Extending models of second order predicate logic to models of second order dependent type theory

Abstract

We describe a method for constructing a model of second order dependent type theory out of a model of classical second order predicate logic. Apart from the construction being of interest by itself, this also suggests a way of proving the completeness of the formulas-as-types embedding from second order predicate logic to second order dependent type theory. Under this embedding, formulas are interpreted as types, and derivability (of a formaula) in the logic should correspond to inhabitation (i.e. the associated type being nonempty) in the type system. This correspondence works in one way (called soundness): if a formula is derivable, then the associated type is inhabited (there is a term of that type). It's an open problem whether the correspondence works in the other direction (called completeness): if the type associated with formula ϕ is inhabited, then ϕ is derivable. Now, the completeness would be proved if any model M of second order logic could be extended to a model S(M) of second order dependent type theory in a faithful way, i.e. for all formulas ϕ, ϕ is true in M if and only if ϕ is inhabited in S(M). Here we prove this equivalence for models M that are full models of classical second order predicate logic. The extension of M to S(M) is constructed by adding to the basic domain of M all λ-terms and defining a smart equivalence relation on the set we thus obtain. The types are then interpreted as subsets of terms of this extended λ-calculus. The models construction is a variant of the one defined in [Stefanova and Geuvers 1996] for the Calcuklus of Constructions.