I know there has always been some debate concerning whether or not a certain logical system (like classical logic) is the correct one, especially when it comes to propositional claims about the external world we inhabit. The Philpapers survey from 2009 shows a clear majority (51.6% classical : 33.1% other : 15.4% non-classical) in favor of classical logic (I'll assume the question refers to which logic they believe to be correct when it comes to reasoning, but there are really several different interpretations of that particular question, which may have influenced the results).
I would also consider myself a proponent of classical logic as the correct logic when it comes to evaluating the world as it is (though not necessarily how it relates to reasoning, especially when it comes to the inconsistent beliefs many individuals hold). I would say that my philosophical worldview (there is a mind-independent external reality that operates on some deep, fundamental principles) heavily influences this, as I don't believe in any inherent fuzziness that might be amenable to a multi or many-valued logic, and I certainly don't believe in true contradictions (whatever the hell that would even mean).
With that being said, what are the best avenues one could take if they wanted to show that classical logic (or another logic, as Alan Ross Anderson was a platonist with respect to relevant logic) was "better" than all the others when applied to uncovering the nature of reality? In addition, has any progress been made when it comes to the program of Universal Logic, which would uncover the foundations or similarities between every logical system conceivable?
Any input on this is greatly appreciated.