Division, Syllogistic, and Science in Prior Analytics I.31

1.1. Division

Plato, in several of his late dialogues (especially the Phaedrus, Sophist, Statesman, and Philebus), develops the method of division in part to solve the longstanding Socratic problem of answering the “What is F?” question. This is a procedure for creating classifications by dividing more general kinds into increasingly specific ones. The most important use of the method for our purposes (although possibly not its only use) is to seek the nature or essence of a target natural kind. In the Statesman, for example, to find out what statesmanship is (i.e., the nature of statesmanship), the interlocutors begin by agreeing that it is a kind of knowledge. Then they distinguish kinds of knowledge in stages, first dividing knowledge into practical and theoretical, then dividing theoretical knowledge in turn, until they arrive at statesmanship.

The method of division promises a way to systematically search for essence by situating the target kind in relation to other kinds in the same domain. By understanding how statesmanship is fundamentally similar to and different from other kinds of knowledge, one can hone in on the thing itself. By setting it down as a kind of theoretical knowledge, for example, Plato can distinguish statesmanship from all the manual arts in one go. The method of division provides a holistic way of searching for essence: if one can “carve nature at its joints” (Phdr 265e), one will find out about the essences of a number of related kinds at once. This procedure allows an inquirer to come to know the essences of kinds which are ontologically fundamental, since the comparison does not aim at reducing the essences to something else.³

1.2. Syllogistic

Aristotle describes his syllogistic method in Prior Analytics I.27–30, where he shows that there is a way to discover syllogisms with the desired conclusions by sorting any set of premises into six different lists. The end result is an algorithm that searches these lists and returns a syllogism with the desired conclusion. Depending on the logical form of the conclusion, only certain lists will be relevant. In what follows, when I use the term “syllogistic”, I am referring to this method of finding and giving syllogisms, although the term is used in the contemporary literature to refer more generally to Aristotle’s logical theory.⁴

To see how the method works, let’s work through an example. Suppose we want to scientifically demonstrate that no human flies. The first thing we need to do is sort true premises into various lists depending on their form. The relevant lists for getting this particular conclusion are:

1. Propositions of the form “No human is X” with different values of X: winged, immortal, etc.

2. Propositions of the form “Every human is X”: mammal, animal, etc.

3. Propositions of the form “No flyer is X”: fish, horse, etc.

4. Propositions of the form “Every flyer is X”: winged, perceiver, vertebrate, animal, etc.⁵

Premises such as “Some human is seated” or “Not every flyer is aquatic” are notoriously not considered at all, and because Aristotle claims that this system is sufficient for finding any deduction, he is committed to the claim that one only needs universal premises for deductions.⁶ Aristotle’s procedure looks for a pair of premises with the same term substituted for X either from lists 1 and 4 or from lists 2 and 3. If there is such a pair, you can construct a sound argument for the conclusion. In the example, we could use the propositions:

1. No human is winged. (list 1)

2. Every flyer is winged. (list 4)

3. No human is a flyer. (desired conclusion)

While this is a trivial case, the method is quite powerful. It is a highly tractable procedure that always finds a syllogism with the desired conclusion whenever there are premises which could produce a syllogism and never outputs an argument that is not a syllogism.⁷ Aristotle himself gets quite excited about his method:

The method is one and the same for all things, both concerning philosophy as well as any skill or learning whatever. (APr I.30 46a3–4)⁸

The method can be applied in any argumentative context. While the arguments that one gives on a particular occasion differ depending on the subject matter (physics, biology) and intent (scientific demonstration, dialectically effective argument), the principles behind these arguments (what makes them valid) and the ways of finding them are the same.

The basic idea is that any scientist, philosopher, or technician inquiring into any subject matter with the aim of getting truth ought to first discover true premises of these particular kinds and then produce syllogisms for the conclusions desired. Demonstrations are syllogisms the possession of which gives us knowledge, since they have true, explanatory premises. This means that the syllogistic method plays an important role in our pursuit of knowledge, even though it is not sufficient for generating scientific knowledge.

1.3. The Puzzle

These two methods seem incredibly different. First, they have very different goals. The goal of the method of division is an account of the essence of a target natural kind, which Aristotle and later philosophers would call a “definition”. The goal of the syllogistic is a deductive argument for a given conclusion. That conclusion need not state the essence of anything at all—in the example above, “not being a flyer” is not a part of human nature, even if it is necessarily true. Most strikingly, division has as its goal something that must be true, which is neither necessary for the conclusion of a syllogism nor for a syllogism itself (which is not even a candidate for truth or falsity).⁹ While we want our syllogisms to have true premises (and hence a true conclusion) in demonstrative contexts, in dialectical contexts, we only want to have premises in accordance with belief. So a perfectly good use of the syllogistic method might conclude a falsehood from falsehoods.

The methods also take very different routes. The method of division begins with a genus and progressively narrows it down until one reaches the target kind. The syllogistic method consists of two steps, neither of which is anything like this. The first step organizes premises into six lists and takes premises for a given deduction from two of these lists. The second constructs a deductive argument for the conclusion with those premises. In neither case does the user of the syllogistic method “narrow down”. Nor does the divider ever seek premises for a deductively valid argument or give such an argument.

