Difference between elementary logic and formal logic

In Kelley book on topology, in the appendix on elementary set theory, he says in the second paragraph, that "a working knowledge of elementary logic is assumed, but acquaintance with formal logic is not essential. However, an understanding of the nature of a mathematical system (in the technical sense) helps to clarify and motivate some of the discussion. Tarski's excellent exposition [here he refers to Tarski's "Introduction to Logic"] describes such system very lucidly and is particularly recommended for general background."

My questions are:

1) What is the difference between elementary logic and formal logic ? Shall I interpret "elementary logic" as those mental processes that enable me do to logic reasoning and inference (for when I deal with strings of symbols that have a mathematical meaning - as when I do when for example I would work with the axioms of ZFC to derive results) ?

2) What is a "mathematical system" ? (I presently don't have the means to look up Tarski's book, to see what Tarski himself wrote there, what it is that Kelley describes as a "mathematical system")

-
I suspect that your answer to question 1 (that is, elementary logic refers to metalogic) is what Kelley had in mind. Concerning question 2 I suggest you look at en.wikipedia.org/wiki/Consequence_operator –  boumol Sep 25 '12 at 7:58
@boumol Metalogic is the mathematical investigation of formal logical systems, and is two steps removed from what Kelley means by "elementary logic". –  Peter Smith Sep 25 '12 at 8:26
@PeterSmith But wouldn't that mean that metalogic is actually formal logic ? –  temo Sep 25 '12 at 8:33
@temo It is useful to distinguish a formal logic like a natural deduction system for quantification reasoning from the mathematical investigation of the properties of such a system. Metatheory is usually done informally like nearly all mathematics. –  Peter Smith Sep 25 '12 at 8:39
@Peter: You are absolutely right that metalogic is in general in a much wider sense. I should have said "elementary logics refers to a part of metalogic" (the easiest part). –  boumol Sep 25 '12 at 9:01

A working knowledge of elementary logic, in Kelley's sense, is a matter of the kind of proficiency with informal deductive reasoning that you pick up early in your mathematical career (and elsewhere of course). E.g., understanding how to refute a proposition by assuming the opposite and deriving an absurdity, understanding how to establish a universal generalization by working with an arbitrary instance, understanding how to establish a conditional by supposing the antecedent is true and deriving the consequent.

Formal logical systems, where we carefully define the syntax, carefully define what can be inferred from what, etc., aim to regiment and formalize such patterns of informal deductive reasoning: as Kelley says, the exercise of regimenting can indeed be illuminating about our ordinary reasoning.

A formal mathematical theory (or system) is what you get when you take a formal logical system and add some specific mathematical axioms, and thereby regiment an informal mathematical theory.

(Metalogic is another step further, being the informal mathematical investigation of such formal logico-mathematical systems treated as themselves objects of mathematical interest)

-