I had a recent conversation with a professional mathematician about the status of relations, functions and predicates. I was arguing that it seems intuitive (to me at least) to classify them in this hierarchy (as to which is more primitive):
- All predicates are functions.
- All functions are relations.
The obvious problem here is that it seems intuitive that unary or even nullary <predicates/relations/functions> are more primitive than their n-ary variants. Is there a way to compose functions as at least unary relations or vice versa (relations as unary functions)? If not, is it possible to order them in such a hierarchy given the binary restriction.
Finally, if there is something more primitive that has a formal definition, then that would do as well. A resource or explanation pointing to how at least functions, relations and predicates are composed using this notion would be helpful.