Most striking, perhaps, is that it might be objected that division and syllogistic seem to yield very different sorts of claims.¹⁰ The method of division investigates definitions, which are a kind of identity claim. Syllogistic, by contrast, investigates quantificational claims. These claims differ significantly in their syntactic structure, but also in their modal status. Definitions require a very strong connection between the definiens and definiendum, much stronger than even the modal propositions of Aristotle’s syllogistic.¹¹ Even if Aristotle talks about definitions in terms of identity (e.g., in Topics I.7) and predication (e.g., Topics I.4–5), he does not talk about them in quantificational terms. That is to say, definitions do not fit into any of the four types of proposition distinguished at the outset of the Prior Analytics.

Aristotle’s comparison of the two methods in Prior Analytics I.31 seems so farfetched that, instead of clarifying their connection, it reinforces the impression that they are fundamentally different. He introduces his discussion by immediately finding fault with the fact that division does not do what the syllogistic does:

It is easy to see that division of genera is a small part of the aforementioned method. For division is a kind of weak syllogism. For, 1) it asks for what it ought to show and 2) it deduces something higher up. This first point eluded all those using division and they tried to persuade us that a demonstration concerning essence and the what it is can come about, with the result that they neither understood what in particular¹² it is possible that those dividing¹³ deduce, nor that it was possible in the way in which we said. (APr 46a31–39)¹⁴

In this passage, Aristotle both announces the substance of his criticism and describes how the users of division got in such trouble. The line of thought Aristotle attributes to the dividers is:

1. Divisions are demonstrations of essence.

2. So, divisions deduce essence.

While Plato does occasionally call divisions “demonstrations” (e.g., Plt 273e, 277a, b), he isn’t obviously using the term in Aristotle’s sense, as introducing a deductive argument of a special sort. The inference relies on something Aristotle believes—namely that all demonstrations are deductions (APr I.1, APo I.2).¹⁵ Aristotle here is just assuming on the basis of the word “demonstration” that the two methods try to deduce something, and then arguing that, where division fails, syllogistic succeeds. But this assumption seems totally unjustified. From the basic descriptions above, division is a method for finding essences, syllogistic a method for finding and giving valid arguments. However, Aristotle doesn’t provide us with any reason to think that one of the methods is “part” of the other, or that a division is a sort of “weak” syllogism. Understanding why Aristotle makes these claims is one major aim of this paper.

In addition to these polemical points, Aristotle has a number of precise complaints against division, first appearing in 1 and 2 above. Importantly, these precise complaints are meant to explain the sense in which division is a weak syllogism and a small part of the syllogistic method, as can be seen from how he links the first three sentences with “for”. Understanding these precise claims, then, offers the surest route to understanding the polemics. Aristotle ends up making four distinct claims in the body of the chapter:

1. Divisions are not deductions of the definitions they seek, because the definition does not follow necessarily from the divider’s assumptions. Instead, the strongest claim deducible from the divider’s assumptions is not of scientific interest. (46a31–46b25)

2. Division cannot demolish claims because it cannot deduce negative propositions (cf. APr I.26). (46b26)

3. Division cannot deduce features that are not definitional, such as accidents, properties, or genera. (46b27–28)

4. Division is useless in solving open problems, such as whether the diagonal of a square is commensurable or not with its side. (46b28–35)

Aristotle only argues for 1 and 4, treating 2 and 3 as common ground with the practitioners of division, who only claim that their method establishes definitions.

His argument for 1 begins with a general description of the problem that division encounters. He had shown earlier (APr I.26) that every syllogism with a conclusion of the form “All A is B” has a middle term that is between A and B in generality (46a39–b3). However, if we look at an arbitrary division, this will not be the case:

For: let A stand for animal, B for mortal, Γ for immortal, and Δ for human, whose account it is necessary to get. Then [the divider] assumes all animal is either mortal or immortal: All this (what would be A) is either B or Γ. Again the one who is always dividing sets down that a human is an animal, so that he assumes A belongs to Δ. On the one hand, there is a syllogism that all Δ will be either B or Γ, so that necessarily human is mortal or immortal, but it is not necessary that it is a mortal animal, but asked. But this was what was necessary to deduce. (APr 46b3–12)¹⁶

While we can conclude something trivial like “every human is either mortal or immortal” on the basis of a division, there is clearly no way to argue deductively for the desired conclusion that every human is mortal. The only thing that the divider can do to get closer to her goal is to ask her interlocutor to agree to the claim that every human is mortal (46a33–34). By asking for this, the divider “begs the question” in the technical Aristotelian sense by assuming the very claim under discussion. By showing that division begs the question for an arbitrary case, Aristotle is entitled to conclude in general that division cannot produce a syllogism of its target. This is the sense in which division is a weak syllogism: it cannot deduce its goal, but only the triviality Δ is B or Γ.

This account of why division is a weak syllogism also explains, as we expected it would, why the method of division is “a small part” of the syllogistic method. Division is a weak syllogism because it can only deduce a triviality. The syllogistic can deduce this triviality, but also much else. So it is straightforward to think that division is a small part of the syllogistic method. We now have an answer for what Aristotle means when he says that division is a small part of the syllogistic and a weak syllogism.

