Enderton's "Mathematical Introduction to Logic": Is he proving second order induction?

Question

I am reading Enderton's "Mathematical Introduction to Logic" and I am puzzled by the following reasoning:

Enderton defines the symbols of propositional logic (sentence letters and connectives) and then defines some "formula-building operations". With these, he states:

Theorem: If $S$ is a set of wffs containing all the sentence symbols and closed under all five formula-building operations, then $S$ is the set of all wffs.

As far as I understand, this seems to me a second order statement of induction. Something which could be formalized as $(\forall S \subseteq \text{Form})(... \rightarrow S = \text{Form})$ where Form would be the set of all wffs. I now have two questions:

i) Is he actually stating and proving induction in second order form or is he doing something slightly different which I am missing?

ii) If he is stating and proving induction in second order form, why is he allowed to do that? Isn't it a problem to have induction in that form while trying to construct a simple propositional logic or, for that matter, a first order theory?

I don't have Enderton's book. How is the set of wffs defined there? — Taroccoesbrocco, Nov 07 '23 at 04:47
It is called structural induction, and yes induction is needed in the meta-theory of every logical calculus. — Mauro ALLEGRANZA, Nov 07 '23 at 07:08
@MauroALLEGRANZA could you please say a bit more? So Enderton is proving a form of second order induction as a meta-theorem and this is standard (even necessary) for every logical calculus? — Promethèus, Nov 07 '23 at 15:20
No, he is using induction. And the meta-theory is "usual" mathematics: thus, it needs a "reasonable" amounts of mathematical concepts: sets, numbers, functions. — Mauro ALLEGRANZA, Nov 07 '23 at 15:29
See e.g. Foundation of Formal Logic and Is logic circular at the basic level? — Mauro ALLEGRANZA, Nov 07 '23 at 15:30
And see When does the set enter set theory? and Is proving metatheorems circular logic?. — Mauro ALLEGRANZA, Nov 07 '23 at 15:32
@MauroALLEGRANZA Thanks. I think I understand the need for a strong enough meta theory, but how can I reconcile this with the fact that he is proving a version of induction which looks 2 order if not by saying that this is a meta-theorem? — Promethèus, Nov 07 '23 at 16:05
@Promethèus who cares if it looks second order? The principle of induction for natural numbers is second order. Do you have any qualms about using it? — spaceisdarkgreen, Nov 07 '23 at 21:40
@Promethèus (And “looks second order” is apropos since this could be a stand-in for some induction schema in some weak arithmetic if you for some reason want to be very parsimonious in how you formalize syntax. Just don’t expect Enderton, or most people, to take that approach… ) — spaceisdarkgreen, Nov 07 '23 at 23:41

score 1 · Accepted Answer · answered Nov 08 '23 at 00:49

When teaching "Intro to Proofs" if we want to go over induction on $\mathbb N$, we'll almost certainly opt to phrase everything in terms of full second-order induction, even though it's doubtful that anything we will prove in class won't be formalizable in PA or much weaker. Why?

The second order phrasing is (superficially) simpler, more intuitive and has less formal baggage.
The second order induction principle is... true, right? Certainly most mathematicians see no problem with it.
There's no rule that says in an exposition that we have to formalize mathematics as parsimoniously as possible, or explicitly formalize it at all.

All three of these considerations apply straightforwardly to Enderton's exposition. The syntactical end of first-order logic can certainly be formalized in PRA or some very weak theory, and if we were explicitly doing it that way for some reason, this theorem would be some schema and there wouldn't be any second-order logic in sight.

But the point of Enderton's book is not to formalize the metatheory as parsimoniously as possible. It's just to teach some mathematical logic. So, like anyone else teaches any other subject of mathematics, Enderton is speaking 'ordinary mathematical English' and not getting bogged down in irrelevant issues of formalization.

Thanks for your answer. So what you are saying is that it doesn't matter what the theorem looks like since this is a meta-theorem and there is no reason to be parsimonious in the meta-theory. Correct? — Promethèus, Nov 08 '23 at 02:22
@Promethèus Correct. The "metatheory" here is just the same place where we do algebraic geometry or complex analysis or number theory. The only reason to call it a metatheorem rather than just a theorem is to avoid confusion with the formal theorems of the formal systems under study. — spaceisdarkgreen, Nov 08 '23 at 02:39

Enderton's "Mathematical Introduction to Logic": Is he proving second order induction?

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