Reasoning About Particulars in Prior Analytics B211

Consider two types of argument originating from the fact that triangles have internal angles adding up to two right angles, or, as I will abbreviate, have 2R.

Type A Type Ball triangles have 2R all triangles have 2Rall isosceles are triangles2 this (i.e., the specific figure before one) is a triangleall isosceles have 2R this has 2R

The propositions the first kind of argument form a syllogism; if one knows both the first and the second, one can draw the third conclusion by a deductively valid argument. We are inclined to think that exactly the same claims can be made about the second. We take the third proposition in type B to be deducible from the first two. And we take such a deduction to require both premises. Before one has encountered this particular figure, one may well know that all triangles have 2R, but surely one has no way of knowing that this triangle has 2R. For one does not know that this is a triangle.

Aristotle, by contrast, sees a sharp disanalogy between type A and type B—or so I will argue. He thinks that if one knows the first proposition in type B, and one does not know the second, there is nonetheless a sense in which one knows the third. And, although he is prepared to allow that one knows the third in another, fuller sense when one knows it by way of the second, he denies that even this case can be characterized as a deductively valid argument (syllogismos). Aristotle discusses the distinctive logical character of type B reasoning in a difficult chapter of the Prior Analytics (Pr.An.). In book two, chapter 21 (henceforth B21) he famously insists that if someone knows that all triangles have 2R, he thereby knows that a particular triangle has 2R—even if he has never encountered that triangle. What is not typically appreciated is that Aristotle denies the parallel claim in relation to a type A conclusion such as “all isosceles have 2R.” Why would Aristotle think that one knows about this triangle by knowing about all triangles, but not—as I will argue—think that one knows about all isosceles by knowing about all triangles?

Let me offer an example to bring out Aristotle’s intuition. Suppose you took a geometry class in which you failed to learn that there are isosceles triangles. Every triangle discussed in the class happened to be equilateral, and the teacher accidentally skipped the section of the textbook defining the kinds of triangles. You could legitimately complain that your teacher never taught you that isosceles have 2R. She failed to impart to you some important piece of geometrical knowledge. You could not make such a complaint in a type B case—it would be absurd to fault the teacher for not having acquainted you with every triangle out there. Indeed, if you tried to complain that “you never taught me that this triangle has 2R,” she might insist that she did exactly that when she taught you that all triangles have 2R. “This has 2R” is a geometrical truth—it could figure, for instance, in a proof, or a construction. If your geometry teacher is to claim to have taught you all the truths of geometry, she must claim to have taught you that this one as well. And Aristotle argues that it is possible for her to make that claim legitimately.

1 [REFERENCE REMOVED]2 We sometimes speak of isosceles trapezoids etc. but Aristotle evidently didn’t, since he gives just this example at Posterior Analytics (Po.An.) A24 86a25-6.

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Aristotle does not wish to deny that there is some kind of cognitive improvement that a person who knows that all triangles have 2R undergoes when she encounters the triangle in question and explicitly acknowledges it as having 2R. He says such a person knows that this triangle has 2R in an unqualified way, whereas the one not in perceptual contact with this triangle knows it merely in a qualified way, “through the universal.” Thus Aristotle grants that there is a sense in which the person without perceptual contact knows that “this has 2R” in an incomplete or defective way. But he denies that this defect is a failure to have knowledge. He wants to resist characterizing the change from knowing merely through the universal to knowing simpliciter as a matter of the acquisition of knowledge. He insists that the subject in question does not know more after encountering the triangle than she did beforehand. Instead, he says that in this kind of case the difference is a matter of how she knows what she knows; she knows more fully or more actively (ὡς τῷ ἐνεργεῖν 67b5). Hence the disanalogy between the two groups of propositions: it is only with type A that one can have the first and yet fully lack the third.

Aristotle is moved to draw the distinction between type A and type B as part of the project of classifying various forms of error. In B21, he seeks to show how it is possible, without contradiction, for someone to both have knowledge about and be in error about the same subject matter. One way in which this can happen is that someone knows one proposition, and has a false belief about a second proposition standing in some proximate deductive relationship to the first. In this kind of case, what one knows differs from what one falsely believes. Since the two propositions are connected via a syllogism, these are type A errors. They comprise the first part of B21. In the second and longer part (67a8ff.), Aristotle turns to cases involving perceptible particulars—type B cases. In these cases, he argues, someone can know and be in error about the very same proposition, because he knows it in one way—through the universal—and fails to know it in another way—through the particular. Aristotle uses the concept of knowing merely through the universal and not through the particular (which I will abbreviate with the phrase “universal knowledge” or “knowing universally” 3) to defend the possibility of a distinctive kind of error that arises in the context of the application of knowledge to sense-experience.

The standard reading of the passage overlooks this defense, because it fails to observe the distinction between type A and type B. The local consequence is that many details of the passage have become obscure; the global one is that we have lost sight of what makes the

3 A note on Aristotle’s terminology. Aristotle’s references to universal knowledge pick it out using a set of very similar phrases, all involving the word “καθόλου”: τὴν καθόλου ἔχειν ἐπιστήμην (67a18, 29), τῇ καθόλου οἶδε/ἐπίσταται/ θεωροῦμεν (66b32, 67a19, 27, 67b5, 11), τῷ καθόλου… ἔχειν ἐπιστήμην (67b3). The contrast class is not favored by such terminological consistency. It appears as knowing ‘through the particular’ (τῇ καθ’ ἕκαστον 67a20), knowing ‘through the knowledge proper to the case’ (τῇ δ’ οἰκείᾳ 67b3,4), and ‘actual’ knowing (τῷ ἐνεργεῖν 67b3,5,9). This variety is explained partly by the fact Aristotle’s focus is on universal knowledge, and partly by the fact that he is developing his conception of the contrast class through the argumentation of B21. He originally introduces knowing through the particular (τῇ καθ’ ἕκαστον 67a20) as the natural contrast to knowing through the universal, but eventually argues that this picks out a subclass of the contrast class, alongside which actual knowing (τῷ ἐνεργεῖν 67b3,5,9) stands as the other subclass. Hence his final word, at 67b4-5, is that we must make a tripartite distinction (τὸ γὰρ ἐπίστασθαι λέγεται τριχῶς). But the distinction between the two forms of non-universal knowledge is only relevant to the point he makes at the end of B21; the deeper distinction, for the passage as a whole, is that between knowing universally and knowing non-universally. This bipartite distinction is the only one present in the less detailed review of the same ground in Po.An.A1, where he contrasts knowing universally (καθόλου ἐπισταται, 71a28) with knowing simpliciter (ἁπλῶς 71a25-30).

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passage philosophically important. One way this emerges is that the standard interpretation has Aristotle making implausible claims about what we “know.” On the standard interpretation I have “universal knowledge” both of conclusions such as “this has 2R” and of conclusions such as “all isosceles have 2R”. More generally, on this reading, I have “universal knowledge” of any proposition entailed by any other proposition that I know. “Universal knowledge,” on the standard interpretation of it, is Aristotle’s way of asserting a form of the closure of knowledge under entailment: if S knows A, and B follows from A by a deductively valid syllogism, S knows B “universally.” It is hard to see the point of ascribing such “knowledge” to S. Hence Jonathan Barnes’ complaint that Aristotle here “foists an utterly unnatural sense upon” the word ‘knows’ (p.88). The natural thing to say about those propositions that are entailed by those I know is that I can come to know them4, not that I do know them.

Setting aside the philosophical implausibility of such a closure principle, the standard reading also rides roughshod over Aristotle’s organization of B21. Aristotle only raises the topic of universal knowledge in what I have called the second part of the passage, when he transitions to the topic of errors that involve perceptible particulars. The standard reading of universal knowledge requires us to read universal knowledge back into the first part of B21, despite the fact that it is not mentioned there.

There are many reasons to think that the organization of B21 is not accidental, and that Aristotelian ‘universal knowledge’ is in fact restricted to perceptual particulars. One of them is that the other explicit discussion of universal knowledge, namely Posterior Analytics A1, likewise elaborates the concept by way of perceptual particulars. In both places, Aristotle exclusively illustrates universal knowledge with examples whose minor premise is of the form ‘this is an X.’ I will argue that Aristotle ascribes universal knowledge to bring out the distinctive epistemic status of applied knowledge and not as a way of implausibly insisting on the closure of knowledge under simple entailment. I replace the standard conception of universal knowledge with an identity conception of universal knowledge: what I know, when I know universally, is the same content as what I know when I know simpliciter. Aristotle is asserting the identity in content between ‘all triangles have 2R’ and ‘this has 2R,’ insofar—but only insofar—as those two propositions are taken as objects of knowledge.

Why, if I am right, has B21 been so misunderstood? For one thing, readers of B21 are bereft of the usual contextual cues. As Ross notes, the chapter has “no close connection with what precedes or what follows” in the Analytics, to the point where one commentator (Maier vol. II p.434-435, n.3) hypothesized that the chapter must originally have been located elsewhere. Nor do we have a wealth of ancient responses to draw on: a single, not especially helpful, commentary has survived from antiquity.5 Another problem lies in the nature of the scholarly attention that B21 has received. B21 has somehow acquired the status of a ‘helper passage,’ a text from which we borrow a sentence or a phrase in order to shed light on a similar passage in the text we are really interested in. We have attempted to mine it for its resources rather than analyzing it as a coherent whole. I believe attention to B21 in its own right is needed to clear up 4 In fact, this is exactly what Aristotle does say when he addresses that topic in Po.An. A24 (See fn. 19 for discussion).5 The commentary on Prior Analytics B collected in Commentaria in Aristotelem Graeca 13.2 has traditionally, but hesitantly, been ascribed to Philoponus: it is more fragmentary, textually problematic, and analytically shallow than Philoponus’ commentary on A. Malink refers to the author as ‘pseudo-Philoponus.’

