Aristotle on Demonstrative Knowledge: Particulars Included

1 Introduction

This paper examines whether Aristotle’s account of scientific knowledge (ἐπιστήμη) through demonstration in Posterior Analytics (An. Post.) restricts all demonstrative knowledge to universals or allows for demonstrations involving definite sensible particulars. Aristotle’s favorite example of a demonstrable truth in An. Post. A is that all triangles have angles equal to two right angles (abbreviated as ‘2R’). The demonstration yielding knowledge of this truth can be roughly reconstructed as follows:

1. All triangles are essentially thus-and-so.

2. Whatever is essentially thus-and-so has 2R.

3. Therefore, all triangles have 2R.

Most scholars would agree that this demonstration yields knowledge of the conclusion by revealing the explanation of why it holds and does so necessarily, thereby satisfying Aristotle’s central requirements for knowledge in An. Post. A 2. Nonetheless, Aristotle also acknowledges demonstrations – or, more cautiously, deductions – that apply this conclusion to subspecies of triangles, such as isosceles, as well as to particular sensible triangles. The latter deduction, envisioned in An. Post. A 1, may be reconstructed as follows:

1. All triangles have 2R.

2. This [figure in the semicircle] is a triangle.

3. Therefore, this has 2R.

The question of whether this deduction counts as a demonstration yielding demonstrative knowledge is significantly more controversial. The prevailing scholarly view leans strongly toward a negative answer and may be summed up as follows: “There can be no demonstrative knowledge, which strictly speaking means no knowledge at all, of particulars”; “There is a long tradition going back to Aristotle, according to which science deals only with the universal”; and “It is only universal propositions which Aristotle will allow into the sciences.”^[1]

This paper challenges the assumption that Aristotle’s account excludes sensible particulars from demonstrative knowledge. I argue that his position is more nuanced and allows for demonstrative knowledge involving particulars. The central focus is a passage in An. Post. A 8, often regarded as evidence that Aristotle rejects demonstrations with definite particulars as subjects. However, I will argue that this passage leaves room for such demonstrations and that they yield knowledge satisfying Aristotle’s requirements, albeit in a qualified way. Accordingly, I propose that Aristotle distinguishes between two types of demonstrative knowledge: unqualified and qualified. While qualified knowledge involving definite particulars falls short of unqualified knowledge, it nevertheless remains genuine demonstrative knowledge.

The remainder of this paper consists of four sections. The first outlines Aristotle’s central requirements for demonstrative knowledge, as specified in An. Post. A 2. The second section examines the controversial passage in An. Post. A 8, arguing that Aristotle distinguishes between unqualified and qualified (‘as if accidental’) knowledge, with the latter involving sensible particulars as subjects. It further explains how qualified knowledge meets Aristotle’s central requirements for knowledge. The third section argues that unqualified knowledge entails potential knowledge of a definite particular, which can be actualized by applying it to a particular at hand. The paper concludes by considering the role of qualified demonstrations in demonstrative science and highlighting the broader implications of this interpretation.

2 Aristotle’s Requirements for Knowledge

Scholarly discussions of Aristotle’s view of scientific knowledge typically center on An. Post. A, where he develops his account of demonstrative knowledge. I set aside the question of whether all scientific knowledge is demonstrative and focus on the converse view to which Aristotle is clearly committed, namely that all demonstrative knowledge is scientific.^[2] In what follows, this is what I mean by ‘knowledge.’ This section outlines Aristotle’s central requirements for knowledge and introduces some considerations regarding the knowability of particulars.

