# Definitions of first-order, second-order, nth-order languages, theories, and logics?

I've heard the terms first-order language, first-order logic, first-order theory, etc., and I'm trying to understand exactly what the definition of an nth-order language, logic, or theory is; and also, what the formal definition of a logic is.

Searching the internet, I've had very little success in finding such definitions, so I'm asking here, in hopes of receiving some clarifying information.

Follow up question: where do, for example, ZFC, and PA fit in all this?

(My motivation for asking: I'm trying to self-study foundations of mathematics, and several theorems, for example, the completeness theorem, or Lowenheim Skolem applies only to first-order theories, as I understand it; but I don't know what a first-order theory is)

In first-order logic you can quantify over individuals (that is, elements of the universe of a structure).

In second-order logic you can additionally quantify over predicates and functions that take individuals as arguments. For example you can write things like $\forall x \forall y \exists P : P(x) \land \neg P(y)$. The $\exists P$ is not allowed in first-order logic.

In third-order logic you can also quantify over meta-predicates (and meta-functions) that take ordinary predicates/functions as arguments.

And so forth.

Generally, higher-order logic allows you to take this to any depth, and you get an entire type theory to play with.

ZFC and PA are both first-order theories. (In ZFC, of course, you can quantify over sets, which looks a lot like quantifying over predicates -- but with the caveat that only some of the possible predicates correspond to sets).

• Can you explain how, e.g. the jump from first-order to second-order, and the jump from second-order to third-order are analogous? Also, what do meta-predicates or meta-functions look like? – Dylan Pizzo Aug 3 '17 at 0:18
• Sorry, but I must ask: Is this your opinion or a widely accepted consensus? Opinions seem to vary as the OP seems to have discovered for him or herself. – Dan Christensen Aug 3 '17 at 1:41
• @DanChristensen - This is the widely accepted meaning of what an $n$-th order logic is, yes. – Malice Vidrine Aug 3 '17 at 3:12
• @DylanPizzo: At each level you're allowed to quantify over predicates/functions that take as arguments anything you could quantify over on the previous level. – Henning Makholm Aug 3 '17 at 12:11