The Discovery of Principles in Prior Analytics 1.30

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 Malink 10.1163/15685284-bja10053 | Phronesis (2022) 1-55 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, 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. . 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 .
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, 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 3 Some manuscripts read μέθοδος instead of ὁδός at 46a3 (n = Ambrosianus L 93, V = Barberinianus 87). This reading is also found in Alexander's lemma of the passage (in APr. 330.31 Wallies) and in Boethius' translation (see Minio-Paluello 1964, 185). Aristotle uses ὁδός rather than μέθοδος throughout 1.27-9 (43a21, 45a21, 45b37; see also 2.1, 53a2). In 1.31, however, he refers to the procedure developed in 1.27-30 as a μέθοδος (46a32; see n. 139 below). Thus, while ὁδός and μέθοδος are not synonymous, Aristotle uses both terms to describe the procedure of 1. 27-30. 4 Striker (2009, 49 and 206) takes μάθημα here to refer specifically to mathematical disciplines.
Rhetoric 2.1, 1378a28; Topics [Top.] 1.14, 105b12-15. See Waitz 1844, 457;Ross 1949, 396;Striker 2009, 206;pace Smith 1989, 158. 6 Similarly, he writes in his discussion of the selection of premises in Topics 1.14: πρὸς μὲν οὖν φιλοσοφίαν κατ' ἀλήθειαν περὶ αὐτῶν πραγματευτέον, διαλεκτικῶς δὲ πρὸς δόξαν, 105b30-1; see Alexander, in Top. 95.19-31 Wallies. 7 See Ferejohn 1991, 142 n. 12. 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, 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 demon- 8 Alexander, I translate ἱστορία as 'collection of facts' (Smith 1989, 49 and 158;Bronstein 2016, 125-6). For Aristotle's use of ἱστορία to denote a collection of facts in a science, see Historia anima lium 1.6, 491a7-14; De partibus animalium 2.1, 646a8-12; De incessu animalium 1, 704b8-11 (cf. Kullmann 1974, 213-15;1990, 336-40;Lennox 1991, 263-70;Bemer 2014, 45-54). 10 The phrase οὗ δὲ μὴ πέφυκεν ἀπόδειξις, τοῦτο ποιεῖν φανερόν does not mean 'for any indemonstrable fact f, to make f evident' (pace Biese 1835, 172), but rather 'for any indemonstrable fact f, to make it evident that f is indemonstrable' (Alexander, Wallies ;Smith 1989, 159). 11 APo. 1.2, 71b17-19. 12 APo. 1.2, 71b19-72a5. 13 APr. 1.1, 24a10-11. 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: 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, 17 Striker 2009, 208. Similarly, McKirahan 1992Striker 1998, 222. 18 Striker 1998, 220-2. 19 Smith 2011 Smith discusses the method of APr. 1.27-30 in a number of places (1989, 158-9; 1991, 51-2; 2009, 62-3; 2011, 420; 2016, 54-5). None of these discussions contains a solution to Striker's problem. 21 Lennox 1991, 268-9. 22 Lennox 1987 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, 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 23 Aristotle employs a conditional with the antecedent in the optative and the consequent in the future indicative (εἰ γὰρ μηδὲν κατὰ τὴν ἱστορίαν παραλειφθείη…, ἕξομεν περὶ ἅπαντος οὗ μὲν ἔστιν ἀπόδειξις ταύτην εὑρεῖν, 46a24-6). In this type of conditional, the antecedent expresses an uncertain possibility, whereas the consequent is asserted categorically and with certainty (Kühner and Gerth 1904, 478 §576b). Thus, while the complete collection of facts described in the antecedent is presented as a possibility that may or may not be realized, our ability to find demonstrations is asserted to follow necessarily if that possibility were to be realized (Kühner and Gerth 1904, 464-5 §571.2iv and §572.1a). 24 For this characterization of propria and accidents, see Top. 1.5, 102a18-22, 102b4-5; 1.8 103b7-19. 25 Barnes 2002, 136 n. 120 (similarly, 130 n. 107 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, Aristotle maintains that there are demonstrable facts of the form 'A holds of all B' in which A is not predicated essentially of B (e.g., 'All triangles have 2R'; see Metaphysics Δ 30, 1025a30-2; De partibus animalium 1.3, 643a27-31). Any such fact is derived from indemonstrable premises 'A holds of all C 1 ' , 'C 1 holds of all C 2 ' ,…, 'C n holds of all B' (n ≥ 1). Since essential predication is transitive (APo. 2.4, 91a18-21), some of these indemonstrable premises are such that the predicate is not predicated essentially of the subject. 27 For the role of this distinction in Aristotle's method, see n. 141. 28 Here, ἀρχαί is used to refer to the premises that serve as indemonstrable principles of a science (Smith 1989, 158-9;Gregorić and Grgić 2006, 20). At 46a10, on the other hand, Aristotle uses ἀρχαί more generally to refer to the premises that serve as the starting points either of scientific demonstrations or of dialectical deductions (see Maier 1900, 304 n. 2). The latter use of ἀρχαί is also found at 1.27, 43a21-2 (Striker 2009, 189 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 29 McKirahan 1992, 262. Similarly, Bemer 2014, 247-8. 30 See Gregorić andGrgić 2006, 17. McKirahan (1992, 307 n. 109) suggests that ἐμπειρία at 46a18 is used differently from the way it is used in Metaphysics A 1, but he does not provide textual evidence for the more demanding kind of ἐμπειρία he attributes to Aristotle in APr. 1.30. 31 Smith 1989, 158;Gregorić and Grgić 2006, 20;Angioni 2019, 173 n. 47;Zuppolini 2020, 39-40. 32 Gregorić andGrgić 2006, 20 n. 38;Hasper and Yurdin 2014, 125 n. 11. 33 Bronstein 2016, 126. 34 Bronstein (2016 takes this to mean that 'what the process of constructing the demonstrations reveals to the inquirer is not so much that the definitions are indemonstrable principles but how' . In my view, this is not a natural reading of οὗ δὲ μὴ πέφυκεν ἀπόδειξις, τοῦτο ποιεῖν φανερόν at 46a27. What is said to be made evident here is not how a given indemonstrable fact is indemonstrable, but rather that it is indemonstrable. 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.
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 35 See Aristotle's description of this collection at 1.28, 44a11-16 (cf. n. 49). The subdivision of predicates at 1.27, 43b6-8 does not give rise to a distinction between demonstrable and indemonstrable affirmative facts (see text to nn. 24-7). Moreover, Aristotle gives no indication in 1.27-8 of distinguishing between demonstrable and indemonstrable negative facts of the form 'A holds of no B' (on which see Section 6 below). 36 The phrase ὑπαρχούσης τῆς ἱστορίας means 'once the collection of facts is present, i.e., has been completed'; Anagnostopoulos 2009, 112;Gotthelf 2012, 271;Bemer 2014, 48-9 and 166. 37 τούτων refers to the facts collected in the ἱστορία; see Lennox 1987, 101;Gotthelf 2012, 271;Bemer 2014, 167;Kullmann 2014, 243. 38 See Lennox 1987, 100-9;1991, 269-70;Gotthelf 2012, 263- 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 nontwinkling 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 39 Kullmann 1974, 81 and 262;2007, 197;2014, 243;Bemer 2014, 167. For περὶ ὧν denoting theses to be established by a deduction and ἐξ ὧν denoting the premises from which they are established, see Top. 1.4, 101b13-16. 40 Kullmann 1974, 263 (see also 268). Similarly, Bemer 2014, 244-5. 41 Angioni 2019, 172-5. 42 This has been argued by Lennox 1991, 267-8. At 1.30, 46a25, Aristotle refers to a collection of τῶν ἀληθῶς ὑπαρχόντων. Presumably, this is the same collection that is described earlier in 1.30 as consisting of τῶν κατ' ἀλήθειαν διαγεγραμμένων ὑπάρχειν (46a8-9). As Angioni (2019, 172-4) admits, the latter collection is identical to the one described in 1.27-8. Hence, the same collection is under consideration at 1.30, 46a22-7. some guidance on this question in Posterior Analytics 2.14 based on the method of division (Section 7).

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).
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, 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, 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 44 Alexander 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.
In the remainder of the passage, Aristotle explains the sense in which scientific deductions proceed from 'what holds' . He does so by distinguishing 45 APo. 1.22, 82b37-83b31. 46 See Irwin 1988, 117 and 528 n. 1. between natural and unnatural predication, as he did in Prior Analytics 1.27. Thus, 'The pale thing is a man' expresses an unnatural predication, whereas its converse expresses a natural predication. Aristotle's point is that scientific deductions, insofar as they proceed from 'what holds' , must employ only natural but not unnatural predications.47 He makes this more explicit in Posterior Analytics 1.22, where he stipulates that demonstrations must not involve any unnatural predications (83a18-21). This assumption plays a vital role in Aristotle's argument for the finitude of demonstration in 1.19-22. As we will see, it is equally important for the heuristic method in Prior Analytics 1.27-30.
In Prior Analytics 1.27, Aristotle concludes his discussion of the three kinds of being by indicating how each of them enters into demonstrations: With respect to these beings [i.e., highest universals, 43a36-7], one cannot demonstrate that something else is predicated of them, except perhaps as a matter of opinion, but only that these are predicated of others. Nor can one demonstrate that particulars are predicated of other beings, but only that others are predicated of them. The intermediate beings, on the other hand, clearly admit of both, since they themselves will be said of others and others of them. κατὰ μὲν οὖν τούτων οὐκ ἔστιν ἀποδεῖξαι κατηγορούμενον ἕτερον, πλὴν εἰ μὴ κατὰ δόξαν, ἀλλὰ ταῦτα κατ' ἄλλων· οὐδὲ τὰ καθ' ἕκαστα κατ' ἄλλων, ἀλλ' ἕτερα κατ' ἐκείνων. τὰ δὲ μεταξὺ δῆλον ὡς ἀμφοτέρως ἐνδέχεται· καὶ γὰρ αὐτὰ κατ' ἄλλων καὶ ἄλλα κατὰ τούτων λεχθήσεται. APr. 1.27, Aristotle is here concerned with our ability to demonstrate (ἀποδεῖξαι, 43a38). He indicates how the availability of demonstrations depends on the presence of natural predications. Thus, there are no demonstrations establishing that a perceptible particular is predicated of something else, or that something else is predicated of a highest universal. There may be dialectical deductions deriving such a conclusion as a matter of opinion (κατὰ δόξαν), but there are no demonstrations of it because these must not involve any unnatural predications.
