The question is, do you think formal logic is a description of (specifically) mathematical reasoning or a formalization of the accepted (intuitive) arguments of ordinary language - or do you have a different interpretation?One could think of a formal logical system as being devised in something like the following way. Some informal arguments are intuitively judged to be valid, others invalid. One then constructs a formal language in which the relevant structural features of those arguments can be schematically represented, and axioms/rules which allow the intuitively approved, and disallow the intuitively disapproved, arguments. This, of course, is at best a very sketchy 'rational reconstruction' and is not intended as detailed, serious history. Still, while I concede that formal logics have sometimes been devised simply out of mathematical curiosity, I think that something like the process I have described was at work when, for instance, Frege devised his Begriffsschrift. Of course, the standard logical languages are now so familiar that one is no longer very conscious of how and why they were first constructed. But the same process can be seen in recent attempts to devise new formalisms for hitherto neglected kinds of argument; see, for example, the procedure adopted by D.K. Lewis 1973 in devising his analysis of counterfactuals.
Food for thought: To see one instance of how badly formal logic represents ordinary language, take material implication (If p then q). In formal logic, this truth function is always true except where p is true and q is false. Thus, the following statements are true in symbolic logic:
If pigs fly (F), then New York is a tiny town (F)
If pigs fly (F), then New York is a Big City (T)
If New York is a big City (T), then 2 + 2 = 4 (T).
