TITLE: Foundation of Formal Logic
QUESTION [5 upvotes]: All the books that I’ve read about formal logic either starts assuming the existence of set theory (they talk of countable sets of symbols and define formulae as sequences of symbols), or follow an axiomatic approach that assumes known the meaning of words like symbol, formula, substitution etc. Is there a way to formalize these concepts (with no reference to set theory, I consider set theory a product of formal logic)?
We could assume as primitive the notion of symbol, but could the concepts of string, formula, composition of formulae be rendered in more precise way that don’t rely on intuition? A book or paper reference on this matter would be appreciated.

REPLY [3 votes]: Here is a possible approach.  Consider the minimalist language Brainfuck. This extremely stripped-down language comprises only eight commands and an instruction pointer.  The first step is to train yourself to be able to execute Brainfuck programs.  This does require some amount of intuition; you need the ability to read and write, at least well enough to be able to read and understand a description of the Brainfuck language.  You need to be able to recognize a Brainfuck program when you see one, parse the symbols, and remember how to execute the commands using whatever writing method you have mastered.   But you don't have to understand what a "symbol" is in general or what a "rule" is in general. You just have to master a very specific set of symbols and rules.
If you get this far, then in principle you have more than enough power to reproduce any formal logical proof, since Brainfuck is Turing complete. Of course, someone still has to code up logical axioms and rules in Brainfuck, but that's their problem, not yours. If they do their part, you can do your part by executing their Brainfuck programs.  In this way, you can formally reproduce all of logic and mathematics, with a minimal reliance on non-formal intuition.  (You won't understand anything, of course, but that doesn't seem to be your question.)