Can logic be traced back to neurons? In our everyday experience, and in both math and philosophy particularly, there exist some binary order in the sense that some statements can be classified as true or false. 

Does this have a specific neural correlation in the brain (classifying a statement as either true or false)? Or in other words, is it possible to correlate the process of assigning true/false with some particular feature of the brain?
Certainly some statements are either true or false. Either it rains or it doesn't. But when you think about many of these statements more carefully, the truth of these statements isn't really binary at all, or the binary fact is largely irrelevant.

Using the weather example, for most people it is not very helpful to know that it doesn't rain when there is a hurricane or it's snowing, both of which wouldn't usually be considered "not rain" by someone who wants to know what to wear. So the answer to "Does it rain?" usually isn't "No", but, for example, "You should take your umbrella because it looks like it might rain later".

The brain also isn't binary. Axons do not either forward an electrical impulse or not, but impulses of different intensities or frequencies. Neurons are often not activated by impulses from a single synapse, but usually by multiple synapses. Synapses do not either send out transmitters or not, but different amounts of them. Functions aren't located in one area, but in multiple areas. And so on.

Logic and maths (and psychological theories) are simplifications of a very complex reality. They provide simple solutions that often work (because life is flexible enough to adapt to a rigid approach), but just as often logic or mathematical (or psychological) approaches to deal with real problems are destined to fail.