I

Distinctions between what is basic and what is derivative run through Aristotelian metaphysics and epistemology. Aristotle’s ontological analyses rest on the distinction between substances and the nonsubstantial beings dependent on them. This distinction is mirrored by (and partially corresponds to)¹ an epistemological distinction between two different kinds of beings studied by the sciences: those kinds that constitute the subject matter of the sciences and the derivative features of those kinds, which are accounted for on their basis. Just as, from the point of view of ontology, substances are ultimate substrates, and nonsubstances ontologically depend on substances, so, epistemologically, certain kinds are taken to be basic by the science that studies them, and scientific understanding of all other beings considered by that science is derived from what the science presumes about those kinds.

APo. 1 sets out the bsics of Aristotle’s theory of demonstration, the form in which scientific explanations can ideally be cast. Demonstrations are syllogisms, of which the minor term is the subject kind under consideration, and of which the major term is the feature belonging to the subject that is being explained. Demonstrations presuppose that certain definitional features belong to the subject kinds. Such features constitute what the subjects of the science are; they are to be distinguished from demonstrated features.

This much is clear enough. But much of the theory of explanation that Aristotle works through in the Posterior Analytics is notoriously complex and obscure. In part this is because Aristotle uses a number of phrases (kath’ hauto, atomos, katholou, heteros aitios, hupokeimenon) in a technical manner, without always clearly distinguishing what sense of a term is being used in a particular passage. Add to this the terse nature of the argumentation, and it is no surprise that for millennia particular passages have admitted of many interpretations. Lingering problems in interpreting the Posterior Analytics have resulted. Although Aristotle’s account is honored as having hit on the notion that explanation involves a kind of inference from the essences of real kinds, it is often considered confused and incoherent in parts, and as containing significant gaps in working through how these explanations are supposed to work.

Aristotle’s account can be acquitted of many such charges if we appreciate that in the Posterior Analytics the terminology by which he distinguishes scientific subjects and demonstrated attributes is more fluid than usual. The distinction between basic kinds and demonstrated features is referred to in a number of ways, since different contexts require focusing on different aspects of the distinction. The following distinctions made by Aristotle, I shall argue, are either equivalent or are closely related, made in different contexts in order to solve different problems.

The relationships among these distinctions have not wholly escaped commentators but they have not been thoroughly worked through in a way that sheds light on otherwise obscure aspects of Aristotle’s theory of demonstration. The present paper focuses on one such issue: what lies behind Aristotle’s declarations that an attribute or feature that is demonstrated to belong to a scientific subject is proper to that subject? The answer is found in APo. 2.8-10, if we understand these chapters as bearing not only on Aristotle theory of definition but also as clarifying the logical structure of demonstration in general. If we identify the basic subjects with things that have no different cause, and demonstrable attributes (the kath’ hauta sumbebēkota) with those that do have ‘a different cause,’ the definitions of the demonstrable attributes necessarily have the minor terms of the appropriate demonstrations in their definitions, for which reason the subjects and demonstrable attributes are coextensive. In this paper I show how and why this is so.

