Abstract
This article intends to reconstruct the textual tradition of the square of oppositions from the earliest textual sources just as treated in Boethius’ commentaries on Aristotle’s De Interpretatione and his treatises on syllogistic, De syllogismo categorico and Introductio ad syllogismos categoricos. The research discovers two different tracks. One way comes from Plato’s Sophist and Aristotle’s De Interpretatione, and the aim is to distinguish contrariety from contradiction. The second influence also starts from Aristotle, but now in connection with his Prior Analytics and its commentaries and treatises on categorical syllogistic, where the aim is to show the square as one of the three main chapters of the complete theory of categorical logic. I suggest that this double ingredient has accompanied the development of the square from the very original beginning of logic.
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- 1.
Boethius names Theophrastus in many places of his commentaries on Aristotle’s De Interpretatione [16]. Here (in Int 2, 12, 3ff.), he characterizes Theophrastus as a critical commentator of Aristotle, able to go beyond Aristotle’s teaching. In another passage, Boethius refers to Theophrastus as a vir doctissimus (in Int 2, 389, 19), a phrase he uses only to refer to the Neoplatonist Porphyry (cf. doctissimus vir: in Int 2, 15, p. 276). In his treatises on syllogistic [18, 19], Boethius refers to Theophrastus indirectly when suggesting that he would be one of the ancient Greeks who “have bequeathed much to posterity in their learned treatises” (translation by Thomsen Thörnqvist [18, p. 102]). But he also refers to his name in connection with the five indirect syllogistic moods Theophrastus added to those ones Aristotle had found in his Prior Analytics [9]. Apuleius in his Peri Hermeneias [6] names Theophrastus twice and both in a more distant relation. One concerns the five indirect moods Theophrastus found in addition (cf. 193, 8). The other relates to the case of a second Darapti in the third direct figure (cf. 189, 26).
- 2.
Boethius is also the author of one treatise on hypothetical or conditional logic. In his De hypotheticis syllogismis [17] Boethius gives an idea of the first Peripatetic developments and later systematization. Boethius and his sources understand categorical logic as the logic of single sentences that can be true or false, without any condition (cf. in Int 2, 186, 13ff.: Categoricas propositiones Graeci vocant, quae sine aliqua condicione positionis pronuntur (…); cf. also in Int 2, p. 112, 10–26), [16].
- 3.
- 4.
- 5.
- 6.
Antiquiores: cf. in Int, p. 58, 18; in Int, p. 59, 18–26; in Int 2, p. 63, 5–14, [16].
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in Int 18–26, p. 59: haec enim differentia quae est non currit et non laborat, quae a verbo puro et simplici distat, nulla apud antiquiores vocabulo nuncupata est differentia. differentiam autem vocavit id quod dicitur non currit et non laborat ab eo quod est currit et laborat. sed quoniam his nullum est ab antiquis nomen impositum, Aristoteles nomen ipse constituit dicens: sit verbum infinitum, [16].
- 8.
Ammonius and all the post-Ammonian commentators refer to certain palaioteroi, but they interpret their existence in the mythical way that Proclus’ commentary on Plato’s Cratylus suggested. Ammonius makes a passing allusion to these ancient authors at in Int, p. 41, 16–20, [4]. His allusion also supposes that they were more ancient than Aristotle, which is clear from his use of the adjective comparative palaioteroi. But Ammonius seems to have understood the existence of these ancient men in a non-historical sense, since when he comments on this point, he says that even though there is an obvious difference between a definite and an indefinite verb, this last name was disregarded (pareóratai: p. 52, 6) by those who have given the names, and some lines later he adds that this difference should have been established by the ‘fathers of the names’ (p. 52, 9). So it is evident that Ammonius’ comments on this point lead to a non-historical interpretation, which was probably influenced by Proclus’ interpretation of Plato’s Cratylus (see Cratylus 389a 2 et passim).
- 9.
Berti [14, p. 122] remarks that Aristotle in Metaphysics Γ, 7, [7], when he explains and defends the validity of the principle of tertium exclusum, maintains in his argument that being and not-being are a contradictory pair (1012a. 5–8). Similarly, some lines later (1012a. 15–18), Aristotle will affirm that not-being is the denial of being. In this passage, as is known, Aristotle says that “when a man, on being asked whether a thing is white, says ‘no’, he has denied nothing except that it is; and its not being is a negation” (translation by Ross [7]). Tricot [10]—whose translation Berti follows—says “et le Non-être est une negation”, which seems to be an assertion of equivalence, and too general. On the other hand, Berti [14] has also observed that Aristotle (Met. Γ, 2, 1004a 2; and I, 3, 1054a 30, [14]) includes ‘the same’ and ‘the other’ in his division of contraries, which implies that Aristotle thinks that they are opposite by contrariety and not by contradiction. Berti has also shown that Aristotle distinguishes clearly ‘the other’ from not-being, because ‘the other’ is one of the forms or species of multiple, and it is opposite to ‘the same’, which is a species of the one.
