Three key elements of Aristotle’s theory of demonstration have Pythagorean antecedents. Demonstration is a revelatory discourse that is 1) inferential, 2) explicitly based on premises that are not themselves demonstrated on the basis of more basic premises, and 3) explanatory, insofar as the premises express those basic facts that are explanatory of the conclusion. The Pythagorean Table of Opposites constitutes a kind of protologic making possible a kind of deduction, which Aristotle would have regarded as a “demonstration,” that reveals the reasons behind Pythagorean commands and prohibitions. Philolaus recognizes ultimate explanatory principles. Both he and Archytas recognize the asymmetry of principles and what is explained on their basis.