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Why the order of the figures of the hypothetical syllogisms was changed
Published online by Cambridge University Press: 11 February 2009
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In chapter 6 of Alcinous’ Handbook of Platonism we find a discussion of categorical, hypothetical, and mixed syllogisms. Alcinous distinguishes three figures of the hypothetical syllogism, and illustrates each figure with a syllogism based on an argument from Plato. Here he remarks in passing that most people called the second hypothetical figure the third and that some called the third figure the second. We may assume that those who called the third figure the second and those who called the second the third were the same. In a parallel passage, Alexander of Aphrodisias advocates the same ordering of figures of hypothetical syllogisms as Alcinous, and reports that Theophrastus, in the first book of his Analytics, had the second and third figure in reverse order. Combining these passages, we can infer that at the turn of the second century A.D. there existed two different views on the ordering of the figures of the hypothetical syllogisms, of which one goes back to Theophrastus, whereas the other presumably was the result of a later change.
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1 ‘And in accordance with the second hypothetical figure (which most people call the third), in which the common term follows the extremes in either premise, he argues thus … And in accordance with the third figure (which some take to be the second), in which the common term precedes the extreme terms, in the Phaedo he argues in effect thus …’ (Alc. Didasc. 159.14–15 and 20 Whittaker, 
2 ‘In the first book of his Analytics, Theophrastus states that the second figure of the wholly hypothetical <syllogisms> is the one in which the premises begin with the same <term>, and end in different ones, and the third figure is the one in which they begin with different ones, and end in the same. But we set them out the other way around.’ (Alex. An.Pr. 328.2–5, Θεóøρασтος μνтοι 
3 Barnes, J., in his ‘Terms and sentences: Theophrastus on hypothetical syllogisms’, Proceedings of the British Academy 69 (1983), 279–326Google Scholar at 297–8, remarks on this fact, but offers no explanation. He assumes that it reflects a philosophical dispute that may go back to the first century B.C. Peripatetics. Later (at p. 311) he suspects that Theophrastus had no reason for his ordering of the figures, whereas Alexander had an analogy to Aristotle's ‘categorical’ syllogisms in mind. Similarly, Invernizzi, G., in II Didaskalikos di Albino e il medioplatonismo, vol. 2 (Rome, 1976), 108Google Scholar, n. 21, suggests that Albinus (i.e. Alcinous) switched the figures in order to make them analogous to the categorical syllogisms. J. Dillon, in his commentary on the passage (Handbook of Platonism [Oxford, 1993], 82) suggests that the reason for the reversal of the second and third figure was the fact that someone from whom Alcinous draws used Plato's Parm. 137d4–9 to show that Plato made use of first figure hypothetical syllogisms, and the immediately following passage Parm. 137e–138al to show that Plato also employed what was up to then regarded as third figure hypothetical syllogisms, and then, in order to achieve symmetry (i.e. between the passages from the Parmenides and the names of the figures) placed what was the third hypothetical figure second. I am not convinced by this account. Bochenski, I. M., in La Logique de Théophraste (Fribourg, 1947), 115Google Scholar, followed by Graeser, A., Die logischen Fragmente des Theophrast (Berlin, 1973), 99–100Google Scholar, argues that the second figure is logically closer to the first than the third, but as Barnes (p. 311) rightly points out, this is not borne out by our evidence.
4 In line with the main tradition, Alexander calls the arguments which Alcinous names ‘hypothetical’ ‘wholly hypothetical’. They are called ‘wholly hypothetical’ since all their premises, and their conclusion, are hypothetical propositions; cf. e.g. Philop. An.Pr. 243.14–16.
5 On pp. 326–8 Alexander throughout speaks of ‘the hypothetical’ ( ὑποθεтιĸòς), and it is unclear whether we should understand ‘syllogism’ or ‘argument’, since he repeatedly voices some doubt as to whether the wholly hypothetical are syllogisms (e.g. An.Pr. 326.12–14).
6 In fact, the last two examples are given as examples for valid arguments in the second and third figure.
7 My main reasons for this assumption are the facts that Alexander, like Alcinous, clearly talks about the relation of (middle and extreme) terms in the passage (e.g. An.Pr. 327.9 т γᾰρ μσος ὼν ρος), and that, when he gives examples for the second and third figures, he replaces the letters by terms such as ‘human being', ‘rational', etc. (An.Pr. 327.11–13 and 18–20). On this point I disagree with Barnes (n. 3, 295), who takes Alexander to refer by ‘middle term’ to the ‘middle hypothesis', i.e. to the component proposition shared by the two hypothetical premises. However, what I say in the following can with some (obvious) modifications be upheld even on the assumption that Alexander used A, B, and C to stand for propositions rather than terms.
8 In Alcinous, the conditionals are understood in a different way, as involving a designator in the antecedent, and a cross-reference to the designator in the consequent. They can be paraphrased as ‘If it (i.e. this thing a) is A, it (a) is B’. (See my ‘Wholly hypothetical syllogisms’, Phronesis 2000 [forthcoming].) For my present purposes, again, this difference is of no importance.
9 Alexander talks about that which precedes (тòγοὺμενον), that which follows (тòέπóμενον), and about the term which precedes or which follows (e.g. тά γᾰρ Γ μέσος ὼν ὅρος…πóμενος, An. Pr. 327.9; ν ἧ ἡγεтαι μέσος…ν ἧ ἕπεтαι μέσος, An. Pr. 327.26–7.)
10 Cf. Alexander's use of οὕтως…ὡς(An. Pr. 326.30) and of νάλογον (An. Pr. 327.3 and 14).
11 Alcinous calls the middle term the ‘common term’(Didasc. 159.3,20–1).
12 οἱ δἰ λων ὑποθεтιĸοἳ, οὕς Θεóøρασтοσ “ĸαтᾰ ναλογἳαν” λέγει(An.Pr. 326.8—9).
13 Alexander provides a different reason why Theophrastus called them ‘by analogy’: their premises are similar (in form) to each other and to the conclusion (An. Pr. 326. 10–12). But this need not have been Theophrastus’ reason for the name. Cf. Barnes (n. 3), 288, n. 2.
14 For Alexander (who when commenting on Aristotle naturally mostly sticks to Aristotle’s formulation), see e.g. An. Pr. 348.17.