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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):

  1. All predicates are functions.
  2. 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.

Vivek Joshy
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  • Relevant: Noah Schweber's answer to Is set theory the most fundamental aspect of mathematics? AND Foundation of Formal Logic. – Dave L. Renfro May 30 '23 at 21:43
  • You have it a bit backwards. All relations are predicates, but unless a 1-ary or 0-ary predicate is a relation, not all predicates are relations. Functions are a special kind of predicate, but they behave as objects as opposed to propositions. For example, $F(a)$ means “$a$ has property $F$”, while $f(a)$ is just whatever object to which $f$ maps $a$. With this in mind, an n-ary function can be interpreted as an n+1-ary predicate. E.g., $f(x)=y$ can be understood as $F(x,y)$. Credit to https://math.stackexchange.com/questions/315936/predicate-vs-function as I used some of their answer. – PW_246 May 30 '23 at 21:58
  • @RW_123 f(a) is an object, but f without an argument to it is definitely not a predicate since the obvious issue is predicates can only return boolean values (T or F) as opposed to functions which can return pretty much anything that is an object. This object could even be a predicate, function or a relation. Now that I think about it, predicates could actually be special case of relations where predicates are relations that only accept terms (functions or objects). – Vivek Joshy May 30 '23 at 22:09
  • Theoretical computer science can be of some help. I advise you in particular to see "lambda-calculus" which has been "implemented" (with some nuances) as LISP language. – Jean Marie May 30 '23 at 22:14
  • @JeanMarie I assume you mean simply typed lambda calculus. If so, I find the concept of types interesting, but I'm versed in it. Is it possible to reduce everything to types? – Vivek Joshy May 30 '23 at 22:20
  • Sorry, I will not answer because I am not here in my domain of expertise. I just wanted to stress the fact that computer science gives a different understanding of the fundamental bricks : for example, a language like Prolog helps to understand the fundamental character of relations... – Jean Marie May 30 '23 at 22:27
  • set, function, number. – Mauro ALLEGRANZA May 31 '23 at 06:51
  • @MauroALLEGRANZA Which set of numbers? Some are constructed using other axioms. So they would be not primitive. In the same vein, a set does not also seem to be primitive at all because of membership and relational constraints decided by axioms. – Vivek Joshy May 31 '23 at 09:23
  • Sorry... I mean natural (aka: counting) numbers. – Mauro ALLEGRANZA May 31 '23 at 11:28
  • "Some are constructed using other axioms." Not exactly: other "numbers" (integers,etc) are constructed starting from naturals and using set theoretic principles. – Mauro ALLEGRANZA May 31 '23 at 11:28
  • "set does not also seem to be primitive at all because of membership and relational constraints decided by axioms." Every "rigorous" theory needs some undefined concepts and some assumed principle (axioms). This is so for number (Peano's axioms) as well as for set theory and category and lambda theories (where in some sense function is primitive). – Mauro ALLEGRANZA May 31 '23 at 11:30
  • Yes, natural numbers can expressed as the application of the successor function, which is well...a function. Perhaps there are some resources that go into whether it even makes sense to have primitive notions. It could be entirely possible that functions aren't actually prior relations or predicates since it is like asking what is north of the North Pole. – Vivek Joshy May 31 '23 at 18:20

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