II

Book 1 tells us that each of the sciences posits certain basic kinds as objects of study. “For every demonstrative science concerns three things, first, those for which it posits the being (this is the kind of which it studies the kath’ hauta attributes). . .” (APo. 1.10 76b11-13).² Each science posits as principles of explanation the definitions of the basic kinds that it studies. “In the case of some things one must already grasp that they are; in the case of others, one must know what that which is referred to (to legomenon) is, and in the case of some things, both . . . In the case of triangle, one must know that [the term] signifies this and in the case of the monad one must know both, what [it is that] is signified and that there is [such a thing] (1.1 71a12-6).³ (This is distinction A.) Aristotle identifies this distinction with that between those items of which the existence is presupposed by the sciences, the primary items, and the derivative items of which it is not: “In the case of the primary items and the things [derivative] from them, what it signifies is assumed, but that they are is assumed for the principles and is proven for the others. For example, what monad and straight and triangle are is assumed, but that they are is assumed for the monad and magnitude, but is proven for the others” (I.10 76a32-6).⁴ The facts that there are such things, and that they have certain essences, are not themselves objects of explanation and accordingly cannot be demonstrative conclusions. The determinations of which basic kinds are actually to be found in the world (expressed in what Aristotle calls hypotheses) and the expression of their explanatorily basic features (in what Aristotle calls definitions)⁵ are not absolute starting points of inquiry, for they are the results of empirical investigation, dialectical discussion, and a process of seeing which purported explanations hold water and which do not. Nonetheless, in the context of demonstration, they are presented as basic.⁶ On the basis of the definitions of the primary things one demonstrates that the derivative entities are predicated of them, and in that way demonstrates their existence.⁷ These derivative entities are accordingly to be understood as demonstrable attributes, which Aristotle elsewhere calls the kath’ hauta sumbebēkota or the pathē kath’ hauta.⁸

Book 1 suggests that there are two kinds of definitions: scientific ones, which are basic to explanation and are accordingly indemonstrable, and pretheoretical accounts of the signification of terms. Book 2 however argues that there is a way, or a sense, in which definitions can be both demonstrable (and are accordingly not basic) yet express essences grasped by science. The key to the apparent discrepancy lies in a proper understanding of Distinctions A and B and showing how implicit appeal is made to them in Book 2.

The account of how there can be a kind of ‘demonstration of the definition,’ is the culmination of a long continuous stretch of argumentation which begins by discussing the kinds of inquiry for which a demonstration provides the answers. Aristotle asserts that the answer to a ‘why is it’ question is the same as the answer to a ‘what is it’ question, and supports this by means of examples (APo. 2.2 90a14-23). The answer to the question ‘why is the moon eclipsed?’ is found in the answer to the question ‘what is an eclipse?’: a privation of light from the moon on account of the interposition of the earth. The two answers differ only in their linguistic formulation.⁹ A second example that Aristotle gives is that of the definition of a concord (what we, though not the Greeks, would call a ‘harmony’) and the answer to the question “why is there a concord?”: A concord is a numerical ratio between high and low; two notes form a concord because there is numerical ratio between them. Accordingly, demonstrations, which are explanatory insofar as their premises reveal the cause of their conclusion, can thereby serve as expressions of the ‘what is it’. Definitions can, in a sense, be demonstrated.

Aristotle’s account of when and how this is so is not clear and even the main lines of his account have always been the occasion for significant controversy. Crucial is how we interpret Aristotle’s assertion that this sort of demonstration is possible only when “there is a different cause” (2.8 93a3-7, 93b19, 2.9 93b21, 25-8). Some commentators have read Aristotle here as talking about kinds of demonstrations and/or definitions different from those discussed in Book 1, while some take him to be talking about different kinds of definienda. Many of those who belong to the first group of exegetes take the antecedent of “what has a different cause” at 2.8 93a5-6, 93b19, and 2.9 93b21-28 to be not a kind of entity, but a kind of definition which is made intelligible on the basis of an alternative, less provisional definition of the same thing. According to this line of interpretation, this more adequate, scientific demonstration is different from the less adequate, more provisional definition, insofar as it has different semantic content, and is the cause (or explanation) of the latter insofar as it renders it intelligible.¹⁰