The threefold classification of beings serves as a basis for the collection of premises described in the remainder of Prior Analytics 1.27.48 For any given being, Aristotle instructs us, one must select the things that follow it and the 47 Zabarella 1608, 893e-895d; Bronstein 2019, 90. 48 Patzig 1968, 6. things which it follows (ἕπεσθαι and ἀκολουθεῖν, 43b3-4).49 Among the things that follow it, one must distinguish between those that are predicated as a matter of opinion and those that are predicated in accordance with truth: Among the things that follow, one must distinguish … those that are predicated as a matter of opinion and those that are predicated in accordance with truth. For the more such things one has available, the sooner one will arrive at a conclusion, and the more they hold in truth, the more one will arrive at a demonstration.
διαιρετέον δὲ καὶ τῶν ἑπομένων … ποῖα δοξαστικῶς καὶ ποῖα κατ' ἀλήθειαν [κατηγορεῖται, 43b8]· ὅσῳ μὲν γὰρ ἂν πλειόνων τοιούτων εὐπορῇ τις, θᾶττον ἐντεύξεται συμπεράσματι, ὅσῳ δ' ἂν ἀληθεστέρων, μᾶλλον ἀποδείξει.50 APr. 1.27, In this passage, Aristotle refers to the distinction drawn earlier between true predication (43a26) and predication as a matter of opinion (43a39). The former is natural predication between the three kinds of being, whereas the latter includes unnatural predications. The more things one is able to collect in either sort of predication, the more likely one is to find a deduction to derive a given conclusion. The resulting deduction may be either dialectical or scientific, depending on what sort of predication is employed in the premises.
Aristotle adds that the more these predications 'hold in truth, the more one will arrive at a demonstration' . He does not explain what he means by predications holding 'more in truth' (ἀληθεστέρων, 43b11). He might be alluding to a doctrine of degrees of truth, according to which a natural predication is true to a higher degree than an unnatural one.51 Alternatively, he may simply have in 49 Aristotle's discussion at 43b22-32 is sometimes taken to imply that one should not select all things that follow a given subject but only those that follow it immediately (i.e., that, for any subject A, one should select only those B which follow A and for which there is no C such that B follows C and C follows A); Barnes 2002, 131  I take ἀποδείξει at 43b11 to be a dative object of ἐντεύξεται (Mignucci 1969, 149;Striker 2009, 193). Alternatively, ἀποδείξει may be construed as a future tense verb (Pacius 1605, 245;Owen 1889, 145;Jenkinson 1928, ad loc.;Smith 1989, 43;Mueller 2006, 79). 51 Aristotle ascribes the highest degree of truth (ἀληθέστατον) to those truths that are fundamental in that they are a cause of truth to other truths but no other truths are a cause of mind that a collection of predications is more or less true depending on how many of these predications are natural as opposed to unnatural. Either way, Aristotle maintains that likely success in constructing demonstrations is tied to how closely one's selection of terms tracks natural predication. Aristotle's discussion in the first half of Prior Analytics 1.27, then, makes it clear that he is concerned with the conditions under which the heuristic method delivers not only deductions but also demonstrations (43b10-11). These conditions are tied to our ability to collect 'true' predications (ἀληθῶς, 43a26; κατ' ἀλήθειαν, 43b10). If a predication is not 'true' , i.e., natural, it cannot be demonstrated (οὐκ ἔστιν ἀποδεῖξαι, 43a38). Aristotle provides a brief description of natural predication at the beginning of 1.27 (43a25-43). He gives a fuller account of it in Posterior Analytics 1.19-22, as part of his proof that all chains of predication are finite. Aristotle notes that the latter result is presupposed by his discussion in Prior Analytics 1.27-30 (νῦν δ' ἔστω τοῦτο κείμενον, 43a37). He chooses not to give a proof of this result in 1.27, perhaps because doing so would distract too much from the exposition of the heuristic method.52 Still, by referring to the result in 1.27 he indicates that his heuristic method, insofar as it aims at demonstrations, relies on the account of natural predication given in the Posterior Analytics.
In the remainder of Prior Analytics 1.27-9, Aristotle discusses the deductive aspects of the heuristic method without focusing specifically on demonstrations. This discussion applies equally to dialectical and scientific deductions and therefore does not presuppose any material from the Posterior Analytics. In chapter 1.30, however, Aristotle returns to the topic of demonstration. He does so by first reminding us of the distinction between predication according to opinion (κατὰ δόξαν) and in accordance with truth (κατ' ἀλήθειαν, 46a8-10). Given his treatment of this distinction in 1.27, predication in accordance with truth can be identified with the relation of natural predication introduced in 1.27. Aristotle appeals to this relation later in 1.30 when he refers to terms that hold in truth (ἀληθῶς, 46a25).53 He claims that, when we have a complete coltruth to them (Metaphysics α 1, 993b23-31; see Broadie 2019, 261). Thus, by calling natural predications 'more true' than unnatural ones, Aristotle might indicate that the former are more fundamental than the latter (i.e., that unnatural predications are true in virtue of certain natural predications, but not vice versa). Cf. also Aristotle's claim in De interpre tatione 14 that belief in a per se predication is 'more true' (μᾶλλον ἀληθής) than belief in an accidental predication (23b17; see Ammonius, in Int. 262.1-3 Busse;Ackrill 1963, 154). 52 This is in line with Aristotle's general practice of assuming a provable claim as a provisional hypothesis and postponing its proof when giving the proof would disrupt the preferred order of exposition in a given context (see Leunissen 2017). 53 I take the collection of τῶν ἀληθῶς ὑπαρχόντων at 46a25 to be the same as that of τῶν κατ' ἀλήθειαν διαγεγραμμένων ὑπάρχειν at 46a8-9 (see n. 42; pace Angioni 2019, 165-75). There The Discovery of Principles in Prior Analytics 1.30 Phronesis (2022)  lection of terms that hold in truth, we will be able to identify all demonstrations and indemonstrable principles of the science under consideration (46a24-7). This is an elaboration of his earlier claim in 1.27 that 'the more [the collected terms] hold in truth, the more one will arrive at a demonstration' (43b10-11).
Since these claims involve the relation of natural predication, we may hope to gain a better understanding of them by considering Aristotle's treatment of natural predication in the Posterior Analytics.

Predication and Demonstration in the Posterior Analytics
For Aristotle, scientific knowledge is primarily obtained through demonstration. Every proposition that falls under the purview of a given science appears either as a premise or as the conclusion of a demonstration. In what follows, I refer to these propositions as 'scientific propositions' .
In the Analytics, demonstrations take the form of deductions in the three syllogistic figures. For example, a universal affirmative proposition AaB ('A holds of all B') is demonstrated by means of a deduction in Barbara, as follows: AaC, CaB, therefore AaB Each of the premises in such a demonstration either is or is not demonstrable. If it is, then, in order to have scientific knowledge of AaB, the demonstrator must have scientific knowledge of this premise through another demonstration.54 This demonstration will again be of the form Barbara, since this is the only way to deduce an a-proposition in Aristotle's syllogistic theory.55 Thus, there is a regress of demonstrations in Barbara.
In Posterior Analytics 1.19-22, Aristotle argues that any regress of demonstrations is finite and terminates in indemonstrable principles. His argument is based on the assumption that every scientific a-proposition corresponds to a natural predication: If we must legislate, let speaking in this way [i.e., natural predication] be predicating, and speaking in the other way [i.e., unnatural predication] are altogether six references to 'truth' in APr. 1. 27-30 (1.27, 43a26, 43b9, 43b11; 1.29, 45b10; 1.30, 46a8, 46a25). The reference in 1.29 is special in that it serves to distinguish direct deductions from those by reductio ad impossibile. If I am correct, all of the other five references to 'truth' serve to demarcate the relation of natural predication. 54 See APo. either not predicating at all or else predicating not simpliciter but acci-dentally…. Let it be assumed, then, that what is predicated is always predicated simpliciter, and not accidentally, of what it is predicated of; for this is the way in which demonstrations demonstrate.
εἰ δὴ δεῖ νομοθετῆσαι, ἔστω τὸ οὕτω λέγειν κατηγορεῖν, τὸ δ' ἐκείνως ἤτοι μηδαμῶς κατηγορεῖν, ἢ κατηγορεῖν μὲν μὴ ἁπλῶς, κατὰ συμβεβηκὸς δὲ κατηγο-ρεῖν…. ὑποκείσθω δὴ τὸ κατηγορούμενον κατηγορεῖσθαι ἀεί, οὗ κατηγορεῖται, ἁπλῶς, ἀλλὰ μὴ κατὰ συμβεβηκός· οὕτω γὰρ αἱ ἀποδείξεις ἀποδεικνύουσιν. APo. 1.22, In this passage, Aristotle states that, if AaB appears as a premise or conclusion of a demonstration, A is predicated of B.56 Since every scientific proposition appears as a premise or conclusion of a demonstration, we have: Given this principle, every regress of demonstrations in Barbara gives rise to a chain of predications. As he promised in Prior Analytics 1.27, Aristotle goes on to prove, based on his theory of natural predication, that there is no infinite chain of predications (83b26-31). He infers that there is no infinite regress of demonstrations (84a29-b2).57 As we have just seen, the principle of NATURAL A-PREDICATION plays a key role in Aristotle's argument for the finitude of demonstration in the Posterior Analytics. In addition, this principle helps to underwrite the heuristic method in Prior Analytics 1.27-30 by providing a criterion for the indemonstrability of scientific a-propositions. To see this, suppose that we have a complete collection of predications in a given science. If AaB is demonstrable, it can be demonstrated from scientific propositions AaC and CaB, which both correspond to natural predications. Thus, we have a sufficient condition for indemonstrability: a scientific proposition AaB is indemonstrable if the collection of predications in the science under consideration contains no C such that A is predicated of C and C is predicated of B. 56 Zabarella 1608, 909f-910a; Mignucci 1975, 454-5;Lear 1980, 18-31;Smith 1991, 48-51;Bronstein 2019, 90. 57 He does so by arguing that any regress of demonstrations gives rise to an infinite chain of scientific a-propositions (APo. 1.19, 82a2-8 and 1.20, 82a21-30; see Smith 1996, 27-30).