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confusions that stand in the way of applying its lessons about ‘universal knowledge’ and ‘induction’ and ‘Meno’s paradox’ to other passages in which these phrases arise6.

But the biggest stumbling block in the way of reading B21 is that it is one of those places in which Aristotle is assuming, as a backdrop to his argumentation, a very different way of carving up the conceptual territory from our own. He assimilates what we are inclined to keep apart, for instance the content of knowledge that all X’s are Y and the content of knowledge that this X is Y. Conversely, he sees deep distinctions where we see none at all, for instance between type A reasoning and type B reasoning. Even in places where we both inclined to draw distinctions, he draws them on a very different basis than we do. Nowhere is this last problem more pointed than in his reference to ‘induction’ at 67a3 Aristotle characterizes the difference between type A and type B in terms of the form of reasoning at stake: reasoning that moves from “all triangles” to “all isosceles” is standard, deductive reasoning (syllogismos), whereas the reasoning that moves from “all triangles” to “this (figure)” is, he says, epagōgē. We translate epagōgē as “induction,” and this translation is thought to be supported by the fact that Aristotle typically uses epagōgē to refer to reasoning that moves from particulars to universals. But the epagōgē referred to in B21, as well as in a few other passages, moves in the opposite direction. Many commentators have puzzled over why Aristotle describes the application of a universal to particulars as ‘induction.’

I offer the following answer. Aristotelian deduction is a movement of the mind from knowing one set of things (the premises) to knowing something else (the conclusion). For Aristotle specifies in the definition of a syllogism that it arrives at something other than what is stated in the premises (ἕτερόν τι τῶν κειμένων Pr.An. A1, 24b19). Induction, by contrast, is a form of reasoning by way of which nothing new is thought. It is inductive reasoning that transitions from knowing universally to knowing simpliciter, and this means that one comes to know that which one knew already in a new way. This is, from a modern point of view, quite surprising. In modern logic we would not tend to think about the distinction between deduction and induction as primarily centered around the question of whether something new is thought, but insofar as we did, we might be inclined to line them up in the opposite way. For we tend to think that it is deductive reasoning that preserves the content present in the premises, and inductive reasoning that ‘adds’ new content or “amplifies” that which is present in the premises. For us, the distinction between induction and deduction is primarily a matter of whether the mind moves up towards what is more general or down towards what is more specific; for Aristotle, it is a matter of whether the mind moves to a new content or a new manner of grasping the same content. It is

6 If we survey the literature on B21, we will find that most interpreters cite it with the hopes of elucidating either Posterior Analytics (Po.An.) A1, or Nicomachean Ethics (NE) VII.3. McKirahan, Mansion, Bronstein, Ferejohn, Fine, Hamlyn, and Engberg-Pedersen are in the first group; Grgic, Price and Gauthier-Jolif are in the second. Morison discusses B21 in relation to both Po.An. A1 and NE VII.3. Leszl reads B21 for insight into yet a third passage, Metaphysics M10. Gifford understands himself as an exception to this trend. Though he does not discuss the second half of the passage, he does actively resist the prevailing tendency to read B21 as a ‘duplicate’ (p.2) of the corresponding discussion in Po.An. A1. Gifford criticizes the ‘duplicate’ approach for attaching a nonstandard sense to epagōgē at 67a23, a move he himself avoids via an unorthodox reading of the first half of B21. Gifford takes that text to aim to validate the existence of universals against empiricists who believe there is only knowledge of the particular. Gifford’s paper, though ingenious, attributes argumentative goals to Aristotle that he never espouses, and ignores ones that he makes explicit. Aristotle’s discussion seems to take the existence of universals for granted; what it explicitly addresses is the question of how certain forms of knowledge are compatible with certain forms of error. Gifford thus does not succeed in his explicit aim to produce a treatment of B21 on “its own terms and within its own argumentative environment.” (p.2)

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in these cases that it is hardest to see eye to eye with Aristotle: his distinctions line up closely enough with some of our own to make us think we have a readily available translation, but the match is imperfect enough for assimilation to produce significant distortion.

Nonetheless, it is possible for us to stop talking past one another, and there are substantial rewards on offer for doing so. Once we start using Aristotle’s terms in his way, it emerges that the identity conception of universal knowledge helps Aristotle make room for knowledge of perceptible particulars. Aristotle is a rationalist: he insists that only universals are proper objects of epistēmē7. But if knowledge of a universal is knowledge of some perceptible particular, then even a rationalist can allow that there is a sense in which particulars are knowable. Or rather, that some of them are. For this is yet another place in which Aristotle draws an unexpected distinction. He does not take all statements about the sensible world to be equally (un)knowable: statements such as “This (triangle) has 2R” or “This (mule) is sterile” have a very different status, for him, from statements such as “This (triangle) is blue” or “This (mule) is three-legged.” B21 promises some guidance on the vexed topic of just how Aristotle took the sensory world to be a proper object of knowledge8.

(I) Two Kinds of Error in B21

B21 addresses the question: given that someone has knowledge, what kinds of error is he nonetheless capable of? After an introductory section in which Aristotle lays out the two ways such error can arise (66b17-34= my T1-2), he proceeds to offer a short discussion of the first way (66b34-67a8= my T3), and a longer discussion of the second way (67a8-67b11= my T4-10) 9.

7 Aristotle’s argument for this claim appears in two stages in the opening chapters of the Posterior Analytics. In chapter two he argues that knowledge requires a demonstration, and in chapter 4 he argues that only what is necessary can be demonstrated, and only universals are necessary.8 The passage usually cited as evidence for knowledge of particulars is Metaphysics M10. Leszl offers a reading of that passage which comes close to having Aristotle there assert what I take him to say in B21. (For an important difference, see fn. 32 below.) M10 presents very thorny interpretative issues of its own; for instance, on an alternate interpretation (see Lear), the ‘tode ti’ that is the object of epistēmē in M10 refers not to perceptible particulars but to forms. I will try to show that, irrespective of our reading of M10, B21 offers us a clear articulation of both the nature of and the motivation for positing epistēmē of particulars. Leszl discusses B21 in passing, but his reading of that passage does not differ in any significant respect from the standard interpretation: he treats “universal knowledge” as entailed knowledge, assimilates type A and type B, and cannot motivate the reference to ‘induction.’ I am not persuaded by Leszl’s reading of M10; and I dispute his reading of B21. Nonetheless, this paper is deeply indebted to Leszl’s, especially as regards the general project of defending Aristotle’s epistemology against a Platonizing reading (represented, in Leszl’s treatment of the problem, by Cherniss and Owens). My focus is much narrower than Leszl’s, and I cannot take on the myriad issues surrounding the supposed “discrepancy between the real and the intelligible” (Cherniss, apud Leszl p.278) for Aristotle. Leszl explains why a view such as the one we want to ascribe to him does not involve Aristotle in absurdities of knowing particulars in all their particularity, and why such knowledge does not contradict Aristotle’s claims that (a) knowledge is of the universal (b) particulars are too singular, complex or transitory to be proper objects of knowledge. What little I have to add to his discussion of these points can be found in note 35, and the corresponding text. 9 My T-numbering corresponds to the order of the text, not the order of my own presentation: T4= 67a8-16, T5= 67a16-21, T6=67a21-26, T7=67a33-38, T8=67a38-b4, T9=67b4-5, T10=67b6-11. I cite all but two passages in B21. 67a27-33, which sits between my T6 and my T7, summarizes points made above. I do not cite it, though I do refer to it in some footnotes. I omit any discussion of the tangential final paragraph of B21 (67b12-26), which discusses the possibility of believing a single, self-contradictory proposition. As Smith notes, “this section does not further the discussion in the rest of B21.” (p.215) ; cf. Maier (part II, p. 367 n.1), according to whom this paragraph does not

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The first way has not received much attention, both because T3 is relatively straightforward, and because Aristotle does not, therein, invoke the ‘universal knowledge’ that is the source of most of our interest in B21. I want, however, to begin by briefly laying out the first way, as presented in T1 and T3. I take it to be no accident that Aristotle does not invoke ‘universal knowledge’ in that context, for my contention is that the difference between the two ways corresponds to the difference between type A and type B. The ‘first way’ is predicated on the fact that a given conclusion can be arrived at by two routes:

T1: Sometimes it happens that, just as we fall into error in connection with the position of the terms, the same error also arises in connection with our beliefs, as, for instance, if it is possible for the same thing to belong to several things primarily and for someone to fail to notice one of these and think the term belongs to none of it, but to know that it belongs to another one. For let A belong to B and to C according to themselves, and these likewise to every D. Now, if someone thinks that A belongs to every B and this to every D, but that A belongs to no C and this to every D, then he will have both knowledge and ignorance about the same thing in the same respect. (66b17-26)10

Aristotle does not offer us an example here, but merely a schema of four true premises (AaB, BaD, AaC, CaD) forming two Barbara syllogisms, each concluding in AaD11. We can fill out Aristotle’s schema using my opening example: A = 2R, B = triangle, D = isosceles. For C, let us introduce the property of having one’s internal angles add up to a straight line. I abbreviate this property as LINE.

S1 S2AaB 2R belongs to all triangle AaC 2R belongs to all LINE BaD triangle belongs to all isosceles CaD LINE belongs to all isoscelesAaD 2R belongs to all isosceles AaD 2R belongs to all isosceles

The subject Aristotle is describing has S1 but not S2. So this subject knows AaD through the middle term B (S1), but errs with respect to AaD through the middle term C (S2). Aristotle considers the possibility that his failure to know AaD through C is due to his having a contrary syllogism, such as S3:

S3AeC 2R belongs to no LINE CaD LINE belongs to all isoscelesAeD 2R belongs to no isosceles

In our example, the subject in question would have to believe that two right angles do not add up to a straight line. Aristotle goes on, in the second part of the introductory section, to set out the other way that error can arise. I will discuss this passage below, but I want now to jump to the subsequent passage, in which Aristotle returns to solve the problem he set out in T1. He explains that having S1 and failing to have S2 due to having S3 is impossible:

belong in the syllogistic at all. 10 Translations of the Prior Analytics are by Smith. All other translations are from Barnes (1984). 11 Though Aristotle does not specify how the four premises are to be organized into syllogisms, these are the only two valid syllogisms that can be formed from this quadruplet of premises.