Aristotle also argues for 4 on the basis of an example. Suppose that we want to know whether the diagonal of a square is commensurable or incommensurable with its side. The divider assumes that the diagonal is a length and then divides length into commensurable or incommensurable. But without having proven the theorem, the divider is stuck: where should she put the “diagonal”? Division seems to offer no guidance on this.

In general, Aristotle’s criticisms in APr I.31 strongly suggest that he thinks division was intended to do what the syllogistic does (produce valid arguments), but give no hint as to why anyone should expect this. These methods seem to be, on their face, far too different to make such a connection. This constitutes the major puzzle of this paper.

1.4. Previous Resolutions of the Puzzle

Previous commentators have failed to motivate Aristotle’s critical comparison. They have taken three different approaches towards resolving this puzzle.

The most common response is Confusion: one party of the debate is simply confused about the relationship between the various methods. Some commentators have pointed the finger at Aristotle, others at the defenders of division.

Ebert and Nortmann claim that the fault lies with Aristotle himself:

Aristoteles’ Kritik dürfte aber auf einer falschen Voraussetzung beruhen, denn die Methode der Dihairesis ist ein heuristisches Verfahren zur Gewinnung einer Definition, nicht aber ein Verfahren, bei dem eine Definition deduziert werden soll. (2007: 794–95)

Diese Kritik des Aristoteles an der Dihairesis scheint unberechtigt. Denn das Verfahren wird bei Platon keineswegs deducktiv gebraucht, sondern immer so, daß ein Gegenstand, dessen Eigenschaften bekannt sind, durch eine fortschreitende Eingrenzung mit einer nächsthöheren Gattung und einem Unterscheidungsmerkmal ausgestattet wird, das ihn von anderen Arten unterscheidet, die ebenfalls unter diese nächsthöhere Gattung fallen. Gerade weil es bei dieser Methode nur um die Gewinnung von Definitionen (i. S. v. Definientia) geht, laufen die hier von Aristoteles erhobenen Vorwürfe ins Leere. (2007: 796)

Among those who think that the mistake resides on the part of defenders of division, there is a disagreement about who exactly is included in that group. Alexander of Aphrodisias thinks that it is everyone in Plato’s circle, including Plato.¹⁷ Striker, on the other hand, lays the blame not on Plato but other Platonists:

Aristotle’s harsh criticism of division in this chapter may be understandable if there were people in the Academy who thought that the method of division was all that they needed in philosophy, and who therefore paid no attention to Aristotle’s innovations. (2009: 209)

So Aristotle’s target was not Plato, but other proponents of the method of division. The earliest such view seems to be that of Philoponus:

He wants to celebrate through these things the method he handed down. For no one, he says, of those before us knew this, but they all used the method of division and through it they thought that they could demonstrate. And they say he is hinting at Plato. And Plato apparently does celebrate the method of division and most of all when it comes to be by contradiction, since it is the most inescapable.¹⁸ However, he at least did not say it was demonstrative. For clearly he knew the difference between the method of division and demonstration. For he says that there are four methods of philosophizing which are instruments of dialectic: division, definition, demonstration, and analysis. Perhaps then he is hinting at others who thought that the method of division was demonstrative. (Phlp in APr 306,31–307,9)¹⁹

Confusion can make sense of the existence of APr I.31, but is otherwise unmotivated and quite uncharitable. There is little evidence that any defender of division was confused. Plato lacked Aristotle’s technical language to talk about deduction, so it is unclear what would even constitute evidence one way or the other. While he does occasionally call a division a proof (“ἀπόδειξις”), he does not use the expression in the same sense as Aristotle (Shorey 1924; Striker 2009: 209).²⁰ The same is true for other members of the Academy: there is no fragment or report in which someone claims divisions are deductions of a definition. There is a text that does seem to bear on the question. Sextus Empiricus reports Xenocrates as supplementing a division with a proof (ἀπόδειξις) of that division:

Xenocrates, however, somewhat unusually compared with the others, and using the singular forms, said, “All that exists either is good or is bad or neither is good nor bad.” And while the rest of the philosophers accepted such a division without proof, he thought it proper also to include a proof, as follows. If there is anything which is distinct from good things and from bad things and from things which are neither good nor bad, that thing either is good or it is not good. And if it is good, it will be one of the three; but if it is not good, either it is bad or it neither is bad nor is good. But if it is bad, it will be one of the three, while if it neither is good nor is bad, it will again be one of the three. Thus everything that exists either is good or is bad or neither is good nor bad. (SE M I.4–5, trans. Bett 1997)²¹

This passage suggests that Xenocrates, at least, does not confuse the two in the way that Aristotle suggests. He is attempting to demonstrate a proposition of the form “Every G is S1 or … or Sn”, where G is a genus and S1… Sn are its species (or at the very least, subclasses). But this is not the problematic step for Aristotle, who rather thinks the problem is demonstrating that the target kind is set under one of the S1–Sn.

