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The Discovery of Principles in Prior Analytics 1.30

In: Phronesis
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Marko Malink Department of Philosophy, New York University USA

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Abstract

In Prior Analytics 1.27–30, Aristotle develops a method for finding deductions. He claims that, given a complete collection of facts in a science, this method allows us to identify all demonstrations and indemonstrable principles in that science (1.30, 46a21–7). This claim has been questioned by commentators. I argue that the claim is justified by the theory of natural predication presented in Posterior Analytics 1.19–22. According to this theory, natural predication is a non-extensional relation between universals that provides the metaphysical basis for demonstrative science.

Abstract

In Prior Analytics 1.27–30, Aristotle develops a method for finding deductions. He claims that, given a complete collection of facts in a science, this method allows us to identify all demonstrations and indemonstrable principles in that science (1.30, 46a21–7). This claim has been questioned by commentators. I argue that the claim is justified by the theory of natural predication presented in Posterior Analytics 1.19–22. According to this theory, natural predication is a non-extensional relation between universals that provides the metaphysical basis for demonstrative science.

1 A Method for Discovering Scientific Principles?

In Prior Analytics 1.27–30, Aristotle develops a method for finding deductions to establish a given thesis from suitable premises. This method, traditionally known as inventio medii or pons asinorum, has been widely studied for centuries. The author of the Byzantine Logica et quadrivium refers to it as ‘the pinnacle of philosophy’ (κολοφὼν τῆς φιλοσοφίας).1 According to the anonymous author, the method rests on ‘a genuinely profound and most scientific theorem of great import, which encompasses in a small space nearly all of philosophy’.2

Aristotle introduces the method in Prior Analytics 1.27 as follows:

It is now time to describe how we ourselves will be well supplied with deductions for any given thesis and by what method we will find the principles concerning each thing. For no doubt one ought not only to investigate how deductions come about, but also to have the ability to produce them.

πῶς δ’ εὐπορήσομεν αὐτοὶ πρὸς τὸ τιθέμενον ἀεὶ συλλογισμῶν, καὶ διὰ ποίας ὁδοῦ ληψόμεθα τὰς περὶ ἕκαστον ἀρχάς, νῦν ἤδη λεκτέον· οὐ γὰρ μόνον ἴσως δεῖ τὴν γένεσιν θεωρεῖν τῶν συλλογισμῶν, ἀλλὰ καὶ τὴν δύναμιν ἔχειν τοῦ ποιεῖν.

APr. 1.27, 43a20–4

The proposed method relies on the syllogistic theory presented in the earlier parts of the Prior Analytics. It starts from a selection of terms in the domain under consideration. Aristotle instructs us that, for any term A, we should select three classes of terms (44a11–17):

  • (i) terms that hold of A

  • (ii) terms of which A holds

  • (iii) terms that cannot hold of A

These classes of terms give rise to a collection of premises regarding A (43b1). In order to establish a given thesis, we need to examine such collections for both its subject term and the predicate term (1.28–9). For example, if we wish to establish that A holds of all B, we must examine class (ii) for A and class (i) for B. If we find a term that appears in both classes, we can use it as a middle term to establish the thesis through a deduction in Barbara (44a17–19).

Aristotle maintains that this method is applicable to any subject matter in science and dialectic. In the opening paragraph of Prior Analytics 1.30, he writes:

The method, then, is the same in all cases, in philosophy as well as in any art and study. For one must discern, for each of the two things [i.e., the subject term and the predicate term of the thesis to be established], the things that hold of it and the things it holds of, and be supplied with as many of them as possible…. And in the pursuit of truth the reasoning proceeds from things that have been listed as holding in accordance with truth, whereas for dialectical deductions it proceeds from premises according to opinion.

ἡ μὲν οὖν ὁδὸς3 κατὰ πάντων ἡ αὐτὴ καὶ περὶ φιλοσοφίαν καὶ περὶ τέχνην ὁποιανοῦν καὶ μάθημα·4 δεῖ γὰρ τὰ ὑπάρχοντα καὶ οἷς ὑπάρχει περὶ ἑκάτερον ἀθρεῖν, καὶ τούτων ὡς πλείστων εὐπορεῖν, … κατὰ μὲν ἀλήθειαν ἐκ τῶν κατ’ ἀλήθειαν διαγεγραμμένων5 ὑπάρχειν, εἰς δὲ τοὺς διαλεκτικοὺς συλλογισμοὺς ἐκ τῶν κατὰ δόξαν προτάσεων.

