¹ Anal. Pr. I, 25b 35, 26b 36, 28a 12.

² Prantl. C., Geschichte der Logik im Abendlande I, 365. The principal source is Alexander of Aphrodisias ad. An. pr. f. 27b, 42b.

³ Prantl. Op. Cit. I, 570–74.

⁴ E.g. Jacques Maritain, An Introduction to Logic (English translation p. 187n). “It is evident that, although the fourth figure is a grammatical figure it is not a distinct logical figure: in that which concerns thought the grammatical predicate of the conclusion is in reality its subject. For this reason every true logician must reject the fourth figure and consider it only as the indirect first.” (Italics in text.) Similarly H. W. B. Joseph (An Introduction to Logic, pp. 330–31) claims: There is no fourth figure. … It must always be remembered that the character of an argument is determined not by the form in which it is thrown in words, but by that which it assumes in our thought. This is our justification for recognizing the figures as distinct types.“

⁵ “It is evident that anyone forcing the concepts into this unnatural position must have forgotten all the fundamental presuppositions of the Aristotelian theory; the need of supplementing the Aristotelian doctrine could only have been felt in a treatment which dealt with the external form alone.” Christoph Sigwart eng. tr. 2nd ed. note on p. 33, vol. 1 elsewhere on page “The so-called fourth figure.”

⁶ Notably W. E. Johnson. See his Logic, II, 88.

⁷ Principally through Averroes, who disagreed with it. See Prantl, Op. Cit. I, 571–572.

⁸ An. Pr. I, Ch. 4, 5 and 6, are on pure syllogisms. Ch. 8 through 11, and 14 through 22 are on modal syllogisms.

⁹ The idea that reduction of a simple syllogism to the first figure constitutes a validation of it, in the sense of settling any real doubts, is based on the assumption that there are people intelligent enough to grasp the validity of a first-figure syllogism and to understand the mechanism of reduction, but stupid enough not to be able to see whether or not syllogisms in other figures are valid. I have found very few such people and see no reason to suppose that their incidence was higher among the Greeks of Aristotle's time.

As a teacher of logic, my experience has been that there are students who cannot ascertain the validity of an “imperfect” syllogism by direct examination. At great pains, some of them can be taught reduction to the first figure with a fair probability of doing it correctly. Having done this, however, they will, in general, be uncertain of the validity of the resulting first-figure syllogism.

¹⁰ Anal. Pr. I, 24b 26.

¹¹ Eth. Nic. II, 39b 18–25.

¹² Anal. Pr. 30b 32 ff.

¹³ Anal. Pr. 35a 3.

¹⁴ Anal. Pr. 32a 18 ff; 33b 22; 33b 29.

¹⁵ Anal Pr. 32a 15–40.

¹⁶ Cf. Anal. Pr. 34b 7 ff.

¹⁷ Meta. 1050b 8; 1049b 23. It is interesting, however, that when posibility is discussed in connection with potentiality, it is possibility in the strict sense and not contingency (Meta. 1019b 23).

To insist on this parallel between logical and metaphysical categories is not to make any assertion about the order in which they were explicitly formulated. After all, William James' essay The Dilemma of Determinism is a clear application of the general doctrine set forth in The Will to Believe. Yet the former appeared in 1884 and the latter in 1895.

¹⁸ Principally in the case of a first figure EAE syllogism where the major premiss is assertoric and the minor problematic. Here Aristotle points out that, with suitable choice of terms, the conclusion may be a statement which is necessarily true and so cannot be contingent. 34b 19 ff.

¹⁹ Cf. 29b 36.

²⁰ 30a 15–24.

²¹ The illustration is not Aristotle's but he would probably admit it, since he quotes both the major (31b 7) and the conclusion (32a 2) as examples of necessary statements. Unfortunately, however, he also states a contrary of the conclusion as a true statement (30b 35).

²² Anal. Pr. I, Ch. 9.

²³ If we allow “→” to stand for implication in any one of its usual senses and “N(p)” to represent “p is necessary” then an E statement involving necessity would be written in the conventional symbolism:

This is, however, open to two objections: 1) It would seem to make the necessity of an E proposition an inferential necessity rather than the absolute necessity that Aristotle demands. What is asserted to be necessary is the connection. 2) In no recognized system of logic would an assertoric minor and necessary major yield a necessary conclusion.

To avoid these difficulties Albrecht Becker (Die Aristotelische Theorie der Moeglichkeitsschluesse, p. 39) suggests the translation:

This avoids the above difficulties, but as Becker notices (p. 42), raises insuperable difficulties in conversion.

