A standard way to approach formalization (judging from my experience) is to set up a kind of informal metatheory with few foundational concepts, such as the metalogical implication $\Rightarrow$, metalogical sets, functions, $n$-tuples, structural induction, etc. All of these must be understood, by definition, in and of themselves, they are like a cornerstone of our proverbial Tower of Mathematics (or some other philosophically useful formal system).
This got me thinking: The only alternative to this ground-up approach that I can conceive of is a circular, self-referential foundation. One simple example of such a foundation in natural language that came to my mind was:
This sentence is true.
I would like to know whether such an approach to the foundations of formal systems has ever been explored, and if so, does it have any mathematical and/or philosophical significance?