Abstract
It is a familiar and much-discussed feature of Aristotle’s syllogistic, that in it the distinction between ‘perfect’ and ‘imperfect’ syllogisms plays an essential part. This distinction divides all valid syllogisms into two classes, the second of which is ‘reduced’ by means of certain logical operations to the first class and thereby proved, while the first class is taken as axiomatic and assumed without proof. Up to now the nature of this distinction has not been properly understood. The ancients debated whether or not Aristotle recognised the validity of the imperfect inferences, and whether, if he recognised them as valid, he had the right to call them ‘imperfect’.1 The 19th century historians of philosophy, who equated the Aristotelian with the traditional syllogism, were no longer able to see the formal peculiarities of Aristotle’s ‘perfect’ inferences in the traditional formulation — in which as a matter of fact they disappear —, and were obliged to seek other grounds for the dichotomy. The fact that all the inferences which Aristotle calls ‘perfect’ belong to the first figure encouraged them to speak of perfect figures rather than of perfect syllogisms, and to suppose that perfect syllogisms were perfect just because they belonged to the perfect, that is to say the first, figure. Aristotle’s reasons for calling this figure the first (it yields conclusions in all forms, a, e,i, and o, whereas the second figure does not allow a or i and the third does not allow a or e as conclusions (A 4, 26b23–33)), and his assertion in the Posterior Analytics (A 14, 79a17–32) that the first figure is the truly ‘scientific’ figure, were taken as reasons for the ’perfection’ of this figure and hence of the syllogisms contained in it.
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Notes
Ross, APPA, pp. 291–292; Lukasiewicz, AS, p. 43; Bochenski, FL, p. 87; HFL, p. 76.
On this cf. H. Scholz, ’Die Axiomatik der Alten’, in Blätter für deutsche Philosophie 4 (1930), 259–278, esp. 265 sqq.
A detailed account of this will be given in Ch. V.
A system of syllogistic based on these formal foundations has been worked out by P. Lorenzen, ’Die Syllogistik als Relationenmultiplikation’, Archiv f. math. Logik 3 (1957), 112–116; cf. also Lorenzen, Formale Logik,1958, § 2.
Aristotle’s proofs of the convertibility of AeB and AiB will be discussed below, 138 sqq.
The symbol “ E ” is introduced by Russell and Whitehead as the sign for inclusion between relations; it is so defined (PM I, 23.01) that R C S holds if and only if xSy holds for all pairs of individuals for which xRy holds.
Philoponus (113, 20) thinks that Aristotle was content to state the first figure pairs alone because they suffice as examples for pairs of the other figures too: aötòç Ltóvov Tfly ttpdnnv dugs cbç Iv ltapaSetyµatt. Ross (APPA, p. 314) writes: “These generalisations are correct, but A. has omitted to notice (sic) that OA in the second figure and AO in the third give a conclusion with P as subject”. Thus neither of these commentators has seen the logical difference between the cases in the first and those in the other figures.
Philoponus (78, 3–9) noticed that the order of the premisses was inverted, and he explained it by observing that the two relational expressions differ only by their axéatç, that is the spatial ordering of their arguments. In the syllogism constructed with “be contained in” what was before the last term becomes the major term, and what was the minor premiss becomes the major premiss. We cannot tell from this passage whether or not Philoponus noticed that this transposition is necessary to preserve the perfection of the syllogism. A little later, however, he gives two formulations of the syllogism AaBandBiC-AiC with concrete terms: “Animal is said of all men, man of some animate things; Conclusion: animal is said of some animate things”; and: “Some animate things are men, all men are animals. Conclusion, some animate things are animals” (78, 15–17). Here it is plain that the second argument attains evidence because its premisses are transposed.
These abbreviations are explained in the Bibliography.
John Locke saw this (Essay concerning Human Understanding,IV, 17, § 8). Without knowing Aristotle’s doctrine of ’perfect’ syllogisms, he asked: “... would not the position of the medius terminus ... show the agreement or disagreement of the extremes clearer and better, if it were placed in the middle between them? Which might be easily done by transposing the propositions ...” of the traditional first figure. The same idea is found in Leibniz, Nouveaux Essais,IV, 17, §§ 4 and 8. — In 1890 E. Schröder hit the nail on the head in his Vorlesungen über die Algebra der Logik,I, pp. 173 sq.: “Aristotle’s choice of the transposed order (sc. of the premisses) is explained by the fact that he attends not to the extensions but to the contents of the terms; then the arrangement ”c is a property of b, b is a property of a, ergo c is a property of a“ seems the more natural”. E. Scheibe brought this passage to my attention (1968).