APr. 1.30, 46a3–10

In the last sentence of this passage, Aristotle contrasts reasoning in pursuit of truth with dialectical reasoning.6 The latter takes the form of dialectical deductions from premises that are based on opinion (δόξα). The former kind of reasoning includes deductions employed in any art and science (περὶ … ὁποιανοῦν … τέχνην τε καὶ ἐπιστήμην, 46a21–2).7 These are based on truth in that they derive true conclusions from premises that have been listed in accordance with truth. While dialectical deductions are discussed in the Topics, scientific ones are studied in the Posterior Analytics. The method developed in 1.27–30 is intended to work for both kinds of deduction.8

In the second half of 1.30, Aristotle focuses on the case of scientific reasoning. Having mentioned the example of astronomy, he states that his method is general in that it helps us to identify scientific deductions in any science whatsoever:

And the same applies to any other art or science [i.e., other than astronomy, 46a19–21]. Thus, when it has been grasped what holds of each thing, at this point it is already in our power readily to exhibit the demonstrations. For if nothing that truly holds of things has been left out in the collection of facts, we will be able, for everything of which there is a demonstration, to find this demonstration and demonstrate it, and for everything of which by nature there is no demonstration, to make that evident.

ὁμοίως δὲ καὶ περὶ ἄλλην ὁποιανοῦν ἔχει τέχνην τε καὶ ἐπιστήμην· ὥστ’ ἐὰν ληφθῇ τὰ ὑπάρχοντα περὶ ἕκαστον, ἡμέτερον ἤδη τὰς ἀποδείξεις ἑτοίμως ἐμφανίζειν. εἰ γὰρ μηδὲν κατὰ τὴν ἱστορίαν9 παραλειφθείη τῶν ἀληθῶς ὑπαρχόντων τοῖς πράγμασιν, ἕξομεν περὶ ἅπαντος οὗ μὲν ἔστιν ἀπόδειξις, ταύτην εὑρεῖν καὶ ἀποδεικνύναι, οὗ δὲ μὴ πέφυκεν ἀπόδειξις, τοῦτο ποιεῖν φανερόν.10

APr. 1.30, 46a21–7

In this passage, Aristotle claims that the method developed in 1.27–30 is useful for finding demonstrations (ἀποδείξεις). A demonstration, for Aristotle, is a deduction through which, when we possess it, we have scientific knowledge.11 As such, a demonstration has a number of features that distinguish it from deductions in general. For example, the premises of a demonstration are true, prior in nature to the conclusion, and explanatory of the conclusion.12 In the opening sentence of the Analytics, Aristotle states that the treatise is concerned with demonstration and demonstrative science.13 Since demonstration is a kind of deduction, he argues, the latter must be studied before the former.14 Thus, most of the Prior Analytics deals with deduction in general, leaving the study of demonstration to the Posterior Analytics. In the present passage from Prior Analytics 1.30, however, Aristotle is concerned specifically with demonstration.

Aristotle claims that we will be able to exhibit demonstrations ‘when it has been grasped what holds of each thing’, that is, when we have collected terms and premises as described in 1.27–8. He has encouraged us to make these collections as exhaustive as possible (43b9–10, 46a6). In the present passage, he makes the stronger assumption that the collection is complete in that ‘nothing has been left out’ (46a24–5). Thus, he assumes that the collection captures all facts of the science under consideration and that no fact has been omitted. Aristotle does not explain how such completeness might be achieved or ascertained, but seems content here to assume that, at least in principle, this can be done.

Among the facts that fall under the purview of a science, some are demonstrable and others are not. The latter are indemonstrable principles of the science, the former are theorems derived from these principles by means of demonstrations. Given a complete collection of facts in a science, Aristotle takes his method to ensure that, for any fact f in this collection:

  • (i) if d is a demonstration of f, we will be able to find d, and

  • (ii) if there is no demonstration of f, we will be able to make it clear that there is no demonstration of f.

