Disjunction elimination

Transformation rules
Propositional calculus
Rules of inference Modus ponens Modus tollens Modus ponendo tollens Biconditional introduction Biconditional elimination Conjunction introduction Simplification Disjunction introduction Disjunction elimination Disjunctive syllogism Hypothetical syllogism Constructive dilemma Destructive dilemma Absorption Rules of replacement Associativity Commutativity Distributivity Double negation De Morgan's laws Transposition Material implication Exportation Tautology
Predicate logic
Universal generalization Universal instantiation Existential generalization Existential instantiation

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:

$A\or B$

$A \to C$

$B \to C$

$\therefore C$

Proof

Proposition	Derivation
$A\or B$	Given
$A\rightarrow C$	Given
$B\rightarrow C$	Given
$\neg A\rightarrow B$	Material implication
$\neg A\rightarrow C$	Hypothetical syllogism
$(A\rightarrow C)\and(\neg A\rightarrow C)$	Conjunction
$A\or\neg A$	Law of Excluded Middle
$C\or C$	Constructive dilemma
$C\,\!$	Tautology

See also

• Disjunction
• Argument in the alternative
• Disjunct normal form