dicate,' &c. A particular quantity is a worse relation than a universal, and a negative quality a worse relation than a positive.1 6. Logicians have laid down special laws of Syllogism which regulate the inferences in the various Figures. All of them emerge on the neglect to give the predicate explicitly the quantity which implicitly belongs to it, and all of them are rendered useless and most of them false as soon as the quantification of the predicate takes effect. The enunciation of the one supreme canon of Syllogism therefore abrogates these special laws.2 7. The supreme canon of categorical Syllogism, since it determines all the varieties of Syllogism, will determine the number of syllogistic Figures. A reference to this canon demonstrates the exclusive possibility of Three Syllogistic Figures, and abolishes the so-called Fourth Figure. The clause which determines the variations of Syllogism with respect to Figure is 'What worse relation of subject and predicate subsists between either of two terms and a common third term,' &c., and this clause determines all the variations possible. These are three for there are only three varieties of relation. relations are: The (1) That in which the common third term is subject of one of the terms, and the predicate of the other. This gives the First Figure in Extension and in Comprehension. For example: In Extension M is contained under P; .. S is contained under P. In Comprehension M comprehends P: (2) That in which the common third term is predicate of both the other terms. This gives the Second Figure, which admits affirmative as well as negative conclusions. For example : Affirmative All P is all M; All S is some M: .. All S is some P. (3) That in which the common third term is subject of both the other This gives the Third Figure, which admits of universal as well terms. The so-called Fourth Figure is a hybrid. Its premises proceed in the whole of Comprehension. For example: i.e. in the premises the two subjects P and M are greater than the two predicates M and S, while in the conclusion the predicate P is greater than the subject S. This Figure is also useless, for reasoning is scientifically complete without it.' 8. Since Syllogism is the result of an act of immediate comparison by which we recognise that two notions stand to each other in the relation of whole and part, through the recognition that these notions severally stand in the same relation to a common third notion, it is not necessary that these notions stand to each other in the relation of subject and predicate. And since the syllogistic variation of Figure depends on the varieties of the relations of subject and predicate, Syllogistic Figure is not an essential variation. There may be Unfigured as well as Figured Syllogism. The UNFIGURED SYLLOGISM is that in which the terms compared do not stand to each other in the reciprocal relation of subject and predicate; the FIGURED SYLLOGISM, that in which the terms compared have this relation.3 But if Figure itself be only an accidental variation of Syllogism, one of the Figures, the first, cannot be in itself the only true and valid form of syllogistic inference, and it is absurd to reduce the others to that form in order to show their validity. Hence the New Analytic abolishes Reduction. 9. But if Reduction be unnecessary, each Figure must have an organic principle of its own on which it proceeds, and, since all varieties of syllogism are to be evolved from its supreme canon, these canons of the separate Figures are to be evolved from that supreme canon. And since the variation of Figure is determined by the varieties of relation of the extremes with the common third term implicitly given in the supreme canon, the canons of each of the Figures will be formed by explicitly enouncing that particular variety of relation which determines each Figure. In the First Figure the common third term is the subject of the one of the terms and the predicate of the other. Hence the canon of this Figure will be: What worse relation of determining (predicate), and of deter Cf. Baynes' New Analytic, pp. 65-69. 2 Cf. above, p. 376. * Cf. Baynes' New Analytic, p. 153; Hamilton's Lect. ii. 404, mined (subject), is held by either of two notions to a third, with which one at least is positively related; that relation do they immediately (directly) hold to each other, and indirectly (mediately) its converse. In the Second Figure, the common third term is the predicate of both of the other terms. Hence the canon of this Figure will be: What worse relation of determined (subject) is held by either of two notions to a third, with which one at least is positively related; that relation do they hold indifferently to each other. In the Third Figure, the common third term is the subject of both of the other terms. Hence the canon of this Figure will be: What worse relation of determining (predicate) is held by either of two notions to a third, with which one at least is positively related; that relation do they hold indifferently to each other.1 10. The syllogistic variation of Mood was seen to be evolved from the supreme canon of Figured Syllogism. The true number of moods is also to be determined by that canon. For the variety of mood depends on the various relations of subject and predicate produced by difference of quantity and quality. Hence the number of moods is to be determined by the number of variations in quantity and quality possible in the premises; and the number of valid moods, by the number of these variations in which both premises