Thus, Aristotle takes his method to be heuristic in that, given a complete collection of facts in a science, the method enables us to identify all demonstrations and indemonstrable principles of the science. As Robin Smith notes:

The object of this procedure [described in APr. 1.27–30] is to find the principles (archai) of a science, the indemonstrable first premises on which that science’s proofs rest. Aristotle claims [at 1.30 46a22–7] that this procedure, applied to the totality of truths concerning any subject matter, will lead to the discovery of which of those truths are the indemonstrable principles.15

Smith 2016, 54

While Smith is no doubt correct about the goal of Aristotle’s heuristic method, it is not clear how the method can in fact achieve this goal. For, Aristotle requires that the premises of a demonstration be not only true, but explanatory of and prior in nature to the conclusion, and he does not indicate how the method might enable us to distinguish deductions that satisfy this requirement from those that do not. Thus, Gisela Striker writes:

[T]hough a collection of facts could show which terms might be suitable as middle terms to derive some conclusions, this would not be enough to tell us which propositions come earlier and which later in the order of explanation (cf. An. Post. A 13).16

Striker 2009, 207

Striker is referring to Posterior Analytics 1.13, where Aristotle distinguishes two kinds of deduction: those that specify the reason why the conclusion holds and those that do not (78a26–b11). An example of the former kind of deduction is:

The planets are near.
Whatever is near does not twinkle.
Therefore, the planets do not twinkle.

In Aristotle’s view, this deduction explains why the planets do not twinkle. It is a ‘deduction of the reason why, since the primary cause (αἴτιον) has been taken’ (78b3–4). By contrast, this is not the case for a deduction such as:

The planets do not twinkle.
Whatever does not twinkle is near.
Therefore, the planets are near.

Although both of its premises are true (78a31–5), this deduction does not explain why the planets are near. As Aristotle puts it, ‘this deduction is not of the reason why (τοῦ διότι), but of the fact (τοῦ ὅτι); for it is not because the planets do not twinkle that they are near, but because they are near they do not twinkle’ (78a36–8). Since the premises are not explanatory of the conclusion, such a deduction is not a demonstration. By contrast, deductions ‘of the reason why’ may constitute demonstrations if they meet the relevant conditions.

Striker’s worry is that Aristotle’s method does not allow us to distinguish demonstrations from mere deductions. Suppose, for example, that the collection of facts in the science of astronomy includes the following truths:

Whatever is near does not twinkle.
Whatever does not twinkle is near.
The planets do not twinkle.
The planets are near.

While Aristotle’s method may help us to identify deductions between these truths, it does not allow us to determine which of these deductions constitute demonstrations. Nor does it enable us to identify indemonstrable principles. For example, the truth ‘The planets are near’ may turn out to be an indemonstrable principle of astronomy even though it can be deduced from other truths in the collection. Thus, Striker contends that, contrary to what Aristotle claims, the principles of a science are ‘obviously not discovered simply by checking whether a given proposition is or is not derivable within the collection’.17 Accordingly, she maintains that Aristotle’s method is only of limited use in demonstrative science and instead more useful in dialectic, in which there is no need to identify demonstrations or scientific principles.18

Against this, Smith argues that Striker’s verdict is ‘too pessimistic’, and that Aristotle’s method succeeds in singling out the indemonstrable principles of a science.19 However, he does not explain how the problem raised by Striker can be solved and, in particular, he does not address the challenges posed by Posterior Analytics 1.13.20 Thus, it remains unclear how Aristotle’s optimism about his method might be justified.

Various responses to this problem have been given by commentators. For example, James Lennox suggests that, contrary to appearances, Aristotle does not claim in 1.30 that a complete collection of facts is sufficient for identifying demonstrations, but only that it delivers a ‘short list’ of candidates for demonstrations.21 His rationale is that ‘what is entirely missing in the APr. [1.27–9] discussion is the identification of the middle term as identifying the cause of the predication to be explained’.22 It is of course correct that there is no explicit discussion of causes and explanatory middle terms in 1.27–9. Still, Lennox’s reading is in tension with Aristotle’s claim in 1.30 that, when the collection of facts is complete, ‘it is already in our power readily to exhibit the demonstrations’ and ‘we will be able, for everything of which there is a demonstration, to find this demonstration’ (46a22–6). Here, Aristotle makes it clear that he regards the ability to identify demonstrations not as something we may or may not have, but as something we definitely have if given a complete collection of facts.23 Since Aristotle is acutely aware of the difference between demonstrations and mere deductions, we should not expect him to make this claim lightly if he thought that his method does not provide any way of ascertaining the key features of a demonstration. Rather, the fact that Aristotle is willing to make this claim indicates that he took the method to provide all the resources needed to distinguish demonstrations from non-explanatory deductions in a complete collection of scientific facts.

Alternatively, Jonathan Barnes suggests that Aristotle’s claim in 1.30 is underwritten by a distinction he draws in 1.27 between three ways in which a predicate may hold of a subject:

Among the things that follow [a subject] one must distinguish those that are predicated in the essence, those that are propria, and those that are predicated as accidents.