I have argued that the alternative strategy, which takes Aristotle to be distinguishing between kinds of being, not kinds of definitions, has its origin in the lost commentary of Alexander of Aphrodisias.¹¹ It was advocated by ^{Ross (1949}), and then lost a bit of favor in the following decades. Nonetheless it has recently gained a number of adherents, myself among them.¹² Those of us who take this line agree that sorts of entities for which demonstrations cannot reveal definitions in the manner described in 2.8, which Aristotle refers to as ‘the things that do not have a different cause,’ are the basic beings posited by the sciences.¹³ For if something has a ‘different cause,’ that is to say, is present in the world on account of something belonging to another kind, that cause would serve as a middle term of a demonstration, the conclusion of which would be the predication of the kind in question of a more fundamental subject. For example, if the kind dog had a cause that was “different” from itself, call it M, one would demonstrate that there are dogs by means of a demonstration of the form “S is M, M is a dog, therefore S is a dog.” The only logically possible candidate for such an S would be a sub-breed of dog (say, miniature schnauzer, where M would be a less specific sub-breed like schnauzer),¹⁴ but being a dog is part of the essence of even such an S. It is what S is, for which reason neither S nor a definitional attribute of S would not be a cause ‘different from’ dog. That S is a dog would be posited in the definition of S, and that this definition refers would be ensured by the ‘hypothesis’ that there is in fact such a thing as S.¹⁵

Accordingly, the canonical form of demonstration is “S is M, M is P; therefore S is P,” where S is a basic subject and P is a predicate that is kath’ hauto sumbebēkos of S, that is, not included in the essence of S although it is demonstrable of S. S, as a basic subject, does not have a cause different from itself. P, as a demonstrable predicate, does have a cause different from itself; this will be M, S, or an intermediate middle term. For this reason, Distinction C turns out to map onto Distinction A and Distinction B. The features of reality treated by each science can be classified into two groups. The first is constituted by certain primary subject kinds, both generic and specific. Although they may be ontologically dependent on other beings (as mathematical entities are quantities ontological dependent on their inherence in substances), the sciences that study them regard them as basic. (It is of no concern to the mathematician qua mathematician that a quantity under consideration is a quantitative aspect of a substance.)¹⁶ That they exist, as having the basic features that they have, is presumed by those demonstrations that show why they have the derivative features that they have.¹⁷ This is why each science considers them as having no cause different from those kinds themselves. The definitional features of these kinds belong to this same group, for, although linguistically and conceptually they can be regarded by themselves (for example, being rational can in a way be conceptually isolated from being human) the positing of the essence and existence of the basic kinds is at the same time the positing of their definitional features.

The other class of beings considered by the science is made up of those things that do have a cause different from themselves. It is natural to read the distinction between things for which the cause is different and those for which it is not to be exhaustive. Given that the things for which there is no different cause are the basic subjects, of which each science assumes both the essence and the existence, we would expect that the things with a different cause (those whose definitions can in a sense be demonstrated), are the items (considered by a science) other than those basic subjects. According to Aristotle’s syllogistic scheme, terms either appear as subjects or predicates. So those things for which there is a different cause will appear in demonstrations as predicates of the basic subjects. This class will exclude definitional predicates, for as noted above, these predicates are what their subjects are, and hence are not different from the subjects. It will be the class of pathē kath’ hauta or kath’ hauta sumbebēkota of the subject kind in question. Distinction C is to be identified with Distinctions A and B.

III

There is a wrinkle here: Aristotle’s theory of demonstration admits of some subject terms that are not basic subjects, insofar as their existence is not presumed by the science that studies them but needs to be demonstrated. Aristotle’s stock example here is that of the triangle. It is demonstrable that triangles are 2RA, but even though it would seem to be the case that triangle is the minor term of this demonstration, triangle is not a simple subject; one must demonstrate that there are triangles (1.1 71a14-16, 1.10 76a31-6). Following ^{McKirahan (1992}) let us call these ‘derivative subjects’. Note that any demonstration that such exist would have to be on the basis of the principles concerning simple subjects. The demonstration that triangles exist would have to resemble the proof of Euclid Elements 1.1 (which constructs an equilateral triangle on a given line) insofar as, on the basis of what we know about lines, we can show that they are such as to be arranged in a way that satisfies what we mean by the term ‘triangle’.¹⁸ ‘Triangle’ then would be a nominalization of the predicate ‘such as to be a side of a triangular figure when arranged in a certain way’, inherent in ‘straight line’ or ‘magnitude’. “Triangles are 2RA” is a short way of saying “lines are such that when arranged in a certain way they form the side of a three sided figure that is 2RA” and so forth.¹⁹ For this reason, Distinction D, between atomic or simple subjects on the one hand, and derivative subjects, on the other, turns out to be a subset of (or identical to²⁰) Distinctions A, B, and C.²¹

