Download Aristotle's Squares of Opposition: Beyond the Traditional Square and more Exams Logic in PDF only on Docsity!
South American Journal of Logic Vol. 3, n. 1, 2017 ISSN: 2446-
Aristotle’s Squares of Opposition
Manuel Correia
Abstract The article argues that Aristotle’s Square of Opposition is introduced within a context in which there are other squares of opposition. My claim is that all of them are interesting and related to the traditional Square of Opposition. The paper focuses on explaining this textual situation and its philosophical meaning. Apart from the traditional Square of Opposition, there are three squares of oppo- sition that are interesting to follow: the square of opposition with privative terms (19b19-24), the one with indefinite-term oppositions (20a20-23), and the modal square (22a24-31), which are all contained in Aristotle’s De Interpretatione 10 and 13. The paper explains that all these squares follow a common plan, which is to demonstrate that every affirmation has its own negation, whatever is the proposition either categorical or conditional, or modal or non-modal, which is a reference to the universal importance of contradiction in logic.
Keywords: opposition, negation, squares of opposition, modality, semantic.
1 The traditional Square of Opposition
The Square of Opposition is a traditional title referring to a didactic diagram designed to distinguish logical relations of opposition between affirmations and negations. No doubt, Aristotle is its author, being not unlikely that he adopted the practice of drawing squares for his own logical purposes. Aristotle developed this Square in his treatise Peri Hermeneias or De Interpretatione, chapter 7 (17a38-18a7), and completed it almost in its total actual shape. Its importance to logic is to have distinguished kinds of opposition between categorical affirmations and negations, which are by definition the simplest propositions. He distinguishes three types of opposition, namely, contradictoriness, contrariness and sub-contrariness.^1 (^1) Soon after, the relation of subalternation should have been added, for already the first western
commentators, as Apuleius in the II AD and Boethius in the VI AD –who tells us to be borrowing from earlier authors–, comment on this last relation, which is not strictly speaking a relation of opposition, but it completes the geometric figure.
2 M. Correia
Aristotle’s intention of drawing a square can be relativized, since he just mentions the horizontal and oblique lines and ignores the vertical lines, since he does not take into account subaltern propositions (namely A-I and E-O relations).
Aristotle’s diagram is completed in its actual shape by the ancient commentators of Aristotle’s logic, in particular those who follow the treatise Peri Hermeneias (later Latinized by De Interpretatione). The oldest textual square in its actual shape is the one attributed to Apuleius, about five hundred years later, even if he does not mention the term “square” either.^2 But even if it is more accurate and literally more attested to referring to a diagram and not a square,^3 the doctrinal evidence is too strong to doubt that Aristotle ignored the Square that tradition brought to study. The aim of this paper is to show that the traditional Square of Opposition developed in chapter 7 falls within a general plan to draw squares of opposition in order to define different kinds of opposition. It will be shown that there are other three square-shaped diagrams in his De Interpretatione (De Int) confirming that Aristotle is fond to explain logical opposition in this way.
(^2) Apuleius or the author of the treatise called Peri Hermeneias (Moreschini (ed.) 1991, pp. 189-215) uses the terms scriptum when referring to the propositions forming a square in the text, a term that could be translated by ‘table’. For instance, when commenting on the traditional Square, the author of the treatise says that “it is easily understood 〈the logical relations of opposition〉 with the help of the propositions written below (facile ostenditur ex ipsis propositionibus infra scriptis). (^3) The square has been reproduced from Alvarez & Correia (2017), p.´ 91. Here the vertical lines (between A and I, and E and O), have been written by discontinuous lines with the purpose I have suggested here, namely, that Aristotle does not mention subaltern relations.