διαιρετέον δὲ καὶ τῶν ἑπομένων ὅσα τε ἐν τῷ τί ἐστι καὶ ὅσα ἴδια καὶ ὅσα ὡς συμβεβηκότα κατηγορεῖται.

APr. 1.27, 43b6–8

In this passage, Aristotle recommends a threefold subdivision of the terms that hold of a given subject (or, ‘follow’ it). The first subclass contains those terms that are predicated of the subject essentially, the second those that are not predicated essentially but are coextensive with the subject (propria), and the third one contains those terms that are neither predicated essentially nor coextensive (accidents).24 With respect to this subdivision, Barnes suggests that ‘the provenance of a middle term will thus indicate whether or not it is suitable material for a demonstrative syllogism’.25 This suggestion has some initial plausibility, given that essential predications play an important role in Aristotle’s theory of demonstration. Yet it is doubtful that Aristotle’s claim in 1.30 relies on this subdivision. Except for the passage just quoted and a similar remark at 43b2, essential predications are not mentioned anywhere in 1.27–30. Instead, the procedure described in 1.28–9 employs only collections in which all terms that hold of a subject, essentially or not, are grouped together in one class. Aristotle notes that ‘all deductions come about through these’ collections (44a37–8). Moreover, Barnes does not explain how the subdivision of predications might help us to construct demonstrations or identify indemonstrable facts. The subclass of essential predications, at any rate, cannot be used to identify all indemonstrable facts of the form ‘A holds of all B’, since Aristotle takes there to be indemonstrable facts of this form in which A is not predicated essentially of B.26 Nor do essential predications allow us to identify indemonstrable negative facts of the form ‘A holds of no B’, which Aristotle includes among the principles of a science. Of course, this is not to say that the distinction between essential and non-essential predications plays no role at all in Aristotle’s heuristic method.27 Still, in the absence of further explanation, the subdivision adduced by Barnes cannot serve as a basis for Aristotle’s claim in 1.30.

Richard McKirahan suggests that Aristotle’s optimism about his method is based on his statement in 1.30 that the indemonstrable principles of a science are given to us by experience:

It is the task of experience to deliver the principles concerning any given subject. I mean, for example, that it is the task of astronomical experience to deliver the principles of the science of astronomy.

διὸ τὰς μὲν ἀρχὰς28 τὰς περὶ ἕκαστον ἐμπειρίας ἐστὶ παραδοῦναι, λέγω δ’ οἷον τὴν ἀστρολογικὴν μὲν ἐμπειρίαν τῆς ἀστρολογικῆς ἐπιστήμης.

APr. 1.30, 46a17–20

According to McKirahan, this passage states that experience (ἐμπειρία) enables us to identify the indemonstrable principles of a science and to distinguish them from demonstrable truths. Thus, he takes experience to constitute ‘a third activity, identifying the principles, which takes place after the facts are recognized and before they are formed into proofs’.29 However, this does not sit well with Aristotle’s usual account of experience and, in particular, with his view that those who have merely experience ‘know the fact (τὸ ὅτι), but do not know the reason why (τὸ διότι)’ (Metaphysics A 1, 981a29).30 For one who does not know ‘the reason why’ presumably is not in a position to identify the indemonstrable principles of a science. Accordingly, the passage from 1.30 just quoted should not be taken to mean that experience enables us to determine which facts are indemonstrable, but only that it provides us with both demonstrable and indemonstrable facts alike.31 Thus, while experience plays an important role in obtaining the collection of facts (ἱστορία), its role does not extend beyond this collection.32 Consequently, experience cannot account for our ability to identify indemonstrable principles in the collection.

David Bronstein argues that Aristotle’s claim in 1.30 is based on the assumption that the indemonstrable facts have already been distinguished from the demonstrable ones at the earlier stage of collecting the facts (ἱστορία, 46a24).33 Thus, when the collection of facts has been completed, there is no longer any need to identify the indemonstrable facts and the only task that remains is to construct demonstrations of the demonstrable ones. However, there is little textual support for this reading of 1.30. Aristotle writes that, when we have a complete collection of facts, we will be able, for any indemonstrable fact, to make it evident that it is indemonstrable (τοῦτο ποιεῖν φανερόν, 46a25–7).34 Thus, for Aristotle, the distinction between demonstrable and indemonstrable facts is not something already evident when we have a complete collection of facts, but something we must make (ποιεῖν) evident after the collection has been completed. Accordingly, the collection described in 1.27–8 does not distinguish between demonstrable and indemonstrable facts, but includes both kinds of fact indiscriminately.35 Nor does Aristotle intend such a distinction when he speaks of a ‘collection of facts’ (ἱστορία) in the biological works. Instead, he states in the Historia animalium that we should try to identify indemonstrable principles only after the collection of facts has been completed:

We should first gather the differences and attributes of every [animal kind]. After this, we must attempt to find their causes. For in this way our method will proceed in the natural order, once the collection of facts has been completed about each kind. For it becomes evident from these collections both about which things and from which things the demonstration must be carried out.

