It seems to me that if a logical paradox demonstrably exists in reality, than logic, logically, cannot be a valid way to analyze reality. I think time may be such a phenomenon (if you can even call it that).
The paradox arises thusly: if I promise to give you a hamburger after infinite time, that is equivalent to saying I will never give you a hamburger. Similarly, saying that time stretches infinitely into the past is nonsensical, because that would place us at the end of an already infinite series of events, the end of which, by definition, cannot exist.
Such would seem to demand that time had a beginning. However, beginnings happen in time. That is part of the definition of a beginning. That time began meant that at one moment, there was no time, and the next moment, there was. Except there could be no moment where there was no time, because moments are a part of time.
It's difficult to get quite the right words for this, but do y'all get what I'm saying? If one of the fundamentals by which we experience violates logic, then thinking of the world logically will lead to fallacies.