Here then are four separate rules laid down, one for each variety of propositions. The rules for the second and third variety are proved by the rule for the first (the Universal Negative), which is thus the basis of all. But how does Aristotle prove the rule for the Universal Negative itself? He proceeds as follows: "If A cannot be predicated of any one among the B's, neither can B be predicated of any one among the A's. For if it could be predicated of any one among them (say C), the proposition that A cannot be predicated of any B would not be true; since C is one among the B's."[13] Here we have a proof given which is no proof at all. If I disbelieved or doubted the proposition to be proved, I should equally disbelieve or doubt the proposition given to prove it. The proof only becomes valid, when you add a farther assumption which Aristotle has not distinctly enunciated, viz.: That if some A (_e.g._ C) is B, then some B must also be A; which would be contrary to the fundamental supposition. But this farther assumption cannot be granted here, because it would imply that we already know the rule respecting the convertibility of Particular Affirmatives, viz., that they admit of being converted simply. Now the rule about Particular Affirmatives is afterwards itself proved by help of the preceding demonstration respecting the Universal Negative. As the proof stands, therefore, Aristotle demonstrates each of these by means of the other; which is not admissible.[14] [Footnote 13: Ibid. p. 25, a. 15: [Greek: ei) ou)=n mêdeni\ tô=n B to\ A u(pa/rchei, ou)de\ tô=n A ou)deni\ u(pa/rxei to\ B. ei) ga\r tini, oi(=on tô=| G, ou)k a)lêthe\s e)/stai to\ mêdeni\ tô=n B to\ A u(pa/rchein; to\ ga\r G tô=n B ti/ e)stin.] Julius Pacius (p. 129) proves the Universal Negative to be convertible _simpliciter_, by a _Reductio ad Absurdum_ cast into a syllogism in the First figure. But it is surely unphilosophical to employ the rules of Syllogism as a means of proving the legitimacy of Conversion, seeing that we are forced to assume conversion in our process for distinguishing valid from invalid syllogisms. Moreover the _Reductio ad Absurdum_ assumes the two fundamental Maxims of Contradiction and Excluded Middle, though these are less obvious, and stand more in need of proof than the simple conversion of the Universal Negative, the point that they are brought to establish.] [Footnote 14: Waitz, in his note (p. 374), endeavours, but I think without success, to show that Aristotle's proof is not open to the criticism here advanced. He admits that it is obscurely indicated, but the amplification of it given by himself still remains exposed to the same objection.]