In An. Post. A 2, Aristotle provides the following definition of what he calls ‘knowledge without qualification’ (ἐπιστήμη ἁπλῶς), or, for short, ‘unqualified knowledge’:

We think that we know (ἐπίστασθαι) each thing without qualification (ἁπλῶς) (and not in the sophistical way, accidentally) when we think that we know the explanation (αἰτία) because of which the thing is, that it is its explanation, and that this [thing] cannot be otherwise.^[3] (71b9–12)

According to this definition, the two central requirements for knowing something without qualification are knowing its explanation and that it cannot be otherwise, which is Aristotle’s way of saying that it must be necessary (An. Post. A 4, 73a21–22; Met. Δ 5, 1015b7–9). He holds that we have this knowledge through demonstration, a ‘scientific deduction’ (An. Post. A 2, 71b17), which consists of premises and a conclusion in subject-predicate form: the predicate term stands for that which belongs to a subject (let us call it an ‘attribute’) and the subject term for that to which the attribute belongs. Thus, the ‘something’ that we know without qualification is the conclusion of a demonstration in which an attribute is predicated of a subject. For a demonstration to yield knowledge of the conclusion, its premises must be explanatory and necessary.^[4] Let us examine these requirements in greater detail.

A demonstration is explanatory in virtue of having premises that are explanatory of the conclusion. Specifically, Aristotle takes the middle term, common to both premises, to be explanatory (αἴτιον, A 13; B 2, 89b37–90a9). A demonstration thus explains why an attribute belongs to a subject via the middle term: S is P because S is M and M is P.^[5] For Aristotle, the middle term expresses the essence of something, and paradigmatically, the essence of the subject.^[6] This is evident in Aristotle’s main example of a demonstrable truth in An. Post. A: all triangles have 2R because they are essentially thus-and-so, and whatever is essentially thus-and-so has 2R.^[7] Here the middle term that explains the connection between the subject and the demonstrable attribute is the essence of triangle. As for the connection between the subject and its essence, Aristotle holds that it is immediate and does not admit of demonstration (An. Post. B 3–7). This suggests that, paradigmatically, the conclusions of demonstrations are truths in which the attribute belongs to the subject non-essentially. Given the necessity requirement, it follows that demonstrations explain why certain non-essential but necessary attributes belong to their subjects based on essential and necessary attributes.

That demonstrations reveal explanations of demonstrable truths from essences is a standard view, defended by Bronstein (2016), Koslicki (2012), and Goldin (1996).^[8] This view implies that there is a distinction between necessary and essential truths, on the one hand, and necessary and non-essential truths explained by essential ones, on the other. The distinction between two types of necessary attribute is present in Topics and also emerges from Aristotle’s discussion of demonstrative necessity in An. Post. A 4:^[9]

One thing belongs to another in itself (καθ' αὑτὸ) if it belongs to it in what it is (τί ἐστιν), e. g., line to triangle or point to line (for their being depends on these and they inhere in the account which states what they are), and also if what it belongs to itself inheres in the account which shows what it is, e. g., straight belongs to line and so does curved, and odd and even to number. […] [C]oncerning things that can be known without qualification, those things which are said [to belong] in themselves, either in the sense of inhering in what is predicated or of being inhered in [by what is predicated], do so on account of themselves and of necessity. (A 4, 73a34–b19)

Aristotle associates demonstrative necessity with two ways in which an attribute belongs to a subject in itself. In the first case, an attribute belongs to a subject in itself, and thus of necessity, if it is part of what the subject is, that is, its essence. For example, if a triangle is defined as a plane figure bounded by three lines, then all the specified attributes belong to the subject essentially and of necessity. Such attributes figure in definitions, accounts revealing or stating the essence (A 4, 73a37–38; B 10, 93b29), which are among the premises of demonstrations. In the second case, an attribute belongs to a subject in itself, and thus of necessity, if the subject is part of the essence of the attribute. For example, number is part of the essence of even. There is considerable controversy over what kind of attributes Aristotle has in mind, but there is broad agreement that this sort of necessity is not restricted to premises of demonstrations. It also applies to conclusions, that is, to attributes that can be demonstrated to belong to their subjects.^[10]

