English:

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

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

3. 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:

${\displaystyle {\begin{aligned}1.\quad &P\to Q\\2.\quad &R\to Q\\3.\quad &P\lor R\\\therefore \quad &Q\end{aligned}}}$

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

Formal notation

The disjunction elimination rule may be written in sequent notation:

${\displaystyle (P\to Q),(R\to Q),(P\lor R)\vdash Q}$ 

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

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

${\displaystyle (((P\to Q)\land (R\to Q))\land (P\lor R))\to Q}$ 

where ${\displaystyle P}$ , ${\displaystyle Q}$ , and ${\displaystyle R}$  are propositions expressed in some formal system.

See also

• Disjunction
• Argument in the alternative
• Disjunct normal form
• Proof by exhaustion