##  [Disjunction Elimination](/disjunction-elimination-0) 

 Definition

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

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

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

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

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

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

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

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

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.