ἵνα πρῶτον τὰς ὑπαρχούσας διαφορὰς καὶ τὰ συμβεβηκότα πᾶσι λαμβάνωμεν. μετὰ δὲ τοῦτο τὰς αἰτίας τούτων πειρατέον εὑρεῖν. οὕτω γὰρ κατὰ φύσιν ἐστὶ ποιεῖσθαι τὴν μέθοδον, ὑπαρχούσης τῆς ἱστορίας36 τῆς περὶ ἕκαστον· περὶ ὧν τε γὰρ καὶ ἐξ ὧν εἶναι δεῖ τὴν ἀπόδειξιν, ἐκ τούτων37 γίνεται φανερόν.

Historia animalium 1.6, 491a9–14

Here, the collection of facts (ἱστορία) consists in gathering the differences and attributes of the animal kinds under consideration. The collection is organized in such a way as to facilitate the discovery of the causes of those differences and attributes.38 Yet, Aristotle is clear that we should attempt to identify these causes only after the collection has been completed. The causes include the things ‘from which’ demonstrations proceed, i.e., the indemonstrable facts of the science, while the demonstrable facts are that ‘about which’ demonstrations are carried out.39 The distinction between these two kinds of fact is not present in the collection of facts, but ‘becomes evident’ (γίνεται φανερόν) after the collection has been completed. Thus, as Wolfgang Kullmann has pointed out, in the collection of facts (ἱστορία) presented in the Historia animalium ‘the facts have not yet been divided into demonstrable and indemonstrable ones’.40

Finally, Lucas Angioni suggests that Aristotle’s optimistic claim in Prior Analytics 1.30 does not pertain to the method developed in 1.27–9, but to a different method not discussed in the preceding chapters.41 In particular, he argues that Aristotle’s claim does not rely on the collection of facts described in 1.27–8, but on a collection of a different sort more suitable for identifying demonstrations. However, since Aristotle does not provide any details regarding the new collection allegedly employed in 1.30, it is unclear how this collection is supposed to work and how it might help to address the problem raised by Striker. Moreover, Aristotle gives no indication of shifting to a new kind of collection in 1.30. Instead, it seems clear that the collection of facts referred to in 1.30 is the same that has been under consideration throughout 1.27–9.42

In what follows, I argue that Aristotle’s optimistic claim in 1.30 can be justified by taking a closer look at the collection of terms in 1.27. This collection is governed by the theory of natural predication developed in Posterior Analytics 1.19–22 (Section 2). According to this theory, any chain of natural predications gives rise to a demonstration in Barbara (Section 3). Moreover, Aristotle states that natural predication is asymmetric. For example, if non- twinkling is predicated of near, the latter is not predicated of the former (Sections 4–5). This allows us to address the problem raised by Striker, for both affirmative and negative propositions. Thus, when applied to a complete collection of scientific propositions, the method of Prior Analytics 1.27–30 succeeds in identifying all demonstrations and indemonstrable principles of the science under consideration (Section 6). Aristotle does not explain in 1.27–30 how we can identify the natural predications in a science, but he provides some guidance on this question in Posterior Analytics 2.14 based on the method of division (Section 7).

2 Predication in Prior Analytics 1.27

In Prior Analytics 1.27, Aristotle introduces the elements of his heuristic method. He begins by distinguishing three kinds of being:

Of all beings, some are such as not to be truly predicated universally of anything else (for example, Kleon and Kallias and what is particular and perceptible), but to have others predicated of them (for each of these is both a man and an animal). Some are themselves predicated of others, but others are not antecedently predicated of them. Lastly, some are both predicated of others and have others predicated of them (for example, man is predicated of Kallias and animal of man).