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T3: Concerning the case mentioned previously, if the middle term is not from the same series then it is not possible to believe both the premises according to each of the middles (for example, to believe that A belongs to every B and to no C and that these both belong to every D)—” [I excerpt the middle lines of T3, in which Aristotle explains that believing all four premises leads to contradiction12. He concludes that—] “It is not possible to believe in this way [i.e., to believe both premises in accordance with both middle terms], but nothing prevents believing only one premise according to each middle term13, or both premises according to <only> one, for example, believing that A belongs to every B and B to every D and that A in turn belongs to no C. (66b34-67a8)

Aristotle’s example of how the subject may avoid contradiction is that he may have S1, and be missing CaD from S3. In this scenario, he will know AaD through S1, and fail to know it through S2. We can illustrated such a partial possession of S3 by putting brackets around the premises the subject does not have:

partial S3AeC 2R belongs to no LINE [CaD] [LINE belongs to all isosceles][AeD] [2R belongs to all isosceles]

The subject who has S1 cannot have S3. He can, at most, have partial S3, which, in turn, entails that he does not know that all isosceles have 2R through S2. Such a person knows that all isosceles have 2R through the middle term triangle, and is deceived about or ignorant of the fact that all isosceles have 2R through the middle term LINE. Why, exactly, does his condition count as being deceived with respect to the fact that 2R belongs to all isosceles (AaD) and not merely the fact that 2R belongs to all LINE (AaC)? Aristotle must hold that if I am in a position to come to a false conclusion about p merely by learning some true proposition q, then in some

12 The passage (b38-a5) runs: “For it results that the first premise is taken as a contrary, either without qualification or partially. For if someone believes A to belong to everything to which B belongs and knows that B belongs to D, then he also knows that A belongs to D. Consequently, if he thinks, in turn, that A belongs to none of what C belongs to, then he thinks that A does not belong to that which B belongs to some of. But to think that that which he thinks belongs to everything to which B belongs does not, in turn, belong to something to which B belongs is contrary, either without qualification or partially.” Aristotle seems to present the contradiction in an unnecessarily convoluted way, perhaps because he is not satisfied with showing that the subject will be lead to contradictory conclusions, but wants to show that her other premises lead her to contradict one she claims to know (AaB). Ross (p.473) does not seem right to insist, against Aristotle’s claim that the premises may be either contrary or contradictory (“either without qualification or partially,” ἢ ἁπλῶς ἢ ἐπί τι ἐναντίον), that they must be the latter and cannot be the former. Ross is right that AoB can be derived from the quadruplet, for it results by felapton from AeD and BaD, with AeD itself being the result, by celarent, of AeC and CaD. AeB is, however, also derivable from the quadruplet: it results by cesare from from CeA and CaB, where CeA is the conversion of AeC.13 Aristotle does not spell out this case, but he wants, for completeness, to point out that there is a second way of resolving the contradiction between S1 and S3: we can withhold from the subject the minor premise from S1 instead of that from S2/S3. He refers back to this case at 67a31-32 as one in which the subject has a belief about both middle terms. Such a person has “belief according to each of the middles” (ἡ καθ’ ἑκάτερον τῶν μέσων ὑπόληψις), since he has one true belief about each of the two middle terms: AaB is his true belief about B and CaD is his true belief about C. By contrast, the person in the example Aristotle lays out has true beliefs only about one of the two middle terms, B. However Aristotle’s focus is understandably on that case, for that is the one in which the subject’s error in respect of S2 contrasted with knowledge (through S1).

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sense I am already—even in advance of learning q—in a position of error with respect to p. His point in this part of B21 would be that this kind of error is compatible with knowing p.

Knowing-and-erring in this way involves two middle terms: we know the proposition through one middle term, and err about it through another. The first part of B21 thus shows that knowledge about p is compatible with error about p when p can be reached by different routes. The second half of the passage discusses the possibility of knowing and erring about a proposition through one, single middle term.

Aristotle’s initial description of this second kind of error is to be found back in T2:

T2: For example, if A belongs to B, this belongs to C, and C to D, but he believes that A belongs to every B and next to no C (for he will at the same time both know that it belongs and believe it does not). Based on these premises, then, would he be doing anything but claiming not to believe that very thing which he knows? For in a way he knows that A belongs to C by means of B (that is, as we know the particular by the universal knowledge (ὡς τῇ καθόλου τὸ κατὰ μέρος)); consequently, what he knows in a way, that he also claims not to believe at all, which is impossible. (66b26-34)

In this kind of case, the subject both knows and does not know AaC through one middle term, B. D is idle here, since we only need a single middle term. The reference to D appears to be a holdover from the first way. For that case involved two middle terms (B and C) and therefore four terms total. When Aristotle reprises the discussion of the single middle term case, having completed (in T3) the discussion of the double middle term case, we see that D has dropped out:

T4: For this sort of error is similar to the way we are deceived in the case of particular premises (περὶ τὰς ἐν μέρει). For example, if A belongs to everything to which B belongs and B to every C, then A will belong to every C. Therefore, if someone knows that A belongs to everything to which B belongs, then he also knows that it belongs to C. But nothing prevents him being ignorant that C exists, as, for example, if A is two right angles, B stands for triangle, and C stands for a perceptible triangle (αἰσθητὸν τρίγωνον): for someone could believe C not to exist14, while knowing that every triangle has two right angles, and consequently, he will at the same time know and be ignorant of the same thing. (67a8-16)

Aristotle introduces the single middle term case by claiming a point of similarity with the already-discussed double middle term case: in both, it is possible, without self-contradiction, to describe someone as knowing and at the same time erring. (He repeats this point of similarity at 67a31-33.) But he also offers us an important difference between the two kinds of cases: the

14 Ross (p.476) is right to puzzle over the fact that Aristotle slides from the idea of not knowing that a triangle exists (ἀγνοεῖν τὸ Γ ὅτι ἔστιν) to believing that it doesn’t exist (ὑπολάβοι γὰρ ἄν τις μὴ εἶναι τὸ Γ 67a14-15). In the corresponding passage of Po.An. A1 (71a17-b8, see esp. 71a32-33 τινα δυάδα ἣν οὐκ ᾤετ’ εἶναι) Aristotle clearly and consistently refers only to the former case. This case makes more sense, because it is hard to see what it means to believe that a specific triangle does not exist. Ross may be right that Aristotle is conflating two cases whose difference is not important at this point: the case in which you misperceive the triangle, and the case in which you fail to perceive it. Indeed, it may be that the difficulty in articulating the misperception case by way of the triangle example is part of what moves Aristotle to invoke the mule example.

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single-middle-term errors under discussion in T4 involve particulars, and they get analyzed with examples as opposed to mere constellations of letters. I have presented the double case by way of the example of knowing that isosceles triangles have 2R, but we saw that Aristotle himself was content to rest with a schematic presentation. When discussing single middle term errors, by contrast, he fills out his letters A, B and C with examples. So he discusses, first, the case in which the subject knows (and doesn’t) that a particular triangle has 2R, and later, that a particular mule is sterile. I suggest that Aristotle moves away from formalism in order to avail himself of expressions such as “this,” which specify particulars.

I am going to propose an interpretation on which T4 represents a crucial turn in the passage—from two middle terms to one—anticipated in T1. I will argue that Aristotle’s distinction between kinds of error not only structures his presentation of his argument, but reflects what he takes to be a deep and important philosophical difference. T5 helps us see why Aristotle’s two ways—two middle terms vs. one—corresponds to my distinction between type A and type B.

T5: For to know of every triangle that it has angles equal to two right angles is not a simple matter, but rather one <way of knowing it> is in virtue of having universal knowledge, and another way is in virtue of having the particular knowledge. In this way, then, i.e., by means of the universal knowledge, he knows C, that it has two right angles; but he does not know it as by means of the particular knowledge; consequently, he will not possess contrary states of knowledge. (67a16-21)

τὸ γὰρ εἰδέναι πᾶν τρίγωνον ὅτι δύο ὀρθαῖς οὐχ ἁπλοῦν ἐστιν, ἀλλὰ τὸ μὲν τῷ τὴν καθόλου ἔχειν ἐπιστήμην, τὸ δὲ τὴν καθ’ ἕκαστον. οὕτω μὲν οὖν ὡς τῇ καθόλου οἶδε τὸ Γ ὅτι δύο ὀρθαί, ὡς δὲ τῇ καθ’ ἕκαστον οὐκ οἶδεν, ὥστ’ οὐχ ἕξει τὰς ἐναντίας.

How is it possible for there to be any errors through a single middle term? T3 showed us that it is possible to know and not know some proposition, without contradiction, by knowing it through one route (one middle term) and erring about it through another route (a different middle term). In T5 Aristotle introduces ‘two ways of knowing’ that can be applied to a single line of reasoning. In T3, we avoid contradiction by invoking multiple middle terms; in T5, we avoid contradiction by invoking multiple ways of knowing. Not every kind of proposition can be known in multiple ways. The distinction between ‘having universal knowledge’ and ‘having particular knowledge’ only applies to cases in which one is reasoning about particulars. Which is to say, it is only applicable to inferences in which the minor premise and the conclusion are of the form, “this is an X.” It is possible for the subject of these inference to know in one way (i.e. by having universal knowledge) and not know in another way (i.e., by having particular knowledge). He has knowledge but, in a way to be specified below, fails to apply it to the world around him. Thus, type B reasoning offers up the possibility of knowing and not knowing through only a single middle term. I will address the nature of this distinction in (III) below. My immediate task, however, is to defend the reading I have just offered of T5. For, on the standard interpretation of the distinction between “having universal knowledge” and “having knowledge simpliciter,” its application is not restricted to type B cases.

