Are there axioms for metalogic? I'm just starting a study of mathematical logic, primarily through 'First Order Mathematical Logic' by Angelo Margaris. In the text, axioms for the statement calculus and the predicate calculus are given and properties of the resulting system are argued. However, it seems a bit circular, in that we are necessarily using logic to argue about logic. This leads to the question, are there axioms for the metalogic used to argue about the logic, and then axioms for the metametalogic, ad infinitum?
 A: Note that you are using the term "logic" here in two separate but related meanings. One meaning is the way we argue, and the other meaning is a formalization of the way we argue.
The first one indeed is more or less a given; it basically consists of all arguments that "are supposed to convince you". This type of logic is largely explored in philosophy. In mathematics, you indeed have to assume that you already have a way to argue.
The second one is what mathematical logic is about: You define a formal system that captures the essential parts of your logical thinking/argumentation process. That is, you formulate a formal language in which you can describe all statements you want to talk about, and manipulation rules where you can convince yourself that when you encode a statement in that language, and apply the manipulation rules, then the resulting statement necessarily will also be the encoding of a valid statement in the formal language.
Why would you do something like that? Well, on one hand, by formalizing,you can make sure that you don't accidentally make a step that is not justified: Your formal system only contains those rule where you are absolutely convinced that they hold; therefore if you follow those rules, you know for sure that if you translate each step back into an argument, that argument will be absolutely convincing. Also, by being conservative what you include, you minimize the risk of later finding that some of your used rules being found problematic. And finally, if you do steps formally, you always know exactly what assumptions you used, so if you later indeed start to doubt some of the used assumptions (for example, you might start to doubt the law of the excluded middle), you can go through your proofs and for each one see whether you used it, and be sure that any proof where you didn't use it remains valid.
Note that you are also creating a formal language, but to describe that language, you still need a language to describe it. Ultimately, you resort to your native language (e.g. English) to do so. There may be several layers between your original language and the formal language (e.g. the first step will likely be a more formalized version of the natural language where you give a more precise definition to common terms), but ultimately you are drawing on your native language, just as you are drawing on your ability to think/argue when defining logic.
A: I haven't thought about this long and I don't know if this is correct.
However it seems to me the point of mathematics is to set about some axioms and an agreed decidable system of logic so that we can explore the consequences of such. 
For example discussing the broad existence of infinity in reality would be philosophy whilst the discussion of the possibilities of infinite classes in a decidable manner, mathematics. Which I believe a specific subset of philosophy. 
Perhaps this can at least explain then how to decide a mathematical approach to solving this problem which satisfies the intuition.
