0

This question is asked in the strictly mathematical sense here: Are all n-ary operators simply compositions of binary operators?

My issue is that all proofs I've seen of this seem to call on set theory in some sense, but it's my understanding that set theory is built "on top of" classical, propositional logic. My question is: Does this statement hold without the notion of a set? That is, is the mathematical notion of a set necessary to prove this, or does it hold as a tautology in propositional logic without any "mathematics"?

I suspect the answer's yes, but proving it without reference to any notion of an arithmetic or a class/set seems difficult.

Any help is greatly appreciated. Thank you!

LL 3.14
  • 12,457
Joseph_Kopp
  • 255
  • 1
    I don't know how it would even be possible to define inductively the language of propositional logic without any notion of set theory... – Taroccoesbrocco Jan 11 '24 at 22:19
  • @Taroccoesbrocco That's exactly the problem I came across when thinking about it. I kept wanting to talk about "the class" of all propositions so that I could refer to a "domain" for logical operators. I'm not sure if there's a way to avoid this or not. I feel there may be but I'm not well versed enough in propositional logic and the calculus or philosophy thereof. – Joseph_Kopp Jan 11 '24 at 22:34
  • For the general theorem that all n-ary operators can be decomposed into binary ones it is very hard to avoid sets. But if the domain is specifically propositional logic (in other words, if the operators are two-valued truth-functions) then you can give arguments like in https://math.stackexchange.com/questions/2646024/can-and-or-and-not-be-used-to-represent-any-truth-table to prove that any truth-function can be decomposed into binary ones. – Bram28 Jan 11 '24 at 23:41
  • @Bram28 Ahh, of course. Thank you! – Joseph_Kopp Jan 11 '24 at 23:43

0 Answers0