IV

I read APo. 2.8 as working through how one can cull a kind of (nonfoundational) definition of demonstrable attribute or derivative subject on the basis of the demonstration concluding that such an entity holds of a basic subject. The demonstrations that are, in a sense, demonstrations of the definitions of such items do not have the definitions in question as their conclusions. That is to say, the demonstration that reveals the definition of P does not have a conclusion of the form “a P is such and such.” The term whose definitions is revealed is not the minor term of the syllogisms in question. Rather, because “in all of these cases it is clear that the what is it and the why is it are the same” (2.2 90a14-5), the middle term of a demonstration that a basic subject S is P will be the cause of P, and once that middle term is revealed, a definitional formula of P can be compiled.²² Aristotle discusses two examples. By demonstrating that the blocking of light that we identify with an eclipse belongs to the moon, by the middle term ‘interposition of the earth’ we, in a sense, ‘demonstrate’ that the (lunar) eclipse is a blocking of light from the moon that results from the interposition of the earth (2.8 94b4-7). By demonstrating that the noise that we identify as thunder belongs to the clouds, on the basis of the middle term ‘quenching of fire’, we, in a sense ‘demonstrate’ that thunder is a noise that results from the quenching of fire in the clouds (2.8 93b8-14).²³

This is possible only if we already have a reliable way of identifying the explanandum, so that we can show that those features by which we identify it follow from and are explained by the essence of basic subjects. This is why in the explanations that Aristotle sketches the explanandum P is initially identified with a complex²⁴ of other terms, say Q and R. This might be done on pre-theoretical grounds.²⁵ The middle term M of the demonstration concluding that S is Q and R, will express why S is Q and R, and one can therefore on the basis of the demonstration formulate a definition of P as “Q and R in S on account of M.” Aristotle’s examples are that we begin with the initial identification of a lunar eclipse with a certain occlusion of light from the moon (or even with the moon’s failure to cast shadows, though full), or of thunder as a certain noise in the clouds. We demonstrate that Q and R belong to S. Given the premise that identifies P with Q and R, we can infer that S is P. So, having demonstrated that a certain occlusion of light belongs to the moon, we can then in a sense demonstrate that the moon is eclipsed, on the basis of the premise that this occlusion is an eclipse.²⁶

V

One might object to this this line of interpretation on the grounds that demonstrable attributes necessarily inhere in their subjects (1.4 73a21, 1.6 75a28-31) and always inhere in them (1.8 75b22-5). But at first sight that does not conform to Aristotle’s own examples here: eclipse and thunder.²⁷ For eclipsing is not a universal, necessary attribute of the moon, since the moon is not always eclipsed. Similarly, clouds do not always thunder, nor do they thunder of necessity. Accordingly Ferejohn has argued that the beings for which there is a different cause are events, and that the middle term of the demonstrations that account for them are the efficient causes of those events.²⁸ On this understanding a demonstration of the sort that explains a lunar eclipse does not conform to the canonical form of demonstration discussed in Book 1, in which the explananda are the facts that certain necessary and eternal features are predicated of kinds studied by the sciences.²⁹

