A metatheorem and constructive procedure which, given a proof that formula A implies formula B in a logic, produces an intermediate formula I (an interpolant) that uses only the nonlogical symbols common to A and B and satisfies A entails I and I entails B.

Principle

When A → B is provable in a logic with the interpolation property, there exists an interpolant built solely from the shared vocabulary; variants (e.g. Lyndon) further constrain preservation of polarities of predicate occurrences.

Demonstration

Propositional example: let A = p ∧ r and B = q ∨ r. Since A implies B (because r true makes q ∨ r true), an interpolant is I = r: A entails r and r entails B, and r uses only the propositional symbol common to A and B.

Misapplication

Asserting an interpolant exists that uses arbitrarily chosen common symbols or that preserves syntactic features (like quantifier structure or polarity) in logics where those stronger forms of interpolation fail.

Consequence

Enables decomposition of proofs and transfer of consequences across theories restricted to shared vocabulary; supports modular reasoning, specification extraction, and some automated verification workflows.

Reversal

The inverse idea would demand an intermediate formula using only nonshared vocabulary or forbidding shared symbols; this typically fails and contradicts the purpose of interpolation.

Boundary

Holds in many propositional and first-order settings but fails in some extensions (certain theories with fixed-point operators, some modal and higher-order logics); equality, polarity, or theory-specific predicates can obstruct existence or require refined variants.

Semantic Tension

Tension between syntactic interpolation (constructing an interpolant from a proof) and semantic interpolation (existence of a formula separating models): some logics admit one notion but not a constructive or polarity-preserving version.

Synthesis

Craig interpolation is the constructive bridge that extracts, from a proof of A → B, a middle formula I expressed only in the common vocabulary that both follows from A and suffices to imply B, enabling modular transfer of information while exposing limitations when vocabulary, polarity, or logic extensions interfere.