M-Set Models

In [1] Andrews studies elementary type theory, a form of Church’s type theory [12] without extensionality, descriptions, choice, and infinity. Since most of the automated search procedures implemented in Tps [4] do not build in principles of extensionality, descriptions, choice or infinity, they are essentially searching for proofs in elementary type theory. In particular, search procedures based on Miller’s expansion proofs correspond to proofs in elementary type theory extended with η-conversion. In [9] a model class Mβη is defined and proven sound and complete with respect a natural deduction calculus corresponding to elementary type theory with η-conversion. One can add extensionality principles to automated search procedures [8, 10, 11] in order to target smaller (more restricted) model classes (as presented in [9]) which better approximate the class of standard models. Alternatively, one can construct interesting models in Mβη which do not satisfy the full extensionality principles. One can then prove results about such models by proving theorems in the weaker logic. Suppose M ∈ Mβη and we want to know if some property P holds for M. Suppose we can find a proposition A such that the property P holds if M |= A. We can conclude P holds if we prove the proposition A in elementary type theory with η-conversion. Category theory can provide a Kripke-style semantics of intuitionistic higher-order logic [14, 15]. In particular, categories of presheaves are Cartesian closed (thus providing a semantics for simply-typed λ-calculus) and contain a subobject classifier (thus providing a semantics for intuitionistic logic). Since a one-object category is simply a monoid, a presheaf over a one-object category is simply a set with a monoid action (anM -set) [13, 15]. From these abstract considerations, we know thatM -sets (for a fixed monoid M) provide a semantics for simply typed λ-calculus and intuitionistic higherorder logic. In this article we consider M -sets as a semantics for simply typed λcalculus and fragments of classical higher-order logic. We can start with