Finally, it seems unlikely that Aristotle is only targeting some members of the Academy because Aristotle is quite explicit that the error he is diagnosing in this chapter is common to everyone who used the method (APr 46a35), so it would be strange if he was not criticizing the first and most prominent user of division: Plato.

Moreover, there is not much reason to think that Aristotle is confused in the way that Ebert and Nortmann suggest. In the Posterior Analytics II.5, a chapter that directly refers to our passage, Aristotle explicitly says that there is “no absurdity” about division making clear a definition without demonstrating or deducing, in exactly the same way that induction is clarifying even though it is not deductive (APo 91b33–4). Later in the same book, he calls division “useful” in the hunt for essence (APo 96b25–26).²² Thus this reading attributes not only confusion in Aristotle, but also inconsistency with his other discussions of division. We should therefore accept Confusion only if all else fails.

Crubellier (2014), by contrast, argues that both division and Aristotle’s method at I.27–29 are aimed at finding scientific propositions, a view which I will call Discovery. Because they share this project, the comparison is justified. Aristotle’s point is that his method is better than division because it is far more general:

Aristote suggère que le projet de la diérèse est en un sens apparenté à celui du Pont aux Ânes – si du moins « la méthode que nous exposons ici » (46a32) vise spécifiquement le Pont aux Ânes et non pas l᾽analytique dans son ensemble. Ce que les deux projets ont en commun, c’est (1) la mise en ordre de séries de termes reliés entre eux par des relations d᾽implication notionnelle ; et (2) l᾽utilisation de ces séries pour produire des propositions scientifiques, ou tout du moins (dans un contexte dialectique) pour obtenir l’assentiment de l’interlocuteur. (Crubellier 2014: 298, cf. 24)

This interpretation has a number of benefits. First, it makes sense of the placement of the chapter directly after the chapters that describe Aristotle’s method for finding syllogisms. Moreover, it gives Aristotle a reasonable point to make that does not attribute any real confusion to any party and fits well with Aristotle’s positive use of division in Posterior Analytics II. Indeed, the notion of “ordering terms” and the use of the series “to produce scientific propositions” are real and important commonalities between division and the method of I.27–29. Finally, the claim that the syllogistic is more general, since it can be used for any kind of proposition, is clearly true. In this way, it is a real improvement on Confusion. Discovery suffers from the problem that it does not have much to do with Aristotle’s arguments against division in this chapter, even though it is certainly right that the “aforementioned method” referred to here is the one described in I.27–30. Aristotle here argues that divisions are not syllogisms of their target claims—he does not argue against their ability to discover any kind of proposition. We can see this by the chain of explanations that begins the passage:

It is easy to see that division of genera is a small part of the aforementioned method. For division is a kind of weak syllogism. (APr 46a31–33)

Aristotle here claims that it is a small part of the aforementioned method because it is a weak syllogism. But if Discovery is right, then this would be a non-sequitur. What reason do we have for thinking that division is an inferior way of discovering truths from the fact that it is an inferior kind of syllogism? Aristotle never claims that his own way of finding syllogistic premises, described in I.27–29, is itself syllogistic. We could only get a plausible line of thought here if we assume that the syllogistic was the only way to discover new propositions. But Aristotle never commits himself to this and claims induction is a non-syllogistic way of learning (APo 71a5–6). Indeed, Crubellier’s account of why division is a weak syllogism shows this:

Il est possible de décrire la diérèse comme un sullogismos parce que, tout comme l’inférence syllogistique s’achève par la récapitulation des prémisses et la production de la conclusion, la diérèse produit la définition recherchée par la récapitulation de ses étapes et que, comme dans la déduction, le répondant n’est pas libre de la refuser. Mais c’est une déduction « sans force » parce que son application se limite aux questions définitionnelles et parce que, même dans ce champ limité, elle ne produit pas réellement de connaissance nouvelle: si l’interrogateur peut conclure, par exemple, que l’homme est un animal mortel, c’est parce que le répondant lui aura déjà accordé que « l’homme est un animal » et que « l’homme est mortel ». (Crubellier 2014: 298)

In this passage, Crubellier tries to reconcile the obvious fact that Aristotle is claiming division is a syllogism with Discovery. But his account of why it is “weak” or “sans force” because it is limited to definitional questions and commits a petitio principii does not fit Aristotle’s own explanation:

For division is a kind of weak syllogism. For, 1) it asks for what it ought to show and 2) it deduces something higher up. (APr 46a32–34)

This explanation of the weakness of division only has to do with the fact that it does not deduce what it claims to deduce, not because it is limited to definitional questions, which is presented by Aristotle as quite distinct criticisms later in the chapter (46b26–28). Thus, while Discovery does point to important commonalities, the line of argument that Aristotle adduces in 46a31–34 does not make reference to division’s limited ability to find anything but only its lack of syllogistic power. The commonality, therefore, plays no explicit role in Aristotle’s argumentation. While this is not sufficient for rejecting Discovery outright, if another interpretation can be found that clearly elaborates Aristotle’s argument without attributing confusion to anyone, that would be preferable.

