Disjunction elimination

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 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.

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 \or R}{\therefore Q}

where the rule is that whenever instances of "P \to Q", and "R \to Q" and "P \or 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 \or 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 \or R in some logical system;

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

(((P \to Q) \and (R \to Q)) \and (P \or R)) \to Q

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

See also


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