Bronstein (2016, ch. 3) argues that demonstrable attributes that are defined partly in terms of their subjects belong to their subjects ultimately because of the subject’s essence.^[11] He distinguishes these from ‘in itself accidents,’ which are explained by the subject’s essence but are not defined in terms of their subjects.^[12] Setting aside the intricate details, I agree that demonstrable attributes belong to the subject (ultimately) because of the subject’s essence and derive their necessity from it. As Barnes puts it, the necessity of the objects of scientific understanding is “ultimately grounded in essential or definitional connections” (Barnes 2002, 120). Accordingly, Aristotle’s view on demonstrative necessity in An. Post. A 4 may be summed up thus: an attribute belongs to a subject of necessity if is part of the subject’s essence or if its belonging to a subject is ultimately explained by the subject’s essence.^[13]

According to this interpretation, the subjects of demonstrative knowledge have essences that explain why they have other necessary but non-essential attributes. Notably, while this interpretation requires subjects to have essences, it does not specify what kind of things can figure as subjects. This leaves open the possibility that the subjects include definite particulars, provided they have essences. Whether Aristotle allows demonstrations with definite particulars as subjects is a separate question, which will be addressed in the next section. But first let us consider whether Aristotle regards particulars as things with essences, focusing on Categories 5, which provides his most extensive treatment of particular things in his logical works.

In Categories 5, Aristotle speaks of ordinary particulars like humans and horses as ‘primary substances’ and argues that all other things are, ultimately, in or said of them as subjects, while they themselves are not in or said of anything (2a10–13; 2a34–b6; 2b15–17). This argument is commonly understood to establish the ontological dependence of other things on particular substances, for Aristotle concludes that “if there were no primary substances it would be impossible for any of the other things to be” (2b5–6).^[14] However, while other things depend on particular substances, there remains an important difference: some reveal what the particular is, whereas others do not. Specifically, Aristotle argues (2b29–36; 2b7–14) that species and genera reveal what the primary substance is, “for if one is to say of the particular human what they are (τί ἐστιν), it will be in place to give the species or the genus (though more informative is to give human than animal)” (2b32–36). Assuming that what something is (τί ἐστιν) indicates its essence, Aristotle can be seen as distinguishing between what is essential to a particular from what is not.^[15] Accordingly, particular substances as the ultimate subjects are not the so-called bare particulars which are essentially no kinds of things at all. Rather, particulars are things with essences: each particular substance is essentially of a certain kind. For example, this human is essentially human, this horse essentially horse, and so forth.^[16]

The next section aims to show that Aristotle allows there to be demonstrations with definite particulars as subjects. To forestall a possible confusion, let me address two objections that no one (as far as I know) has stated in quite these terms but that might help to explain why Aristotle’s views are thought to involve a difficulty regarding the knowability of particulars.

One might object that the alleged demonstrative knowledge of particulars as things of a certain kind is not about particulars as such, in all their uniqueness and particularity.^[17] I concede that demonstrations involving particulars explain why they have necessary attributes that belong to all things of the same kind, and they do not explain the presence of non-necessary or unique attributes (if we call such unique aspects of particulars ‘attributes’ at all). Hence, if the claim that there is no demonstrative knowledge of particulars means that there is no knowledge of them in all their uniqueness, then this is true: as such, particulars are objects of sense-perception, rather than knowledge. Nonetheless, to insist that this rules out demonstrative knowledge of particulars presupposes that they can only be viewed in their full particularity and uniqueness. Yet that is not the view that emerges from Categories 5, where Aristotle treats particulars as things of a certain kind. He characterizes a particular substance as a ‘this something’ (τόδε τι, 3b10–13), and his examples ‘this human’ and ‘this horse’ (2a13) suggest that ‘something’ indicates the species under which the particular falls.^[18] Thus, for Aristotle, particulars are not just unique ‘thises’ but also ‘somethings,’ and this is what matters for demonstrative knowledge.