Can Aristotle’s theory account for the demonstrations that explain occasional particular phenomena like eclipses or thunder? The proper object of scientific knowledge is not the particular event of an eclipse or thunder; it is rather the necessary, eternal fact that the moon is such as to eclipse, and the clouds are such as to thunder. The middle term, explaining what it is about the subject that make them subject to such events, could be employed to reveal why the events occur when and how they do. (For example, by showing why the moon is such as to have an orbit that results in the blockage of the sun’s light from its surface, one could account for a particular occurrence of an eclipse, but that occurrence would not be a proper object of scientific explanation.³⁰) Just as Aristotle says that in a sense the one who knows that the interior angles of a triangle add up to two right angles knows it about some particular triangle (1.1 71a31-b6), so too the one who understands the workings of lunar eclipses in some way understands why and how a particular eclipse occurs. To use an Aristotelian locution absent from the Posterior Analytics, the actual astronomer has the potentiality for rendering intelligible the particular eclipse. There is no need to suppose that the only sort of entity such as to have its definition revealed through demonstration in the manner accounted for in 2.8 is an event for which one can demonstrate that it occasionally or regularly holds of a basic subject.

VI

I now turn to consider the nature of demonstrable predicates and the role that they play in Aristotle’s general theory of demonstration.

Commentators often write as though Aristotelian demonstrations are a matter of unpacking the genus-species relations implicit in the definition or ‘what is it’ of a minor term. For example, if the minor term S is its definition “G and D”, and G itself is defined as G′ and D′, the form of an Aristotelian demonstration would be “S is G, G is G′; therefore S is G.′” An example is “Flounder are fish, fish are animals; therefore flounder are animals.”³¹ (Note that on the account argued for above, none of these terms would be ‘different from’ the minor term.) This view has several advantages. First, both premises are definitional, so it is easy to see how it is the case that a demonstration is based on unmediated premises that are first principles of the sciences. Second, Aristotle’s own argument in 1.22-23 that no demonstration is of infinite length itself presupposes that all demonstrative premises are of this sort. But even if we grant that some demonstrations do indeed take this form, there are severe problems in interpreting Aristotle as holding the view that they all do. First, demonstrative conclusions would thereby be limited to tautologies (at least for those who have mastered the appropriate science and for that reason have command of the relevant definitions). Second, it is hard to see how the middle term of such a demonstration could serve as identifying the cause of conclusion. Being a fish is not the cause of a flounder’s being an animal in any straightforward way, as being animal is simply what a fish is. (Being an animal “has no other cause”.) Third, Aristotle’s own examples, such as 2RA, belie this view. On no plausible interpretation can we interpret Aristotle as taking this predicate to be part of the definition of triangle.

