The Craig Interpolation Theorem states that if a first-order logical formula implies another, then there exists an intermediate formula, using only the non-logical symbols common to both, that is implied by the first and itself implies the second. Named for William Craig, it is a foundational result of model theory with applications throughout logic, including to definability and to modular reasoning in automated verification.
Facts
StatementIf a formula implies another formula and the two share at least one atomic variable symbol, then there is an interpolant formula whose non-logical symbols occur in both, which the first implies and which implies the second. 1 Classification
Statement Form Statement Form Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Craig's interpolation theorem, Wikipedia
Sources
1. Craig's interpolation theorem, Wikipedia
Lead paragraph
Roughly stated, the theorem says that if a formula φ implies a formula ψ, and the two have at least one atomic variable symbol in common, then there is a formula ρ, called an interpolant, such that every non-logical symbol in ρ occurs both in φ and ψ, φ implies ρ, and ρ implies ψ.
Lead paragraph, sentence on first proof
The theorem was first proved for first-order logic by William Craig in 1957.
In Branch: Logic and Foundations, Lead sentence
In mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories.
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