Is this condition also necessary? In other words, if the collection of predications contains a C such that A is predicated of C and C of B, does it follow that AaB is demonstrable? If the answer were no, Aristotle's heuristic method would fail to deliver all indemonstrable a-propositions; for, when the collection contains such a C, we would not be able to tell whether or not AaB is demonstrable. Fortunately, however, this is not the case. For Aristotle makes it clear in Posterior Analytics 1.22 that the answer to the question is yes. Having argued that any demonstration in Barbara gives rise to a chain of natural predications, he goes on to argue the converse: that any chain of natural predications gives rise to a demonstration in Barbara (83b32-84a4). In the course of this argument, he asserts that, if the collection of predications in a science contains a chain of predications from A to B, then AaB is demonstrable:58 If things are so related to a subject that some things are predicated of it prior to them, there is a demonstration of these.59 ὧν πρότερα ἄττα κατηγορεῖται, ἔστι τούτων ἀπόδειξις. APo. 1.22, Aristotle considers a predicate, A, that is predicated of a subject, B. He supposes that some Cs are predicated of B 'prior to' A. By calling the Cs 'prior to' A he means that they appear before A on an ascending chain of predications leading from B to A. If there are such Cs, he claims, there is a demonstration of AaB. This demonstration will proceed via Barbara using the Cs as middle terms. Pacius illustrates Aristotle's point with an example in which A is the predicate living, B the subject man, and C animal:60 Any mediate proposition can be demonstrated. A proximate predicate is predicated of a given subject prior to a remote predicate. For example, animal is predicated of man prior to living. Consequently, the proposition 'Man is living' is mediate and can be demonstrated as follows: every animal is living, every man is an animal, therefore, every man is living. Thus, Aristotle takes the collection of predications in a science to obey the following principle: DEMONSTRATIVE PREDICATION: If A is predicated of C 1 , C 1 is predicated of C 2 ,…, and C n is predicated of B (n ≥ 1), there is a demonstration of AaB using C 1 ,…, C n as middle terms.
With this principle in place, we have a necessary and sufficient condition for the indemonstrability of a-propositions: a scientific proposition AaB is indemonstrable just in case there is no C such that A is predicated of C and C of B.61 Given a complete collection of predications in a science, this criterion allows us to identify all and only indemonstrable a-propositions.
Moreover, we will be able to find all demonstrations of demonstrable a-propositions in a complete collection of predications. To see this, recall that, in Aristotle's syllogistic theory, the only way to demonstrate AaB is from scientific a-propositions forming a chain AaC 1 , C 1 aC 2 ,…, C n aB (n ≥ 1). Given the principles of Natural a-predication and Demonstrative predication, any such chain gives rise to a demonstration of AaB using the middle terms C 1 ,…, C n . This demonstration will proceed by sequential applications of Barbara. Aristotle instructs us to begin the demonstration with C 1 and then move down the chain of middle terms, using C n in the last step to infer AaB.62 In this way we will be able to construct all demonstrations of demonstrable a-propositions. Now, Aristotle acknowledges that some natural predications fall outside the purview of demonstrative science.63 These include cases in which an accident holds of a subject merely by chance and therefore is not demonstrable of it.64 When Aristotle states the principle of DEMONSTRATIVE PREDICATION in the passage just quoted, he excludes such predications and considers only those natural predications that fall under the purview of a science. With respect to these predications, Aristotle maintains that A is predicated of B just in case AaB is a scientific proposition.
In Prior Analytics 1.30, Aristotle assumes that 'nothing that truly (ἀληθῶς) holds of things has been left out in the collection of facts (κατὰ τὴν ἱστορίαν)' (46a24-5). This means that the affirmative part of the collection 61 Similarly, Smith (1982, 126)  contains all and only natural predications in the science under consideration. Thus, the collection of facts (ἱστορία) demarcates exactly the class of scientific a-propositions. It does not distinguish between demonstrable and indemonstrable a-propositions, but contains all of them indiscriminately. Nonetheless, by means of the principles of NATURAL A-PREDICATION and DEMONSTRATIVE PREDICATION, we are able to determine which of these a-propositions are indemonstrable, and to construct demonstrations of the remaining a-propositions from chains of indemonstrable ones.65 Aristotle does not explain in Prior Analytics 1.27-30 how we might be able to identify the natural predications in a science, but he postpones this question to the Posterior Analytics (see Section 7 below). Accordingly, Aristotle's concern in Prior Analytics 1.30 is not with how we can obtain a complete collection of natural predications in a science. Instead, his focus in 1.30 is on the conditional claim that, if we were to have such a complete collection, we will be able to find all demonstrations of a-propositions and identify all indemonstrable a-propositions in that science. While this is a substantive claim, it turns out to be correct, justified by the principles of Natural a-predication and DEMONSTRATIVE PREDICATION. Both of these principles are asserted by Aristotle in Posterior Analytics 1.22, the chapter to which he refers when introducing natural predication in Prior Analytics 1.27 (43a36-7). At the same time, however, the two principles do not come for free, but have substantive consequences for his theory of demonstration. In particular, as we will see, they give Aristotle reason to deny the possibility of mutual predication.

Counterpredication
According to the principle of Demonstrative predication, any chain of predications gives rise to a demonstration in Barbara. It is important to note that this principle imposes constraints regarding the possibility of mutual predication between terms. Suppose, for example, that A and B are predicated of one another, and that both are predicated of C. Thus, we have the following two chains of predications: A is predicated of B, B is predicated of C B is predicated of A, A is predicated of C Given Demonstrative predication, each of these chains gives rise to a demonstration, one deriving AaC from BaC and the other deriving the latter from the former. Now, Aristotle holds that every premise of a demonstration is prior in nature to the conclusion.66 Hence, the existence of the two demonstrations entails that BaC is prior in nature to AaC and vice versa. But this is impossible since priority in nature is asymmetric (nothing can be prior in nature to something prior in nature to it).67 Thus, Aristotle's commitment to Demonstrative predication is in tension with the existence of terms that are mutually predicated of one another. Aristotle resolves the tension in Posterior Analytics 1.19-22 by denying the existence of such terms. In these chapters, he refers to mutual predication as 'counterpredication' (ἀντικατηγορεῖσθαι). Thus, A and B are counterpredicated of one another just in case A is predicated of B and B is predicated of A.68 Aristotle considers several putative cases of counterpredication and argues that they are not admissible in his theory of natural predication (1.22, 83a36-b12). He first notes that A and B cannot be counterpredicated of one another in such a way that one is a quality of the other and vice versa (83a36-9). More generally, he asserts that nothing is counterpredicated of an accident which is in the category of quality or in any other non-substance category (83b10-12). Moreover, Aristotle rejects cases in which A and B are counterpredicated of one another essentially, or, 'as substance' (83a39-b10). Taken together, these claims amount to a general rejection of counterpredication. Thus, it is widely agreed that at 83a36-b12 Aristotle excludes the possibility of any counterpredication.69 This is confirmed by Aristotle's treatment of what he calls 'primitive' (πρῶτον) predication in 1.19-22. By a 'primitive' predication he means one that is immediate in the sense that there is no chain of predications leading from the predicate to a term that is predicated of the subject. If a predication is primitive, the corresponding a-proposition is indemonstrable. Aristotle maintains that all demonstrations in Barbara are based on chains of primitive predications. In any series of demonstrations by Barbara, he writes, 'it is necessary for there to be a thing of which something is predicated primitively (εἶναί τι οὗ 66 APo. 1.2, 71b19-72a5; 1.25, 86a39-b5. 67 APo. 1.3, 72b27-8; see also 2.15, 98b16-21. 68 Ross 1949, 578;Hamlyn 1961, 119-20;Lear 1980, 31;Barnes 1993, 178. See also Bostock 2004, 149-51. 69 Waitz 1846Mure 1928, ad loc.;Ross 1949, 578-9;Hamlyn 1961, 120;Barnes 1975, 169-70;Lear 1980, 31;Tricot 2012, 119-20;Zingano 2017, 18;Malink 2020, 121-3. πρῶτόν70 τι κατηγορεῖται), and something else of this, and this must come to a stop ' (1.22, 83b28-30). At the same time, Aristotle denies that there are any such chains of primitive predications among counterpredicated items: Among counterpredicated things, there is none of which any is predicated primitively or of which it is predicated last; for in this respect at least every such thing is related to every other in the same way.
οὐ γὰρ ἔστιν ἐν τοῖς ἀντικατηγορουμένοις οὗ πρώτου κατηγορεῖται ἢ τελευ-ταίου· πάντα γὰρ πρὸς πάντα ταύτῃ γε ὁμοίως ἔχει. APo. 1.19, In a class of counterpredicated terms, every term is predicated of every other. Thus, as far as predication is concerned, 'every such thing is related to every other in the same way' . Aristotle infers from this that there is no basis for distinguishing primitive (or, immediate) from mediate predications, and hence that there are no primitive predications among such terms.71 By the same token, counterpredicated terms are not predicated primitively of anything and nothing is predicated primitively of them. Thus, as Smith points out, by taking all demonstrations in Barbara to be based on chains of primitive predications, Aristotle in effect excludes counterpredication from the domain of demonstration.72 In this way, Aristotle is able to maintain his commitment to Demonstrative predication: if there are no counterpredicated terms, every chain of predications can give rise to a demonstration in a coherent manner, without subverting the order of priority in nature between scientific propositions.
On the other hand, it is true that Aristotle accepts counterpredication (ἀντικατηγορεῖσθαι) in the Topics. According to the Topics' theory of predication, both definitions and propria are counterpredicated of their subjects.73 For example, if biped animal is a definition of man, and if capable of learning grammar is a proprium of man, both are counterpredicated of man. In Prior 70 For πρῶτον indicating primitive (or, immediate) predication, see Pacius 1605, 474;Owen 1889, 294;Barnes 1993, 179. 71 Waitz 1846Mendell 1998, 200-1. 72 Smith (1986 takes 82a15-17 to show that 'propositions involving convertible [i.e., counterpredicated] terms are somehow disqualified because of the lack of any priority' (similarly, Mendell 1998, 200-1). Smith (1984, 593-4;1991, 50;1996, 27) argues that Aristotle's rejection of counterpredication plays an important role in his argument for the finitude of demonstration in APo. Analytics 1.27, Aristotle states that definitions and propria should be included in the collection of terms that serves as the basis of his heuristic method (43b1-8). Thus, it might be thought that he accepts counterpredication in the heuristic method, thereby ignoring the prohibition imposed in the Posterior Analytics. This, however, is not the case-or so I shall argue. For, as we will see, the kind of counterpredication Aristotle accepts in the Topics is different from the one he rejects in the Posterior Analytics.
In the Topics, counterpredication is introduced as follows: A proprium is what does not reveal the essence of the thing but holds of it alone and is counterpredicated of it. For example, it is a proprium of man to be capable of learning grammar; for if something is a man, it is capable of learning grammar, and if it is capable of learning grammar, it is a man.

