then, as soon as we can ascertain that A is B, we can conclude that P is Q; or, if we can shew that P is not Q, we know that A is not B. But if we find that A is not B, we can infer nothing; for the preceding does not assert that P is Q _only_ when A is B. And if we find out that P is Q, we can infer nothing. This conditional syllogism may be converted into an ordinary syllogism, as follows. Let K be any ‘case in which A is B,’ and Z a ‘case in which P is Q’; then the preceding assertion amounts to ‘Every K is Z.’ Let L be a particular instance, the A of which may or may not be B. If A be B in the instance under discussion, or if A be not B, we have, in the one case and the other, Every K is Z Every K is Z L is a K L is not a K ———————— ————————————— Therefore L is a Z No conclusion Similarly, according as a particular case (M) is or is not Z, we have Every K is Z Every K is Z M is a Z M is not a Z ————————————— ———————————— No conclusion M is not a K That is to say: The assertion of an hypothesis is the assertion of its necessary consequence, and the denial of the necessary consequence is the denial of the hypothesis; but the assertion of the necessary consequence gives no right to assert the hypothesis, nor does the denial of the hypothesis give any right to deny the truth of that which would (were the hypothesis true) be its necessary consequence. Demonstration is of two kinds: which arises from this, that every proposition has a contradictory; and of these two, one must be true and the other must be false. We may then either prove a proposition to be true, or its contradictory to be false. ‘It is true that Every A is B,’ and, ‘it is false that there are some As which are not Bs,’ are the same proposition; and the proof of either is called the indirect proof of the other. But how is any proposition to be proved false, except by proving a contradiction to be true? By proving a necessary consequence of the proposition to be false. But this is not a complete answer, since it involves the necessity of doing the same thing; or, so far as this answer goes, one proposition cannot be proved false unless by proving another to be false. But it may happen, that a necessary consequence can be obtained which is obviously and self-evidently false, in which case no further proof of the falsehood of the hypothesis is necessary. Thus the proof which Euclid gives that all equiangular triangles are equilateral is of the following structure, logically considered. (1.) If there be an equiangular triangle not equilateral, it follows that a whole can be found which is not greater than its part.[1] Footnote 1: This is the proposition in proof of which nearly the whole of the demonstration of Euclid is spent.