- 10.
Met. 1089a. 4–6. “They <the Platonics> thought it necessary to prove that what is not is; for only in this way—from being and from something else—would it be possible for there to be many existing things” (translation by Annas [5, p. 119]).
- 11.
For example, Ammonius in his commentary on Aristotle’s De Int argues so. Cf. in Int, p. 86, 26–p. 87, 7; p. 159, 24–29; p. 214, 6–24, [4].
- 12.
For the history of this diagram, see Correia [21, pp. 41–56].
- 13.
We follow later tradition of naming the universal affirmative proposition an A, the universal negation an E, the particular affirmation an I, and the particular negation an O.
- 14.
As I will explain in detail in the second part of this paper, the most ancient Peripatetic tradition of commentaries think that Aristotle would determine the truth values of opposition by supposing that every proposition has a signification given by the matter of the proposition. The matter of the proposition is an ancient doctrine also accepted by Neoplatonic commentators of Aristotelian logic. Cf. Correia [22, p. 397, n. 22].
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In other words, Ammonius Hermeias in his commentary on Aristotle’s De Int [4] illustrates this with the position of the negative particle in the proposition that will be denied: “Now, that the negation arises when the affirmation takes on the negative particle, is clear. But where in the affirmation one must place it, in order to make the negation, and why this is so, we must specify. I say, therefore, that one must not join it to the subject, but to the predicate; first, because the predicate is more important, as has been said (70, 4f.), and prior to the subject, which is also why the whole sentence is called ‘predicative’ (so, if we want to destroy the affirmation and make a negation, we must not attach the negative particle, which is the cause of the destruction, to the less important of the parts, but to the more important, since in animals too, more than in other living things, the whole does not perish if just any part is destroyed, but only if one of the more important part <is destroyed>” (Blank’s translation, [15]). This biological comparison can stem from Dionysius Thrax (Scholia, p. 516, 28–36), [31], where the name and the verb are compared to the brain and the heart in the human body, and the other parts of the phrase to such parts as a hand or a foot: a man can live without a hand, but not without brain and heart.
- 17.
In De syllogismo categorico 96, 6ff., Boethius defends this doctrine by saying “And we should not be disturbed because in some syllogisms the conclusion is opposed to the contrary proposition, while in others to the contradictory proposition. Since the error will be committed in both cases: either accepting both contrary propositions or accepting both contradictory propositions” [18].
- 18.
Boethius, De syllogismo categorico 101, 6–11: “And I took over these things from an introduction to the Categorical Syllogisms, by following Aristotle and borrowing certain things from Theophrastus and Porphyry, as much as the brevity of an introduction allowed me” (my translation from [18]).
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- 20.
This is the reason why the most likely author of this division is Theophrastus. Indeed, before Boethius, Alexander of Aphrodisias already assumed this division in his comment on Aristotle’s Prior Analytics [1] and Apuleius [6] also gives textual evidence of being dependent of this division in his Peri Hermeneias.
- 21.
Indeed, the division is in Boethius’ De syllogismo categorico 17, 10ff., [18], but it is neither in the commentaries on De Int [16] nor at any other ancient commentary on this treatise, but a similar division takes place in Introductio ad syllogismos categoricos 26, 7–8, [19]; in Apuleius’ Peri Hermeneias 183, 9–19, [6]. Also “the procedure is closely paralleled”, as Thomsen Thörnqvist [18, xxii] has recently corroborated it, in the commentaries on Analytics by Alexander (in An Pr 45, 10ff.), [1], Ammonius (in An Pr 35, 36ff.), [3], and Philoponus (in An Pr 40, 31ff.), [25]. This certainly proves that syllogistic treatises by Boethius belong to the tradition of commentaries on Aristotle’s Prior Analytics [9], rather than to the commentaries on Aristotle’s De Int. See also Lee [24, 65–74].
- 22.
Cf. Correia [22, p. 395, n. 14].
- 23.
Thomsen Thörnqvist ([18, p. 164] and [19, pp. 140–143]) and Correia [22, p. 394, n. 13] have been independently stated that this remark by De Rijk [23, p. 18] is not convincing. What really happens is that the account of pairs of propositions having one term in common anticipates the three figures of the categorical syllogism.