Another related objection is that knowledge of particulars as falling under kinds is not about the particulars themselves, but about the kinds under which they fall. For example, when we know why Socrates qua human is grammatical, our knowledge is not about Socrates but about the species human. Indeed, Aristotle holds (at least in his logical works) that particulars of the same kind share the same essence, along with necessary and non-essential attributes explained by their essence.^[19] However, this does not mean that their essences are not their own, as it were. Some authors associate the denial that particulars have essences with Platonism, as Aristotle understands it.^[20] On this view, particulars are what they are in virtue of something else. For example, Socrates is human because he participates in the Form of Human. For Aristotle, Socrates is human (not in virtue of something else but) in himself.^[21] Consequently, our knowledge about Socrates qua human is knowledge about Socrates, given that being human is part of what he is. Thus, demonstrative knowledge about particulars as things of a certain kind can be said to be about particulars, insofar as they are things of a certain kind.

Without further ado, let us examine what I will call a ‘target passage’ in An. Post. A 8. This passage is often regarded as a crucial piece of evidence against the proposal that Aristotle allows for demonstrative knowledge with definite particulars as subjects. However, I will argue that this passage does not rule out such knowledge but rather makes a distinction between unqualified and qualified demonstrative knowledge.

3 Unqualified and Qualified knowledge

Let us first present the target passage from An. Post. 8 and outline the key issues that this passage raises:

It is also clear that if the premises from which a deduction proceeds are universal, then the conclusion of such a demonstration – of a demonstration without qualification – must be eternal. There is therefore no demonstration or knowledge of perishables without qualification, but only as if accidentally because it [the predicate] does not belong [to the subject] universally but only at a certain time and in a certain way. Whenever there is [such demonstration and knowledge], one premise must be non-universal and perishable – perishable because when it holds, the conclusion too will hold, and non-universal because it will hold of one [instance of the subject] to which [the major premise applies] but not of others – so it is impossible to deduce [a] universal [conclusion], but that [the conclusion holds] now.^[22] (75b21–30)

This passage invokes two issues with implications for the knowability of particulars. First, Aristotle asserts that the premises, and so also the conclusion, of an unqualified demonstration – presumably, a demonstration yielding unqualified knowledge – are universal. This invokes the question of what he means by a ‘universal’ (καθόλου), and whether he requires the premises of demonstrations to be universal. I will argue that whereas unqualified demonstrations proceed from universal premises, definite particulars can figure in ‘as if accidental’ demonstrations yielding demonstrative knowledge that meets the requirements for knowledge, albeit in a qualified way. Second, Aristotle makes use of temporal notions, describing the conclusion of an unqualified demonstration as ‘eternal’ and contrasting it with the conclusion of an ‘as if accidental’ demonstration that holds ‘at a certain time’ and ‘now.’ I will examine these temporal notions in connection with his account of necessity and reject a line of interpretation which requires the subjects of necessary truths to exist eternally.

What does Aristotle mean by a ‘universal’ in Posterior Analytics? In A 4, he introduces two relevant notions, which we may regard as notions of a universal:^[23]

I say that something belongs to all instances of the subject (κατὰ παντός) if it does not belong to some instances and not to others, nor at some times and not at others. (73a28–29)

I mean by a universal (καθόλου) that which belongs to all instances of the subject and in itself (καθ' αὑτὸ) and as itself (ᾗ αὐτό). (73b25–26)

The first notion of a universal is defined in terms of belonging to all instances of the subject, which is contrasted with belonging to some instances and at some times. In the target passage, Aristotle contrast the conclusions of an unqualified and an ‘as if accidental’ demonstrations in similar terms: the latter holds of one instance of the subject and at a certain time. This suggests that Aristotle takes the conclusions, and so also the premises, of unqualified demonstrations to be universal at least in the sense that the attribute belongs to all instances of the subject and at all times. We may thus conclude that unqualified demonstrations are deductions in the universal affirmative mood Barbara, with premises and conclusions of the form ‘all S’s are P’ or ‘P belongs to all S.’ Nonetheless, Aristotle does not think that all demonstrations are in Barbara. He describes deductions in the universal affirmative mood as ‘especially scientific’ (A 14, 79a24) but also acknowledges demonstrations in other moods (A 21, 23–25).