4 M. Correia
is a two-term proposition, the negative particle must be placed before the verb. And if the affirmation is a three-term proposition, the negative particle must be placed before the verb ‘to be’. And if it is a modal proposition, the negative particle must be placed before the modality. For instance:
Two-term propositions:
‘S eats’ is denied by ‘S does not eat’^6
Three-term propositions:
‘S is P’ is denied by ‘S is not P’
Two-term modal proposition:
‘S necessarily eats’ is denied by ‘S does not necessarily eat’
Three-term modal proposition:
‘S is necessarily P’ is denied by ‘S is not necessarily P’
To this ancient doctrine we should add the case of the quantified propositions, which are denied when the negative particle is placed before the quantifier. This is the reason why Aristotle draw the Square of Opposition in Chapter 7, and the reason why he claims (De Int 20a23-30) that sometimes we can deny a proposition with an affirmation. For instance, if someone asks whether Socrates is wise, then other answers that he is not. You can say, Aristotle says (20a30), ‘Socrates is not-wise’. But this is an exception, for in quantified propositions, if someone who has asked us whether is true that every man is wise, we are not to deny by saying ‘No, every man is not wise’ (because this is the contrary) but ‘no, not every man is wise’ (because this is the contradictory). Now, as said, in De Interpretatione there are three squares of opposition, besides the Square of Opposition of chapter 7, deserving to be highlighted, because of their significance and level of complexity. The first square is the one relating simple propositions with indefinite predicate (‘S is not-P’) propositions (which will be called the indefinite predicate square: 10, 20a20-23). The second square is that relating modal propositions (which will be called the modal square: 13, 22a 24-31). The third square is that relating privative, indefinite and simple propositions (which here will be called the privative square: 10, 19b19-24).
refers to this doctrine and states that: for two-term propositions the more important part is the predicate, which is the verb (cf. in Int p. 87, 8 ff.). For three-term propositions the more important part is the verb ‘to be’, i.e. ‘is’ (cf. in Int p. 160, 14-15). For modal propositions, the mode (cf. ibid., p. 218, 8-9). Also Boethius (VI AD) refers to this doctrine in his second commentary on the same Aristotelian treatise: in Int 2, 18-2, pp. 377–8; and 23-27, p. 378. (^6) Here, the verb ‘to eat’ represents any verb which is not the verb ‘to be’.
Aristotle’s Squares of Opposition 5
Some time ago, the first and the second square also called attention to the British logician A.N. Prior. Indeed, Z. Rybaˇr´ıkov´a (2016), pp. 3473, reminds us that Prior in one of his unpublished papers, titled “Aristotle on logical squares”, in plural, “in his attempt to define Greniewski’s 2 operator, also considers squares which are introduced in the 10th and 12th chapters of De Interpretatione, i.e., the square that comprises indefinite names and the modal square.” (p. 3473).^7 The reason why Prior remarks these two other Aristotelian squares in discussing with H. Greniewski on the Square of Opposition is that these squares are outstanding in the reading of De Int.
2 The indefinite-predicate square
As to the first square, its discussion has been done elsewhere in more detail,^8 and I will only point out its main characteristics. Aristotle deals with denying the universal affirmation having an indefinite predicate (‘Every S is not-P’). And he maintains the ancient thesis that the negative particle takes the main part of the proposition, which in quantified propositions is the quantifier. Accordingly, the corresponding contradictory is ‘Not every S is not-P’. So the square is the following:
The diagonals are also contradictory, and their truth values are always different from one another. This square presents some difficulties relating the vertical relations, for Aristo- tle says that ‘No man is Just’ follows from (akolouthousi de hautai: 20a16) ‘Every man is not-just’, but not that they follow each other (akolouthousi allellais: Ammonius, in Int p. 181, 27-28) as the ancient Neoplatonic commentators took it.^9 Behind this Neo- platonic interpretation is the Canon of Proclus and the theory of formal equipollences later called obversion (Bain 1870).^10 Current logic follows Neoplatonic interpretation, as it accepts that ∀x(M x → ¬Jx) ↔ ¬∃x(M x ∧ Jx) and (p → ¬q) ↔ ¬(p ∧ q).
(^7) Rybaˇr´ıkov´a (2016), pp. 3473, mentions the modal square in chapter 12 but actually it is in chapter 13. (^8) Correia (2002), pp. 71-84. (^9) Boethius in Int 2, 25-29, p. 330 translates Aristotle sequuntur vero hae but he comments on to it
that the propositions sequuntur sese sibique. (^10) Correia (1999), pp. 53-63.
Aristotle’s Squares of Opposition 7
The problem now is how to arrange privative, indefinite and simple propositions in a square. Boethius’ commentaries on Aristotle’s De Int contain a complete historical report of the ancient solutions to this obscure passage,^15 which is only testimony of the opinions by Herminus, Alexander of Aphrodisias and Porphyry. According to Herminus (almost unknown commentator but probably Alexander of Aphrodisias’ master), the logical relation between simple, privative and indefinite propositions, is the following:
Herminus’ interpretation of Aristotle’s De Int 10 is a square. But, according to Boethius it is a wrong and incomplete exegesis of Aristotle’s words: it is wrong, for indefinite subject propositions has nothing to do here,^16 and it is also incomplete, for Herminus does not explain what the meaning of secundum consequentiam is.^17 However, his square was influential in Alexander of Aphrodisias’ opinion, for Alexander arranges the same propositions and maintains the columns. Alexander’s idea is to take privative and indefinite propositions as logically equivalent, so that both indefinite and privative affirmations are opposed to the simple affirmation. In the other column, the negative propositions behave in the same way. Thus, he draws another square.
