Conservativity between logics and typed λ calculi

Abstract

When looking at systems of typed A calculus from a logical point of view, there are some interesting questions that arise. One of them is whether the formulasas-types embedding from the logic into the typed A calculus is complete, that is, whether the types that are inhabited in the typed A calculus are provable (as formulas) in the logic. It is well-known that this is not a vacuous question: the 's tandard' formulas-as-types embedding from higher order predicate logic into the Calculus of Constructions is not complete. (See [Berardi 1989], [Geuvers 1989] or [Geuvers 1993].) Another interesting issue is whether the typed )~ calculus approach can help to solve questions about the logics or vice versa. An example of such a fruitful interaction is the proof of (strong) normalization for the Calculus of Constructions, which has as corollary in higher order predicate logic that cut elimination terminates. In this paper we want to treat questions of conservativity between systems of typed A calculi (and hence between the logical systems that correspond with them according to the formulas-as-types embedding). On the one hand this is an issue of interest for the typed A calculi themselves. (Can new type forming operators create inhabitants of previously empty types?) On the other hand, however, this is a nice example of how the formulas-as-types embedding can help to solve questions about logics by making use of typed A calculi and vice versa. If one sees a typed A calculus as a logical system, one takes one specific universe ( 'sort ' in the terminology of Pure Type Systems) to be interpreted as the universe of all formulas. Let's call this universe Prop. Now suppose that $1 is a system of typed A calculus containing the universe Prop, and suppose that $2 is a system that extends $1.