The second notion of a universal (referred to in the above passage as ‘universal’) is more demanding: it requires that the attribute belongs not only to all instances of the subject but also in itself and as itself. While it is not immediately clear what belonging as itself adds to belonging in itself, I will follow a well-established interpretation that associates belonging as itself with co-extensiveness or convertibility.^[24] This interpretation finds support in Aristotle’s explanation in A 4 (73b32–74a3), where he states that 2R belongs to triangles as triangles but not to figures as figures (which is too broad) or to isosceles as isosceles (which is too narrow: “it does not apply to isosceles universally but extends further,” 74a2–3). This suggests that an attribute belonging to a subject as itself is convertible with that subject. For example, if something has 2R, then it is a triangle and if something is a triangle, it has 2R. Scholars often refer to such universals as ‘commensurate universals.’

It is controversial whether Aristotle requires demonstrations to have commensurately universal premises and conclusions. Some scholars maintain that he does, whereas others do not consider this a requirement for all demonstrations.^[25] In what follows, I will proceed on the assumption that unqualified demonstrations have premises and conclusions in which the attribute belongs to all instances of the subject, in itself and as itself, which entails convertibility. A paradigmatic example is a demonstration establishing that all triangles have 2R, where all terms (minor, major, and middle) convert. My aim is to show that even if we take Aristotle to have such universals in mind when stating in A 8 that an unqualified demonstration proceeds from universal premises, this does not rule out demonstrations involving definite particulars. After stating that such a demonstration has an eternal conclusion (more on this shortly), he adds that “there is therefore no demonstration or knowledge of perishables without qualification, but only as if accidentally” (75b25–26). This suggests that demonstrating without qualification is not the only way to demonstrate: there can be demonstrations of perishables ‘as if accidentally.’

The translation ‘as if accidentally’ (ὥσπερ κατὰ συμβεβηκός) follows McKirahan’s translation ‘as if incidentally’ (1992, 181). However, it is common to omit the qualifier ‘as if’ (ὥσπερ, ‘as it were’) and translate the phrase simply as ‘accidentally’ or ‘incidentally,’ as seen in Barnes (2002, 13). This translation becomes problematic if one assumes that accidental knowledge is not genuine knowledge.^[26] In the target passage, Aristotle does not deny the possibility of demonstrations about perishables but rather the possibility of unqualified demonstrations. If there are demonstrations about perishables, they must yield demonstrative knowledge, though not unqualified knowledge. Accordingly, I propose that the phrase ‘as if accidentally’ introduces a qualification on knowledge: such knowledge meets Aristotle’s central requirements for knowledge, albeit in a qualified way. Insofar as it satisfies these requirements, it counts as genuine demonstrative knowledge. I will henceforth refer to ‘as if accidental’ demonstrations and knowledge as ‘qualified’ demonstrations and knowledge.^[27] Qualified knowledge remains distinct from unqualified knowledge but also from what Aristotle in A 2 calls accidental (or sophistical) knowledge, which is not genuine knowledge.