(II) The Entailment Conception of Universal Knowledge

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Benjamin Morison offers the following definition of universal knowledge: “X knows universally that A belongs to C just in case (i) X knows that A belongs to all B, and (ii) B belongs to C. This definition leaves open15 that C is a particular thing (this triangle) or a universal (as it might be, isosceles triangles)”, p.35. Morison understands universal knowledge as knowledge of entailed propositions. (He only discusses cases in Barbara, but offers no principled reason to deny universal knowledge in any of the other moods or figures.) Morison’s view is also the standard interpretation: the secondary literature on B21 draws on the assumption, not always made explicit, that I know a conclusion universally when it is entailed by a major premise that I know, taken together with a minor premise that I may not know16. On this view, I know that same conclusion “through the particular” when I know it by knowing both the major premise and the minor premise, and actually drawing the conclusion in question.

On the entailment conception, someone who knows AaB thereby has “universal knowledge” of AaC for every C which is such that BaC. Barnes (1976) complains that this is an “unnatural” (p.88) use of the word “knowledge.” We can illustrate Barnes’ worry with an example: suppose that I know that all mammals give birth to live young, but do not know that dolphins are mammals. And suppose, in addition, that I go around saying things like “dolphins lay eggs, since they’re not mammals.” It does not seem true to say that I know that dolphins give birth to live young. It seems, rather, that I do not know this, though I could know it. 17

15 Morison suggests (p.30) that the insertion of this last clause is supported by the way Aristotle introduces the topic of universal knowledge at 67a9-12: “If A belongs to everything to which B belongs and B to every C, then A will belong to every C. Therefore, if someone knows that A belongs to everything to which B belongs, then he also knows that it belongs to C.” Here Aristotle appears to allow that the conclusion take a universal form (A will belong to every C). I think that Aristotle is here adopting a mode of speaking about this case that is colored by the previous case (in much the way that his original statement of it included an idle reference to ‘D.’) Moreover, as I note below, his logic does not offer him an alternative way of shoehorning universal knowledge into formal notation. (There is no such sentence as ‘A belongs to all this.’) The trajectory of Aristotle’s discussion suggests a sensitivity to just this problem—for it is exactly at this point that he departs from using a schematic example. The topic of universal knowledge calls for filling out the letters (A, B, C, D) of the schematic example with concrete details so as to specify a case involving perceptible particulars. 16 Clear statements of the entailed knowledge conception can be found in Bronstein p.125-6, Fine p. 65, Grgic 345-6, Ross p.473-474, Leszl p.295 (both of the latter describe it as ‘implicit knowledge’). The only clear statement of an alternative is in Ferejohn, who does not discuss the passage in detail. Ferejohn identifies universal knowledge with a knowledge of the universal premise, and particular knowledge with knowledge of the particular premise (see fn. 18).17 Some commentators have hoped to fill out the entailment conception by invoking an epistemic version of the distinction between de re and de dicto knowledge. But it is far from obvious how to turn what is basically a semantic distinction between ways of interpreting a proposition into a distinction between forms of knowledge. Confusions about how to bring the distinctions into contact might explain why one commentator (Fine, p.67) identifies universal knowledge as de re knowledge, while another (Ferejohn, p.106) identifies it as de dicto knowledge. Morison (pp.40-42) plausibly suggests that we take the phrase ‘de dicto knowledge’ to pick out a subject’s knowledge that, e.g., the lady he sees before him is sitting in the room, whereas he can be said to know de re, of the queen of England, that she is sitting in the room—despite the fact that he does not know that the lady he sees before him is the queen of England. On this interpretation, de re knowledge is produced by substituting an extensional equivalent into (the intentional context of) someone’s de dicto knowledge. This is, I think, the best we are going to do in terms of generating an epistemic de dicto/de re distinction in the proximity of Aristotle’s distinction. But, as Morison points out, it is not good enough. Morison gives a number of reasons why the two distinctions don’t line up, but the central one is that de re knowledge presupposes some kind of epistemic contact with the item in question (in the example above, the queen). We need a proposition, such as ‘this lady is in the room,’ to substitute into. But the universal knower in the triangle example has no such contact with the triangle in question: he doesn’t know it exists, or have any beliefs about it. Though the universal knower in the mule example

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Morison offers the following example by way of defending ‘universal knowledge’ against Barnes’ objection: suppose Angela, well-behaved child that she is, knows that she shouldn’t eat cookies before dinner. Pointing to the cookies on the table, you ask Angela’s mother, “does Angela know she shouldn’t eat those before dinner?” Even if Angela’s mother knows that Angela hasn’t yet seen those particular cookies, since she just bought them, it seems intelligible for her to answer “yes.” Suppose we grant to Morison that the major premise in this case, “I shouldn’t eat cookies before dinner” is a possible object of scientific knowledge. I agree with Morison that, on this assumption, Aristotle would say that Angela has universal knowledge that “I shouldn’t eat these cookies before dinner” even before encountering the cookies in question. But notice that Morison’s example of universal knowledge is an example of a type B case; he has not justified the application of the distinction to type A cases. And I think we have reason not to analyze the Angela example in Morison’s way: it is not true to say that the reason Angela has universal knowledge of the conclusion “I should not eat these before dinner” is that it follows deductively from the major premise known to her (“I should not eat cookies before dinner.”) For this interpretation of the Angela story cannot fit the text of T5.

On Morison’s interpretation, only conclusions of syllogisms can be known universally. It would be a category mistake to apply the phrase ‘universal knowledge’ to anything other than the conclusion of a given syllogism. The problem is that Aristotle himself does so, and he does so by way of introducing the category of universal knowledge. Recall this line from T5: “For to know of every triangle that it has angles equal to two right angles is not a simple matter, but rather one <way of knowing it> is in virtue of having universal knowledge, and another way is in virtue of having the particular knowledge. In this way, then, i.e., by means of the universal knowledge, he knows C, that it has two right angles.”18 In this sentence Aristotle first asserts that it is possible to have universal knowledge of the major premise that every triangle has 2R and then he transitions seamlessly to the claim that one thereby has universal knowledge of the conclusion ‘this has 2R.’ That is, he applies his two ways of knowing to the major premise and to the conclusion. If the major premise of a syllogism can be the object of universal knowledge—and not because it is itself the conclusion of another syllogism—then universal knowledge cannot be a matter of knowledge by entailment of a conclusion.

It may at this point be mysterious how the standard reading even got going, given that Aristotle introduces universal knowledge, in T4, as relevant to how we are deceived “in the case of particular premises (τὰς ἐν μέρει).” For it is striking that Aristotle never19 raises the topic of

does come into epistemic contact with the mule (he falsely believes, ‘this is pregnant’), that fact is clearly an incidental feature of the example. If it were an illustration of de re/de dicto, Aristotle’s point with the mule example would not be that the subject knows, universally, that this mule is sterile; rather, his point would have to be that the subject believes, de re, of a sterile animal, that is pregnant. But the latter belief-ascription is clearly far from Aristotle’s mind.18 Ferejohn has the opposite problem to Morison with this line. For he takes universal knowledge to be knowledge of the universal, and particular knowledge to be knowledge of the particular. On his view, only the first half of the sentence makes sentence: Aristotle should only be allowed to say that the subject has universal knowledge of the major premise. 19 There is one passage that has sometimes (though not by Morison) been taken to do just this. At Po.An. A24 86a22-30, Aristotle says that someone who knows that all triangles have two right angles knows potentially that the isosceles has two right angles, though the reverse does not hold. Unlike Gifford (p.7), Fine (p.67) and Bronstein (who is somewhat more cautious, p.126, fn.32), I do not identify the knowledge under discussion here with the

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universal knowledge except with reference to particulars. The proponent of the entailment conception might hope to save his view by invoking a systematic ambiguity in the way Aristotle uses both of the words for “particular premise” (ἡ καθ’ ἕκαστον, ἡ κατὰ μέρος) that show up in this passage. “Particular,” in Aristotle, sometimes means only relatively particular, and in this sense can describe a (universal) minor premise by contrast with the more universal major. Not every particular is what we might call an ultimate particular, a “this.” But it turns out that we cannot read Aristotle’s language of ἡ καθ’ ἕκαστον, ἡ κατὰ μέρος in T4 as indicating merely relative particularity. For Aristotle has two ways of clearly indicating ultimate particularity, and he makes use of both of them when discussing universal knowledge.

The first is that ultimate particulars are objects of perception; and Aristotle uses perceptual language to discuss universal knowledge in B21. He describes the original triangle as a perceptible triangle (αἰσθητὸν τρίγωνον 67a14)20 and specifies that moving past universal knowledge to knowledge simpliciter involves coming to see (ἐιδῶμεν21) the triangle. Moreover, in a later passage, Aristotle emphasizes the way in which universal knowledge is tied to a distinctive feature of perceptible knowledge:

T8: And this is just what the relation is of universal to particular knowledges. For we do not know any perceptible thing when it is outside our perception, not even if we happen to have perceived it before, except as in virtue of possessing universal knowledge, or in virtue of possessing, but not exercising, its peculiar knowledge.

