Empty Types in Polymorphic Lambda Calculus

Albert R. Meyer, John C. Mitchell, Eugenio Moggi, Richard Statman

Abstract: The model theory of simply typed and polymorphic (second-order) lambda calculus changes when types are allowed to be empty. For example, the “polymorphic Boolean” type really has exactly two elements in a polymorphic model only if the “absurd” type ∀t.t is empty. The standard β-ε axioms and equational inference rules which are complete when all types are nonempty are not complete for models with empty types. Without a little care about variable elimination, the standard rules are not even sound for empty types. We extend the standard system to obtain a complete proof system for models with empty types. The completeness proof is complicated by the fact that equational “term models” are not so easily obtained: in contrast to the nonempty case, not every theory with empty types is the theory of a single model.