TASC: The Transcendental Argument for Square-Circles

Hitherto it has been commonly supposed by the ill-informed that the non-existence of Square-Circles could be casually taken for granted. But demonstrating their non-existence has always proven problematic. After all, proofs are useful in demonstrating a positive, while proving a negative has always been notoriously difficult if not dubious. How exactly would one draw the conclusion that Square-Circles do not exist without begging the question or committing some other informal fallacy? What would one point to as evidence for their non-existence when everything we observe aligns so conclusively with the presupposition that Square-Circles exist? Wouldn’t one need to be omniscient to know that there are no Square-Circles existing somewhere in the universe beyond the reach of mere mortal sensibilities in order to proclaim definitively and with confidence that there are in fact no Square-Circles anywhere at all whatsoever? How would unSquare-Circulers account for logic, science and morality?