An inference rule that allows one to infer a disjunction A ∨ B from a proof of one of its disjuncts (from A infer A ∨ B, or from B infer A ∨ B).

Principle

A proof that a particular disjunct holds suffices to establish the disjunction by adding an alternative that may or may not hold; disjunction introduction preserves truth by weakening.

Demonstration

From a derivation of A, apply disjunction introduction to obtain A ∨ B. Example: from 'It is raining' infer 'It is raining ∨ it is snowing.'

Misapplication

Using disjunction introduction to justify choosing a specific disjunct from a disjunction (inferring A from A ∨ B), or introducing irrelevant or misleading disjuncts in a way that obscures constructive information needed later in a proof assistant context.

Consequence

Enables expansion of possibilities, creation of case analyses, and preparation for proof by cases; in programming languages corresponds to injecting a value into a sum type (left or right injection).

Reversal

Contrasts with disjunction elimination: introduction creates an ambiguous alternative from a definite fact, whereas elimination requires resolving a disjunction by separate case analysis to derive a common consequence.

Boundary

Valid in classical and intuitionistic logic as a basic structural rule; in constructive/type-theoretic settings one must supply the appropriate injection (left or right) as constructive evidence, and in some contexts arbitrarily adding disjuncts may lose computational content.

Semantic Tension

There is tension between the weakening character of disjunction introduction (it makes assertions less informative) and the need in constructive settings to preserve evidence; disjunction introduction is sound but may be non-informative for later constructive steps.

Synthesis

Disjunction introduction is the mechanism that converts a definite fact into a broader, truth-preserving alternative statement by adding an extra possible disjunct; it is a lightweight way to prepare for case-based reasoning.