Top. 1.5, 102a18-22
A proprium holds of its subject alone and is counterpredicated of it. In the present passage, counterpredication is characterized by reference to the particulars of which the two terms are predicated: A is counterpredicated of B just in case, for any particular X, if X is A, then X is B, and vice versa. On this definition, counterpredication means being coextensive in the sense of being predicated of the same class of particulars.75 Crucially, this notion of counterpredication is different from mutual predication. Thus, Brunschwig comments on the passage just quoted: The word ἀντικατηγορεῖσθαι [at Topics 1.5 102a19] does not designate the legitimacy of the inversion of the subject and the predicate, but that of a reciprocal substitution of two predicates in relation to the same concrete subject…. In other words, we can say that a predicate P is 74 Here, τοῦ πράγματος denotes the subject (e.g., man) of which the predicate (e.g., capable of learning grammar) is a proprium. See Barnes 1970, 137- counterpredicated of a subject S, not when we have 'S is P and P is S' , but when we have 'for any concrete subject X, if X is S, X is P, and if X is P, X is S' . Brunschwig 1967, 122 n. 1 In the Topics, Aristotle holds that, if A is a proprium of B, then A is predicated (κατηγορεῖσθαι) of B.76 At the same time, he warns that, if A is coextensive with B, it does not follow that B is predicated of A. For example, if lightest body is coextensive with fire, this does not mean that the latter is predicated of the former: In attack, see if the opponent has given the subject as a proprium of that which is said to be in the subject; for then what has been stated to be a proprium will not be a proprium. For example, since someone who has given fire as the proprium of lightest body has given the subject as a proprium of its predicate, fire will not be a proprium of lightest body.

Top. 5.4, 132b19-24
According to this passage, fire is a subject of which lightest body is predicated. Aristotle infers that lightest body is not a subject of which fire is predicated, and hence that the latter is not a proprium of the former. Fire is not predicated of lightest body even if the latter is a proprium of the former and the two are coextensive. This shows that, for Aristotle, predication is non-extensional in that it is not determined by the extension of the terms involved. Two terms may be coextensive while one of them is predicated of the other but not vice versa.77 76 Top. 1.8, 103b7-12 with κατηγορούμενον at 103b8. See Smith 1997, 62;Malink 2013, 115-17. 77 By contrast, Barnes (1970, 150) takes predication at 132b19-24 to be extensional: A is predicated of a subject B just in case A is predicated of every particular of which B is predicated. On this reading, the passage would imply that, if A is a proprium of B, then B is not predicated of every particular of which A is predicated, contradicting the fact that every proprium is coextensive with its subject. Thus, Barnes concludes that Aristotle's reasoning at 132b19-24 is 'absurd' and 'a muddle which cannot be interpreted away' . The problem, however, is not with Aristotle's reasoning, but with Barnes's assumption that predication is extensional. Once this assumption is dropped, Aristotle's reasoning is coherent, indicating that, in addition to being coextensive with its subject, a proprium must satisfy the The passage implies that the Topics' relation of being a proprium is not symmetric: lightest body is a proprium of fire, but not vice versa.78 Alexander argues that, more generally, the passage implies asymmetry: for any A and B, if A is a proprium of B, then B is not a proprium of A.79 Accordingly, Striker maintains that 'the predication-relations Aristotle discusses in the Topics are all asymmetrical' .80 But whether or not predication in the Topics is asymmetric in all cases, it is clear that it is not symmetric for coextensive terms.
Thus, Aristotle's treatment of propria in the Topics is consistent with his rejection of counterpredication in the Posterior Analytics. In both treatises he accepts coextensive terms, but denies that coextensiveness entails mutual predication. The apparent conflict between the treatises is due to an ambiguity in Aristotle's use of the verb 'counterpredicate' (ἀντικατηγορεῖσθαι). In the Topics he uses it to express coextensiveness, whereas in Posterior Analytics 1.19-22 he uses it to express mutual predication.
In Posterior Analytics 1.13, Aristotle regards the terms near and non-twinkling as coextensive. In fact, he states that they are counterpredicated of one another (ἀντικατηγορεῖσθαι, 78a28). Smith argues that this conflicts with Aristotle's rejection of counterpredication in Posterior Analytics 1.19-22.81 But the conflict may be only apparent. For, when Aristotle speaks of 'counterpredication' in 1.13, he need not have in mind mutual natural predication. After all, the framework of natural predication is not yet on the table at this stage of the Posterior Analytics and will not be introduced before 1.19. Instead, 'counterpredication' in 1.13 may be understood in the weaker sense familiar from the Topics, indicating coextensiveness but not mutual predication.
Aristotle takes the a-proposition 'Whatever is near does not twinkle' to serve as a premise in a demonstration explaining why the planets do not twinkle. Given his commitment to Natural a-predication, this means that non-twinkling is predicated of near. The converse, therefore, does not hold: further condition of being predicated of the subject (see Smith 1997, 61-2). For further arguments that predication in the Topics is non-extensional, see Malink 2009, 125-9. 78 By contrast, Aristotle countenances a wider notion of proprium (ἴδιον) which is symmetric at APo. 2.4, 91a15-18 (see Reinhardt 2000, 31 and 188-9; Barnes 2003, 211-12). 79 Alexander, in Top. 392.29-30 Wallies. 80 Striker 1994, 47. A possible exception is Aristotle's claim in Topics 4.6 that ὄν is predicated of ἕν and vice versa (127a24-31). Crucially, however, these are not instances of natural predication as described in the Analytics. For Aristotle requires that every natural predication be in one of the ten categories (APo. Although the premise 'Whatever does not twinkle is near' does not correspond to a natural predication, it is true in an extensional sense: near is predicated of every particular of which non-twinkling is predicated. This dovetails with Aristotle's remark that this premise is 'obtained through induction or through perception' . Induction, for Aristotle, proceeds by way of considering particulars and particular cases (71a8-9). Similarly, the paradigmatic objects of perception are particulars (43a25-7). Thus, by adding the qualification that the premise is obtained through induction or perception, Aristotle seems to indicate that this premise makes an extensional claim about the particulars that fall under the terms near and non-twinkling without asserting a natural predication between the terms themselves. By contrast, Aristotle does not add such a qualification for the converse premise, 'Whatever is near does not twinkle' (78a39-b3). There is no need for him to do so since this premise is not only true extensionally but corresponds to a natural predication. As we have seen, the relation of natural predication underlying Aristotle's theory of demonstration is asymmetric. Consequently, if AaB is a scientific proposition, its converse, BaA, is not.83 For example, if 'All triangles have 2R' is a scientific a-proposition, its converse is not. Such converse a-propositions are not among the principles or demonstrable theorems of the science under consideration, and hence will not be included in the science's collection of facts (ἱστορία). Thus, if the collection captures all and only natural predications in the science of geometry, 2R will be included among the terms that follow triangle, but not vice versa. 82 τοῦτο refers to the major premise, 'Whatever does not twinkle is near' , but not to the minor premise, 'The planets do not twinkle' (Zabarella 1608, 837f-838b;Detel 1993, 298;pace Owen 1889, 175;Zuppolini 2021, 128 n. 34). 83 This has been argued by Mendell 1998, 200-1. Of course, Aristotle acknowledges that it is true to say that triangle is predicated of every particular of which 2R is predicated. Such extensional truths, however, are not represented by demonstrable or indemonstrable a-propositions in the relevant science. Instead, Aristotle introduces an alternative way of registering such extensional truths in the course of gathering the facts of a science. As we will see, he does so in Prior Analytics 1.29 by means of a special mode of reasoning 'from a hypothesis' .

Extensional Truths
Aristotle maintains that every deduction is either direct or 'from a hypothesis' (ἐξ ὑποθέσεως, APr. 1.23, 40b23-9). Direct deductions proceed through the syllogistic moods in the three figures, whereas deductions from a hypothesis involve an extra inferential step not covered by these moods. In Prior Analytics 1.28, Aristotle describes how his heuristic method can be used to construct direct deductions. In 1.29, he explains how it can be used to construct deductions from a hypothesis. He begins his discussion in 1.29 by considering the most prominent case of such deductions, those by reductio ad impossibile (45a23-b15). He goes on to comment on other kinds of deduction from a hypothesis, noting that these must be investigated further (45b15-20). While Aristotle does not undertake such an investigation in 1.29, he does discuss one of the other kinds of deduction from a hypothesis in some detail. This is a mode of reasoning by which a universal thesis can be established 'through the examination of particular cases': While each kind of problem can be proved in the way described [previously in 1.28-9], it is also possible to deduce some of them in another manner. For example, universal problems may be deduced from a hypothesis through the examination of particular cases. For if C and G were the same and E were assumed to hold of the Gs alone, then A should hold of all E…. It is evident, then, that one should also examine in this way.
