[Footnote 45: Analyt. Prior. II. xxi. p. 67, b. 23: [Greek: a)ll' i)/sôs e)kei=no pseu=dos, to\ u(polabei=n tina\ kakô=| ei)=nai to\ a)gathô=| ei)=nai, ei) mê\ kata\ sumbebêko/s; pollachô=s ga\r e)gchôrei= tou=th' u(polamba/nein. e)piskepte/on de\ tou=to be/ltion.] This distinction is illustrated by what we read in Plato, Republic, v. pp. 478-479. The impossibility of believing that one contrary is identical with its contrary, is maintained by Sokrates in Plato, Theætetus, p. 190, B-D, as a part of the long discussion respecting [Greek: pseudê\s do/xa]: either there is no such thing as [Greek: pseudê\s do/xa], or a man may know, and not know, the same thing, ibid. p. 196 C. Aristotle has here tried to show in what sense this last-mentioned case is possible.] Whenever (Aristotle next goes on to say) the extremes of a syllogism reciprocate or are co-extensive with each other (_i.e._ when the conclusion being affirmative is convertible simply), the middle term must reciprocate or be co-extensive with both.[46] If there be four terms (A, B, C, D), such that A reciprocates with B, and C with D, and if either A or C must necessarily be predicable of every subject; then it follows that either B or D must necessarily also be predicable of every subject. Again, if either A or B must necessarily be predicable of every subject, but never both predicable of the same at once; and if, either C or D must be predicable of every subject, but never both predicable of the same at once; then, if A and C reciprocate, B and D will also reciprocate.[47] When A is predicable of all B and all C, but of no other subject besides, and when B is predicable of all C, then A and B must reciprocate with each other, or be co-extensive with each other; that is, B may be predicated of every subject of which A can be predicated, though B cannot be predicated of A itself.[48] Again, when A and B are predicable of all C, and when C reciprocates with B, then A must also be predicable of all B.[49] [Footnote 46: Ibid. II. xxii. p. 67, b. 27, seq. In this chapter Aristotle introduces us to affirmative universal propositions convertible _simpliciter_; that is, in which the predicate must be understood to be distributed as well as the subject. Here, then, the quantity of the predicate is determined in thought. This is (as Julius Pacius remarks, p. 371) in order to lay down principles for the resolution of Induction into Syllogism, which is to be explained in the next chapter. In these peculiar propositions, the reason urged by Sir W. Hamilton for his favourite precept of verbally indicating the quantity of the predicate, is well founded as a fact: though _he_ says that in _all_ propositions the quantity of the predicate is understood in thought, which I hold to be incorrect.