universal as described in B21 and Po.An. A1. As I noted above (fn.3), while Aristotle is not terminologically fastidious about picking out knowledge ‘through a particular,’ he is consistent throughout B21 and A in picking out the defective case as knowledge ‘through a universal,’—and not, for instance, as ‘potential knowledge.’ The latter phrase is more comprehensive than the category of universal knowledge. Po.An. A24 does contain a contrast between the universal and the particular, but this contrast is tangential to ours. First, it is a contrast between universal and particular demonstration, e.g., proving all triangles have 2R for triangle as such vs. proving it for isosceles, equilateral and scalene separately. Aristotle’s point is simply that if you had the former proof, you could also produce one of the more particular proofs—that is, you know those potentially. Second, the passage as a whole argues to the conclusion that the universal form of demonstration is superior, as knowledge, to the particular form. The epistemic priority is thus reversed from our passages, in which knowledge through the particular is identified as knowledge simpliciter (ἁπλῶς Po.An. 71a25-30). Universal knowledge is not under discussion in Po.An. A24, and so the isosceles example cannot be offered as support for the entailment conception.20 Morison discusses this phrase at in his appendix. He is right that ‘αἰσθητὸν τρίγωνον’ here must pick out a particular triangle, as opposed to the class of sensible triangles generally. But it is, nonetheless, a particular sensible triangle to which Aristotle points us. Morison wants to downplay this fact and translate αἰσθησις as ‘awareness’: “αἰσθησις here need not be understood as perception….really term C should be understood as ‘triangle which is grasped by awareness… Thus, ‘sensible triangle’ is in fact another way of saying ‘particular triangle’.” (pp.56-7) If I am right, Aristotle adverts to sensation to bring out the fact that his topic is knowledge as applied to the world of experience. This may also contribute to the explanation of why he goes on to deviate from his favored example of triangles to that of someone misjudging the mule he sees before him. In discussing that case, Aristotle does not have to go out of his way to establish that the object under discussion is sensible. And in that context, when Aristotle uses the same word, (τῆς αἰσθήσεως 67b2), we must translate with ‘perception.’ Morison’s undertranslation of αἰσθησις (at 67a14) as awareness thus resists the tenor of Aristotle’s discussion.21 With Bekker, but against Ross, I read ἐιδῶμεν instead of ἰδωμεν at 67a25. (Smith seems to translate ἐιδῶμεν (“see”), but he fails to list it among his deviations from Ross’s text (p.238)). ἐιδῶμεν is attested in the main manuscripts; ἰδωμεν appears only in corrections to manuscripts, scholia and (pseudo) Philoponus’ commentary. I suspect that it is the entailment conception of universal knowledge that has motivated the ‘correction’ of ἐιδῶμεν (see) to ἰδωμεν (know). For generalizing Aristotle’s point beyond the case of perception serves to support the entailment conception.

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ὅπερ ἔχουσιν αἱ καθόλου πρὸς τὰς κατὰ μέρος ἐπιστήμας. οὐδὲν γὰρ τῶν αἰσθητῶν ἔξω τῆς αἰσθήσεως γενόμενον ἰσμεν, οὐδ’ ἂν ᾐσθημένοι τυγχάνωμεν, εἰ μὴ ὡς τῷ καθόλου καὶ τῷ ἔχειν τὴν οἰκείαν ἐπιστήμην, ἀλλ’ οὐχ ὡς τῷ ἐνεργεῖν. τὸ γὰρ ἐπίστασθαι λέγεται τριχῶς, ἢ ὡς τῇ καθόλου ἢ ὡς τῇ οἰκείᾳ ἢ ὡς τῷ ἐνεργεῖν, ὥστε καὶ τὸ ἠπατῆσθαι τοσαυταχῶς. (67a38-b4)

In T8, he seems to be explaining universal knowledge by reference to features of the vagaries of perception: because we know perceptible things only while we are perceiving them, we can be said to have universal knowledge of them even when we are not. It is a familiar theme in Aristotle that active knowledge of particulars is restricted to the moment of perception. In Meta Z15 he says:

For perishing things are obscure (ἄδηλά) to those who have knowledge of them (τοῖς ἔχουσι τὴν ἐπιστήμην), when they have passed from our perception (ὅταν ἐκ τῆς αἰσθήσεως ἀπέλθῃ); and though the formulae remain in the soul unchanged (σωζομένων τῶν λόγων ἐν τῇ ψυχῇ), there will no longer be either definition or demonstration. 1040a2-5

Given that we cannot take direct perceptual contact ‘with us’ in our souls, Aristotle identifies an attenuated element of such knowledge that is preserved in the soul: the ‘logoi.’ These ‘logoi’ alleviate the tension in the claim that those who have knowledge (τοῖς ἔχουσι τὴν ἐπιστήμην) about some objects are in a state of unclarity (ἄδηλά) with reference to those very objects. Α parallel passage at Z1022 licenses our linking the logoi of Z15 with universal knowledge in B21 (τῷ καθόλου λόγῳ, 1036a8), and this yields a reiteration of T8: there is a special form of knowledge which exists in order to, so to speak, combat the vagaries of perception. This form of knowledge allows us to know sensible things even as they enter and exit our perceptual acquaintance with them.

My ultimate aim is not to use Aristotle’s comments about the qualified knowability of the sensible world in the Metaphysics to shed light on B21, but rather the reverse. For I think that B21 explains exactly what the qualification in question consists in. But my immediate point is not to offer any such explanation, and merely to note the connection of ideas. If universal knowledge is introduced in order to ease the tension between the stability of knowledge, on the one hand, and the instability of our contact with the perceptual world, on the other, we must indeed restrict it to type B cases. For the knowledge of nonperceptible things is not subject to this worry—we are fully able to ‘take it with us.’ Thus no similar rationale is available for universal knowledge of what is non-perceptual.

22 “But when we come to the concrete thing, e.g. this circle, i.e. one of the individual circles, whether sensible or intelligible (I mean by intelligible circles the mathematical, and by sensible circles those of bronze and of wood), of these there is no definition, but they are known by the aid of thought or perception; and when they go out of our actual consciousness it is not clear whether they exist or not; but they are always stated and cognized by means of the universal formula (τῷ καθόλου λόγῳ)” (1036a6-8). cf. Topics V 3, 131b19-33: “For every perceptible attribute, once it passes beyond the range of perception, becomes obscure. For it is not clear whether it still belongs, because it is known only by perception. This will be true in the case of any attributes that do not always and necessarily follow.” See also De Anima 3.4, 429b10-18 for the distinction between perceiving something and discerning the essence of that same thing. What I perceive in perceiving flesh—hot and cold—is the very thing of which flesh is a certain ratio. I perceive enmattered form, while what is knowable is the form (logos, 16) without the matter.

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Besides speaking of perceptibles, Aristotle has one other way of specifying absolute as opposed to relative particularity. He can use the language of the Categories to pick out an ultimate subject of predication. Ultimate particulars are those things that cannot be predicated of anything further; and ultimately particular predications involve predicating something of such a subject. So, for instance, Aristotle says in the Categories that what is ‘not said of’ anything further is e.g., an individual man such as Socrates, or the individual knowledge of grammar in Socrates’ soul (Cat. 1a20-b9). Aristotle does not use language indicating an ultimate subject of predication in B21, but he does use such language in the other passage in which talk of universal knowledge arises. He says in the Posterior Analytics that we speak of universal knowledge “in cases dealing with what are in fact particulars and not said of any underlying subject (ὅσα ἤδη τῶν καθ’ ἕκαστα τυγχάνει ὄντα καὶ μὴ καθ’ ὑποκειμένου τινός 71a23-4).”

But this passage of the Posterior Analytics offers more than evidence against the standard view. It also helps us answer the ensuing questions as to what universal knowledge is and why it is restricted to particulars. Having concluded our survey of problems with the entailment conception of universal knowledge, I propose we turn to Posterior Analytics A1. I will compare the discussion of universal knowledge there with that in B21 to articulate an alternative conception of universal knowledge that I call the identity conception. My universal knowledge of a universal proposition such as “all X’s are Y’s” is, in a sense to be elaborated, identical to my universal knowledge of the corresponding particular proposition “This is a Y.” Aristotle’s claim is that I only know one thing, and that one piece of knowledge can be represented either as the (universal) knowledge that “all Xs are Ys” or as the (universal) knowledge that “this is a Y.”

(III) The Identity Conception of Universal Knowledge

Aristotle opens the Posterior Analytics by investigating how it is possible for knowledge to come into being. This question takes its bearings from Meno’s paradox, which raises the worry that knowledge-acquisition is impossible: someone who already knows has nothing to learn, and someone who does not has no way of coming to learn. Aristotle denies that any teaching or learning needs to come out of sheer ignorance. He grants that reasoning is predicated on some kind of prior knowledge (γνῶσις, 71a2 et passim), but wants to insist that cognitive progress—learning—is nonetheless possible. I do not want to investigate the details of how we motivate Meno’s paradox or the question of whether Aristotle provides a satisfactory solution to it23. My attention will be on how Aristotle organizes his solution into two parts.

For Aristotle divides learning into two types, both of which proceed on the basis of prior knowledge, but in different ways. He separates the deductive reasoning which is most familiar to us from what he calls a form of ‘inductive reasoning’ that he describes as being about ultimate particulars (cf. Po.An. I.18). In the case of this second kind of reasoning, he describes learning as a matter of transitioning from “knowing merely universally” to “knowing simpliciter.” He does not make explicit the contrast with the deductive case, but I think the implied contrast is that in the case of deductive reasoning, the transition in question is from knowing one thing (the premises of an argument) to knowing another thing (the conclusion of the argument). For Aristotle’s definition of a syllogism is that of reasoning in which “certain things being stated,

23 On this topic, see Bronstein.

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something other than what is stated (ἕτερόν τι τῶν κειμένων, Pr.An. A1, 24b18-20) follows of necessity from their being so.” That is, inductive reasoning represents a transition in ways of knowing, whereas deductive reasoning represents a transition in objects of knowledge. Learning, contends Aristotle, is possible both because it is possible to come to know something new from what one knew before and because it is possible to come to know that which one knew before in a new way. The second case concerns (ultimate) particulars, because the ‘new way’ is a matter of applying the knowledge to sense-experience.

