the latter term belongs to every object of the former. other premise, it must fulfil the function of a subject. general formula of all argument must be Hence, in the M is P S is M S is P; which is to be understood in this sense, that the terms of every syllogistic argument fulfil functions of subject and predicate as here indicated, but not that the argument can be grammatically expressed in this way. PART II. § 1. Of Apagogical Forms. If is true when P is, then P is false when C is. Hence it is always possible to substitute for any premise the denial of the conclusion, provided the denial of that premise be at the same time substituted for the conclusion.* Hence, corresponding to every syllogistic argument in the general form, The apagogical forms make it necessary to consider in what way propositions deny one another. If a proposition be put into the general form, S is P, its contradictory has, 1st, as its subject, instead of S, "the S now * This operation will be termed a contraposition of the premise and conclusion. VOL. VII. 34 meant "* or some S"; and has, 2d, as its predicate, instead of P, that which differs from P or "not P." From these relations of contradictories, from the necessities of the logic of apagogically related arguments, therefore, arises the need of the two divisions of propositions into affirmative and negative on the one hand, and into universal and particular on the other. The contradictory of a universal proposition is particular, and the contradictory of an affirmative proposition is negative. Contradiction is a reciprocal relation, and therefore the contradictory of a particular proposition is universal, and that of a negative proposition is affirmative. The contradiction of particular and negative propositions could not be brought under the general formula, were the distinctions of affirmative and negative absolute and not merely relative; but, in fact, not-not-P is the same as P. And, if it is said that "what is now meant of the part of S meant at another time, is P," since the part of S meant at another time is left to be determined in whatever way the proposition made at another time may determine it, this can only be true if All S is P. Therefore, if one man says some S is not P," and another replies, "some of that same S is P," this second person, since he allows the first man's some S, which has not been defined, to remain undefined, in effect says that All S is P. 66 Whether contradictories differ in other respects than these wellknown ones is an open question. § 3. Of Barbara. Since some S means "the part now meant of S," a particular proposition is equivalent to a universal proposition with another subject; and in the same way a negative proposition is equivalent to an affirmative proposition with another predicate. therefore, as well as representing propositions in general, particularly represents Universal Affirmative propositions; and thus the general form of syllogism M is P; S is M; S is P, represents specially the syllogisms of the mood Barbara. *What S is meant being generally undetermined. § 4. Of the First Figure. Since, in the general form, S may be any subject and P any predicate, it is possible to modify Barbara by making the major premise and conclusion negative, or by making the minor premise and conclusion particular, or in both these ways at once. Thus we obtain all the modes of the first figure. It is also possible to have such arguments as these :— and Some M is P, S has all the common characters of that part of M (whatever that part may be, and therefore of each and every M), but as the theory of apagogical argument has not obliged us to take account of these peculiar modifications of subject and predicate, these arguments must be considered as belonging to Barbara. In this sense the major premise must always be universal, and the minor affirmative. Three propositions which are related to one another as though major premise, minor premise, and conclusion of a syllogism of the first figure will be termed respectively Rule, Case, and Result. § 5. Second and Third Figures. Let the first figure be written thus: Then its two apagogical modifications are the second and third figures. It is customary to enumerate six moods of the third figure instead of four, and the moods Darapti and Felapton appear to be omitted. But a particular proposition is asserted (actually and not merely virtually) by the universal proposition which does not otherwise differ from it; and therefore Darapti is included both under Disamis and Datisi, and Felapton both under Bocardo and Ferison. (De Morgan.) The second figure, from the assertion of the rule and the denial of the result, infers the denial of the case; the third figure, from the denial of the result and assertion of the case, infers the denial of the rule. Hence we write the moods as follows, by allowing inferences only on the straight lines: The symmetry of the system of moods of the three figures is also exhibited in the following table. Enter at the top the proposition asserting or denying the rule; enter at the side the proposition asserting or denying the case; find in the body of the table the proposition asserting or denying the result. In the body of the table, propositions indicated by italics belong to the first figure, those by black-letter to the second figure, and those by script to the third figure. If, as the denial of the result in the second and third figures, we put the form "Any N is N," we have These are the formulæ of the two simple conversions. Neither can be expressed syllogistically except in the figures in which they are here put (or in what is called the fourth figure, which we shall consider hereafter). If, for the denial of the result in the second figure, we put "No not-N is N" (where "not-N" has not as yet been defined) we obtain In the same way, if we put "Some Nis some-N" (where some-has not been defined) for the denial of the result in the third figure, we have Some N is some-N |