The demonstrable attributes are the kath’ hauta pathē and the kath’ hauta sumbebēkota. That they are not definitional presumably accounts for their being said to be accidental (sumbebēkos).³² But in what sense are they kath’ hauta? We can presume that Aristotle takes them to be kath’ hauta in at least one of the four senses of the term that he delineated just a few lines above.³³ We have already noted that they cannot be definitional. This excludes the first sense of kath’ hauto distinguished in 1.4 (which I refer to as kath’ hauto 1): “Those things are kath’ hauta, first, which belong in the ‘what is it’ [of their subject] (73a34-5).”³⁴ Two other possibilities have emerged (not mutually exclusive). The first is that the relevant sense is the fourth which Aristotle outlines at 1.4 73b10-11: “that which belongs to each thing because of itself (di’ hauto) is kath’ hauto”. This formulation itself suggests that Aristotle is saying that whenever S is P because of S we can say that S is P is kath’ hauto. But the example that Aristotle proceeds to give (the animal’s death is kath’ hauto in respect to its being sacrificed), shows that, even if the exact predicative form of the connection is unclear, the entities that Aristotle takes to be kath’ hauta in this sense are particular events, not kinds of event or regular or necessary features of being subject to a kind of event (for example, being such as to be subject to eclipse). This suggests that entities that are kath’ hauta in this sense will fall outside the scope of demonstration, though, to be sure, demonstrations can be employed to make sense of them. (The situation is parallel to how demonstrations concerning triangle in general can be employed to render intelligible some feature of a particular triangle, cf. 1.1 71a29-72b8). This objection is not decisive however, as Aristotle could be understood as identifying a kind of intelligible connection between kinds of events, and these are predicates. It might be an incidental feature of Aristotle’s examples that they are cases in which the predicates that stand in relation are kinds of temporally localizable events.³⁵ There is however another problem with understanding the kath’ hauta sumbebēkota as kath’ hauta in the sense of 1.4 73b10-11. At Topics 2.3 110b22-5 Aristotle says 2RA is sumbebēkos in respect to isosceles but not triangle. At APo. 1.4 73b38-74a3 we are told that 2RA is kath’ hauto in respect to triangle, not isosceles triangle. This is because a term definitional of ‘triangle’ as such (and not its genus or a subordinate species) must feature as a middle term of the demonstration that triangles are 2RA.³⁶ Showing that isosceles triangles are 2RA is properly accomplished only by means of an application of that demonstration.³⁷ Any demonstration that proceeds to show that isosceles triangles are 2RA on the basis of features unique to isosceles triangles misses the point insofar as it fails to identify the more basic feature that is ontologically responsible for the derivative feature.³⁸ But in the cases described at 1.4 73b10-11 all that is necessary is that the kath’ hauto feature arises on account of (dia) that of which it is kath’ hauto, as its effect.³⁹ There is no requirement that the latter be referred to at the appropriate level of specificity. In the very case that Aristotle presents as an example, we lack that proper specification. Whether to sphattesthai refers to sacrifice or throat cutting, the term is too specific to be cause of death as such.⁴⁰ I suggest that this sense of kath’ hauto is not relevant for demonstrative science. Aristotle is bringing up the term for the sense of completeness.⁴¹

By elimination, we can conclude that the kath’ hauta sumbebēkota are those that are such that their subjects are present in their definitions (1.4 73a37-b3), which I call kath’ hauta 2. Aristotle is explicit that all demonstrated predicates are kath’ hauta in one of the first two senses: that of having the predicate present in the subject and that of having the subject present in the predicate (1.4 73b16-18, 6.74b5-12, 22.84a11-14). Since the first, third, and fourth senses of kath’ hauto are not the relevant ones, the ‘natural reading’ is to take the second to be the relevant sense. On this account, the distinction between kath’ hauta 1 predicates and kath’ hauta 2 predicates, which (grouping among the kath’ hauta 1 predicates the subjects for which they are definitional) is Distinction E, turns out to map onto Distinctions A, B, C, and D.⁴²

VII

This reading has encountered resistance because in many cases, including Aristotle’s own favored example of 2RA, Aristotle’s description of kath’ hauta 2 predicates simply does not hold. The definition of ‘having angles equal to two right angles,’ whatever it may be⁴³ does not contain ‘triangle’.⁴⁴ This is so even though no other plane figures have their interior angles summing to two right angles.

Zuppolini addresses this objection by appeal to a distinction familiar in the secondary literature on Aristotle, that between ‘linguistic predication’ and ‘metaphysical predication’. Linguistic predication is a relation between terms in a logos: whenever we say that S is P we are saying “S is P”. Metaphysical predication is a relation that holds among realities. In the case in which ‘S’ signifies shoe and ‘P’ signifies pink, the linguistic predication “S is P” indicates that pink is metaphysically predicated of shoe. As this example shows, linguistic predication often mirrors metaphysical predication, like true propositions in a Russellian ‘logically perfect language’.⁴⁵ Logos depends on this kind of isomorphism (DI 1 16a3-9). But the isomorphism is often merely apparent. Categories shows how linguistic predications that are formally indistinguishable express metaphysical predications of very different kinds.⁴⁶ On one influential account of the central books of the Metaphysics,⁴⁷ Aristotle re-employs the distinction between linguistic and metaphysical predication as a way of resolving ontological perplexities brought on by his hylomorphic analysis of substance: in spite of the truth of the linguistic proposition “Socrates is human,” the form ‘human’ is metaphysically predicated not of the composite substance Socrates but of Socrates’ matter. Even if Aristotle does not envisage a logically perfect language as such, Aristotle shares with Russell the convictions that metaphysical perplexities arise through being misled by the surface grammar of linguistic statements, and that ontological analysis largely consists in showing how this is so and through formulating (in a technical vocabulary) how things actually stand. That means that Aristotle thinks that it is in principle possible to formulate linguistic predications isomorphic to the metaphysical predications in the world.

