Epagōgē as a way of grasping a syllogism
Joshua Mendelsohn - Graduate Student Conference on Inquiry
The Posterior Analytics opens with the statement that “all teaching and learning
of an intellectual sort” comes about from “what we already know.”1 But in the
next paragraph, Aristotle goes on to qualify this claim:
It is possible to acquire knowledge when you have acquired knowledge of some items earlier and get knowledge of the others at the
very same time (e.g. items which in fact fall under a universal of
which you possess knowledge). Thus you already knew that every
triangle has angles equal to two right angles; but you got to know
that this figure in the semicircle is a triangle at the same time as
you were performing epagōgē (ἐπαγόμενος).(71a.17–21)2
Not all learning takes place on the basis of strictly prior knowledge. Sometimes,
learning takes place on the basis of facts that you only come to know at the very
same time you learn what follows from them. As an example, Aristotle describes
someone – a student of geometry, say – who already knows that the internal
1
πᾶσα διδασκαλία καὶ πᾶσα μάθησις διανοητικὴ ἐκ προϋπαρχούσης γίνεται γνώσεως (71a.1–
2)
2
Ἔστι 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 ἴσας, προῄδει· ὅτι δὲ τόδε τὸ ἐν τῷ ἡμικυκλίῳ τρίγωνόν ἐστιν, ἅμα ἐπαγόμενος
ἐγνώρισεν. For the Posterior Analytics, I provide Ross’s text (Ross 1949) and Barnes’s
translation (Barnes 1993), in places modified.
1
angles of any triangle sum to two right angles (71a.20), a property which I will
refer to following the literature as “2R”. The geometer examines or constructs a
geometrical diagram.3 On the basis of her knowledge that all triangles have 2R,
and the fact that the diagram displays a triangle in a semicircle, the geometer
learns that this particular triangle has 2R. Her learning therefore takes place on
the basis of two facts. One of these, the universal proposition that all triangles
have 2R, is known strictly prior to the episode of learning. The particular
proposition that there is a triangle in the semicircle, however, is not something
of which the student has prior knowledge. Rather, she learns this fact at the
very same time that she learns something else on its basis: That this triangle,
in particular, has 2R.
Aristotle goes on to compare the mental state of the student before and after
she learns:
Before you perform epagōgē (ἐπαχθῆναι), i.e. before you get a deduction (λαβεῖν συλλογισμὸν), you should perhaps be said to understand
it in one way–but in another way not. If you did not know whether
3
Aristotle does not make clear here whether the person initially fails to be aware of the
triangle because she has not yet constructed it (and hence because it does not yet exist),
or whether it is already drawn but, perhaps owing to complexity of the diagram, hiding
in plain sight. However, if Aristotle has in mind the same construction as Met. Θ.9,
then he must have in mind the former, since he makes clear there that if the triangle
“had been constructed (ἦν διῃρημένα)”, it would have been “evident (φανερὰ)” (1051a.23;
see also 1051a.25–26). This counts against the reading of McKirahan (1983) (see below),
who takes Aristotle’s case to be one in which the geometer sees the triangle, but fails
to see it as a triangle, since his reading presupposes that the triangle has already been
drawn and therefore visible but not yet recognized as such. Heath (1949, 39), however,
takes Met. Θ.9 to be referring to a different construction. He notices that Aristotle
emphasizes in Post. An. Α2 (but not Met. Θ.9) that the person does not first learn
that something is a triangle and then infer from this that it has 2R but instead learns
that it has 2R at the very same time as recognizing that it is a triangle. He provides a
construction in which this takes place. I have included this construction as an appendix.
It is to be borne in mind that Heath’s proposal is fairly speculative. Nevertheless, his
construction will provide a useful working model for the sort of thing Aristotle might
have had in mind. For a different construction, see Byrne (1997, 109–12).
