##  [Interpolation](/interpolation-0) 

 Definition

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

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

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

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

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

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

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

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

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.