Finally, according to Robin Smith, the real issue has to do with obtaining Knowledge:

Aristotle is sometimes criticized for treating this procedure as a rival in some way to his theory of deductions: after all, it may be urged, the two have quite different objectives, and therefore it is no more a valid criticism of division that it fails to prove than it would be to object that deductions or demonstrations fail to define. But there is a deeper point. Aristotle’s real complaint is that division is not a method which leads to the acquisition of knowledge: at each step, the divider must ‘ask for the initial thing.’ Thus, the method cannot produce understanding. (1989: 160)

Knowledge is, very broadly speaking, on the right track. The two methods seem to play important roles in Plato’s and Aristotle’s respective conceptions of scientific knowledge, so it would be no surprise if the different views about division resulted from different conceptions of science. But this on its own is too vague to justify Aristotle in making the comparison, since after all, they could play quite different roles in the acquisition of knowledge. According to Knowledge, the gap is filled by the requirement that knowledge cannot be acquired from question-begging arguments. The reason why, it seems, is that division doesn’t give the inquirer anything beyond what she started with.

This diagnosis, however, is still in tension with Aristotle’s positive claim that division can produce knowledge:

But it [division] is still not a syllogism, even if it produces knowledge in another way. And this is not absurd, since someone performing an induction does not demonstrate, but nevertheless makes something clear. (APo 91b32–35)²³

The basic point here is that there are many non-deductive and non-demonstrative ways of making things clear. Induction clearly extends our knowledge in this way. And here in APo II.5, which directly refers to APr I.31 (91b12–15), Aristotle clearly still thinks that division begs the question, but says that it is not absurd that it produces knowledge.²⁴ If Aristotle thought division could not give us new facts, which were previously unknown, this would also be problematic for his own view of demonstration, since he claims that one will often first obtain the facts and then later seek their explanations (APr I.30, APo II.1). In such cases, the conclusion of the demonstration was something already known. What was not known was its cause. Thus, if the requirement is that division produces knowledge of new facts, that would seem to be just as problematic for what Aristotle has to say about demonstration as it is for division.

This point could be finessed by saying that the sort of knowledge produced by division is a different sort of knowledge (γνῶσις) than that provided sometimes by syllogistic (ἀποδεικτικὴ ἐπιστήμη). Perhaps Aristotle’s point is that what syllogisms, at least when they are demonstrative, get you is more intellectually rewarding than what division gets you. The scientific understanding that comes with possessing a demonstration is just a much more significant achievement than the knowledge one has from division. This would be adequate if there were demonstrative knowledge of the thing that division aims at: definitions. For then we could see how the syllogistic does something that division does not. But Aristotle denies that there are demonstrations of these kinds of definitions. Knowledge, in the end, seems to undermine Aristotle’s justification for comparing the two methods, since division and syllogistic are, after all, aimed at acquiring different things. Division will be aimed at getting γνῶσις of essences through definitions, while demonstrative syllogisms will aim at getting scientific understanding of other propositions.²⁵ So, while Knowledge was right to look at the respective conceptions of science, it still failed to justify the comparison.

In what follows, I will reconsider the Platonic texts concerning division and use them to provide a new interpretation of Aristotle’s comparison. This interpretation will make sense of both Aristotle’s critical remarks about division in APr I.31 (unlike Discovery) and be consistent with his positive remarks about division as a method (unlike Knowledge and Confusion). First, I will make clear how both methods are intended to be rigorous and general ways of obtaining scientific knowledge concerning the part-whole relations among kinds. Contextualizing this chapter in this way justifies Aristotle’s criticism by showing how they have relevantly similar features that could form the basis of a meaningful comparison. Then I will argue that Aristotle’s point in comparing the methods is to show the importance of valid arguments in this scientific enterprise. Instead of replacing division with syllogistic, Aristotle is showing that syllogistic is a necessary part of a scientist’s repertoire that is not already covered by division.

In short, Aristotle thinks of division as a means of obtaining definitions, which are the starting points of demonstrations, or scientific syllogisms. In order for division to play this role, it is important to see how it is not a demonstration, since the starting points of demonstration are indemonstrable. What will emerge is a considerable continuity in the Platonic and Aristotelian projects. Division is not playing a sort of second fiddle to demonstration. Instead, the methods are compared because doing so will shed light on how they both contribute to a common Platonic and Aristotelian conception of what scientific understanding is and how to achieve it. This is a common way for Aristotle to deal with his predecessors: he puts their views into his framework in order to show where they go wrong or miss something. While such a procedure might sometimes lead to distortion of their views, in this case at least, I will argue that all of Aristotle’s critical points can be appreciated even if division is not thought to be deductive.

In §2, I show that both methods have the properties of epistemic generality and rigor. I argue in §3 that the methods both investigate predications conceptualized in terms of part and whole, thus suggesting that the contrast drawn in §1.3 between identity claims and quantificational claims is not as stark as it initially seemed.²⁶

2.1. Epistemic Generality

Both Platonic division and Aristotelian syllogistic were intended to have a certain kind of generality that has historically been associated with logic. The sort of generality that I have in mind is that the theory is meant to apply to any sort of science one might engage in, whatever the subject matter.²⁷ Moreover, division and syllogistic are supposed to be general ways of coming to know about any subject matter whatsoever through reasoning, or “giving accounts”. This property might seem to be too abstract and vague to count as an important similarity. But it is not. There are many scientific methodologies which have no such aspirations. Methods of medical inference from symptom to underlying cause, as was common in the Greek “rationalist” tradition, have no place in geometry. Nor does geometrical analysis have any obvious place in medicine. That is to say, neither of these methodologies aspire to be general, since there are scientific domains in which they are not applicable.²⁸ If Platonic division and Aristotelian syllogistic are (intended to be) general in this sense, that would be a striking point of similarity.

