Spiritus Asper versus Lambda: On the Nature of Functional Abstraction

Abstract

The spiritus asper as used by Frege in a letter to Russell from 1904 bears resemblance to Church’s lambda. It is natural to ask how they relate to each other. An alternative approach to functional abstraction developed by Per Martin-Löf some thirty years ago allows us to describe the relationship precisely. Frege’s spiritus asper provides a way of restructuring a unary function name in Frege’s sense such that the argument place indicator occurs all the way to the right. Martin-Löf’s alternative approach shows that this is only half of what lambda does. The other half is the deletion of the argument place indicator, resulting in what Frege would have called an isolated function name.