Before expanding on these distinctions, let us consider plausible candidates for examples of qualified demonstrations:

In the target passage, Aristotle says that in a qualified demonstration one of the premises, and so also the conclusion, is “non-universal because it will hold of one [instance of the subject] to which [the major premise applies] but not of others.”^[28] The term ‘non-universal’ covers both subordinate species such as isosceles triangles, as well as particular triangles. The first example (A) is a reconstruction of a demonstration about isosceles that Aristotle mentions frequently in An. Post.^[29] The second example (B) is a reconstruction of a deduction outlined in An. Post. A 1 (71a17–29), which will be discussed in the next section. Here it will suffice to say that this example fits well with Aristotle’s characterizations of a non-universal premise and a conclusion as ‘perishable’ and as holding ‘now,’ assuming that this triangle is a sensible, perishable object.^[30] This characterization does not fit as well with example (A), since the conclusion that all isosceles have 2R holds at all times. This suggests that while Aristotle recognizes demonstrations about subspecies, in the target passage he likely has in mind demonstrations about definite particulars, such as this sensible triangle here and now.

In what follows, I turn to the question of how knowledge provided by qualified demonstrations meets Aristotle’s central requirements for knowledge. My focus is on demonstrations with definite particulars as subjects, where it is less clear how these requirements are satisfied. Both types of qualified knowledge – as exemplified by (A) and (B) above – meet the requirement concerning explanation in a qualified way, and the same reasoning applies to both cases. As far as the necessity requirement is concerned, however, qualified knowledge involving subspecies meets it without qualification, whereas it may be unclear whether qualified knowledge involving particulars meets it at all. In what follows, it will be argued that it does meet the necessity requirement, albeit in a qualified way.

4 Applying Unqualified Knowledge to Particulars

Let us start by presenting the passage in An. Post. A 1 which presents a deduction previously used as an example of a qualified demonstration:

It is possible to come to know by knowing some things beforehand and getting knowledge of the others at the very same time, namely of whichever things fall under a universal which one knows. Thus, the person already knew that all triangles have two right angles, but they come to know that this [figure] in the semicircle is a triangle at the same time as they draw the conclusion [that it has two right angles] […] Before drawing a conclusion or grasping the deduction, they should perhaps be said to know in one way, but in another way not. If they did not know without qualification (ἁπλῶς) whether there is [such-and-such a thing], how could they have known without qualification that it had two right angles? Yet it is clear that they know in this way: they know it universally (καθόλου), but do not know it in an unqualified way. (71a17–29)

Assuming that we are dealing here with an example of a qualified demonstration, this passage sheds light on the ‘mechanism’ of such demonstrations. It suggests that the major premise must be known beforehand, whereas the minor premise becomes known simultaneously with the conclusion.^[45] For example, knowing in advance that all triangles have 2R, the geometer recognizes that this figure in the semicircle is a triangle and immediately infers that it has 2R. More importantly for our purposes, this passage offers insight into the knowledge expressed in the major premise. Assuming that this premise expresses unqualified knowledge, this passage suggests that unqualified knowledge entails potential knowledge of a given particular.

Aristotle argues that when we know the major premise but have not yet drawn the particular conclusion, we know the conclusion in one way but not in another. There is a sense in which having prior knowledge is compatible with ignorance of the conclusion: we may have this knowledge without being aware of the particular instance to which it applies. For example, we may know that all triangles have 2R without realizing that this applies to some particular triangle we have never come across. However, there is also a sense in which we do have knowledge of the conclusion. In the passage above, Aristotle says that we know it universally (καθόλου). In An. Post. A 24, he says we that know it potentially (δυνάμει).^[46] Thus, when we know that all triangles have 2R, we know – universally or potentially – that this triangle has 2R. This, then, suggests that unqualified knowledge entails potential knowledge of a given particular.

I will focus on two questions raised by this suggestion, starting with the question of what ‘potential’ might mean in this context. To address this, it becomes relevant to consider an aspect of Aristotle’s definition of knowledge in An. Post. A 2 that we have not yet addressed: this definition characterizes a cognitive state of a person possessing knowledge (a scientist, if you wish). The state of having unqualified knowledge is achieved by knowing the explanation of something as being its explanation and that it cannot be otherwise.^[47] To say that a person with unqualified knowledge has potential knowledge of a given particular does not mean that they have a mere capacity to acquire such knowledge.^[48] Rather, they already possess actual knowledge which entails potential knowledge of that particular. Here it is helpful to invoke Aristotle’s distinction between two ways of being actual, commonly called ‘first’ and ‘second’ actuality: first actuality is a kind of potentiality with respect to its exercise, while second actuality corresponds to its exercise or use – a distinction “analogous to the possession of knowledge and the exercise of it” (De An. B 1, 412a23; see also B 5, 417a21–b2).