Aristotle defines P (APr. A 13, 32a18–20) as the modal operator which a proposition or the predicate of a proposition has when neither it nor its negation has the operator N. In modern modal logic this operator is called two-sided possibility; one-sided possibility is also found in Aristotle: it is present if a proposition satisfies only the second condition for P,i.e. if its negation does not have N.
PI is the symbol for one-sided possibility (“not impossible”).
This answers to Aristotle’s statement: “It is clear that what is necessary is also actual” (de Int. 13, 23a21 sq.). Becker introduces this law as T 18 (ATM, p. 15).
Philoponus gives this operator the appropriate and pleasing name “half-possibility” (tòÉvS£xoµévou in APr. 163, 16 et passim).
This follows from Becker’s T 19b: “p -÷ Ei p” (ATM, p. 15).
Cf. Becker’s formula F20 (ATM, p. 15); Bochenski, Ancient Formal Logic (hereafter AFL), p. 57.
AFL, p. 54; cf. also Bochenski, La Logique de Théophraste,p. 59. Bocheríski’s conjecture that A 7 is a later addition and that Aristotle did not have time to systematize his discoveries, is reported and accepted by Lukasiewicz (AS, p. 27).
Ammonius expressly refers to this difference (in APr. 32, 33).
Note here (a) that Philoponus, like the other commentators (and Aristotle himself in most cases) inverts the order of the premisses to preserve the evidence of the first figure in spite of the “B is A” formulation; (b) that Philoponus formulates his syllogisms no longer as propositions but as rules,the predominant practice since Apuleius and Boëthius (ob. 525 AD).
Sextus Empiricus too (3rd century AD) offers as a “Peripatetic syllogism” an argument in Barbara with transposed premisses: “The just is beautiful; the beautiful is good; therefore the just is good” (Pyrrh. Hyp. II, 163).
Philosophia rationalis sive Logica 3 (1744), § 388, p. 317: “Quoniam syllogismi primae figurae non sunt nisi applicatio Dicti de omnibus et nullo, istud autem Dictum per se evidens est, ... quivis in prima figura syllogismus suam fert evidentiam.” So too in J. Jungius, Logica Hamburgensis (1638), p. 139 of R. Meyer’s new edition, Hamburg, 1957.
ib. § 400, p. 327: “Figura perfecta dicitur, in qua omnes propositiones inferri possunt, imperfecta contra, in qua non omnes inferre licet.”
ib. § 393, p. 322: “Ceterum apparet in his modis syllogismorum ... medium terminum non continere rationem, unde intelligitur, cur praedicatum conveniat subjecto.”
Examples could be multiplied. Prantl (I, p. 348) laments the process, completed in late antiquity, of “releasing logic from the bonds with which, in Aristotle, it had in general been bound to philosophy”; he talks (I, p. 402) of the “Platonic and Aristotelian principle of a logic closely conjoined with philosophy” ; he stresses (I, p. 136) “the real metaphysical side of Aristotle’s logic”; he speaks (I, p. 104) of “the inseparable unity of logic and metaphysics” in Aristotle.
Trendelenburg, Logische Untersuchungen 3 (1870), I, p. 32, writes: “Aristotle’s fine discussion showing that the middle term of a true syllogism corresponds to the ground of the fact, has been pushed to one side and ignored by formal logic”; ib.: “Kant blotted out the last traces of its metaphysical origin”.
Maier (SdA II, 2, p. 85): “Inference-theory can never dispense with its epistemological and metaphysical foundations”. Further (p. 386): “The metaphysical background of Aristotle’s original logic, to which it owes its real synthetic power, has been forgotten since the Stoic plague”. Solmsen follows Maier (Die Entwicklung der aristotelischen Logik and Rhetorik,1929, p. 54): “Thus the reduction of all the other figures to the first ... in fact means precisely their reduction to the ontologically rational line of terms”.
Another recent scholar to state and emphasize this interpretation of Aristotle’s logic is N. Hartmann (Grundzüge einer Metaphysik der Erkenntnis,2nd ed., 1925, p. 22): “There can be no doubt that Aristotle thought of logic from an ontological point of view, and that he meant it to prepare the way for ’first philosophy’ or ’the science of being as being’ ”.
Cf. the discussion of the passage in J. König, ‘Bemerkungen über den Begriff der Ursache’, in Das Problem der Gesetzlichkeit,Hamburg, 1949, esp. pp. 109–117.
E. Zeller, Die Philosophie der Griechen II,22, 1862, p. 166.
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Patzig, G. (1968). Perfection. In: Aristotle’s Theory of the Syllogism. Synthese Library, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0787-9_3
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