- 24.
- 25.
Boethius [16] follows Porphyry (in Int 2, 2–6, pp. 279–283) in using the doctrine of the matter of propositions for proving results.
- 26.
Boethius names Syrianus on this concern in: in Int 2, p. 322, 29–p. 323, 13. There is no clarity concerning whether Syrianus is distinguishing matter from mode. The vagueness of the expression used by Boethius in his report of Syrianus, namely, qualitates propositionum, makes difficult to decide whether Syrianus was thinking of what Ammonius called later the matters of the proposition, or of the modal propositions. Zimmermann [32, p. lxxxix, n. 3] thinks that Syrianus refers here to modalities and not to matters. Lack of textual material here is decisive.
- 27.
- 28.
Ammonius Hermeias [3], in Int, p. 88, 12–23, gives an eloquent confirmation of the importance and interest of this doctrine for the development of ancient logic when he says: “I am talking about the relation according to which the predicate term either always holds of the subject term, as when we say the sun moves or man is an animal, or never holds <of it>, as when we say ‘The sun stands still’ or ‘Man is winged’, or sometimes holds and sometimes does not hold, as when we say Socrates walks or reads. Those who care about the technical treatment of these things call these relations the matters (hulai) of the propositions, and they say that one of them is necessary (anagkaia), another impossible (adunatos), and the third contingent (endekhomenê). The reason for these names is obvious, but they decided to call these relations ‘matters’ in the first place because they are seen together with the things which underlie (hupokeimena) the propositions and are not obtained from our thinking or predicating, but from the very nature of the things” (Blank’s translation, [15]).
- 29.
In Peri Hermeneias VI, 4–6, p. 182, [6], Apuleius adds that the doctrine consists of verifying each proposition, with the help of the significations of terms (significationes), whether the truth and falsity keep identical once the proposition is converted. And he lists the significations for, he says, they are not innumerable (innumerae): the property, the genus, the difference, the essence and the accident. These are transformed into necessary, impossible and contingent matters already in Ammonius.
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in Int, p. 215, 9–16, [18]. It is worth noting that Philoponus’ commentary on Aristotle’s Prior Analytics, by following Ammonius, says that Aristotle established this distinction between matter (hulai) and mode (tropos) in De Int. Cf. Philoponus in An Pr, pp. 44–45, [25]: “It must be said that we consider the matters <of the propositions> in the very nature of the facts stated in the proposition (because it is necessary that what is predicated of the subject either always belongs to it, or never, or sometimes belongs and sometimes does not, and this is the reason why we call these ‘matters’: for without these the principle of understanding of propositions does not subsist either. On the other hand, we say that the modes are put in the very expression that we make and that they are put from outside to the terms that complete the proposition and maintain what is predicated.”
- 31.
Barnes [13, p. 79] says that “An argument—I conjecture—concludes methodically provided that there is some formal rule which validates it. And there will be such a formal rule just in case the argument is valid in virtue of a logical form. (For any logical form will determine a formal rule and any formal rule will conversely determine a logical form.) Hence methodically concluding arguments are, in effect, formally valid arguments; and unmethodically concluding arguments are materially valid arguments.”
- 32.
Sullivan [30, pp. 71–75].
- 33.
For example, in the case of impossible matter, Boethius uses the formula: At si id de subiecto praedicetur quod vel numquam subiecto valeat convenire, ut lapis homini. (“But if <we take> this <matter> that never is true when predicated of the subject, as ‘stone’ of ‘man’, then (…)”, cf. Introductio ad syllogismos categoricos, p. 53, 10–15.) Similar formulae are used to describe the notion off the other matters of proposition, [19].
- 34.
Diagrams are common in Boethius’ writings on Aristotelian logic. They are both descriptions and tools for the memory. For example, in in Int 2, p. 324, 15, [18], he calls a diagram a subsidium memoriae (a tool for the memory); also in Int 2, p. 152, 7–10, [19]. He uses descriptio in De syllogismo categorico, p. 21, 1, [18].
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Acknowledgements
This article was made possible thanks to Fondecyt N° 1095206. I am also grateful to Prof. Pierre Simonnet for his great help and Prof. Pieter Seuren for his comments on the oral version of this paper at the Second World Congress on the Square of Opposition, June 17–20, Corsica. I am also very grateful to the editor and the referees for their comments on my paper.
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Correia, M. (2012). Boethius on the Square of Opposition. In: Béziau, JY., Jacquette, D. (eds) Around and Beyond the Square of Opposition. Studies in Universal Logic. Springer, Basel. https://doi.org/10.1007/978-3-0348-0379-3_3
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