Unqualified knowledge corresponds to ‘first actuality’ knowledge – an acquired knowledge that, for example, all triangles have 2R. This knowledge entails potential knowledge of a definite particular and can be exercised by applying it to a particular of the relevant kind. When we recognize something as a particular instance of that kind and combine it with unqualified knowledge, we come to know that this particular is thus-and-so, which corresponds to ‘second actuality’ knowledge. In An. Post. A 1, Aristotle claims that as soon as we recognize a particular member of the relevant kind, we come to know ‘without qualification’ that it has the attribute belonging to things of this kind. When we know that all triangles have 2R and recognize this figure as triangle, we come to know ‘without qualification’ that this triangle has 2R.

This claim raises a second question: why does Aristotle refer to knowledge of the particular conclusion as ‘unqualified knowledge’ when such knowledge is described in A 8 as knowledge as if accidentally or in a qualified way? One might propose that knowledge applied to particulars is knowledge without qualification.^[49] My position is more moderate: I agree with the prevailing view that, for Aristotle, demonstrative science is primarily concerned with explaining universal truths rather than applying them to particular cases. Yet this does not mean that particulars fall outside the scope of science or that applying universal knowledge to particulars fails to yield scientific knowledge. If unqualified knowledge is scientific and entails potential knowledge of a given particular, then, reasonably, applying it to a particular at hand yields scientific knowledge too – it is a way to exercise or use unqualified knowledge. My position thus occupies a middle ground between the view that knowledge applied to particulars is knowledge without qualification and the view that such knowledge plays no role in science. How, then, should we reconcile the apparent tension between A 1 and A 8?

My preferred answer was introduced already in the previous section as part of my interpretation of An. Post. A 24. I argued that the best explanation for why this triangle has 2R holds of it as triangle: it has 2R because it is a triangle and all triangles have 2R. This, I propose, is what Aristotle has in mind in A 1 when he speaks of unqualified knowledge: it is the best or most appropriate knowledge we can have of demonstrable truths involving definite particulars. In An. Pr. B 21, Aristotle distinguishes between universal and particular knowledge (ἐπιστήμη) and states that “we contemplate particulars by universal [knowledge], without knowing them by [knowledge] appropriate (οἰκεία) to them” (67a28–30). It is reasonable to associate universal knowledge with unqualified knowledge and knowledge appropriate to particulars with qualified knowledge.^[50] Qualified knowledge would provide a more appropriate knowledge of a given particular than unqualified knowledge, which we may possess without being aware of the existence of that particular.

The above interpretation suggests that Aristotle’s use of the term ‘unqualified knowledge’ varies depending on whether he speaks of the requirements for knowledge or of knowledge involving definite particulars. With respect to the requirements for knowledge, knowledge of the universal conclusion (e. g., ‘all triangles have 2R’) satisfies the requirements without qualification. When this knowledge is applied to a given particular, we acquire knowledge of a particular conclusion (e. g., ‘this triangle has 2R’) that satisfies these requirements in a qualified way. With respect to knowing a particular conclusion, however, qualified knowledge counts as the most appropriate knowledge. Thus, by having knowledge that satisfies requirements for knowledge in a qualified way we have unqualified knowledge of a given particular. By having knowledge that satisfies these requirements without qualification, we have potential and less appropriate knowledge of that particular.^[51]

This concludes my exposition of Aristotle’s view on demonstrative knowledge, showing that his position is more nuanced than commonly assumed and does not exclude knowledge involving sensible particulars. This could be further strengthened by examining additional evidence, such as demonstrations with premises and conclusions that hold ‘for the most part’ or those involving singular events, like eclipses or thunder.^[52] But I hope my interpretation of An. Post. A 8 has removed an important obstacle to acknowledging demonstrative knowledge of definite sensible particulars. In the final section, I will summarize my interpretation and highlight some of its implications.

