When the notion of figure is taken into account [Srttootsu ; Reta vtoa), its force and meaning is beet seen by stating the combination of relation in the different figures. Thus when we say a species of a species is a specie, we speak in the first figure, and compare tho minor with the major by the relation which the minor stands in to the middle and the middle to the major. When we read it thus—species and genus of the same are species and genus of one another,—we use the second figure, and compare both major and minor with the middle. When we say that those to which the same is genus and species are species and genus of one another, we use the third figure, and compare the middle term with both major and minor. And when we speak thus,—that to which another is species of species is a genus of that other,—we use the fourth figure, that is, the first figure with the con cluding relation inverted.
We now come to the exemplar form, already spoken of. Lct x) and signify any one x : let )x and X ( signify sonic one X, it not being known, even if true, that we may say any one.
The list of propositions is as follows. Each universal is followed by its contradicting particular :— c. x )( T Any one x is any one Y. This means that there is but one x and one 1', and the x is the Y.
r. x ( • ) T Some one x is not some one Y. Either x and T are not both singular : or if both singular, not identical r. x)) T Any one x is some one T ; or every x is Y.
P. X ( • (T Some one x is not any one Y ; or some xe are not ye. r. x ( ( T Some one x is any sole T ; or every v in x. P. x )• )1' Any one x is not seine one Y ; or some vs aro not as. r. x )• ( T Any one x is not any one v ; or no x is v.
P. x T Some one x is some one ; or some an are vs.
Omit the word one throughout, for any read all when tho word is in italics, and we have I lamilton's • system, provided the first pair be not read as contradictions.
There are 30 valid forms of syllogism, both in the exemplar reading and In Hamilton's reading. All that in required is one affirmative premise, and one total occurrence of the middle term. In the exemplar system, the rule of inference is as above described for the cumular system. In Hamilton's system there is this modification : when the middle terms are of different quantities, a concluding term which is total, with a spicule turning a different way from those of the middle term, must be altered from total to partial, if its proposition be affirmative. Thus, ) ( ( ) In the exemplar system gives ) ), but in Hamilton's system it gives ( ). Or, " Any one x is any one T, and
some one T is some one z7 gives " Any one x in some one z :" but " All x is an Y. and some Y is some z," gives " Sonic x in some a." Again, )( (•(, in Hamilton's system, gives (• (, not )• (, as in the exemplar system. The reason of this difference will easily appear, on con suleration.
The exampler proposition is often used, as distinguiehed from the cumular. It in, for example, the proposition of geometry. When Euclid proves that all Isosceles triangles are of equal angles at the balm, be shows that any Isosceles triangle is so, and he demonstrates only to those who can see that nothing is assumed in his demonstration which limits the selection.
The numerical syllogism leads to a species of syllogism which Is occasionally used, but cannot be reduced to a common syllogism. It is the syllogism of transposed quantity, in which one of the concluding terms enters with the whole quantity of the other, or of its contrary. Fur instance, " For every man iu the house there is a person who is aged ; some of the men are not aged; therefore some of the persons in the house are not men." Nothing has yet been printed on the laws of this syllogism of transposed quantity : the following brief rules will enable the reader to detect the cases, which are sixty-four in number.
Let a particular proposition which has a term of the whole quantity of an external term—that. is, in another proposition—be trilled an e.rternal unirersal, the ordinary universal being called internal. The proposition has a rewiring term; the other proposition has an importing term. Let the receiving term be distinguished by a hyphen below its spicule ; the imparting term by a hyphen above. Thus x )) ) •) z means as follows :—Every x is Y ; for every x there is a z which is not Y. But there is no valid syllogism when a term taken totally imparts its quantity : in the case before us the only chance of a valid inference arises from allowing x, which enters partially, to impart its quantity ; as in " Every x is Y ; for every x there in a a which is not T." This being understood, and also that in forming the conclusion every imparting term must change its quantity, the rules for detecting valid syllogism are precisely those of the cumular syllogism already described. That is, any two universals, each of either kind, give a conclusion ; and one universal of either kind and one partieular,.if the middle terms have different quantities in the two.