(^15) cf. in Int 3-5, p. 132. huius sententiae multiplex expositio ab Alexandro et Porphyrio, Aspasio quoque et Hermino proditur. (^16) Boethius in Int 2, 31-3, pp. 275–6: “These things, however, Herminus [says]. He, misunderstanding badly the complete sense of the phrase [19b. 22-24], introduced these propositions, namely, that with both [terms] indefinite and that with an indefinite subject.” (^17) According to Boethius, Herminus does not explain Aristotle’s expression secundum consequentiam
(which is the translation of Boethius for kata to stoikhoun) and therefore it is not clear which are the two propositions that must be disposed secundum consequentiam in accordance with the privative propositions (in Int 2, 3-8, p 276).
8 M. Correia
According to Boethius, Porphyry criticizes Alexander by saying that Alexander takes the propositions only in a syntactic way.^18 By contrast, Porphyry proposes to read Aristotle’s phrase secundum consequentiam (kata to stoikhoun) in a semantic way:
Porphyry changes the columns. His reading is partially correct in taking the simple affirmation entailing the indefinite negation and the indefinite affirmation entailing the simple negation, for it is what Aristotle says in An Pr I, 46. However, he is wrong in making the indefinite propositions logically equivalent to the privative ones, for Aristotle explains in Categories 11b38-12a25 that the privative proposition is equivalent to the indefinite proposition only when the predicate is necessary and not incidental to the subject. Indeed, this is an essential passage in many respects:
“If contraries are such that it is necessary for one or the other of them to belong to the things they naturally occur in or are predicated of, there is nothing intermediate between them. For example, sickness and health naturally occur in animals’ bodies and it is indeed necessary for one or the other to belong to an animal’s body, either sickness or health; (... ) But if it is not necessary for one or the other to belong, there is something intermediate between them. For example, black and white naturally occur in bodies, but it is not necessary for one or the other of them to belong to a (^18) Boethius in Int 16-20, p. 134. Quod autem ait ad consequentiam, tamquam si dixisset ad simili-
tudinem, ita debet intelligi.
10 M. Correia
It is possible to be It is not possible to be It is contingent to be It is not contingent to be It is not impossible to be It is impossible to be It is not necessary to be It is necessary to be
It is possible not to be It is not possible not to be It is contingent not to be It is not contingent not to be It is not impossible not to be It is impossible not to be It is not necessary not to be It is necessary not to be
Aristotle in the previous chapter teaches how to make a negation by following the rule that the negative particle must be placed before the more important part of the proposition. Since in modal propositions the more important part is modality, in any modal proposition either a two- or three-term proposition, the negative particle will be placed before modality. This way to deny shows what later is called the dictum or the modalized part of the proposition. In fact, in any modal proposition M(P),^20 both M and P can be affirmed or denied. For instance: M(P), M(¬P), ¬M(P), ¬M(¬P).^21 Accordingly, he draws this new square in De Int 13 to declare his statement about negation. One of the characteristics of this square is the re-definition of possibility. It is first defined by the negation of necessity, i.e., ¬N(P). But, since he also accepts that if something is necessary, then it is possible, i.e., N(P) → P(P), then the conclusion will be that N(P) → ¬N(P), which is a contradiction. So he realizes (22a15 ff) that P(P) is defined by the negation of the necessity of non-P, i.e., ¬N(¬P). As I take it, if non-x is necessary, then x will not be able to exist. Then he rearranges the square by inter- changing the fourth of the first group and its corresponding negation by the fourth of the third group and its corresponding negation. As a result, the new square is modified by this new definition of possibility:
(^20) The categorical proposition which is modalized or the dictum is always in parenthesis: (P). The
modality, M, comes always from outside. (^21) M stands for any modality (necessary, contingent, possible, impossible). N stands for the modality
of necessity, and P for possible, I for impossible. Later, Aristotle introduces A, assertoricity o non- modality to recall the two- and three-terms categorical non-modal proposition (i.e., S is P).
