##  [Disjunction Property](/disjunction-property-0) 

 Definition

A metatheoretical property of a deductive system stating that whenever the system proves a disjunction A ∨ B (typically a closed formula) it proves A or it proves B individually; commonly required of constructive systems.

 

 

 

 

 

 





## Principle

Principle

Constructive proofs of a disjunction must contain information selecting which disjunct holds: proving A ∨ B must effectively yield a proof of A or a proof of B rather than merely excluding their joint falsity.

 

 

 

 

 





## Demonstration

Demonstration

In intuitionistic propositional logic or in certain constructive type theories, a derivation of A ∨ B arises from a proof term that is either an injection into the left or right disjoint sum, so the term itself witnesses which disjunct is provable and the system therefore proves that disjunct.

 

 

 

 

## Misapplication

Misapplication

Assuming the disjunction property holds in classical systems or for arbitrary metatheoretic encodings; or interpreting a derivable meta-level disjunction (provable 'either A or B') as immediate provability of one disjunct without an internal constructive witness.

 

 

 

 

 





## Consequence

Consequence

When a system has the disjunction property one can extract definite algorithmic information from disjunctive proofs, improve program extraction, and ensure that proofs of alternatives are not merely non-constructive eliminations by contradiction.

 

 

 

 

## Reversal

Reversal

A system lacking the disjunction property can prove A ∨ B while proving neither A nor B, typically by resorting to non-constructive principles such as excluded middle or classical reasoning.

 

 

 

 

 





## Boundary

Boundary

Concerns provability in the object theory and often only for closed or arithmetical formulas; it does not assert that every meta-theoretic disjunction lifts to an internal witness in every formalization and can fail in extensions that add classical axioms.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension with the law of excluded middle: excluded middle allows proving A ∨ ¬A without supplying which side holds, directly opposing the constructive content required by the disjunction property.

 

 

 

 

 





## Synthesis

Synthesis

The disjunction property captures the constructive demand that proofs of alternatives be informative: if the system proves A ∨ B then it must—within the system—prove one of the disjuncts, ensuring disjunctive proofs carry a concrete selection rather than mere non-contradiction.