In philosophy, free will pretty much has to be taken as an axiom. It can't really be proven, but it's got to be assumed to exist. Without free will, we wouldn't be free to reject illogical conclusions. Would we be rejecting them only because we were predetermined to do so? If so, the whole enterprise of reason would be a futility, and without free will we would be so programmed that we wouldn't be able to know the difference. To cut through this unresolvable thicket, so that we can continue thinking with confidence that we are more than mere calculators reaching pre-determined results, it's essential to posit free will as an axiom. Thus, free will is a "genuine" axiom, embraced out of absolute necessity, and not for an isolated arbitrary purpose (such as "axiomatically" accepting the existence of the Olympian gods in order to practice the Hellenic religion).
How can logic admit the acceptance of illogical conclusions? To my admittedly inadequate mind, logic appears to be a closed and unfree system. The problem with logic is that it operates on tokens, which are only approximations of reality. Reality is complex, chaotic, indeterminant (turtles all the way down).
Very little of our minute to minute behavior is governed by logic (in the formal sense). So what are we doing, if it isn't logic? Answer that and you would be on your way towards artificial intelligence.