δείκνυται μὲν οὖν ἕκαστον τῶν προβλημάτων οὕτως, ἔστι δὲ καὶ ἄλλον τρόπον ἔνια συλλογίσασθαι τούτων, οἷον τὰ καθόλου διὰ τῆς κατὰ μέρος ἐπιβλέψεως84 84 I take τῆς κατὰ μέρος ἐπιβλέψεως to mean 'the examination of particular cases' (see Alexander,in APr. 328.19 Wallies). Alternatively, the phrase might be rendered as 'the method of investigation proper to the corresponding particular conclusion' (Tredennick The 'problems' referred to in this passage are theses to be established by a deduction.87 In Aristotle's example, the thesis to be established is the universal affirmative proposition 'A holds of all E' . C and G are terms gathered in the original collection of terms (44a12-17). C is in the class of terms of which A holds, and G in the class of terms of which E holds. C and G are assumed to be the same. As Aristotle explained in 1.28, this allows us to deduce the particular proposition 'A holds of some E' (44a19-21). In the present passage, he makes an additional assumption that allows us to infer that A holds, not only of some, but of all E. The additional assumption is that E 'holds of the Gs alone' (μόνοις τοῖς Η ὑπάρχειν). This means that there is no particular of which E holds but G does not hold. Hence, E and G are coextensive. Given this assumption, Aristotle notes, we may infer the universal thesis 'A holds of all E' .88 Alexander illustrates this mode of reasoning with an example in which E is the term able to laugh and G the term human.89 The former is a proprium of the latter.90 In the Topics, Aristotle characterizes propria as predicates that hold of their subject 'alone' (μόνῳ ὑπάρχειν, 1.5, 102a18). There, the condition of holding of the subject 'alone' means that a proprium is coextensive with its 1938, 355;similarly, Mure 1928, ad loc.;Striker 2009, 49). The latter rendering would be more natural if κατὰ μέρος were nominalized by a definite article. 85 The qualification ἐξ ὑποθέσεως indicates that the mode of reasoning described in this passage is not a direct deduction but a deduction from a hypothesis (Kilwardby,contra Philoponus, Wallies). In the Prior Analytics, all occurrences of the phrase ἐξ ὑποθέσεως, including those at 1.29, 45b15-20, refer to deductions from a hypothesis (see APr. 1.23 and 1.44 passim). 86 Reading τὸ Γ καὶ τὸ Η (Ross 1949) instead of τὰ Γ καὶ τὰ Η (Bekker 1831, Waitz 1844). While both readings have support in the manuscripts (Williams 1984, 26), the former has the advantage of making it clear that what is taken to be the same is not the class of terms of which A holds and the class of terms of which E holds, but instead a member of the first class and a member of the second class (Alexander, in APr. 328.16-28 Wallies;Ross 1949, 395 subject.91 Similarly, in the passage from 1.29 just quoted, the coextensiveness of E and G is expressed by the condition that E 'holds of the Gs alone' (μόνοις τοῖς Η ὑπάρχειν). Thus, the mode of reasoning described by Aristotle in this passage is of an extensional nature, proceeding from the assumption that E is coextensive with G just as a proprium is coextensive with its subject. Despite the fact that E and G are coextensive, Aristotle apparently does not include G in the class of terms that hold of E. For otherwise he could simply establish the desired thesis, 'A holds of all E' , by means of his usual method described in 1.28 through a direct deduction in Barbara, using G as a middle term that is included both in the class of terms of which A holds and in the class of terms that hold of E (44a17-19). In this case, there would be no need for Aristotle to introduce a new mode of reasoning 'from a hypothesis' to establish that A holds of all E. Thus, Striker argues that the new mode of reasoning is superfluous because G should have been included in the initial class of terms that hold of E: It is puzzling that Aristotle lists this [i.e., the new mode of reasoning described at 45b21-8] as a different way of finding premisses instead of pointing out … that the relevant premisses could have been found without the additional hypothesis. Striker 2009, 204-5 Striker makes an important observation. It is true that Aristotle's discussion of the new mode of reasoning would be puzzling if he took the collection of terms in 1.27 to be based on purely extensional criteria, so that a term would be included in the class of terms that hold of E whenever it is predicated of every particular of which E is predicated. In my view, though, the fact that Aristotle introduces the new mode of reasoning shows that he did not take the collection of terms to be extensional in this way. Given that the collection aims to track the asymmetric relation of natural predication, G cannot be included in the class of terms that hold of E, since the latter is predicated of G. Hence, the usual way of establishing the universal thesis 'A holds of all E' is not available. Nevertheless, Aristotle wishes to find a way in the framework of his heuristic method to register the extensional truth that A is predicated of every particular of which E is predicated. He does so by introducing a new mode of reasoning 'from a hypothesis' distinct from the usual direct way of establishing 91 Barnes 1970, 137. More precisely, in the Topics, A holds of B alone (μόνῳ ὑπάρχει) just in case A is both predicated of B and coextensive with B (see Smith 1997, 62). a-propositions. Unlike the direct way, the new mode does not establish natural predications, but merely extensional truths. Aristotle emphasizes that the new mode of reasoning proceeds in 'another manner' (ἄλλον τρόπον, 45b22) from the modes of direct reasoning described in 1.28.92 Thus, he indicates that the a-propositions obtained by means of this mode should be listed separately from the a-propositions gathered in the original collection of terms based on natural predication. For example, if the a-proposition 'All triangles have 2R' corresponds to a natural predication, it will be included in the collection described in 1.27. Its converse, 'Whatever has 2R is a triangle' , will therefore not be included in this collection. Nor can this converse be deduced from the collection by means of a direct deduction or by means of reductio ad impossibile. Instead, it can be derived from propositions in the original collection by means of the new mode of reasoning 'from a hypothesis' .93 However, the converse proposition should not simply be added to the a-propositions in the original collection, but should be listed separately. In this way, Aristotle is able in Prior Analytics 1.27-9 to distinguish a-propositions that correspond to a natural predication from those that are true merely extensionally. If he did not do so but included both kinds of a-proposition indiscriminately in the collection of facts, his heuristic method would not allow us to discern demonstrations from deductions, or indemonstrable propositions from demonstrable ones. The success of Aristotle's method thus relies on distinguishing these two kinds of a-proposition in the collection of facts (ἱστορία).
There is evidence that a similar distinction was drawn by some ancient mathematicians in their treatment of what they called 'converting theorems' , that is, pairs of propositions of the form 'All As are B' and 'All Bs are A' . A prominent example of converting theorems is found in Euclid's Elements 1.5 and 1.6: Proposition 1.5. In isosceles triangles the angles at the base are equal to one another…. 92 The new mode of reasoning is expressed by means of a potential optative (παντὶ ἂν τῷ Ε τὸ Α ὑπάρχοι, 45b25). This is the only occurrence of a potential optative in APr. 1.27-30. Thus, the new mode of reasoning at 45b21-8 is presented with an element of caution and uncertainty that is absent from Aristotle's exposition of the other modes of reasoning in 1.27-9 (cf. Kühner and Gerth 1904, 477 §576a). 93 This is in accordance with Mendell's (1998, 200-1) observation that, for Aristotle, the converse of a scientific a-proposition cannot be demonstrated by means of a deduction in the three syllogistic figures but requires an alternative method of proof 'which is not syllogistic' . However, the specific method of proof suggested by Mendell differs from the one described by Aristotle at APr. 1.29, 45b21-8.
Proposition 1.6. If in a triangle two angles are equal to one another, the sides subtending the equal angles will also be equal to one another. τῶν ἰσοσκελῶν τριγώνων αἱ πρὸς τῇ βάσει γωνίαι ἴσαι ἀλλήλαις εἰσίν….
ἐὰν τριγώνου αἱ δύο γωνίαι ἴσαι ἀλλήλαις ὦσιν, καὶ αἱ ὑπὸ τὰς ἴσας γωνίας ὑποτείνουσαι πλευραὶ ἴσαι ἀλλήλαις ἔσονται. Euclid, According to Proclus, the first proposition is of the form 'All As are B' , where 'A' stands for isosceles triangle and 'B' for having two equal angles; the latter proposition is the converse of the former and takes the form 'All Bs are A' .94 With respect to this pair of propositions, Proclus writes: Among converting theorems, they [i.e., geometers, 252.5-6] are accustomed to call some 'leading theorems' and the others 'converses' . For when they posit a genus and demonstrate its property, they call this a leading theorem; but when, in reverse order, they make the property a hypothesis and the genus of which the property holds a conclusion, they call such a theorem a converse. 'Every isosceles triangle has its base angles equal' is a leading theorem, for its subject is what is by nature prior, namely, the genus itself, isosceles triangle. But 'Every triangle having two angles equal also has the subtending sides equal and is isosceles' is a converse, for it exchanges the subject and its attribute, positing the latter and proving the former from it. αὐτῶν δὲ τῶν ἀντιστρεφόντων θεωρημάτων τὰ μὲν εἰώθασι καλεῖν προηγούμενα, τὰ δὲ ἀντίστροφα. ὅταν μὲν γὰρ ὑποθέμενοί τι γένος ἀποδεικνύωσι τὸ περὶ αὐτὸ σύμπτωμα, προηγούμενον τοῦτο λέγουσιν, ὅταν δὲ ἀνάπαλιν ὑπόθεσιν μὲν ποιῶνται τὸ σύμπτωμα, συμπέρασμα δὲ τὸ γένος, ᾧ τοῦτο συμβέβηκεν, ἀντίστροφον τὸ τοιόνδε προσαγορεύουσι. πᾶν ἰσοσκελὲς τρίγωνον ἴσας ἔχει τὰς πρὸς τῇ βάσει, τοῦτο προηγούμενον· ὑπόκειται γὰρ τὸ τῇ φύσει προηγούμενον, λέγω δὴ τὸ γένος αὐτὸ τὸ ἰσοσκελὲς τρίγωνον. πᾶν τρίγωνον δύο γωνίας ἴσας ἔχον καὶ τὰς ὑποτεινούσας πλευρὰς ἴσας ἔχει καί ἐστιν ἰσοσκελές, τοῦτο ἀντιστρέφον· ἐναλλάττει γὰρ τὸ ὑποκείμενον καὶ τὸ τούτου πάθος, καὶ τὸ μὲν ὑποτίθησι, τὸ δὲ ἀπὸ τούτου δείκνυσι. Proclus, In this passage, Proclus states that, by nature (τῇ φύσει), the genus isosceles triangle is a subject (ὑποκείμενον) whereas the property of having two equal angles is an attribute (πάθος). Euclid's 1.5 is a leading theorem on the grounds that it conforms to the natural order of subject and attribute, proving the attribute of the subject. By contrast, 1.6 is not a leading theorem but a converse because it inverts this natural order, establishing the subject on the basis of the attribute.95 This account of leading theorems and converses resembles Aristotle's distinction between natural and unnatural predications. Aristotle maintains that a statement 'A is B' expresses a natural predication only if A is a genuine subject (ὑποκείμενον). Thus, for example, while 'The log is pale' expresses a natural predication with log as a subject, its converse, 'The pale thing is a log' , does not express a natural predication because the pale thing is not a genuine subject (ὑποκείμενον).96 Similarly, for Aristotle, fire is a subject of which lightest body is predicated but not vice versa.97 In the same vein, triangle is a subject of which 2R is predicated but not vice versa.98 Thus, although Aristotle does not use Proclus' terminology of 'hypothesis' and 'conclusion' in his account of natural predication, he agrees with Proclus that Euclid's 1.5 corresponds to a natural predication whereas its converse does not.
Proclus attributes the distinction between leading theorems and converses to some geometers. We do not know who these geometers were and what role the distinction played in their work. Still, Proclus' remarks show that some ancient mathematicians distinguished between leading theorems and converses in a way akin to Aristotle's distinction between natural and unnatural predications. The distinction is crucial to Aristotle's heuristic method, which relies on a collection of scientific facts that includes leading theorems but not their converses.