Zuppolini employs this distinction between linguistic and metaphysical predication in his solution of the problem of how to find a place for the kath’ hauta sumbebēkota in the typology of kath’ hauta predications given in APo. 1.4. To say that S is P is kath’ hauto 2 is not necessarily to say that S is linguistically predicated in the definition of P. It might be (as it is in the case of “number is odd”) but it need not be. “The fact that the subjects of per se 2 predications ‘inhere in’ the λόγος of the predicates does not make them linguistic items. Since the word ‘λόγος’ denotes here the definition of an entity (and not of a term), neither are the predicates (i.e. the corresponding definienda) linguistic items. Therefore, per se 2⁴⁸ connections are also world-world relations (or metaphysical predications).”⁴⁹ On this account, “triangle is 2RA” is a kath’ hauto 2 predication because triangle (or triangularity?) is metaphysically predicated in the attribute of 2RA. But what does this mean?

Zuppolini argues that a kath’ hauto 2 predication need not have the subject linguistically present in the definition of the predicate, even if the definition is a scientific one, formulated with as much accuracy and perspicuity as possible. For in Metaphysics Δ 18 1022a 29-31 Aristotle gives as an example of a kath’ hauto 2 predication “the surface is white.” Here the subject is not surface, understood universally, but a particular surface, and surely Aristotle is not suggesting that the particular surface is even metaphysically present in the definition of white. After all, the particular surface will come and go, but the definition of white abides eternal. For this reason, Zuppolini suggests the following as what Aristotle has in mind: “a subject S ‘inheres in’ (ἐνυπάρχει) the definition of a predicate P if the relevant kind-term ‘K’ applying to S appears in the definition of P”.⁵⁰ Zuppolini argues that this allows us to understand why “triangle is 2RA” is kath’ hauto 2. Triangles are a variety of rectilinear figure, and it is rectilinear figures that are (metaphysically) present in the definition of 2RA. For to be 2RA is to have interior angles, and even if this or that account of what it is to have interior angles does not mention ‘rectilinear figure’, the notion is implicit, for which reason the attribute of being a rectilinear figure is metaphysically present in the definition of 2RA.

To be sure, even if Aristotle’s writings sometimes seem to be subject to use/mention confusion, he is aware both of the distinction between a state of affairs and the linguistic account that expresses it, as well as of the philosophical confusion that can arise from reading ontological truth directly off of the structure of a linguistic predication. Definitions and demonstrations are both linguistic entities. A definition cannot reveal what something is, and a demonstration cannot reveal why something is the case, if that which it expresses does not correspond to reality. Terms need to be disambiguated and basic assumptions need to be made explicit. Accordingly, a demonstration can only do its job of isolating and explicating metaphysical relations if the premises and conclusion of the demonstration are formulated in a perspicuous way.

So I see no reason to think that Aristotle would ever deny that what is metaphysically present in the essence of a kind will be linguistically present in an adequate scientific definition of the kind, though it may well not be present in the initial grasp of the meaning of the term denoting the kind in question.⁵¹ The “natural reading” according to which the subject is (linguistically) predicated in the definition of the demonstrated attribute, is the correct one. ⁵² But how?

