Indeed it may be argued that everything the human brain does best is based not on traditional logic but something much fuzzier and incomparably more efficient.
Take the problem of identifying two things as the same. Russell's predicate calculus - the reduction of logic to mathematics - deals at great length with the mathematico-philosophical implications of equating two separately-named concepts. Simply to state the equivalence of two things, represented by the Greek letters phi and psi in the equation, takes a long and apparently messy (but decidedly unfuzzy) equation. In the language of mathematical logic it simply expresses the fact that the dyadic relation of identity involves not only expressing an equivalence between that thing which is phi and that which is psi, but also establishing that phi and psi are themselves well-defined in the sense that exactly one thing is phi (or psi, as the case may be). Though it must be admitted that Russell's theory of descriptions, from which that equation is derived, is still mildly controversial.
Human (fuzzy) logic does not bother with such details. A quick glance at phi, a brief look at psi, a pause for thought and the conclusion: 'They look much the same to me' is quite enough for most people. This method may lead to a few mistakes, but it saves a good deal of time and pairs off the phis and psis quicker than Russell could have hoped to.
Our ability to recognise faces is a powerful example. Despite a great deal of recent research, the question of how we can pick out a familiar face in a crowd is still baffling. Even if we have seen a photograph only once, we may identify the same face. Using traditional logic, the original image would need to be scaled to the correct size, projected from two to three dimensions, and compared with the new mental image. With dozens of factors - nose, brows, eyes, ears, lips etc - contributing to making up an individual face, and most of them changeable according to expression, the necessary computations involved would appear to be formidable. Yet we perform our mental Identikit in a fraction of a second.
Computers, using Aristotelian logic, can now make a fair guess at whether a face is male or female, but any more complex discrimination task is beyond them. Our own pattern recognition skills, which must be based on a fuzzier form of thinking, are undoubtedly more powerful, but it remains to be seen to what extent the fuzzy logic of modern mathematics can emulate the fuzziness of human brains.