ἁπάντων δὴ τῶν ὄντων τὰ μέν ἐστι τοιαῦτα ὥστε κατὰ μηδενὸς ἄλλου κατηγορεῖσθαι ἀληθῶς καθόλου, οἷον Κλέων καὶ Καλλίας καὶ τὸ καθ’ ἕκαστον καὶ αἰσθητόν, κατὰ δὲ τούτων ἄλλα, καὶ γὰρ ἄνθρωπος καὶ ζῷον ἑκάτερος τούτων ἐστί· τὰ δ’ αὐτὰ μὲν κατ’ ἄλλων κατηγορεῖται, κατὰ δὲ τούτων ἄλλα πρότερον οὐ κατηγορεῖται· τὰ δὲ καὶ αὐτὰ ἄλλων καὶ αὐτῶν ἕτερα, οἷον ἄνθρωπος Καλλίου καὶ ἀνθρώπου ζῷον.

APr. 1.27, 43a25–32

The three kinds of being are characterized by means of the relation of true universal predication (κατηγορεῖσθαι ἀληθῶς καθόλου, 43a26). Beings of the first kind are not predicated of anything, but others are predicated of them. These include perceptible particulars such as Kallias. Beings of the second kind are highest universals which are predicated of others but have nothing predicated of them.43 Beings of the third, intermediate kind are both predicated of others and have others predicated of them. These include universals such as the species man and, presumably, the genus animal. Thus, the beings countenanced by Aristotle include both particulars and universals, with the relation of predication obtaining between them.

Aristotle goes on to add a clarification concerning particulars:

It is clear, then, that some beings are by nature such as not to be said of anything. For as a rule every perceptible being is such as not to be predicated of anything except accidentally; for we sometimes say that this pale thing is Socrates, or that what is approaching is Kallias.

ὅτι μὲν οὖν ἔνια τῶν ὄντων κατ’ οὐδενὸς πέφυκε λέγεσθαι δῆλον· τῶν γὰρ αἰσθητῶν σχεδὸν ἕκαστόν ἐστι τοιοῦτον ὥστε μὴ κατηγορεῖσθαι κατὰ μηδενός, πλὴν ὡς κατὰ συμβεβηκός· φαμὲν γάρ ποτε τὸ λευκὸν ἐκεῖνο Σωκράτην εἶναι καὶ τὸ προσιὸν Καλλίαν.

APr. 1.27, 43a32–6

Aristotle notes that we sometimes make statements such as ‘This pale thing is Socrates’. He argues that, contrary to what might be thought, such cases do not contradict his claim that particulars by nature are not predicated of anything. This is because, in his view, such statements do not express genuine predications but improper (or, as he puts it, ‘accidental’) predications that do not conform with the nature of the beings involved. Since antiquity, these improper predications have been called ‘unnatural’, and the genuine ones ‘natural’.44 In the passage just quoted, Aristotle makes it clear that the trifold classification of beings in 1.27 is based on natural predication, with unnatural predication excluded from consideration. Following Aristotle’s lead, I will simply speak of ‘predication’ to denote natural predication.

Having excluded unnatural predication, Aristotle asserts that any ascending chain of predications terminates:

But that the progression of predications also comes to a stop at some point in the upward direction we will explain later; for the present let this be posited.

ὅτι δὲ καὶ ἐπὶ τὸ ἄνω πορευομένοις ἵσταταί ποτε, πάλιν ἐροῦμεν· νῦν δ’ ἔστω τοῦτο κείμενον.

APr. 1.27, 43a36–7

In this passage, Aristotle considers a sequence of beings A, B, C, … in which every member (except A) is predicated of its predecessor. He claims that any such ascending chain is finite and terminates in a highest universal, a being of the second kind. Aristotle here takes this claim for granted and promises to prove it later. He does not fulfill this promise in the Prior Analytics, but supplies the requisite proof in Posterior Analytics 1.22.45 This proof is part of a larger argument in 1.19–22 to the effect that any regress of demonstrations is finite and terminates in indemonstrable principles. Aristotle’s argument in these chapters is based on an elaborate theory of predication (83a1–23). He introduces this theory in Posterior Analytics 1.19 as follows:

When deducing according to opinion and only dialectically, it is clear that we need only inquire whether the deduction proceeds from the most reputable premises possible; so that, even if there is not in truth any middle term between A and B, but there is thought to be one, someone who deduces through this middle term has deduced dialectically. But when aiming at truth, we must inquire on the basis of what holds. It is as follows: there are things which themselves are predicated of something else non-accidentally—by accidentally I mean this: we sometimes say, e.g., that the pale thing is a man, and we do not then make the same sort of statement as when we say that the man is pale; for whereas the man is pale not by being something else, the pale thing is a man because being pale is an accident of the man.

κατὰ μὲν οὖν δόξαν συλλογιζομένοις καὶ μό