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APPENDIX. 



ON SYLLOGISMS OF TRANSPOSED QUANTITY. 



In my Formal Logic, and in my recently published Syllabus of a proposed System of 

 Logic, I gave instances of the syllogism of transposed quantity: that is, the syllogism in 

 which the whole quantity of one concluding term, or of its contrary, is applied in a premise 

 to the other concluding term, or to its contrary. As in the following: — Some Xs are not 

 Ys; for every X there is a Y which is Z: from which it follows, to those who can see it, that 

 some Zs (the some of the first premise) are not Xs. 



Such syllogisms occur in thought and in discussion. It also happens that the premises 

 and conclusion are stated independently, and their connexion not seen. It may also happen 

 that the premises are stated simultaneously with the contrary of the conclusion. The fol- 

 lowing sentences, though they will not pass current in a paper on logic which produces them 

 as an example of fallacy, would be very likely to slip through without detection, as part of 

 an ordinary page of writing : — 



To say nothing of those who achieved success by effort, there were not wanting others of 

 whom it may rather be said that the end gained them than that they gained the end: for they 

 made no attempt whatever. But for every one who was more fortunate than he deserved to be, as 

 well as for every one who used his best exertions, one at least might be pointed out who aban- 

 doned the trial before the result was known. And yet, so strangely are the rewards of persever- 

 ance distributed in this world, there was not one of these fainthearted men but was as successful 

 as any one of those who held on to the last. 



Might not many educated logicians pass this over, supposing it presented without warning, 

 as containing nothing but what might be true, without seeing that, except under forced 

 interpretation, it combines in one the assertions that all are and that some are not ? 



The syllogism of transposed quantity is essentially a case of the numerically definite 

 syllogism, though the number of instances is in every case of the indefinite, or rather unspe- 

 cified, character of the algebraical letter : and the same may be said of every onymatic syllo- 

 gism. Those who have commented upon the arithmetical syllogism have for the most part 

 missed this point : they have not seen that the numerical definiteness of the premises is the 

 definiteness of general, not of particular, symbols. That is, they have not caught the dis- 

 tinction between the form and the matter of arithmetical definition. The following slight 

 account of the numerical syllogism will be sufficient for the present purpose. 



Let us understand by »wXY that m or more Xs are Ys; and by »»:XY that m or 

 fewer Xs are Ys. Then by mXY we also mean, if x and y be the whole numbers of Xs 

 and Ys in the universe, both {x — m):Xy and (y — m):xY. Let u be the number of in- 

 stances in the universe ; .r, y, x, the numbers in X, Y, Z ; and «', y, z, the numbers in 

 X, y, z. Then mXY is (ij?-»n):Xy, ot (y - .?? + m) xy, or (m •)- « - ir - y) xy. And 

 mXy is (m + u - x - u +y) xY, or (w + y - x) xY. 



45—5 



