A logical construction that, given formulas A and B with A logically entailing B, produces a formula I (the interpolant) that is implied by A, entails B, and mentions only the nonlogical symbols common to A and B.

Principle

When A ⊨ B holds in a logic with the interpolation property, there exists an I built from exactly the shared vocabulary such that A ⊨ I and I ⊨ B; the interpolant separates the information contributed by A and B along the frontier of their common language.

Demonstration

In propositional logic, let A be (p ∧ q) and B be (p ∨ r). Because A entails B, an interpolant is I = p: A ⊨ p and p ⊨ B, and I uses only the symbol p that appears in both A and B.

Misapplication

Forcing the interpolant to contain symbols not shared by A and B, or assuming an interpolant exists in a logic that lacks the interpolation property (for example certain non-classical or extended first-order systems) leads to incorrect separation claims.

Consequence

Interpolation provides a modular witness that isolates the shared information between premises and conclusions; it underpins compositional verification, modular ontology alignment, and some model-checking decompositions.

Reversal

The absence of a separating formula using only common symbols (failure of interpolation) reverses the guarantee: one cannot syntactically isolate the shared content though semantic entailment A ⊨ B may still hold.

Boundary

Holds in propositional and standard first-order classical logic (Craig's theorem) but can fail in extensions or fragments (some modal, fixed-point, or second-order systems); interpolation refers to shared nonlogical vocabulary and excludes logical connectives and quantifiers from the restriction.

Semantic Tension

Interpolants are simultaneously syntactic objects (formulas built from symbols) and semantic separators (they split models of A from countermodels of B); tension arises when a semantic separator exists but no syntactic formula in the shared language can express it.

Synthesis

Interpolation is the process of extracting from an entailment A ⊨ B a formula I that syntactically captures exactly the shared vocabulary separating A's support from B's consequence, enabling modular reasoning and information hiding while acknowledging that some logics do not admit such a syntactic separator.