Aristotle’s Squares of Opposition 11
It is possible to be It is not possible to be It is contingent to be It is not contingent to be It is not impossible to be It is impossible to be It is not necessary not to be It is necessary not to be
It is possible not to be It is not possible not to be It is contingent not to be It is not contingent not to be It is not impossible not to be It is impossible not to be It is not necessary to be It is necessary to be
This square contains logical equivalences and oppositions. One can find equivalences in each corner group and oppositions in front, but its oppositions can also be simplified and arranged in correspondence with the traditional Square in order to find the con- tradictory oppositions (A-O) and (E-I), thus:
A E It is not possible not to be It is not possible to be (=It is necessary to be) (=It is necessary not to be)
I O
It is possible to be It is possible not to be (=It is not necessary not to be) (=It is not necessary to be)
It is evident that the modal square intends to show that there is only one modal nega- tion for one modal affirmation, and in general the use of squares of opposition gains importance in Aristotle’s logic not only because it confirms that every affirmation has only one negation, but also because it defines the types of opposition in modal cate- gorical propositions and their corresponding truth logical values. Lukasiewicz (1951), p. 137, identified this Aristotle’s modal square and its implicit formulae with “a basic modal system” and takes it as “the foundation of any system of modal logic”. These Aristotle’s intuitions “(... ) are the roots of our concepts of necessity and possibility”.^22 Aristotle’s implicit modal formulae are of two kinds: what he assumes and what he defines. He assumes that A(P) → P(P), but not vice versa, i.e., if something is true, then something is possible, or if something exists, then something is possible to exist.^23 He does not accept the converse, for something possible could not come to exist. He
(^22) However, Aristotle’s intuitions in modal logic “do not exhaust the whole stock of accepted modal
laws” (Lukasiewicz, 1951, p. 137). (^23) Where A stands for assertoricity or the inesse quality of categorical propositions, that is, the
quality of being not modalized, but true in the existence).
Aristotle’s Squares of Opposition 13
References
[1] J.L. Ackrill. Aristotle’s Categories and De Interpretatione. Translation with Notes. Oxford: Oxford University Press, 1963.
[2] Ammonius Hermeias. Ammonii In Aristotelis De Interpretatione Commentarius. In: A. Busse (Ed.), Commentaria in Aristotelem Graeca, vol. iv, 4.6, Berlin: Reimer, 1895.
[3] Apuleius. Apuleius Platonici Madaurensis opera quae supersunt. Vol. III. De philosophia libri, C. Moreschini (ed.), pp. 189–215. Stuttgart: Teubner, 1991.
[4] Aristotle. Aristotelis Categoriae et Liber de Interpretatione. L. Minio-Paluello (Ed.), Oxford: Oxford University Press, 1949.
[5] Aristotle. Tricot, J., Aristote. Organon. Translation with notes. Vols. i-iii, Paris: J. Vrin, 1977.
[6] A. Bain. Logic. London: Longman, Green, Reader and Dyer, 1870.
[7] Boethius. Anicii Manlii Severini Boetii Commentarii in Librum Aristotelis PERI ERMHNEIAS. Prima et secunda editio. C. Meiser (Ed.), Leipzig: Teubner, 1877-
[8] M. Correia. ¿Hay equivalencias en la l´ogica de Arist´oteles? Seminarios de Filosof´ıa, 12-13, pp. 53–63, 1999-2000.
[9] M. Correia. ¿Es lo mismo Ser No-Justo que Ser Injusto? Arist´oteles y sus Comen- taristas? M´ethexis, Vol. 19, pp. 41–56, 2006.
[10] M. Correia. El Canon de Proclo y la idea de l´ogica en Arist´oteles. M´ethexis, Vol. 15, pp. 71–84, 2002.
[11] H. Greniewski. The square of opposition - a new approach. Studia Logica 1, 1, pp. 297–301, 1953.
[12] M. Kneale and W. Kneale. The development of logic. Oxford: Oxford Univ. Press,
[13] J. Lukasiewicz. Aristotle’s syllogistic from the standpoint of modern formal logic. Oxford: Oxford Univ. Press, 1951.
[14] Z. Rybaˇr´ıkov´a. Prior on Aristotle’s Logical Squares. Synthese 193, 11, pp. 3473– 3482, 2016.
14 M. Correia
Manuel Correia Department of Philosophy Pontifical Catholic University of Chile Av. Libertador Bernardo O’Higgins 340, Santiago, Chile E-mail: mcorreia@uc.cl
