A strengthening of interpolation that requires, for each formula φ and each chosen sublanguage (set of non-logical symbols) Σ, the existence of a formula ψ using only symbols from Σ such that ψ captures exactly the Σ-consequences of φ; ψ is uniform in the sense that it works for all entailments from φ to formulas in the sublanguage.

Principle

The organising principle is symbol-elimination by a single interpolant: rather than producing an interpolant tailored to a specific consequence, uniform interpolation demands a canonical formula that uniformly represents all consequences in the restricted vocabulary.

Demonstration

Example: propositional logic admits uniform interpolation via variable elimination (forgetting): given a formula φ and a set of propositional variables to keep, one constructs a formula ψ in the smaller vocabulary that characterises exactly the consequences of φ in that vocabulary.

Misapplication

Confusing uniform interpolation with Craig interpolation: Craig interpolation guarantees an interpolant for each entailment pair φ ⊨ χ but does not ensure the existence of a single ψ that captures all Σ-consequences of φ; assuming uniform interpolation from mere Craig interpolation is a misuse.

Consequence

When a logic has uniform interpolation it enables robust modular reasoning, componentwise specification and effective symbol elimination; it often implies related definability properties and supports algorithmic abstraction procedures.

Reversal

Failure of uniform interpolation means there exist formulas whose consequences in a sublanguage cannot be captured by any single formula in that sublanguage, obstructing uniform symbol-elimination and complicating module composition.

Boundary

Property depends sensitively on the logic and the allowed language fragments: some modal and propositional systems have uniform interpolation, many first-order fragments do not, and the presence of additional operators or quantifier patterns can break uniformity.

Semantic Tension

There is tension between uniform interpolation and expressive strength: richer logics can express more distinctions but may lose uniform interpolants; uniform interpolation trades off expressivity for modular compressibility of information.

Synthesis

Uniform interpolation formalises a uniform form of symbol-forgetting: it guarantees that for any formula one can extract a canonical representation of its consequences in a restricted vocabulary, providing a powerful tool for modularity, definability, and automated abstraction whenever it holds.