²⁴ The negative premiss is converted, yielding a first-figure syllogism with assertoric major and apodictic minor. Hence the conclusion is assertoric. 30b 19.

²⁵ 36a 7–15. He says that, in addition, the conclusion “It is contingent that no S is P” may be drawn.

²⁶ 36a 17–25; 35b 38 ff.

²⁷ In the case of Baroco, Aristotle indicates by an example that the conclusion cannot be necessary. (31a 17). The example may or may not be correct, but if it is it merely proves that his doctrine is inconsistent, since the following proof shows the conclusion to be necessary. The given premisses are “All P is M” and “Necessarily some S isn't M.” Let X be the group of S's which necessarily are not M (or alternatively, let X be one such S). Then we have: Necessarily, no X is M which converts to: Necessarily, no M is X. With “All P is M,” this yields the conclusion: “Necessarily no P is X.” (30a 16). Hence, “Necessarily, no X is P” and “Necessarily, some S isn't P.“

In the case of Datisi, Aristotle converts the necessary minor premiss and so arrives at a first-figure syllogism with assertoric major and apodictic minor, yielding an assertoric conclusion. (31b 22). This is the strongest result obtainable by this method. The substitution of “Necessarily, all X is S” for “Necessarily, some M is S”, together with a prosyllogism proving “All X is P” and its converse “Some P is X”, provides the basis for a conclusion “Necessarily, some S is P“.

Aristotle's preference for direct reduction causes him to miss the point. Similar criticisms apply to his treatment of Disamis (31b 22) and Bocardo (31b 39) when the major is apodictic and the minor assertoric.

²⁸ He offers it as a procedure alternative to indirect reduction in the case of Bocardo with pure premisses. (28b 20). He relies on it exclusively for both Baroco and Bocardo where both premisses are apodictic (30a 6 ff) since at this point he has not yet developed the mechanism which would make indirect reduction possible. But these are the only uses.

It is regrettable that Aristotle did not make more extensive use of exposition since it would have provided a simple means of reducing particular syllogisms of a given figure to universal syllogisms of the same figure and materially shortened the enumeration of types of syllogisms.

²⁹ Aristotle states that a syllogism in Celarent with assertoric major and contingent minor has a possible conclusion. His reasoning uses the following involved and invalid procedure. 1) Assume the minor to be assertoric. This may lead to a false conclusion but cannot lead to an impossible one. 2) Combine the new minor with the contradictory of the conclusion: It is necessary that some S is P and deduce the contradictory of the major premiss. Aristotle argues that since the contradiction arose in the second step, the assumption of the second diction arose in the second step, the assumption of the second step must be incorrect and the original conclusion is thus validated. Though this procedure has been defended by Alexander (Commentaria in Aristotelem Graeca, II pars. 1, 175 and Maier (Die Syllogistikdes Aristoteles 2. Teil. 139), the fallacy is shown by Tredennick (Loeb edition of Aristotle's Organon I, 270 note d) and more fully by Becker (Op. Cit. 51 ff).

The point of this note is that had Aristotle given the proper analysis of Datisi with assertoric major and necessary minor (see note 27), this procedure would have been unnecessary. He might have reasoned as follows:

No M is P

It is contingent that All S is M

∴ It is possible that No S is P for, assume the contradictory of this conclusion and combine it with the major

No M is P

It is necessary that Some S is P

∴ It is necessary that Some S isn't M

This is incompatible with the given minor.

It is only because Aristotle does not see his way clear to drawing the necessary conclusion of the last syllogism that he is forced to adopt the more cumbersome—and invalid—procedure described before.

³⁰ It should be noted incidentally that, while all perfect syllogisms are in the first figure, not all first-figure syllogisms are perfect—only those in which the conclusion is immediately evident. For example, whenever the minor premiss expresses contingency and the major does not, the syllogism is imperfect. 33b 27; 35b 23 ff.

³¹ Baroco and Bocardo often require special treatment such as indirect reduction or exposition.

³² I am using the term “mood” to refer to a pattern of syllogism, reserving “mode” to refer to the modality of a proposition.

³³ Bramantip, Camenes, and Dimaris.

³⁴ Fesapo and Fresison.

³⁵ 29a 21.

³⁶ E.g. Festino itself is not always equipollent with Ferio, but only when the major premiss is not contingent. Cf. 25b 15, 36b 35 ff. Hence there is no universally applicable method of treating this type of syllogism.

³⁷ 25b 14.

³⁸ Cf. Prantl. Op. Cit. 1370–375. This shift of doctrine is attributed to Theophrastus and Eudemus by Alexander of Aphrodisias ad An. pr. f. 49a.

³⁹ 36b 26.

⁴⁰ 36b 35 ff.