Disjunction elimination

An inference rule that derives a conclusion from a disjunction by showing that the conclusion follows from each disjunct separately.

Disjunction: A compound proposition that is true when at least one of its component propositions is true. The rule starts from a disjunction and considers its alternatives.
Proposition: A declarative statement that can be assigned a truth value. Each disjunct and the conclusion are propositions.
Modus ponens: An inference rule that derives a consequent from a conditional and its antecedent. It applies a conditional directly, rather than splitting a disjunctive premise into cases.
Proof by cases: A proof method that establishes a conclusion by proving it separately in each exhaustive case. Mathematical arguments use this pattern whenever alternatives cover all possibilities.
Natural deduction: A proof system that represents reasoning with introduction and elimination rules for logical connectives. Disjunction elimination is one of its connective-specific elimination rules.
Logical connective: An expression that combines propositions or modifies one to form a compound proposition. Disjunction elimination operates on the connective joining its two cases.
Modus tollendo ponens: A traditional name for disjunctive syllogism, which rejects one disjunct to infer the other. It uses a disjunction and a negation, not two separate proofs of one conclusion.
Case analysis: A reasoning method that handles a problem by examining its distinct possible cases. It applies the rule’s structure to definitions, classifications, and exhaustive alternatives.
Disjunction introduction: An inference rule that allows a proposition to be inferred from that proposition alone as one side of a disjunction. It introduces a disjunction; this rule reasons outward from one already given.
Assumption (logic): A proposition temporarily accepted in a proof, often discharged when a conditional or inference is established. A separate temporary assumption opens each disjunct’s case.

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