We will see that this way of characterizing the contrast between the two forms of reasoning receives confirmation from the parallel discussion in B21; but first, let us look at the portion of A1 in which he gives an example to illustrate the ‘inductive’ form of knowledge acquisition:

But you can become familiar by being familiar earlier with some things but getting knowledge of the others at the very same time—i.e. of whatever happens to be under the universal of which you have knowledge. For that every triangle has angles equal to two right angles was already known; but that there is a triangle in the semicircle here became familiar at the same time as the induction (ἐπαγόμενος). For in some cases learning occurs in this way, and the last term does not become familiar through the middle—in cases dealing with what are in fact particulars and not said of any underlying subject. (71a17-24) 24

Aristotle imagines a subject who knows ‘in advance’ (πρότερον, 17) that all triangles have 2R, but he draws the conclusion—that this triangle has 2R—at the same time as he sees that a certain figure inscribed in a semicircle is a triangle. Aristotle is pointing to a difference between ‘all triangles have 2R’ and ‘this is a triangle’: only the first is known in advance of the conclusion ‘this has 2R.’ In some way, the knowledge that constitutes the conclusion is arrived at through the major premise alone. Thus he says that “the last term does not become familiar through the middle.”

Aristotle is encouraging us not to hear “all triangles have 2R, this is a triangle, this has 2R” as a standard, deductive syllogismos. If it were a syllogism, then the subject would have to know both the major and the minor premises in advance of the conclusion; and this is because he would have to know the conclusion by using middle term to generate something new (ἕτερόν τι) from the two premises. By contrast, in the ‘inductive’ case, I do not know the conclusion via a middle term. What can Aristotle mean by insisting that in these kinds of cases, I know the conclusion ‘directly’ or ‘immediately’ by way of the major premise? Let us turn to B21, where

24Ἔστι δὲ γνωρίζειν τὰ μὲν πρότερον γνωρίσαντα, τῶν δὲ καὶ ἅμα λαμβάνοντα τὴν γνῶσιν, οἷον ὅσα τυγχάνει ὄντα ὑπὸ τὸ καθόλου οὗ ἔχει τὴν γνῶσιν. ὅτι μὲν γὰρ πᾶν τρίγωνον ἔχει δυσὶν ὀρθαῖς ἰσας, προῄδει· ὅτι δὲ τόδε τὸ ἐν τῷ ἡμικυκλίῳ τρίγωνόν ἐστιν, ἅμα ἐπαγόμενος ἐγνώρισεν. (ἐνίων γὰρ τοῦτον τὸν τρόπον ἡ μάθησίς ἐστι, καὶ οὐ διὰ τοῦ μέσου τὸ ἔσχατον γνωρίζεται, ὅσα ἤδη τῶν καθ’ ἕκαστα τυγχάνει ὄντα καὶ μὴ καθ’ ὑποκειμένου τινός.) There are two ways we might construe ἅμα ἐπαγόμενος ἐγνώρισεν (21a). What does the subject know at the same time as having drawn what proposition by ‘induction’? McKirahan (p.5) takes ἐπαγόμενος to govern the minor premise, and ἐγνώρισεν to govern the conclusion; Bronstein (fn.18) takes it in the opposite way, and points out that parallelism with 71a19-20 favors his reading. I follow Bronstein, in holding that what the subject reasons his way to must be the conclusion, whereas what he realizes simultaneously, i.e. rather than prior to, drawing this conclusion is the minor premise.

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we find the same association among universal knowledge, induction, a “simultaneous” kind of learning and ultimate particulars:

T6 And the argument in the Meno that learning is being reminded is also similar: for it never results that people know the particular in advance, but rather that they get the knowledge of the particulars at the same time (ἅμα), by means of the induction (τῇ ἐπαγωγῇ), like those who recognize something. For there are some things which we know right away (for example, we know that something <has angles equal> to two right angles, if we see (ἐιδῶμεν) that it is a triangle, and similarly also in the other cases).

In this passage, as in the Po.An.A1 passage, Aristotle describes the move from universal to unqualified knowledge as involving one step, instead of two. Instead of saying that the subject who has the major premise first acquires the minor and then draws the conclusion, Aristotle maintains that the latter two events happen together. What does Aristotle mean by insisting on the simultaneity of the minor with the conclusion? One natural interpretation would be temporal. Perhaps, if you have the major premise, you acquire the conclusion at the instant that you acquire the minor. But this cannot be what Aristotle means, given the way in which B21 proceeds25. For Aristotle is going to go on to offer an example in which someone knows the major premise, has the minor, and still fails to draw the conclusion. He imagines someone who knows that all mules are sterile, and perceives that the animal before him is a mule, but fails to put these premises together and therefore believes this animal to be pregnant.

In describing the conclusion as ‘simultaneous’ (ἅμα) with the minor premise, Aristotle is not saying that the presence of the minor entails the presence of the conclusion, but rather that if the conclusion is arrived at, it is arrived at in an unmediated way. Aristotle’s point about induction is best understood with his conception of deduction as a contrastive backdrop. The middle term is the explanatory engine of a deduction (94a20-23), and the simultaneity of inductive premises likewise contrasts with the priority of deductive premises. For Aristotle specifies that “the premises of demonstrated knowledge must be…prior (πρώτων) to the conclusion, which is further related to them as effect to a cause.” (71b18-22) Aristotle’s ἅμα in connection with knowledge through induction is no more temporal than his πρώτων in connection with knowledge through deduction. The deductive conclusion is a further (third) proposition beyond those already in play, arrived at from the major through the intermediacy of the minor. The inductive conclusion in some sense is the major premise. Recall the feature of Aristotle’s introduction of universal knowledge that was so problematic for the entailment conception of universal knowledge: in T5 Aristotle spoke, indifferently, of universal knowledge as a way of knowing the major premise, and as a way of knowing the conclusion. He says that one can know the universal through the universal, and then seems to be filling out the same thought (οὕτω μὲν οὖν) by saying that one can know the conclusion through the universal. “Universal knowledge of the major premise” is a category mistake on the entailment conception of universal knowledge. But it makes perfect sense if universal knowledge is knowledge where the conclusion is, in a sense, identical to the major premise. On the identity conception of universal knowledge, we can say that I know ‘this is a Y’ by way of ‘all Xs are Ys’ and ‘this is an X’, but really, I know this is a Y directly, simply in virtue of knowing ‘all Xs are Ys’.

25 See Cat. 13 for Aristotle’s catalogue of non-temporal uses of ἅμα, e.g. simultaneity by nature and the simultaneity of co-ordinate species of a genus.

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One significant payoff of the identity conception of universal knowledge is that it sheds light on the darkest feature of both of B21 and A1: Aristotle’s association between universal knowledge and a form of reasoning he calls “inductive.” That Aristotle intends to contrast universal knowledge with deduction is clear, for in both of the places in which he refers to the simultaneity of the minor premise and the conclusion, he also characterizes the knowledge attained as inductive. I have taken pains to avoid characterizing type B reasoning as ‘syllogizing,’ because Aristotle takes such pains himself. He is careful to contrast type A and type B cases not by speaking of two kinds of syllogisms, but instead by speaking of errors involving one (as opposed to two) middle terms. And it is the single-middle term (type B) kind of error that moves him to adopt this circumlocution; for he refers back to the double-middle-term (type A) case as involving a συλλογισμός at 67a32.

The references to induction in B21 and A1 have long puzzled commentators who take their bearings on epagōgē from passages where induction seems to get us from experience of the sensible world to knowledge in the form of universals26. For Aristotle, as for us, induction can mark the move from particulars to universals. But the terms ‘ἐπαγόμενος’ in Pr.An. A1 (71a21) and ‘τῇ ἐπαγωγῇ’ in B21 (67a23) pick out a move from universals to particulars rather than the reverse. In the face of the strangeness of characterizing the triangle or mule example as “inductive,” commentators have adopted a number of different strategies. The most obvious, but least promising, is to tried to read ‘induction’ in these passages as, contrary to appearances, a move from particulars to universals27. Another strategy is to set aside these uses as ‘non-technical.’28 But, unless the exceptional character of the context can be well-motivated, this amounts to giving up on the project of providing a unified account of epagōgē. The remaining alternative is to achieve unity by abstraction: so Hamlyn speaks of epagōgē as a process of “com[ing] to see the application of a general principle to a case” (p.171), McKirahan of “coming to see individuals… as individuals of a particular sort.” 29 On these views, moves in either direction can count as epagōgē.

Surveying Aristotle’s many references to epagōgē, let alone all the problems assimilating these into a coherent whole, is beyond the scope of this paper. I cannot here provide an analysis of Aristotelian induction. What I can do is offer some support for the general strategy adopted by Hamlyn and McKirahan. Their approach is problematic for the same reason as it is promising: the account of epagōgē they offer is vague. When we give up on the idea that epagōgē must

26 See Po.An. B19 and Nicomachean Ethics 1139b28 (epagōgē as archē tou katholou). Though Hamlyn’s classic paper argues that even in those passages, epagōgē is better characterized as a form of argumentation than a process of discovery.27 Gifford adopts this strategy, though only for B21. See LaBarge for an extended response to Gifford.28 Ross, p.481-483, Engberg-Pedersen: “Nothing, then, seems implied in the use of epagōgē and epagein in these passages besides the simple idea of being led to see some particular point.” p.303-4.29 And McKirahan’s as well, though he takes himself to disagree with Hamlyn. Hamlyn thinks that epagōgē must point us to better ways of understanding of the universal, so that in cases where epagōgē is a matter of applying a universal to a particular, these applications have a propaedeutic character. They pave the way for a subsequent superior understanding of the universal (see esp. p.171). McKirahan argues that there are cases of terminal, applied epagōgē. On the interpretation offered here, the dispute dissolves: epagōgē can be, as McKirahan thinks, a terminal, i.e. nonpreparatory, application of an already known universal to a particular, and nonetheless offer up a new way of knowing that same universal.

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move from particulars to universals, we seem to lose the basis of the contrast with syllogismos that animates a number of Aristotle’s discussions of induction.

