We can go over each of the premisses in each of the moods, and find out what conclusion logically follows. But this is done in the works on logic; most simply and clearly I believe in “Jevon’s Logic.” As we are only concerned with a formal presentation of the results we will make use of the mnemonic lines printed below, in which the words enclosed in brackets refer to the figures, and are not significative:—
- Barbara celarent Darii ferioque [prioris].
- Caesare Camestris Festino Baroko [secundae].
- [Tertia] darapti disamis datisi felapton.
- Bokardo ferisson habet [Quarta insuper addit].
- Bramantip camenes dimaris ferapton fresison.
In these lines each significative word has three vowels, the first vowel refers to the major premiss, and gives the mood of that premiss, “a” signifying, for instance, that the major mood is in mood a. The second vowel refers to the minor premiss, and gives its mood. The third vowel refers to the conclusion, and gives its mood. Thus (prioris)—of the first figure—the first mnemonic word is “barbara,” and this gives major premiss, mood A; minor premiss, mood A; conclusion, mood A. Accordingly in the first of our four cubes we mark the lowest left-hand front cube. To take another instance in the third figure “Tertia,” the word “ferisson” gives us major premiss mood E—e.g., no M is P, minor premiss mood I; some M is S, conclusion, mood O; some S is not P. The region to be marked then in the third representative cube is the one in the second wall to the right for the major premiss, the third wall from the front for the minor premiss, and the top layer for the conclusion.
It is easily seen that in the diagram this cube is marked, and so with all the valid conclusions. The regions marked in the total region show which combinations of the four variables, major premiss, minor premiss, figure, and conclusion exist.
That is to say, we objectify all possible conclusions, and build up an ideal manifold, containing all possible combinations of them with the premisses, and then out of this we eliminate all that do not satisfy the laws of logic. The residue is the syllogism, considered as a canon of reasoning.
Looking at the shape which represents the totality of the valid conclusions, it does not present any obvious symmetry, or easily characterisable nature. A striking configuration, however, is obtained, if we project the four-dimensional figure obtained into a three-dimensional one; that is, if we take in the base cube all those cubes which have a marked space anywhere in the series of four regions which start from that cube.
This corresponds to making abstraction of the figures, giving all the conclusions which are valid whatever the figure may be.
Fig. 57.
Proceeding in this way we obtain the arrangement of marked cubes shown in [fig. 57]. We see that the valid conclusions are arranged almost symmetrically round one cube—the one on the top of the column starting from AAA. There is one breach of continuity however in this scheme. One cube is unmarked, which if marked would give symmetry. It is the one which would be denoted by the letters I, E, O, in the third wall to the right, the second wall away, the topmost layer. Now this combination of premisses in the mood IE, with a conclusion in the mood O, is not noticed in any book on logic with which I am familiar. Let us look at it for ourselves, as it seems that there must be something curious in connection with this break of continuity in the poiograph.