Is Logic based on Belief ?

I was thinking yesterday after TOK class about the way we reason and what we call logic. In my opinion i think we practice binary logic. To illustrate this lets take for example i say if something is "non-existent" then it is not "existent" (It is either this or that). This is used in indirect proof (reductio ad absurdum). But i was thinking, what if there were other possibilities like say it was both or neither (I think in Artificial Intelligence this is called fuzzy logic). All these questions were aroused by our talk on Godel's Incompleteness Theorem in class. Godel has made us classify some statements as neither true nor false(they are referred to as Undecidable). All along (until Kurt Godel came along), we thought statements were either true or false; That something could only be one of two opposites. I think this is particularly interesting in Quantum Electro Dynamics (QED) where we have a host of counterintuitive results. For example, in Young's double slit experiment, the question as to which slit the photon passed through is still unresolved because it appears to have entered neither, both and yet still only one. What a conundrum !!! But to propose fuzzy logic in my opinion has far reaching consequences for indirect proof which relies on something either being true or false. So, is logic dependent on belief ????