5 Conclusion

According to the interpretation developed in this paper, the target passage in A 8 does not contrast demonstrative knowledge with what is not real knowledge but rather distinguishes between unqualified and qualified demonstrative knowledge. This raises the question of how much this interpretation alters the prevailing view that knowledge is restricted to universals and there is no knowledge of sensible particulars. One could argue that these claims still hold, with the modification that they apply only to unqualified knowledge. Nonetheless, I maintain that this modification is significant: it challenges the assumption that unqualified knowledge exhausts the domain of demonstrative knowledge. I have argued that knowledge involving particulars meets Aristotle’s requirements for knowledge, albeit in a qualified way: it involves knowing the explanation of why the attribute belongs to a subject as of a certain kind (though not as such), and it involves knowing that this truth holds of necessity (though not at all times). Although it falls short of unqualified knowledge, it is genuine demonstrative knowledge.

Further, I have argued that unqualified knowledge entails potential knowledge of a given particular, and qualified demonstrations actualize this knowledge by applying it to a particular at hand. Admittedly, applying unqualified knowledge to particulars is not the only way scientists can exercise their knowledge. As Heinaman (1981, 65–67) argues, universal knowledge can be exercised by contemplating or thinking about it.^[53] Nonetheless, while scientists may reflect on their knowledge from an armchair, applying it to sensible particulars remains an important way of using it. McKirahan (1992, 184) emphasizes that application arguments are “the key to applying demonstrative knowledge to the world,” and I agree. Insofar as demonstrations yield knowledge about the world, qualified demonstrations have a role to play: they enable us to apply scientific explanations to the particulars around us. For example, they allow a geometer with demonstrative knowledge of triangles to have knowledge of a triangle drawn on the chalkboard or an ornithologist with knowledge of birds to apply it to a bird flying by.

Qualified demonstrations can be especially valuable in the context of learning by students. Aristotle holds that in learning a new domain, we should proceed from what is better known to us – sensible particulars – to what is better known by nature – universals.^[54] Qualified demonstrations involving particulars can facilitate this transition. Similarly, Lennox argues that A-type explanations are “a crucial stage in the acquisition of understanding about a domain” (Lennox 2001, 11). Further, qualified demonstrations may play a role in the acquisition of knowledge by the scientists themselves. While it has been argued that scientists do not acquire knowledge by demonstrations but rather use demonstrations to set forth knowledge already gained, Bronstein (2016) challenges this view.^[55] One of his arguments is that scientists can learn new facts through demonstrations involving definite particulars; for example, “an expert geometer, who has scientific knowledge of the fact that all triangles have 2R, learns that and why this particular figure has 2R upon learning that it is a triangle” (Bronstein 2016, 41). My interpretation can be seen as supporting Bronstein’s view by showing that such cases involve genuine demonstrations.

This interpretation also bears on a more fundamental question regarding the consistency of Aristotle’s philosophy. There has been a long-standing worry that Aristotle’s philosophical commitments are inconsistent. He holds that sensible particulars are fully real – indeed, ontologically fundamental – and assumes that knowledge is of what is real. Yet he restricts knowledge to universals.^[56] His account of knowledge, which receives its most rigorous treatment in An. Post. A, is often seen as the main source of this tension, seemingly isolating demonstrative knowledge from the world of sensible particulars. However, this presupposes that Aristotle excludes particulars from demonstrative science. Since my interpretation challenges this assumption, it helps to alleviate this worry. At the very least, it shows that demonstrative knowledge does not remain isolated from the world of sensible particulars.