Plato claimed the method of division had this very special kind of generality. For instance, in the Philebus, when Socrates introduces the method of division (albeit with some unusual language, calling kinds “ones” that are discovered along the way from the genus and calling a division complete when the inquirer “knows how many the original one is”), he makes a strong statement about how widely his method is meant to apply:

Since these things are organized in this way, with regard to everything we must always look for a single form after positing it in each case—for we will find it because it is there. And once we have grasped it, we must look for two, as the case would have it, or if not, for three or some other number. And we must treat every one of those further unities in the same way, until it is not only established of the original unit that it is one, many and unlimited, but also how many kinds it is. […] The gods, as I have said, have left us this legacy of how to inquire and learn and teach one another. (Phlb 16c–e, emphases mine)²⁹

So far, we just have heard that this method is a divine way to inquire, learn, and teach. But soon after this, Socrates strengthens the claim:

So at the same time they [the first music theorists] made us realize that one should investigate about every one and many in this way. For whenever you have mastered these things in this way, then you have acquired expertise there, and whenever you have grasped the unity of any of the other things there are, you have become wise about that. (Phlb 17d–e)³⁰

This passage suggests that the method of division is a way to inquire (or learn or teach) in any domain and that success in the method means that you have acquired wisdom. Those who ignore it, on the other hand, are said to be eristics simply grouping things “in a chance way” (Phlb 16e). The same is said in the Phaedrus, where division is the ability in any instruction “to cut up any kind according to its species along its natural joints” (Phdr 265e). In the Sophist, dialectic is defined by the Eleatic stranger as the quite general skill of being able to “divide things by kinds and not to think that the same form is a different one or that a different form is the same” (Sph 253d1–2) and, in the Statesman, he claims to be teaching Theaetetus and Young Socrates about division in order to make them “better dialecticians in relation to all subjects” (Plt 285d5–6). We see just how serious Plato is about generality in his uncontroversial applications of the method to domains as diverse as music theory, political philosophy, and phonology.³¹ The enormous range of application further motivates the thought that Plato took division to be perfectly general, in the sense that it applies to every subject matter.

Aristotle considers his syllogistic to have a similar scope. Recall that in the introduction of the method in APr I.30, he claims:

The method is one and the same for all things, both concerning philosophy as well as any skill or learning whatever. (46a3–4)

Aristotle claims that his syllogistic method works in the same way for all things. An argument will be a syllogism regardless of the epistemic status or subject matter of the premises and conclusion, and our method of looking for syllogisms and deducing validly will also be the same. Here we see a close analogy with what Plato claims for division. The similarity is clear when we look at how closely the passage above is paralleled by the end of I.31: “this way of investigation [division] is neither suitable for every inquiry, nor even useful in those very cases in which it appears to be most appropriate”. It seems that Aristotle denies for division almost exactly what he affirmed about syllogistic.

While Aristotle does not think that every syllogism produces knowledge, a special kind of syllogism, demonstration, will. A syllogism requires special kinds of premises in order to be demonstrative: true, primary, immediate, better known than the conclusion, prior to the conclusion, and explanatory of the conclusion (APo I.2). Coming to know, then, requires the scientist to do two things: to have at hand premises of this sort and to construct syllogisms of the propositions to be demonstrated from those premises. The syllogistic is one of the crucial elements in Aristotle’s theory of science as described in the Posterior Analytics. The theory of science there is completely general and meant to apply to any scientific or technical domain whatsoever. Thus the syllogistic too must be general.

We have found that the project of division and syllogistic are, despite appearances, very close in this respect. Both are meant to play a central role in every scientific enterprise.

2.2. Rigor

Plato and Aristotle wanted these procedures to be not only general but also rigorous. While generality regards what subject matters the procedure can be applied to, rigor is a matter of how that procedure is applied. There is no agreed upon analysis of rigor available.³² However, I will assume here that a procedure is rigorous just in case it minimizes error in pursuit of its goal: applications of the method should result in errors as little as possible. For some types of problem there might be infallible methods, but in most situations there will inevitably be residual error. A method, for the purposes of this discussion, is a procedure that outputs an answer to a certain kind of question and (possibly) some fixed set of parameters. An application of a particular method requires specifying a particular question and the parameters (for instance, with statistical methods one must supply both the hypothesis under investigation as well as parameters such as data points and the p-value). An application of a method results in an error when the answer returned by the method is false. So a method is rigorous if it minimizes the proportion of false answers to the input questions with parameters. Thus, while a non-rigorous method might give a correct answer sometimes, it does not reliably give correct answers.

