In propositional logic disjunction elimination, or proof by cases, is the inference that, if "A or B" is true, and A entails C, and B entails C, then we may justifiably infer C. The reasoning is simple: since at least one of the statements A and B is true, and since either of them would be sufficient to entail C, C is certainly true.

For example:

It is true that either I'm inside or I'm outside. It is also true that if I'm inside, I have my wallet on me. It's also true that if I'm outside, I have my wallet on me. Given these three premises, it follows that I have my wallet on me.

Formally:

See also

• Disjunction
• Argument in the alternative
  Argument in the alternative
  Originating in the legal profession, argument in the alternative is used to pre-empt objections by arguing on multiple strategies simultaneously, showing that regardless of interpretation there is no reasonable conclusion other than the advocate's. Bart Simpson's classic "I didn't do it, no one saw...
  
  
• Disjunct normal form