The line of thought I have been pursuing here helps us see how ‘seeing a particular as an instance of a universal’ (McKirahan, p.11) constitutes a distinctive form of reasoning. Unlike deductive reasoning (syllogismos), in which I come to know something other than the premises with which I begin, ‘inductive’ reasoning can be characterized as that of coming to know the very thing that I already knew before. Recall that the subject, even before he encountered the particular premise, had (universal) knowledge of it. Thus, when he comes to have knowledge of it simpliciter, that knowledge has not been arrived at by deduction. There is a common formal principle underlying induction that moves to universals and induction that moves from them. Aristotle is articulating this common principle in our two passages by way of the contrast with deductive reasoning30. Movements of the mind are called ‘inductive’ because they are movements to a different way of understanding the same content; deductive movements, by contrast, offer us the same way of understanding a new content. For us, the distinction between induction and deduction is a matter of the ‘direction’ (upward or downward); for Aristotle the distinction has a different basis, and this difference makes it possible for him to describe the application of a universal to sense-experience as ‘induction.’ We can take in Hamlyn and McKirahan’s insight about the bidirectionality of epagōgē while also recovering the sharpness of Aristotle’s distinction between ‘syllogismos’ and ‘epagōgē’.

But is induction genuine reasoning? This is a question to which the remainder of B2131 is addressed. Having denied that type B reasoning is deductive, Aristotle needs to do some work to show that it is nonetheless reasoning. Aristotle is clear that in order to know the major premise not only universally but also simpliciter, the subject needs the minor premise as well. But pointing to the necessity of the presence of the minor premise for the activation of the major does not suffice to characterize such activation as reasoning. Consider a parallel. Suppose it is true that anyone who knows what Simmias looks like, will, upon seeing Cebes, be reminded of Simmias. Being reminded of someone is a different way of grasping his appearance than simply knowing what he looks like. This example thus fits a certain schema: knowing A + seeing B knowing A in a new way. And yet when someone is reminded of Simmias, this is not an inference. Aristotle wants to defend the rational status of induction, and so he needs to argue that the movement to the third proposition in a type B case is legitimately characterized as a case of (non deductive) inference. To that end, he offers us another example:

T7: And nothing prevents someone who knows both that A belongs to the whole of B, and that this, in turn, belongs to C, from thinking that A does not belong to C (for example, knowing that every female mule is infertile and that this is a female mule but

30 Aristotle does, sometimes, speak of inductive syllogisms e.g. 68b15. He even does so in contexts where he is engaged in distinguishing deduction from epagōgē, i.e., at A1, 71a24. ‘Syllogismos’ thus has a sense in which it is restricted to deduction, and then another, wider sense, in which induction qualifies as a species of ‘syllogismos.’ (For discussion, see Hamlyn p. 169.)31Contrast McKirahan: “Had he been asked what kind of reasoning was found in a successful epagōgē, I am doubtful whether he would have given a specific answer,” (p.12) “What, if any, hidden intellectual processes are taking place on those occasions, is obscure to me. (p.13)” McKirahan’s skepticism may be tied to the fact that he is addressing the question of validity for epagōgē as such, whereas I am only claiming that there is a subset of epagōgai—those constituting applications of known universals to perceptible particulars—whose legitimacy Aristotle is motivated to vouchsafe.

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thinking that this is pregnant): for he does not know that A belongs to C, if he does not simultaneously reflect on the term related to each one (μὴ συνθεωρῶν τὸ καθ’ ἑκάτερον_. Consequently, it is also clear that if he knows one but does not know the other, then he will be in error. (67a33-38)

Here we have another variant of the single-middle term (i.e. type B) case: someone who knows both that all mules are sterile, and that this is a mule might, nonetheless, believe that this animal is pregnant (i.e., non-sterile). Aristotle explains that the subject who knows both premises might fail to ‘see them together’ (συνθεωρῶν). Someone who knows that all mules are pregnant and who sees a mule before him might fail to incorporate these two contents into a single or unified realization (theōria). Aristotle then points out that the triangle case falls out as a kind of a fortiori consequence of the mule case: since the person in mule case errs, the person who doesn’t even have knowledge of the minor will err as well. The mule case reveals that the structure of applied knowledge is not bipartite, but tripartite:

T9: For 'to know' can be used with three meanings: as knowing by means of universal knowledge, knowing by means of the peculiar knowledge of something, or as knowing by means of exercising knowledge; and consequently 'to be in error' also has the same number of meanings. (67b4-5)

In addition to erring by lacking universal knowledge, one can err both by lacking the particular, and by failing to put it together with one’s knowledge of the universal in such a way as to have active knowledge (which Aristotle also glosses as knowledge simpliciter). Active knowledge entails perception (τῆς αἰσθήσεως 67b2), but perception does not entail active knowledge.

The mule scenario is designed to illustrate the possibility that I can be in perceptual contact with the object specified in my universal, but nonetheless fail to know, perceptually, that it is as the universal says it must be. Moreover, Aristotle wants to characterize the failure in this case as one of thinking two things together: συνθεωρῶν τὸ καθ’ ἑκάτερον. Aristotle is arguing that the person in the mule scenario fails to put two and two together. Such failures can be nothing other than failures of reasoning. Thus despite the fact that the minor premise does not serve as that by way of which the subject can draw a new conclusion, it can still be thought together with the major.

In the kind of reasoning at play in the application of knowledge to experience, I perform a mental act of fitting together the universal ‘all X’s are Y’s’ with the particular ‘this is an X’—but I do not do so in order to generate a new proposition. Instead, this mental act serves to raise the knowledge I already have to a full state of actuality. When I apply my knowledge to experience, I use my power of seeing the connections between thoughts, not for the sake of acquiring new knowledge, but rather to raise to full actuality knowledge I already had. The mule example shows that the application of knowledge to experience nonetheless constitutes a form of reasoning, and learning (ἡ μάθησίς Po.An. 71a22). Aristotle brings out the presence of such reasoning by pointing to the possibility of its absence, noting the distinctively rational breakdown of a failure to bring premises together. He concludes by pointing to how such error is possible:

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T10: Nothing then prevents someone both knowing and being in error about the same thing (although not contrarily), which is also what happens to the man who knows a premise according to each kind of knowledge and has not previously examined them (καὶ μὴ ἐπεσκεμμένῳ πρότερον): for in believing that the female mule is pregnant, he does not have knowledge in the sense of exercising it (οὐκ ἔχει τὴν κατὰ τὸ ἐνεργεῖν ἐπιστήμην), nor indeed does he have the error contrary to the knowledge as a result of his belief (διὰ τὴν ὑπόληψιν ἐναντίαν ἀπάτην τῇ ἐπιστήμῃ) (for the error contrary to universal knowledge is a deduction (συλλογισμὸς γὰρ ἡ ἐναντία ἀπάτη τῇ καθόλου)). 67b5-11

Aristotle makes the possibility of a failure to put ‘mules are sterile’ and ‘this is a mule’ together conditional on not having, at some earlier time, considered the question of the sterility of some particular mule. Presumably, Aristotle’s assumption is that previous thought on the matter would ensure that someone readily assimilates the universal and the particular. This may explain why he illustrates the point by shifting examples from the triangle to the mule. Someone who knows that mules are sterile may, in fact, never have had occasion to apply that information to a particular mule; whereas the parallel scenario with triangles is harder to imagine. I want to concentrate on Aristotle’s obscure final claim that the contrary to universal knowledge has a deductive form.

On the identity conception, Aristotle’s point is that the contrary of universal knowledge is the deductive and not the inductive reasoning to the contrary. It is only deductive, and not inductive error that is incompatible with knowing. Knowledge of the proposition “all mules are sterile” is knowledge of the proposition “this is sterile” (when “this” is a mule). Which is to say, one knows those two propositions in one single state of knowledge. But when we turn to specifying the error that is contrary to that single state of knowledge, Aristotle directs us to formulate it on the basis of “all mules are sterile” rather than “this is sterile.” The contrary error is not “this is pregnant” but rather “some (or all) mules are pregnant” And so the reasoning which is precluded by my knowledge that all mules are sterile/this is sterile is not the inductive reasoning leading to the conclusion “this is pregnant” but rather the deductive reasoning leading to the conclusion “some (or all) mules are pregnant.”

I hope it has emerged just how deeply we misread B21 if we assimilate type A and type B. It is worth noting that Aristotle’s resistance to our modern inclination towards such assimilation is backed by his syllogistic. We might say that I know that this has 2R because I deduce it from the major ‘all triangles have 2R’ and ‘this is a triangle’. But Aristotle’s logic requires that an affirmative proposition take either the form P belongs to all S or P belongs to some S. Given that “this is an X” takes neither form, the inference would not, strictly speaking, count as deductively valid for Aristotle. And my claim has been that B21 is best read as Aristotle’s insistence that we do indeed speak strictly here, which entails denying the validity of the inference in type B as a deductive inference. Ultimately, the most basic problem with the entailment conception of universal knowledge is that Aristotle does not think that the conclusions in type B are entailed by the premises at all32. ‘This has 2R’ simply cannot be validly deduced (syllogizesthai), using Aristotelian logic, from ‘all triangles have 2R’ and ‘this is a triangle’.

32 Leszl tries to get around this problem by reformulating the major premises of type B syllogisms as universally quantified conditional sentences: “2R belongs to all triangle” becomes “For all X, if X is a triangle, it has 2R.” See Ebrey for one reason why this revision is unAristotelian.

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(IV) Epistēmē of Particulars?

I want to end by considering the motivation for the identity conception of universal knowledge: why would Aristotle think that the knowledge that “all triangles have 2R” is the same as the knowledge that “this (triangle) has 2R”? Note, first, that Aristotle is not identifying every cognition that “This has 2R” as a cognition that “All triangles have 2R.” The identity conception applies specifically to knowledge. So, for instance, someone might have no idea that all triangles have 2R but measure the angles of a particular triangle and assert “This has 2R.” His belief does not simultaneously consist in a belief that all triangles have 2R. Indeed, this is exactly why his belief is not knowledge33. The only way in which a proposition about a particular can constitute knowledge is when it is an activation of (i.e., identical in content to) a more universal proposition.

