In propositional logic, disjunction elimination[1][2] (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 {\displaystyle P} implies a statement Q {\displaystyle Q} and a statement R {\displaystyle R} also implies Q {\displaystyle Q} , then if either P {\displaystyle P} or R {\displaystyle R} is true, then Q {\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.
An example in English:
It is the rule can be stated as:
where the rule is that whenever instances of " P → → --> Q {\displaystyle P\to Q} ", and " R → → --> Q {\displaystyle R\to Q} " and " P ∨ ∨ --> R {\displaystyle P\lor R} " appear on lines of a proof, " Q {\displaystyle Q} " can be placed on a subsequent line.
The disjunction elimination rule may be written in sequent notation:
where ⊢ ⊢ --> {\displaystyle \vdash } is a metalogical symbol meaning that Q {\displaystyle Q} is a syntactic consequence of P → → --> Q {\displaystyle P\to Q} , and R → → --> Q {\displaystyle R\to Q} and P ∨ ∨ --> R {\displaystyle P\lor R} in some logical system;
and expressed as a truth-functional tautology or theorem of propositional logic:
where P {\displaystyle P} , Q {\displaystyle Q} , and R {\displaystyle R} are propositions expressed in some formal system.
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