Reflections on a Theorem of Henkin

Abstract

The λδ-calculus is the λ-calculus augmented with a discriminator which distinguishes terms. We consider the simply typed λδ-calculus over one atomic type variable augmented additionally with an existential quantifier and a description operator, all of lowest type. First we provide a proof of a folklore result which states that a function in the full type structure of [n] is λδ-definable from the description operator and existential quantifier if and only if it is symmetric, that is, fixed under the group action of the symmetric group of n elements. This proof uses only elementary facts from algebra and a way to reduce arbitrary functions to functions of lowest type via a theorem of Henkin. Then we prove a necessary and sufficient condition for a function on [n] to be λδ-definable without the description operator or existential quantifier, which requires a stronger notion of symmetry.