For Aristotle, the converse of 'All triangles have 2R' does not have the status of a scientific proposition. But this still leaves open the question of whether this converse is true. The answer will depend on what the truth-conditions of a-propositions are taken to be. While Aristotle does not give a detailed account 95 In Euclid's text, the distinction between leading theorems and converses may be reflected in the fact that he uses different formulations in 1.5 and 1.6, expressing the former by a simple declarative sentence (τῶν ἰσοσκελῶν τριγώνων αἱ πρὸς τῇ βάσει γωνίαι ἴσαι ἀλλήλαις εἰσίν) and the latter by a conditional (ἐὰν τριγώνου αἱ δύο γωνίαι ἴσαι ἀλλήλαις ὦσιν, καὶ αἱ ὑπὸ τὰς ἴσας γωνίαις ὑποτείνουσαι πλευραὶ ἴσαι ἀλλήλαις ἔσονται). The same difference appears in the formulations of the leading theorem in 1.47 and its converse in 1.48. 96 APo. 1.22, 83a1-14. 97 Top. 5.4, 132b19-24 (discussed in Section 4 above). 98 APo. 2.2, 90a12-14; see Bronstein 2016, 173-7. of their truth-conditions, he offers a brief explanation of what he means by phrases such as 'holds of all' in the first chapter of the Analytics (24b26-30). This explanation, which is known as the dictum de omni, is open to different readings. On one reading, the truth-conditions of a-propositions are extensional: AaB is true just in case A is predicated of every particular of which B is predicated. On this reading, the a-proposition 'All triangles have 2R' and its converse are both true, but the latter is excluded from the collection of facts in Prior Analytics 1.27 because it does not correspond to a natural predication. Thus, there are true a-propositions constructed from the terms of a science that are not included in the science's collection of facts, and hence do not have the status of a scientific proposition. They lack this status even if they are true of necessity. On another reading of Aristotle's dictum de omni, known as the 'heterodox' reading, the truth-conditions of a-propositions are not extensional but more abstract.99 Thus, AaB is true just in case A stands to B in relation R, where R is some reflexive and transitive relation between terms.100 This condition is abstract in that it allows for different choices of R in different applications of Aristotle's syllogistic theory. For example, in one application, R might be taken to be the reflexive and transitive closure of natural predication.101 In this case, the a-proposition 'All triangles have 2R' is true, whereas its converse is false. All direct deductions and deductions by reductio discussed in Prior Analytics 1.28-9 preserve truth on this version of the heterodox reading. But the new mode of reasoning 'from a hypothesis' introduced in 1.29 does not preserve truth, since the a-propositions inferred by it may be of an extensional nature not corresponding to a natural predication. In order for this mode of reasoning to preserve truth, Aristotle would need to adopt an alternative version of the heterodox reading, e.g., one in which R is taken be the relation of extensional inclusion.102 Whatever the truth-conditions of a-propositions are taken to be, though, it should be noted that the collection of facts in Prior Analytics 1.27 does not include all true a-propositions constructed from the terms of the science under consideration. This is because Aristotle takes all reflexive a-propositions to be 99 On the 'heterodox' reading of the dictum de omni, see Barnes 2007, 386-412;Morison 2008, 212-15;2015, 131-9;Malink 2013, 45-72. 100 Smyth 1971Malink 2013, 73-85;Vlasits 2019, 2-11. 101 Malink 20092013, 128-30. 102 This would also account for Aristotle's claim in APo. 1.13 that the a-proposition 'Whatever does not twinkle is near' is true (ἀληθές, 78a32). true: 'A holds of all A' is true for any A.103 But the relation of natural predication is asymmetric and hence irreflexive: nothing is predicated of itself. Or, as Philoponus puts it, in any natural predication the predicate must be distinct from the subject (in APo. 246.18-24 Wallies). Hence, reflexive a-propositions do not have the status of scientific propositions and are not included in the collection of facts in Prior Analytics 1.27, even if they are true of necessity. This is as it should be. For, every scientific proposition serves either as a premise or as the conclusion of a demonstration; but Aristotle makes it plain that reflexive a-propositions cannot play either role in a demonstration.104 Thus, natural predication does not demarcate the class of true a-propositions constructed from the terms of a science, but the class of scientific a-propositions (i.e., the class of those a-propositions that are either indemonstrable premises of the science or demonstrable from such premises in Aristotle's syllogistic theory). Natural predication is a relation between beings (ὄντα).105 Smith argues that these beings can be individuated extensionally in Aristotle's theory of demonstration: if A and B are coextensive, they are identical.106 Thus, for example, the being signified by 'triangle' is identical with the being signified by '2R' . Such a view, however, cannot be plausibly attributed to Aristotle. For, as we have seen, he regards these beings as distinct, the former being predicated of 103 Aristotle holds that every reflexive o-proposition is false, and hence that every reflexive a-proposition is true; APr. 2.15, 64b7-13 together with 64a4-7, 64a23-30 (see Thom 1981, 91-2). Similarly, Alexander, in APr. 34.18-20 Wallies. 104 If AaA were a premise of a demonstration, it would be a premise of a deduction in the three syllogistic figures. The conclusion of this deduction would be identical with the other premise, violating Aristotle's requirement that the conclusion of a deduction be distinct from each of the premises (APr. 1.1, 24b19; Top. 1.1, 100a25-6; see Alexander,Ammonius, Wallies; Barnes 2007, 487-90;Striker 2009, 80). On the other hand, if AaA were the conclusion of a demonstration, it would be demonstrable via Barbara through a middle term B distinct from A. Given Natural a-predication, A and B would be predicated of one another, violating Aristotle's prohibition on counterpredication. Accordingly, Aristotle regards the question 'Why is it the case that AaA?' as trivial and hence not a suitable subject of inquiry in a demonstrative science: 'to inquire why a thing is that thing is to inquire into nothing at all' (τὸ μὲν οὖν διὰ τί αὐτό ἐστιν αὐτό, οὐδέν ἐστι ζητεῖν, Metaphysics Z 17, 1041a14-15). 105 Aristotle refers to them as ὄντα (APr. 1.27,43a25,43a32;1.30,46a16) or πράγματα (1.27,43b12;1.30,46a25). 106 Smith 1984, 594;1986, 63;1996, 27. Smith adopts this view in order to exclude cases in which there are distinct beings A and B such that both AaB and BaA are scientific propositions. On the present account, such cases are excluded, not by identifying coextensive beings, but by treating them as distinct relata of an asymmetric, non-extensional relation of predication. the latter but not vice versa.107 This distinctness is crucial to his principle of Demonstrative predication, according to which every chain of predications gives rise to a demonstration.108 Thus, the beings countenanced by Aristotle are not individuated extensionally but in a more fine-grained way.
They are universals that cannot be reduced to their extension, but have a nature in their own right that goes beyond the particulars of which they are predicated. Aristotle hastens to add in the Posterior Analytics that they are not Platonic forms (1.11, 77a5-6; 1.22, 83a32-4).109 Still, he is clear that his theory of demonstration requires the existence of universals (77a6-9).110 It is their non-extensional nature that enables these universals to support the asymmetric relation of natural predication in Aristotle's theory of demonstration.

Negative Demonstration
Aristotle's heuristic method is designed to work not only for affirmative propositions but for negative ones as well. In Prior Analytics 1.27, he instructs us to collect, for any given subject A, those terms which 'cannot hold of it' (43b4-5). These terms give rise to a collection of universal negative propositions BeA ('B holds of no A'). In the context of a science, this collection includes all and only scientific e-propositions.
Among scientific e-propositions some are demonstrable and others are not. Indemonstrable e-propositions are discussed by Aristotle in Posterior Analytics 1.15, where he characterizes them as 'atomic' in the sense that there is no middle term through which they can be demonstrated:111 Just as it is possible for A to hold of B atomically, so it is also possible for it atomically not to hold. By atomically holding or not holding I mean that there is no middle term between them; for, in this case, they no longer hold or do not hold in virtue of something else.112 ὥσπερ δὲ ὑπάρχειν τὸ Α τῷ Β ἐνεδέχετο ἀτόμως, οὕτω καὶ μὴ ὑπάρχειν ἐγχωρεῖ. λέγω δὲ τὸ ἀτόμως ὑπάρχειν ἢ μὴ ὑπάρχειν τὸ μὴ εἶναι αὐτῶν μέσον· οὕτω γὰρ οὐκέτι ἔσται κατ' ἄλλο τὸ ὑπάρχειν ἢ μὴ ὑπάρχειν. APo. 1.15, Aristotle goes on to provide a criterion for distinguishing demonstrable scientific e-propositions from indemonstrable, or 'primitive' , ones: When either A or B is in something as in a whole, or both are, then it is not possible for A to primitively [i.e., indemonstrably] not hold of B. For, let A be in C as in a whole. If, then, B is not in C as in a whole…, there will be a deduction that A does not hold of B. For if C holds of all A and of no B, then A holds of none of the Bs. Similarly, if B is in something as in a whole, e.g., in D; for D holds of all B and A of none of the Ds, so that A will hold of none of the Bs through a deduction. ὅταν μὲν οὖν ἢ τὸ Α ἢ τὸ Β ἐν ὅλῳ τινὶ ᾖ, ἢ καὶ ἄμφω, οὐκ ἐνδέχεται τὸ Α τῷ Β πρώτως113 μὴ ὑπάρχειν. ἔστω γὰρ τὸ Α ἐν ὅλῳ τῷ Γ. οὐκοῦν εἰ τὸ Β μή ἐστιν ἐν ὅλῳ τῷ Γ…, συλλογισμὸς ἔσται τοῦ μὴ ὑπάρχειν τὸ Α τῷ Β· εἰ γὰρ τῷ μὲν Α παντὶ τὸ Γ, τῷ δὲ Β μηδενί, οὐδενὶ τῶν Β τὸ Α. ὁμοίως δὲ καὶ εἰ τὸ Β ἐν ὅλῳ τινί ἐστιν, οἷον ἐν τῷ Δ· τὸ μὲν γὰρ Δ παντὶ τῷ Β ὑπάρχει, τὸ δὲ Α οὐδενὶ τῶν Δ, ὥστε τὸ Α οὐδενὶ τῶν Β ὑπάρξει διὰ συλλογισμοῦ.114 APo. 1.15, 79a36-b4 111 Similarly, APo. 1.23, 84b24-31;cf. Malink 2017, 176-81. 112 Translations from APo. 1.15 follow Barnes 1993, 22, with minor modifications. 113 Here, πρώτως is used interchangeably with ἀτόμως (79a33) to denote indemonstrable (or, immediate) scientific propositions; Philoponus, in APo. 187.11-13 Wallies;Pacius 1605, 451;Tredennick 1960, 94-5. Cf In this passage, Aristotle uses the phrase 'A is in C as in a whole' to express the a-proposition CaA, and the phrase 'B is not in C as in a whole' to express the e-proposition CeB.115 He asserts that AeB is demonstrable if there is either a C such that CaA and CeB are scientific propositions or a D such that AeD and DaB are scientific propositions. For if there is such a C or D, then AeB is derivable from these propositions 'through a deduction' , i.e., through a demonstration.116 Thus Aristotle endorses a principle for scientific e-propositions analogous to the principle of Demonstrative predication for scientific a-propositions. Just as any pair of scientific propositions AaC and CaB gives rise to a demonstration of AaB, any pair of scientific propositions AeC and CaB (or CaA and CeB) gives rise to a demonstration of AeB: Demonstrative e-predication: If AeC and CaB, or CaA and CeB, are scientific propositions, there is a demonstration of AeB using C as a middle term.