So let us restrict ourselves to those particular premises which constitute applications of known universals. Even in these cases, one might think one could distinguish between the content of the particular and the content of the universal. Why would Aristotle disagree? In T6 and in Po.An. A1 we see Aristotle struggling to accommodate the sense in which we do, and that in which we don’t, have a kind of fore-knowledge (προεπίστασθαι 67a22) of particulars. Aristotle wants to represent the necessity of the minor premise for the full, active knowledge of the conclusion, while nonetheless restricting the contribution of the minor premise to the conclusion. The minor premise makes possible a kind of perceptual awakening (“if we see,” ἐιδῶμεν) in which the very same conclusion is present in an activated form. The minor premise, in these cases, does not represent more information that the subject acquires on his way to the conclusion, as it does in type A cases. Rather, it represents a kind of application-condition of the conclusion. Aristotle’s insistence on universal knowledge is a response to the fact that the application of knowledge to experience does not fill out or add to what one knows. He introduces the category of universal knowledge to avoid assimilating all forms of cognitive achievement as taking the form of acquisition.

The motivation for avoiding this assimilation is nothing other than the possibility of knowledge of particulars. In order to see why this is a problem for Aristotle in the first place, let us remind ourselves of some peculiarities in his conception of knowledge. Since Miles Burnyeat’s influential paper, “Aristotle on Understanding Knowledge,” it has been generally accepted that ‘understanding’ is in many respects34 a better translation of ‘epistēmē’ in the Analytics than ‘knowledge.’ Burnyeat shows that Aristotle’s epistemic concerns in the Analytic are not with

33 Aristotle often expresses this point by saying that the particular, as such, cannot be known. See passages cited in fn. 22 above, and the corresponding text. If I am right to understand what Aristotle is saying in all of these passages as a matter of contrasting the particular unknowability of the proposition with a knowability of that very same proposition through the universal, then Burnyeat (p.113 and fn.34) and Hintikka (pp.75-6) are wrong to think that they serve as evidence for the conclusion that the particular cannot in any sense be known. See also fn 35.34 I retain “knowledge” as a translation not only in deference to tradition, but also because “knowledge” is an honorific. By translating epistēmē as ‘understanding’ we cannot help but demote Aristotle’s enterprise in the Analytics. Though I am persuaded by Burnyeat’s argument that the thing Aristotle calls “epistēmē” corresponds to the thing we call “understanding,” I am also moved by the fact that when Aristotle uses the word “epistēmē,” he means to accord the kind of importance to the thing he describes with that word that English speakers allot by way of the word “knowledge.” Finally, I think we ought, if we can, preserve the dialectic between Aristotle and others in the tradition who have offered theories of knowledge—epistemologists—even if that comes at the cost of ascribing to Aristotle a very atypical epistemology.

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what we might call the evidence-justification-knowledge complex but rather with the demonstration-explanation-understanding complex. Thus it is that when something is not susceptible to an explanation of why it couldn’t be otherwise, I cannot have epistēmē of it. In ordinary English it would sound strange to say I cannot know that my eyes are brown, or that the dolphins at Sea World were born in captivity. I might insist that I feel sure that I know both of those facts, on the basis of the fact that I have excellent evidence for them. Aristotle will point out that I cannot, nonetheless, have a kind of systematic understanding of such facts. Epistēmē requires that I be “well-acquainted or thoroughly familiar with something in an intellectually principled way; as when a man is said to have knowledge of, say, mononucleosis or the turnip,” (Burnyeat, p.106). “Understanding is constituted by knowing the explanation of necessary connections in nature.” (p.110) One might take it to follow from such a conception of epistēmē that one cannot have epistēmē of either accidental or particular facts—and indeed Burnyeat does takes this to follow. Burnyeat holds that Aristotle allows that I can cognitively grasp particulars in a variety of ways—gnōsis, aesthesis, memory etc. But he has Aristotle denying that I have the highest kind of cognitive grasp—epistēmē—of particulars35.

But it is a mistake to conflate an accidental fact, such as the accidental particular fact that I have brown eyes, or the accidental universal fact that all the dolphins at Sea World were born in captivity, with a particular but necessary fact, such as the fact that I am rational, or that this (triangle) has 2R, or that this mule is sterile. If not all facts about ultimate particulars are accidental facts, then some particular propositions may be able to be known. Burnyeat might offer the following objection to saddling Aristotle with a commitment to the existence of what we might anachronistically call “a posteriori necessary truths.” One such truth is that this mule is sterile and another such truth is that that mule is sterile. It looks as though there are an infinite number of truths of this kind, even within the restricted arena of knowledge-of-mules. And thus it looks like I cannot have what Aristotle called epistēmē—and what we, following Burnyeat, might call a complete, systematic understanding—of mules, let alone of equines, or mammals, or biology. For where the evidence-certainty-justification-knowledge invokes the centrality of the concept of certainty, the proof-explanation-understanding complex will place the emphasis on completeness. Epistēmē reflects a kind of systematic mastery of a subject matter that is not

35 See Leszl for an account of how it is possible for Aristotle to countenance the possibility of applied knowledge, given his commitment to knowledge being of universals. I would add to Leszl’s excellent discussion only a few points stemming from our disagreements about B21. First, the identity conception makes it easier to defend knowledge of particulars, since what one knows in knowing a particular is the same in content as what one knows in knowing a universal. Thus, on my interpretation, it is easy to see how Aristotle can meaningfully contrast his view with one on which particulars, as such, are knowable. Second, my view faces an additional challenge Leszl’s does not: on Leszl’s view, the knowledge one has of particulars is standard, deductive knowledge. On my view, it is inductive knowledge. And thus my view, and not his, may seem to conflict with Aristotle’s claims that epistēmē must be the result of a demonstration, since demonstrations are deductions. But I do not deny that epistēmē presupposes deduction. As noted above, one only has epistēmē of a particular if one has epistēmē of the relevant universal, and the latter requires a deduction. See Meta. M10 1086b33-37: "This is clear both from demonstrations and from definitions; for we do not conclude (οὐ γὰρ γίγνεται συλλογισμὸς) that this triangle has its angles equal to two right angles, unless every triangle has its angles equal to two right angles, nor that this man is an animal, unless every man is an animal." Here Aristotle seems to allow that ‘this triangle has 2R’ is an item of knowledge precisely when the universal—‘every triangle has 2R’—is known. (I acknowledge that it is unfortunate that Aristotle did not, in this passage, invoke the distinction between syllogismos and epagōgē. But, as I note in fn. 30 above, he commonly uses ‘syllogismos’ in a more generic way, to mean ‘reasoning.’)

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compatible with not knowing an infinite number of the propositions in the field in question36. And this is exactly where the identity conception of universal knowledge comes in.

Consider Burnyeat’s paraphrase of Po.An. A1 71a19-21: “I see that this figure in the semicircle is a triangle and immediately infer that it has angles equal to two right angles. The case illustrated involves perception of a particular and simultaneous inference to new information…” (p.119, my emphasis) Contra Burnyeat, Aristotle never claims that the subject in the triangle scenario arrives at new information. We have seen that Aristotle makes efforts to insist that what such a subject arrives at is what he already knew before arriving at it. And the reason he must make this remarkable claim is that, given his conception of epistēmē as understanding, there is no other way for him to secure epistēmē of the empirical world. Burnyeat observes, correctly, that “Demonstration can and induction cannot ἐπιστήμην ποεῖν,” (p.119). He wrong, however, to conclude that this speaks against the possibility of inductive epistēmē. For the point of B21 is to insist that in addition to the distinction between knowing one thing and knowing another, we must distinguish between knowing merely universally and knowing simpliciter. Induction might be impotent to effect a transition of the first kind, but empowered to effect a transition of the second kind. Coming to know needn’t be a matter of having epistēmē generated (poiēthenai) in one by one’s previous knowledge. It can also be a matter of knowing what one knew before in a new way.

Aristotle’s commitment to universal knowledge of particulars doesn’t entail a commitment to the existence of an infinite set of a posteriori necessary propositions over and above the universals they instantiate. For Aristotle insists that when I know that this triangle has 2R, I do not know (or think or grasp) something over and above the proposition that all triangles have 2R. We might be inclined to think that all sentences of the form “this is an X” must share the same epistemic status. Aristotle demurs, identifying as special that subset of such propositions constituting conclusions of type B reasoning. Those conclusions are a place where the knowability of an ultimate particular does not conflict with the claim that knowledge is ‘of’ universals.

Aristotle is a rationalist, in that he takes knowledge to be of necessary and unchanging truths and not of how this or that changeable object happens to be. And yet he sees he must allow science to touch down in the world around us: studying geometry ought to enable us to know about the angles on this triangle, studying biology ought to allow us to know about the sterility of this mule. For, not to put too fine a point on it, it is Plato, and not Aristotle, who posits the unknowability of our sensible world. B21 helps us articulate how Aristotle can be a rationalist

36 We must be careful how we specify this completeness, so that it does not fly in the face of Aristotle’s acknowledgement, in T3, that someone can know a universal and make a mistake with respect to a closely related universal. That passage is consistent with Burnyeat’s conception of knowledge as unified mastery if we make clear that the latter should be understood as an aim of all partial acts of knowing. It is the potential completability of the project of knowing with respect to some subject matter on which any knowledge of that subject matter must hang.

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without being a Platonist37. For what we see there is Aristotle carefully working out a way in which knowledge of some universal can be knowledge of some perceptible particular.

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37 Frede also ascribes to Aristotle a rationalist yet anti-Platonic epistemology. Where I am concerned to show how rationalism can have an empirical output, his interest is in allowing for empirical input. For he argues that “for Aristotle, reason is not something with which we are born,” (p.169) and that “reason develops out of our ability to discriminate perceptually and to remember.” (p.170) Like Frede, I think a thoroughgoing rationalism as regards the justification of knowledge can be combined with acknowledging the empirical antecedents or consequences of that same knowledge.

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