VIII

I have argued that those items said to “have another cause” in 2.8 and 9 are none other than the kath’ hauta sumbebēkota. Understood in such a way, APo. 2.8 shows exactly how it is the case that the kath’ hauta sumbebēkota, even those like 2RA, do indeed have their subjects present in their definitions.

Recall that for the demonstration of the definition of P to be the vehicle by which that definition becomes known, that very definition cannot be presupposed by the demonstration in question; otherwise the demonstration would be question begging. Nonetheless, there is an explanandum, the attribute or phenomenon in question, for which one has at least a preliminary understanding. Call F what is identified by such an account of P. (This may well be a conjunction of attributes.) To show how the features by which P is identified follow the essence of subject S⁵³ one must demonstrate that S is F by means of the middle term M. On this basis, we can extract a definition of P as an F in S, by virtue of M.

If all demonstrations of kath’ hauta sumbebekota are indeed of this form, there are two important results to which I would like to draw special attention. First, we can see why it is the case that demonstrable characteristics include their subjects in their definitions. For if the scientific definition of P is “F as predicated of S, by virtue of M” then S necessarily is present in the definition of P. (Because F is predicated of S, whatever is F is S. Accordingly, whatever is P is S.)

As noted above, the demonstration that explains what thunder or eclipse is will not primarily account for the individual events of thunder or eclipse (for such are temporally localizable particulars) but the universal necessary character of being such as to undergo such an event. And this, by definition, will have the subject of that character predicated within its definition. Further, we can see why it is the case that an attribute, scientifically understood, will convert with its subject; the thesis that a demonstrable predicate is a ‘commensurate universal’.⁵⁴ P not only follows from the essence of S (and other premises); S follows from the essence of P.^{55 56}

This sheds light on the theory of demonstration as laid out in Book 1. Ferejohn has argued that there is a tension between the formal requirements of demonstration as laid out in 1.2 and in 1.5.⁵⁷ For in 1.2, all that is required for a demonstration to qualify as a demonstration is that its premises be unmediated principles that are intrinsically intelligible and express the causes of the conclusion (71b20-23). In 1.5 we are told that there is epistēmē of a fact only if the demonstration that we have of that fact has a conclusion that is ‘universal’ in the sense delineated in 1.4 73b25-7,⁵⁸ according to which the predicate of a universal proposition must hold of its subject in every case and primitively and as such.⁵⁹ Demonstrations that satisfy the first but not the second conditions are said to allow for epistēmē only in a sophistic sense (1.5 74a25-30). For example, a demonstration that isosceles triangles have 2RA would fail to qualify as a true demonstration, for it is insofar as it is a triangle that the figure is 2RA.

I have shown how to resolve the tension that arises in Ferejohn’s reading of Aristotle’s demonstrative theory. If one can demonstrate that S is P, one will be able to define P as a certain feature present in (and thereby caused by) the subject S. S will accordingly be predicated of P in its definition.⁶⁰ That would mean that if one has a legitimate demonstration that isosceles triangles are 2RA, isosceles would be present in the definition of 2RA. But the feature of 2RA as present in scalene triangle is the same feature as is present in isosceles. The two features do not have different definitions. Because their ‘what is it’ is the same, their ‘why is it’ is the same. It follows that the ‘demonstration’ that isosceles triangles are 2RA does not, as such, reveal the ‘why is it’ of 2RA. This is why a definition of 2RA could be culled not from it⁶¹ but from the proof that triangles are 2RA.⁶²

The account I have given considerably tidies up Aristotle’s philosophy of science. The kath’ hauta sumbebēkota (which include derivative subjects) are the same as the kath’ hauta 2 predicates. In order to understand how this is so we need not import the distinction between metaphysical and linguistic predication and put it to a use to which Aristotle himself does not put it. We are able to make good sense of the doctrine of the commensurate universal, and to show how the long discussion of whether and how a definition can be demonstrated forms an integral part of Aristotle’s account of the logical structure of scientific explanation.