Conversely, if an e-proposition is demonstrable, there are scientific propositions of the form just described, since this is the only way to derive an e-conclusion in Aristotle's syllogistic theory.117 Thus, we have a necessary and sufficient condition for the indemonstrability of e-propositions: a scientific proposition AeB is indemonstrable just in case there is no C such that either AeC and CaB or CaA and CeB are scientific propositions. Given a complete collection of scientific e-and a-propositions, this criterion allows us to identify all and only indemonstrable e-propositions.118 Once the indemonstrable e-propositions have been identified, we can construct demonstrations of the demonstrable ones. Any two-premise deduction from indemonstrable e-and a-propositions is a demonstration. If the deduction has more than two premises, the middle terms will have to be employed in a certain order. For example, to derive AeB from indemonstrable premises AeC, CaD, and DaB, Aristotle may instruct us to use C before D, deriving AeD before inferring the desired conclusion, AeB.119 Following these instructions, we will be able to produce all demonstrations of demonstrable e-propositions.
In addition to universal propositions, Aristotle's syllogistic theory deals with particular propositions of the form AiB ('A holds of some B') and AoB ('A does not hold of some B'). The collection of premises described in Prior Analytics 1.27 does not contain any particular propositions but only universal ones (43b11-17). Aristotle explains in 1.28 how particular propositions can be derived from the universal premises gathered in the collection.120 This fits with his account of demonstrative science in the Posterior Analytics. As we have seen, Aristotle includes both a-and e-propositions among the indemonstrable premises of a science. He does not mention any indemonstrable i-or o-propositions in the Posterior Analytics, although he acknowledges that they can appear as demonstrable theorems.121 Presumably, he thinks that all scientific i-and o-propositions can be demonstrated from a-and e-propositions. He does not specify how exactly these demonstrations proceed and there are different ways of spelling out the details; but once those details have been spelled out, we will be able to construct demonstrations of all scientific i-and o-propositions from indemonstrable a-and e-premises.
When Aristotle's heuristic method is applied to a science, the complete collection of facts (ἱστορία) includes all and only scientific a-and e-propositions. Given that natural predication is asymmetric, the collection does not contain any pair of a-propositions AaB and BaA. More generally, it does not contain any cycles of the form AaB 1 , B 1 aB 2 ,…, B n aA.122 Consequently, the collection does not give rise to any deductive cycles in which propositions are deduced from explaining) is what makes us know that they [the indemonstrable principles] are principles' . However, there is no textual evidence for such a reading of 46a17-27. On the present account, the method described by Aristotle in APr. 1.27-30 allows us to identify all indemonstrable a-and e-propositions in a complete collection of scientific facts without constructing any demonstrations. 119 Cf n. 62. 120 APr. 1.28, 43b43-44a2, 44a9-11, 44a19-21, 44a28-30. 121 APo. 1.21, 82b21-8;1.23 85a7-12;see Malink 2020, 103 n. 43. 122 This follows from the asymmetry of predication by means of the principles of Natural a-predication and Demonstrative predication. one another. More precisely, the collection does not contain any propositions φ and ψ such that (i) φ can be deduced from ψ and some other propositions in the collection and (ii) vice versa. Instead, whenever one proposition in the collection can be deduced from others, each of the latter propositions is prior in nature to the former.123 Thus, deductions between propositions in the collection proceed in the order of priority in nature: in every deduction between propositions in the collection, the premises are prior in nature to the conclusion. This allows us, finally, to respond to Striker's objection to Aristotle's method. Recall, 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). Striker 2009, 207 Striker's criticism may be correct for a collection of premises in dialectic. But it is not correct for a collection of scientific a-and e-propositions. In such a collection, deducibility does tell us which propositions are prior and which posterior in the natural order of explanation. In particular, if the collection is complete, the indemonstrable principles of the science are exactly those propositions that are not deducible from other propositions in the collection. In a collection of scientific a-and e-propositions, every deduction from indemonstrable premises either is a demonstration or can be turned into a demonstration simply by changing the order of the middle terms. Thus, anyone deducing in such a collection from indemonstrable principles has succeeded in giving an explanatory scientific deduction. This is corroborated by Aristotle's remarks in Rhetoric 1.2 on the selection of premises. If the premises selected by a dialectician or rhetorician happen to be principles of a science, Aristotle writes, the deductions constructed from them will no longer be dialectical or rhetorical, but will converge upon scientific demonstrations:124 The better one selects the premises, the more one will have unwittingly produced a kind of knowledge different from dialectic and rhetoric. For whenever one hits upon the principles of a science, it will no longer be dialectic or rhetoric but the science of which one has the principles. ταῦτα δὲ ὅσῳ τις ἂν βέλτιον ἐκλέγηται τὰς προτάσεις,125 λήσει ποιήσας ἄλλην ἐπιστήμην τῆς διαλεκτικῆς καὶ ῥητορικῆς· ἂν γὰρ ἐντύχῃ ἀρχαῖς, οὐκέτι διαλεκτικὴ οὐδὲ ῥητορικὴ ἀλλ' ἐκείνη ἔσται ἧς ἔχει τὰς ἀρχάς. Rhetoric 1.2, 1358a23-6 In this passage, Aristotle considers a reasoner who selects premises (ἐκλέγηται τὰς προτάσεις) to deduce a given conclusion. Similarly, the heuristic method presented in Prior Analytics 1.27-30 is a tool for selecting premises (τὰς προτάσεις ἐκλέγειν, 1.30 46a28-9).126 In both passages, Aristotle maintains that, when the premises selected by the reasoner are indemonstrable scientific propositions, the resulting deduction will converge upon a demonstration. As we will see, the same view is expressed by Aristotle in his discussion of the selection of premises in Posterior Analytics 2.14.

Division
The application of Aristotle's method in Prior Analytics 1.30 is based on a collection of natural predications in a science. The heuristic value of the method therefore depends on the extent to which we are able to identify these predications. Given that predication is not extensional, how can we tell whether a given term is predicated of another? Aristotle is largely silent on this question in the Prior Analytics, as his focus in this part of the treatise is on the deductive aspects of the method. However, he offers some guidance on the question in Posterior Analytics 2.14, where he advises us to select premises on the basis of genus-species trees obtained by division. In the opening sentence of the chapter, Aristotle writes: In order to deal with problems, one must select from dissections and divisions.
The 'problems' referred to in this passage are theses to be established by a scientist. Aristotle's aim is to lay out a procedure for selecting premises to establish such theses.128 Thus, the discussion in Posterior Analytics 2.14 pertains directly to the heuristic method developed in Prior Analytics 1. 27-30.129 Since the focus in 2.14 is on scientific problems rather than on dialectical ones, the premises selected ought to be useful for establishing not only that a given thesis holds, but why it holds (διὰ τί, 98a7, 11, 18).130 Aristotle encourages us to select such premises by considering 'dissections and divisions' . As Pseudo-Philoponus puts it, 'in order to have the causes, i.e., the middle terms through which problems are demonstrated, one must select these causes from dissections and divisions .
The requisite divisions (διαιρέσεις) are those by which a genus is divided into species by means of differentiae. Aristotle has discussed these divisions in the preceding chapter, highlighting their usefulness for the discovery of definitions and providing rules for the correct division of a genus (2.13, 96b25-97b6). In the present chapter, Aristotle relies on this account of division.131 Scholars differ as to whether the 'dissections' (ἀνατομαί) are identical to these divisions or whether they constitute a different kind of scientific analysis.132 At any rate, Aristotle's focus in the remainder of 2.14 is on division. Given a division of a genus into species and subspecies, Aristotle instructs us to select premises as follows: 128 The phrase ἔχειν τὰ προβλήματα at 98a1 may be taken to refer to our ability to solve problems by finding suitable premises to establish the thesis under consideration (Pseudo-Philoponus, in APo. 417.1-6 Wallies; von Kirchmann 1878, 173; Barnes 1993, 250). Alternatively, the phrase might be taken to refer to our ability to formulate problems properly (Zabarella 1608(Zabarella , 1223aRoss 1949, 662-3). Quarantotto (2017, 106-7) argues that APo. 2.14 offers guidance on both the proper formulation and the solution of problems. Whatever the precise meaning of ἔχειν τὰ προβλήματα at 98a1, it is clear that the bulk of APo. 2.14 provides procedures for solving problems (Lennox 2015, 50-1;Stein 2021, 40-1). 129 Lennox 1987, 97-8;1991, 272 and 276;2015, 44-6;2021, 51-6;Ferejohn 1991, 29-31;Barnes 1993, 250. The connection between APo. 2.14 and APr. 1.27-30 is corroborated by the fact that these are the only parts of the Analytics in which Aristotle uses the terms ἐκλέγειν and ἐκλογή (see nn. 126, 127, and 133). 130 Accordingly, the scientific problems (προβλήματα, 98a1) under consideration in APo. 2.14-15 may take the form of why-questions, 'Why P?' , where P is the thesis to be established in a given context (98a24-34; see Lennox 1994, 58-70;, 44-51). 131 Ferejohn 1991Charles 2000, 239-41. 132 For the first option, see Pacius 1605, 536;Ross 1949, 663-4;Mignucci 2007, 294; for the latter option, see Mure 1928, ad loc. Some commentators take ἀνατομαί at 98a2 to refer specifically to anatomical descriptions of animals based on dissections (Lennox 1987, 97 n. 16;2021, 52 n. 27;Barnes 1993, 250). However, this reading seems too narrow, as Aristotle's point in APo. 2.14 is not limited to biology, but is meant to apply to all sciences (Ross 1949, 663-4).