Before going on to Plato and Aristotle, let me give a few examples to illustrate rigorous and non-rigorous methods. Something like “Believe the truth” isn’t a method, since it does not actually output any answer to a question. A very non-rigorous method is something like “Believe only what is in your horoscope”. The sieve of Eratosthenes, an algorithm for finding prime numbers, is rigorous in this sense: following it guarantees that you get the right answer. While algorithms (in the technical sense used in computer science) are an important class of rigorous procedures, rigor is not only manifested in algorithms. The user of a rigorous but non-algorithmic method may need some ingenuity, but that ingenuity is either sufficiently constrained by the method or there are parts of the method that prevent errors, perhaps because there is an algorithm for checking the answer that one has.³³ The ancient method of geometrical analysis, for instance, does not work without input from the inquirer, but is rigorous in that the technique does not lead to errors. Stephen Menn puts this quite clearly:

The method of analysis had enormous prestige, in antiquity and down to the days of Descartes and Fermat, because it was seen as the basic method of mathematical discovery: not simply a way for a student to discover and assimilate for himself propositions already known to his teachers, but also a way for a mature geometer to discover previously unknown propositions. While analysis is a method with clear rules for step-by-step work (though it is not a mechanical method—the geometer must apply the rules intelligently in order to succeed), it terminates when something unpredictably “clicks”; then, if and when this happens, the geometer must again proceed methodically (again, not mechanically) by the method of synthesis, to confirm what has been discovered by analysis; if this succeeds, then the newly discovered proposition may be presented with a demonstration in the usual highly stylized form given in the classic Greek mathematical texts. (Menn 2002: 194–95)

What Menn is suggesting by calling analysis “step-by-step” but “not … mechanical” is exactly what I am intending by a rigorous but non-algorithmic method. The problem here is that sometimes the “steps” are not sufficiently precise that they can be run by a computer. But that does not mean that the steps don’t work to reduce human error when applied with intelligence. Nevertheless, algorithms have a distinct advantage over rigorous, non-algorithmic, methods like geometrical analysis: they always terminate. Deterministic algorithms (and a special class of non-deterministic algorithms) have the further advantage that they always terminate with the correct answer. A non-rigorous method, by contrast, can attain its goal, but not in virtue of the method alone—it also requires luck or the inquirer’s knack.

According to Plato, divisions should be done rigorously. In the Philebus, he gestures at the need for rigor by pointing out defective divisions:

The gods, as I have said, have left us this legacy of how to inquire and learn and teach one another. But these days the wise guys make a one in a chance way and a many faster and slower than they ought. After the one they go directly to the unlimited and the intermediates escape them, which are what determines whether our speaking with each other is dialectical or eristic. (Phlb 16e–17a)³⁴

According to Socrates, dividing incorrectly results in an eristical account, while his preferred method leads to a dialectical account. The difference between the dialectical and eristical accounts is not that the former must be true and the latter false. Rather, Socrates objects to the way that the eristic gets her account: by collecting in a chance way or dividing too quickly or slowly. I will argue below that the problem here is precisely one of rigor: it is not about getting the correct answer on a given occasion, but using a method that minimizes the possibility of error generally.

In the Statesman, we get some explicit reasoning why going too fast is problematic when Young Socrates divides herd-rearing too quickly into rearing of human herds and beast herds. This is exactly the same kind of mistake mentioned in the Philebus passage. The Eleatic Visitor explains the problem:

Visitor: Ah, yes. You’ve made a very zealous and courageous division! However, we should try not to have this happen ever again! … For it is most fine to separate off the object of inquiry from the others straightaway, should you do it correctly, just as a moment ago, when you thought that you had the division, you rushed the account, seeing that it was headed towards humans. But my friend, it is not safe to do such fine work. Instead, cutting through the intermediate stages is safer and one would more encounter ideas. This makes all the difference in investigations. (Plt 262a–b)³⁵

We could formalize such a norm that the Visitor mentions in the following way. Assume that we start from kind K and are aiming to define a target kind T. Then, he would be saying, of these two divisions:

one should prefer the former to the latter if n>m. That is, one should prefer a division with more “intermediates”. As defined here, this norm is non-trivial and gives significant advice to the one dividing. While it certainly does not amount to anything like an algorithm, it does provide concrete actionable steps for an inquirer to take. If an inquirer makes several attempts at a division, for example, she could look at them all and reject those that have fewer or no intermediates. Indeed, this norm my seem to be too strong to be plausible. As will be seen below, however, the Visitor does have an argument for accepting it.

To illustrate the mistake, the Visitor gives two other divisions that reach the same end point but do so more quickly (266c9–11) and more slowly (265b8–e2). In introducing them, he claims that the faster route will resemble the problematic division of Young Socrates:

Now it seems that there are two routes to be seen stretching out in the direction of the part towards which our argument has hurried, one of them quicker, dividing a small part off against a large one, while the other more closely observes the principle we were talking about earlier, that one should cut in the middle as much as possible, but is longer. (Plt 265a1–5)³⁶

Since the slower, longer route satisfies the rule of not chopping off little bits while the other does not, it seems that the faster one is problematic. This suggests we should prefer that longer route because it is “safer”.³⁷ The safety of the Visitor’s method is meant to guard against a particular sort of error that the fast method could not: missing ideas, the explanatory natural kinds that should be in one’s definitions. This constitutes evidence that Plato is arguing that slow division is more rigorous than fast division in the sense that it minimizes error. Indeed, it seems that the notion of “safety” in these passages should be understood precisely as rigor in the sense defined above.

