| If anything is P it is M (or is not M), | |
| Certain subjects S are not M (or are M), | |
| therefore, | They are not P. |
This scheme may be expressed in the following dictum,—“If a certain attribute can be predicated, affirmatively or negatively, of every member of a class, any subject of which it cannot be so predicated does not belong to the class.”[368] This dictum may, like the dictum de omni et nullo, claim to be axiomatic, and it is related to the valid syllogisms of figure 2 just as the dictum de omni et nullo is related to the valid syllogisms of figure 1.[369]
[368] The dictum for figure 2, sometimes called the dictum de diverso, is expressed in the above form by Mansel (Aldrich, p. 86). It was given by Lambert in the form, “If one term is contained in, and another excluded from, a third term, they are mutually excluded.” This is at least expressed loosely, since it would appear to warrant a universal conclusion, if any conclusion at all, in Festino and Baroco. Bailey (Theory of Reasoning, p. 71) gives the following pair of maxims for figure 2,—“When the whole of a class possess a certain attribute, whatever does not possess the attribute does not belong to the class. When the whole of a class is excluded from the possession of an attribute, whatever possesses the attribute does not belong to the class.”
[369] Lambert is usually regarded as the originator of the idea of framing dicta that shall be directly applicable to figures other than the first. Thomson, however, points out that it is an error to suppose that Lambert was the first to invent such dicta. “More than a century earlier, Keckermann saw that each figure had its own law and its own peculiar use, and stated them as accurately, if less concisely, than Lambert” (Laws of Thought, p. 173, note). Distinct principles for the second and third figures are laid down also in the Port Royal Logic, which was published in 1662.
271. Scheme of the Valid Moods of Figure 3.—Dealing with figure 3 in the same way as we have done with figure 2, we get the following scheme, summing up the valid moods of that figure:
| Some (or all) A is not C (or is C), | (Denial of Result) | |
| All (or some) A is B, | (Case) | |
| therefore, | Some B is not C (or is C). | (Denial of Rule) |
It is not easy to express this scheme in a single self-evident maxim.[370] Separate dicta of an axiomatic character may, 338 however, be formulated for the affirmative and negative moods respectively of figure 3, namely, “If two attributes can both be affirmed of a class, and one at least of them universally so, then these two attributes sometimes accompany each other,” “If one attribute can be affirmed while another is denied of a class, either the affirmation or the denial being universal, then the former attribute is not always accompanied by the latter.”[371]
[370] Lambert gave the following dictum de exemplo for figure 3:—“Two terms which contain a common part partly agree, or if one contains a part which the other does not, they partly differ.” This maxim is open to exception. The proposition “If one term contains a part which another does not, they partly differ” applied to MeP, MaS, would appear to justify PoS just as much as SoP, or else to yield an alternative between these two. Mr Johnson gives a single formula for figure 3, namely, “A statement may be applied to part of a class, if it applies wholly [or at least partly] to a set of objects that are at least partly [or wholly] included in that class.” This is correct, but perhaps not very easy to grasp.
[371] These dicta (or dicta corresponding to them) are sometimes called respectively the dictum de exemplo and the dictum de excepto.
272. Dictum for Figure 4.—The following dictum, called the dictum de reciproco, was formulated by Lambert for figure 4:—“If no M is B, no B is this or that M ; if C is (or is not) this or that B, there are B’s which are (or are not) C.” The first part of this dictum is intended to apply to Camenes, and the second part to the remaining moods of the fourth figure; but the application can hardly in either case be regarded as self-evident. Several other axioms have been constructed for figure 4; but they are, as a rule, little more than a bare enumeration of the valid moods of that figure, whilst at the same time they are less self-evident than these moods considered individually. The following axiom, however, suggested by Mr Johnson, is not open to these criticisms: “Three classes cannot be so related, that the first is wholly included in the second, the second wholly excluded from the third, and the third partly or wholly included in the first.” This dictum affirms the validity of two antilogisms; in other words, it declares the mutual incompatibility of each of the following trios of propositions: XaY, YeZ, ZiX ; XaY, YeZ, ZaX ; and it will be found that these incompatibles yield the six valid moods of the fourth figure.[372]