One must select as follows: positing the genus common to all things, e.g., if the objects of study are animals, select those things that hold of all animal. Having gathered these, next select those that follow all of the first of the remaining kinds, e.g., if this is bird, select those things that follow all bird; and in this way select always the things that follow the proximate kind. It is clear that we will now be able to state why the things that follow hold of those that fall under the common kind, e.g., why they hold of man or of horse. Let A be animal, B what follows all animal, and C, D, E the particular animals [i.e., the species of animal]. Then it is clear why B holds of D: it does so because of A. Likewise, it will be clear why it holds of the others. And the same account will apply in the case of the subordinate kinds.
APo. 2.14, 98a2-12 In this passage, A is the genus under consideration, D is one of its subspecies obtained by division, and B is one of things that hold of all A.134 Thus, B is predicated of A, and the latter is predicated of D. Aristotle states that any such choice of terms gives rise to a deduction in Barbara establishing not only that B holds of all D, but also explaining why it does. If B's holding of A is not indemonstrable, a full explanation of BaD will appeal to further middle terms explaining why B holds of all A.135 In any case, there will be a demonstration of BaD using A as a middle term (possibly along with other middle terms).136 Thus Aristotle is here relying on the principle of Demonstrative 133 Reading ἐκλέγειν at 98a2 (Bekker 1831, Ross 1949) rather than διαλέγειν (Waitz 1846). 134 It is sometimes thought that Aristotle's aim in APo. 2.14 is to identify predicates that are coextensive with their subject (Kullmann 1974, 197;Lennox 1991, 272-5;1994, 64 n. 21;Quarantotto 2017, 107 Lennox 2015, 47-8). 135 Tugendhat 2003, 128. 136 Pseudo-Philoponus, in APo. 418.8-11 Wallies;Ferejohn 1991, 30-1 and 125-6;pace Lennox 2015, 47. predication, according to which any chain of natural predications gives rise to a demonstration in Barbara. As in Prior Analytics 1.30, this principle underlies his heuristic method for selecting premises that yield not only deductions but explanatory demonstrations. Since any genus is predicated of its species, the division of a genus into species and subspecies gives rise to a structure of natural predications (1.22, 83a39-b5). In the passage from 2.14 just quoted, Aristotle explains how these genus-species structures can be used in a systematic manner to construct demonstrations answering why-questions. In doing so, he relies on what David Charles calls an 'explanation-involving conception of genus, species, and differentiae' .137 This explanation-involving conception ensures that the natural predications obtained by means of division obey the principle of Demonstrative predication.
Of course, the method of division cannot deliver all natural predications needed in a science. For example, if B is not a genus or differentia of A but one of its non-essential demonstrable attributes, such as 2R of triangle, the fact that B is predicated of A cannot be obtained by division. Aristotle is aware of this limitation. He points it out in his critique of division in Prior Analytics 1.31, when he notes that 'by means of the method of division it is not possible to reason about accidents and propria' (46b26-7).138 Nor is this the only shortcoming of the method of division in Aristotle's view, as he contends that it lacks the force of deductive necessity and therefore cannot account for how demonstrable theorems follow necessarily from indemonstrable principles (46a31-b25). Thus, Aristotle writes at the beginning of Prior Analytics 1.31: It is easy to see that division by means of genera is only a small part of the method just described; for division is a sort of weak deduction. ὅτι δ' ἡ διὰ τῶν γενῶν διαίρεσις μικρόν τι μόριόν ἐστι τῆς εἰρημένης μεθόδου, ῥᾴδιον ἰδεῖν· ἔστι γὰρ ἡ διαίρεσις οἷον ἀσθενὴς συλλογισμός. APr. 1.31, Here, 'the method just described' is the heuristic method developed in Prior Analytics 1. 27-30.139 Aristotle regards this method as superior to that of 137 Charles 2000, 239-43. 138 See Alexander, in APr. 334.8-10, 338.11-15 Wallies;Philoponus, in APr. 307.35-308.4, 313.6-15 Wallies;Striker 2009, 211;Crubellier 2014, 299;2017, 115;Vlasits 2021, 276. 139 Pacius 1605Crubellier 2017, 108 n. 13;Vlasits 2021, 281. Alexander (in APr. 333.18-21 Wallies) entertains the possibility that the μέθοδος referred to at 46a32 is not the heuristic method of APr. 1.27-30, but more generally the syllogistic theory developed in 1. 1-26. division. Still, he acknowledges that the latter is a part of the former, albeit a small one. While he does not explain in 1.27-30 what role division might play in the heuristic method, he does so in Posterior Analytics 2.14. There he makes it clear that division contributes to the heuristic method by delivering at least some natural predications, namely, essential predications of genera and differentiae, and also by structuring the process of collecting the other natural predications. This is what he means when he refers to division as a 'part' of the heuristic method in Prior Analytics 1.31.140 Presumably this is also why he finds it fitting to conclude his presentation of the heuristic method with a discussion of division in 1.31.
In Prior Analytics 1.30, Aristotle considers an application of the heuristic method in which the collection of facts is complete in that 'nothing has been left out' (46a24-5). As we have seen, division can deliver some but not all natural predications in this collection. Aristotle does not specify a procedure for gathering the remaining natural predications. Doing so would require a precise demarcation of what counts as a natural predication. Aristotle's discussion of natural predication in Posterior Analytics 1.22 is intricate and subject to considerable scholarly debate. For example, David Bostock takes Aristotle in 1.22 to maintain that, if A is the subject of an essential predication, then A is predicated essentially of every particular of which A is predicated.141 Other scholars attribute to Aristotle a more liberal conception of natural predication on which this condition does not hold.142 It is beyond the scope of this paper to enter into a discussion of these issues. What is important for present purposes is the central role natural predication plays in Aristotle's heuristic method based on the principles of Natural a-predication and Demonstrative predication.
When Aristotle refers to a complete collection of facts in a science, he seems to regard this as an ideal case that is in principle possible but not likely to occur in practice.143 Consider, for example, the collection of zoological facts presented in the Historia animalium. It is generally agreed that this collection is organized in accordance with the guidelines laid down in Prior Analytics 1.27-30 and Posterior Analytics 2.13-14.144 At the same time, it is clear that the collection in the Historia animalium is incomplete and falls short of the Analytics' model in various respects.145 This does not mean, however, that Aristotle's heuristic method is not applicable to this collection. For, when we are dealing with an incomplete collection of scientific facts, we may still use the method to identify preliminary principles that cannot be derived from propositions included in this collection, and then further scrutinize them as to whether or not they are in fact indemonstrable. Alternatively, we may use the method in a more indirect manner by hypothetically adding a candidate scientific proposition to an incomplete collection of facts for the purpose of testing what demonstrations it gives rise to. If it gives rise to a putative demonstration that we have independent reasons to reject, perhaps because we believe that it is not explanatory of the conclusion, we will exclude the candidate proposition from the collection of facts. Similarly, if we are confronted with a deficient collection that contains both an a-proposition and its converse, the method helps us to determine which of these two propositions should be eliminated, by clarifying which demonstrations each of them gives rise to.
Of course, Aristotle does not discuss such applications of the heuristic method in Prior Analytics 1.27-30. His aim in these chapters is to give a general description of the method, and he does so largely by abstracting from the specific epistemic contexts in which it may be applied. He chooses to illustrate the power of the method by considering the ideal scenario of a complete collection of facts (46a24-7). Nonetheless, the method can be employed in a wide variety of other, less ideal epistemic scenarios. The application will be more complicated in such cases, but its core remains the same, consisting of the three principles stated by Aristotle in the Posterior Analytics: Natural a-predication, Demonstrative predication, and Demonstrative e-predication. 143 This is suggested by his use of the optative εἰ γὰρ μηδὲν κατὰ τὴν ἱστορίαν παραλειφθείη at APr. 1.30, 46a24-5 (see n. 23). 144 Lennox 1987, 100-9;1991, 263-88;Charles 2000, 326-30. For contrasting views on the extent to which the Historia animalium conforms to different aspects of the Analytics' model, see Charles 1990and Lennox 1990. 145 See, e.g., Balme 1987Charles 2000, 326 and 330.

Conclusion
I began this article with a quotation from the Byzantine Logica et quadrivium, according to which Aristotle's heuristic method rests on 'a genuinely profound and most scientific theorem of great import, which encompasses in a small space nearly all of philosophy' (1.39, 32.8 Heiberg). Whether or not the method encompasses 'all of philosophy' , it spans large portions of Aristotle's logical theory. It is based on the deductive theory of the categorical syllogism, which is put to work in Prior Analytics 1.28-9. The method is applicable in both dialectic and demonstrative science. The latter application relies on Aristotle's theory of natural predication, briefly introduced in 1.27 and further elaborated in Posterior Analytics 1.19-22. Some of the requisite natural predications can be obtained by means of division, as indicated in Posterior Analytics 2.14 and Prior Analytics 1.31. All of these elements contribute to the method's success as a tool for finding demonstrations and identifying the indemonstrable principles of a science.
A crucial element of the method is the principle of Demonstrative predication, according to which every chain of natural predications gives rise to a demonstration in Barbara. Aristotle states this principle in Posterior Analytics 1.22 and reaffirms it in 2.14. A variant of this principle, Demonstrative e-predication, is stated in 1.15. Both principles are based on the assumption that natural predication is asymmetric. Together with Natural a-predication, they allow us to identify all indemonstrable propositions in a complete collection of scientific a-and e-propositions.
Natural predications are, as Jonathan Lear puts it, 'predications which reveal metaphysical structure' .146 They do not obtain between linguistic expressions but between beings (ὄντα, 43a25). Crucially, these beings include not only particulars but universals of a non-extensional nature not reducible to the class of particulars of which they are predicated. In addition to natural predication, the particulars and universals countenanced by Aristotle are related by what we may call natural incompatibility, the relation that obtains between beings A and B just in case AeB is a scientific proposition. Thus, the metaphysical structure underlying Aristotle's theory of demonstration consists of a domain of beings endowed with the relations of natural predication and natural incompatibility. The task of demonstrative science is to map this metaphysical structure. For, as Myles Burnyeat has noted, a demonstration, for Aristotle, 'is not just a preferred ordering of humanly constructed knowledge, but a mapping of the structure of the real' .147 In a complete collection of scientific propositions, the relations of natural predication and incompatibility are represented by a-and e-propositions, respectively. Thus, the language of Aristotle's theory of demonstration is metaphysically perspicuous in that a-and e-propositions correspond exactly to the two relations that characterize the underlying structure of beings. Moreover, Aristotle's heuristic method ensures that this language is metaphysically perspicuous in yet another respect: given a complete collection of scientific a-and e-propositions, we will be able to determine what is fundamental and what explains what. In other words, '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' (1.30, 46a25-7).148