An inference rule (proof by cases) that allows deriving C from A ∨ B together with a derivation of C under the assumption A and a derivation of C under the assumption B. Formally: from A ∨ B, [A ⇒ C], [B ⇒ C] infer C.

Principle

If a conclusion follows from each disjunct separately, and at least one disjunct holds, then the conclusion holds unconditionally; disjunction elimination transfers casewise consequences to a global consequence.

Demonstration

Given A ∨ B, assume A and derive C; assume B and derive C; then discharge the assumptions to conclude C. Example: from 'It is sunny ∨ it is cloudy', and proofs 'if sunny then picnic' and 'if cloudy then picnic', conclude 'picnic'.

Misapplication

Applying disjunction elimination without providing valid separate derivations for each disjunct, or using it with an infinite or ill-formed disjunction without addressing how cases cover all possibilities; also misusing it to infer case-specific extra information not common to all cases.

Consequence

Enables rigorous reasoning by cases and the elimination of alternatives once a common consequence is shown; central to structured proofs, program pattern matching, and theorem-proving tactics.

Reversal

Contrasts with disjunction introduction: the latter creates a disjunction from a single fact, while elimination resolves a disjunction into a single consequence; an incorrect reversal would try to derive disjuncts from the global consequence without justification.

Boundary

Valid in classical and intuitionistic proof systems; in constructive settings one must supply explicit constructions for the derivations from each disjunct, and in logics with infinitely many disjuncts or with non-exclusive alternatives extra care is required to ensure completeness of the cases.

Semantic Tension

Tension appears between disjunction elimination and non-deterministic or probabilistic readings of disjunction: proof-by-cases requires coverage of all logical alternatives, whereas other interpretations may treat disjunction as a tentative or uncertain choice.

Synthesis

Disjunction elimination is the rule that collapses a provable alternative into a single conclusion by demonstrating that each possible case entails that conclusion; it operationalizes reasoning by cases and is indispensable for combining case analyses into unconditional results.