are not negative, and in which the quantifications of the middle term, whether as subject or predicate, exceed the quantity of the term taken in its whole extent (i.e. where the middle term is distributed). When some others are excluded which have particular conclusions where universal are competent, there remain in all thirty-six valid Moods, twelve affirmative and twelve negative. These Moods are all evolved from the supreme canon of Figured Syllogism, and are independent on the variation of the various Figures. Hence they are valid in every Figure. Each one of them may be evolved from the supreme canon without reference to the others, and therefore they are mutually independent. The variation of moods in the different Figures under the Old Analytic was caused by the confusion created by not quantifying the predicate. When that confusion is cleared up the moods are seen to be virtually identical or relatively equivalent throughout every variety of schematic difference. 11. In the Second and Third Figures, where the extremes both hold the same relation to the middle term, there is not, as in the first, an opposition and subordination between a term major and a term minor, mutually containing and contained, in the counter wholes of Extension and Comprehension. In the First Figure, since the major term is pre 1 Hamilton's Lect. on Log. p. 350; cf. Discussions, pp. 654, 655. dicate of the one premise and the minor term subject of the other, in the whole of Extension the major is greater than the middle, and therefore much greater than the minor term; while in the whole of Comprehension the minor is greater than the middle, and therefore much greater than the major term. But in the Second Figure both major and minor terms are subjects in the premises, and in the Third Figure they are both predicates in the premises; and the mutual subordination cannot arise. Consequently, in the Second and Third Figures there is no determinate major and minor premise, and there are two indifferent conclusions. For example, we may equally well say- Whereas in the First Figure the premises are determinate, and there is a single direct conclusion. For example, we must say— MP and since every proposition is an equation, we may have also the indirect conclusion P S. 12. In the First Figure, Comprehension and Extension are in equilibrium, the major and minor terms are reciprocally whole and part, and this Figure is, therefore, common to Induction and Deduction, indifferently. In the Second Figure, Extension is predominant, for the predicate is naturally the greater, and this Figure is, therefore, more appropriate to Deduction, which proceeds from the universal to the particular, from what is true of a class to what is true of individuals. In the Third Figure, Comprehension is predominant, for the subject is naturally the greater, and this Figure is, therefore, more appropriate to Induction, which proceeds from the particular to the universal-from what is true of the individuals which make up a class to what is true of the class itself. § 9. III. The scheme of logical notation is meant to show, with mechanical simplicity, all the propositional and syllogistic forms. Sir W. Hamilton' intends this scheme to be wholly different in principle and perfection from those which have been previously proposed, but, as Archbishop Thomson says, ' many of the different elements are not new.' This notation can represent any relation of the terms, any order of 1 Lect. on Logic, ii. 251. the proposition, any extent of quantity. The terms are represented by letters-the extremes by the letters C and F, which are each the third letter in its respective alphabet, and the middle term of the syllogism by the letter M, their quantity by the points, and the propositions by the lines with the letters.' 'Definite quantity (all, any) is indicated by the sign (); indefinite quantity (some), by the sign (, or,). The horizontal tapering line ( -) indicates an affirmative relation between the subject and predicate of the proposition. Negative is marked by a perpendicular line crossing the horizontal (). . . . In Extension, the broad end of the line denotes the subject, the pointed end the predicate. In Comprehension this is reversed; the pointed end indicating the subject, the broad end the predicate. . . . A line beneath the three terms M denotes the relation of the extremes of the conclusion. . . . In the Second and Third Figures,—a line is inserted above as well as below the terms of the syllogism, to express the double conclusion in those figures. The symbol (~~) shows that when the premises are converted, the syllogism remains in the same mood; the symbol (X) shows that the two moods between which it stands are controvertible into each other by conversion of their premises.' The mood Barbara is thus expressed by Hamilton's notation : This New Analytic is almost entirely based upon the doctrine of the Quantification of the Predicate, and stands or falls with that doctrine. For a criticism of the doctrine, cf. Appendix B, p. 579. Archbishop Thomson, who does not accept all the results given above, gives the most complete exposition and application of the doctrines in his Outlines of the Laws of Thought. § 10. The doctrines contained in this New Analytic of logical forms lead directly to the theories of Boole and Jevons. A leading characteristic of the Doctrine of the Quantification of the Predicate, and other [recent] theories of a similar kind, is the attempt 1 Baynes' New Analytic, p. 151. 2 Hamilton's Lect. on Log. ii. 473. |