Plato not only wants a rigorous method of division, he also proposes two norms to promote it: 1) divide slowly and 2) divide into the smallest number of subkinds possible (Plt 287b–c, Phlb 16d). We saw his statement of the first norm above. The second is made explicit later on in the dialogue:

Visitor: So do you recognize that it is difficult to cut them into two? The cause, I think, will become more evident if we proceed. … Then let’s divide them limb by limb, like a sacrificial animal, since we can’t do it into two. For we must always cut into the nearest number so far as we can. (Plt 287b10–c5)³⁸

The last sentence of this passage makes clear another norm, which is that one should minimize the number of proximate subkinds that one divides into. Although he does not think that all good divisions need to be dichotomous, he still holds on to a weaker norm that, if one is presented with two divisions of some kind K:

the former should be preferred to the later if n<m. This too is paralleled in the Philebus, where Socrates says “after the one, one ought to seek two, if that’s now many there are, or else three or some other number” (Phlb 16d3–4).³⁹ Here we seem to get the advice to first look for two subkinds, then three, and so on. Such a procedure would be equivalent to looking always for the smallest number of dividentia.

Moreover, in the first case, he explicitly defends the norm by arguing that it minimizes the possibility of error (Plt 261e–264b). He shows this by arguing that, if one does not follow the norm, there is a possibility of error that would have been excluded by following the norm. This is made clear in the examples he uses following Young Socrates’ division. Young Socrates was not rigorous in his division of animals into humans and beasts because it was possible, using the same kind of reasoning, to divide humans into Greeks and Barbarians or numbers into 10,000 and not-10,000 (Plt 262c–e). Such divisions, the Visitor claims, obviously would have been erroneous, since they would have missed fundamental kinds along the way—in the case of numbers, odd and even and in the case of humans, male and female.⁴⁰ While he does not claim that Young Socrates himself missed a fundamental kind in his division, he objected to the way he was dividing because it could lead to errors. Here the Visitor is showing concern not just for the truth, but at arriving at the truth “safely”. He claims that dividing slowly, by which he means dividing with many intermediate steps, reduces the possibility of this kind of error. That is because there could be a kind more general than the kind that you divided into that was passed over in the fast division and this will not be part of the account, as you will already be dealing with smaller classes. Thus you will not meet your goal of having a complete account of the essence. Applying these norms helps make division rigorous by preventing some important kinds of errors, even if they are not sufficient for eliminating every possible kind of error.⁴¹

One might worry that dividing too quickly does not just risk an error, but is itself an error. Here we should distinguish between the idea that there is an error in the output from the idea that there is an error or mistake in the method. I am claiming that Young Socrates does make a mistake in the method that does not, in this one case, lead to a mistake in the output. So much seems to be demanded by the very language of safety, which does not by itself imply that there was an error in the resulting definition, but only greater possibility of such an error. But how is it that Young Socrates didn’t make a mistake in the output if he skipped over a kind? He did not make an error in the output because the long conjunction of descriptions “two-footed, non-interbreeding, hornless, terrestrial, tame animal” is just a further spelling out of “human”. Thus, when Young Socrates said that statecraft was a kind of herding of humans, he was not saying anything inconsistent with the Visitor’s definition. Nothing essential was missed in the account because Young Socrates got lucky when using the term “human”, which builds in the long description given by the Visitor. In the case of the number 10,000, by contrast, one would have missed the essential property of evenness.

Rigor is manifested in Aristotle’s syllogistic at two points in the method. Recall that the goal of his method is to produce a syllogism of a desired conclusion. As was mentioned above, the first stage of the method (discovering the syllogism) is algorithmic. If a set of premises contains a syllogism, Aristotle’s procedure is guaranteed to find it and following the procedure will never lead to one giving an argument that is not a syllogism. This method is completely rigorous. Not only is the method for finding syllogisms rigorous, so too is the syllogism itself.⁴² Valid argumentation is plausibly more reliable than invalid argumentation, simply because, whereas the only way for a valid argument not to result in a true conclusion is if one of the premises is false, invalid argumentation may not result in a true conclusion both when one of the premises is false and when they are true. These two points together make the entire procedure described in APr I.31 rigorous, since that method consists in both the search for syllogisms and giving those syllogisms.

In both cases, we are demanding of the methods that they minimize the possibility of error. However, while there is residual error in Plato’s method, Aristotle’s has two advantages: his method always terminates and does so with the correct answer, so long as the inputs were themselves correct.⁴³ Because they are different methods it will turn out that rigor is manifested differently in each case, but they have a similar spirit. Moreover, very few methods are both rigorous and general. Most rigorous scientific methods with which we are familiar have a very limited domain of applicability, and likewise very general methods have, on the whole, a large room for error. For instance, the so-called “Socratic elenchus” can be used on any question whatsoever, but has a wide margin of error, if one seeks positive results with it.