2
there was such-and-such a thing simpliciter, how could you have
known that it had two right angles simpliciter ? Yet it is plain that
you do understand it in this sense: you understand it universally–but
you do not understand it simpliciter. (71a.24–30)4
Aristotle is distinguishing between two senses of ἐπίσταται (“to know”) in order
to explain the sense in which a person does and doesn’t know that the particular triangle has 2R before she learns that it exists. My interest here will not
be in this distinction, however. Instead, I am interested in the appearance of
of the language of epagōgē, usually translated as “induction”, in these passages
(ἐπαγόμενος, 71a.21, ἐπαχθῆναι, 71a.24). But despite the peculiarity of involving two facts being learned simultaneously, this episode of learning seems, as
some scholars have noted,i to look like a straightforward instance of syllogistic
reasoning. The student, that is, appears to be reasoning as follows:
1. Every triangle has 2R (premise)
2. This figure in the semicircle is a triangle (premise)
3. Therefore, this figure in the semicircle has 2R (conclusion)
This is a syllogism in Barbara, the only quirk being that the second premise and
the conclusion are learned simultaneously. So why does Aristotle describe the
reasoning involved as epagōgē at 71a.21? This is the first puzzle. To complicate
matters further, the second passage then commences with: πρὶν δ’ ἐπαχθῆναι
ἢ λαβεῖν συλλογισμὸν (“Before you perform epagōgē or get a syllogism”).5 This
time, both syllogism and epagōgē are mentioned explicitly. But Aristotle usually
4
πρὶν δ’ ἐπαχθῆναι ἢ λαβεῖν συλλογισμὸν τρόπον μέν τινα ἴσως φατέον ἐπίστασθαι, τρόπον
δ’ ἄλλον οὔ. ὃ γὰρ μὴ ᾔδει εἰ ἔστιν ἁπλῶς, τοῦτο πῶς ᾔδει ὅτι δύο ὀρθὰς ἔχει ἁπλῶς; ἀλλὰ
δῆλον ὡς ὡδὶ μὲν ἐπίσταται, ὅτι καθόλου ἐπίσταται, ἁπλῶς δ’ οὐκ ἐπίσταται.
i
TODO: Reference needed.
5
I discuss the meaning of λαβεῖν here in section ?? below.
3
contrasts epagōgē and syllogismos. Epagōgē is an ascent from the particular to
the universal; syllogismos is a descent from the universal to the particular.ii
How can this episode of simultaneous learning be describable both as λαβεῖν
συλλογισμὸν and as ἐπαχθῆναι? This is the second puzzle.
In the face of such “grievous philological difficulties”, Gifford (2000, 166) has
taken the step of declaring Posterior Analytics A1 to be corrupt. In this paper, I
argue that we needn’t be driven to such a conclusion – at least, not by the puzzles
I raised above.6 We can understand why Aristotle feels drawn to the language of
epagōgē in this passage, and why he simultaneously describes the passage using
the language of syllogismos. In doing so, I will not aim to give a general account
of epagōgē, which would require a much longer paper than this one. Instead,
I wish to show that if we distinguish between συλλογίσασθαι and λαμβάνειν
συλλογισμὸν, then epagōgē can be understood in the sense that Aristotle defines
it as having in this chapter, and satisfies the core feature of epagōgē, that it is a
form of argument which ascends from particulars to universals. If these can be
shown, then the appearance of ἐπαγόμενος and ἐπαχθῆναι in this passage gives
no occasion for doubting the integrity of this chapter. First let us review the
present state of scholarship on these issues.
Previously proposed solutions
Ross and Barnes propose to solve these puzzles by denying that the language
of epagōgē is being used with its technical meaning here. Barnes takes Aristotle
to be using ἐπαχθῆναι in its non-technical sense of being “led on” (Barnes 1993,
ii
6
TODO: Reference needed.
Gifford (2000) provides other lexical reasons for doubting the authenticity of Post. An.
A1. I will not address these here. My aim is only to show that ἐπαχθῆναι and λαμβάνειν
συλλογισμὸν provide no reason to sound the alarms.
4
85), while Ross takes it to be employed in a non-standard way as a synonym
for “syllogizing” (Ross 1949, 506). In this non-standard or non-technical usage,
Ross and Barnes claim, ἐπαχθῆναι is compatible (Barnes) or even identical (Ross)
with syllogizing. They read the ἢ at 71a.25 as an epexegetic, and maintain that
Aristotle is describing a case of ordinary syllogistic reasoning, despite the fact
that Aristotle describes it in terms of epagōgē.7
Perhaps this solution would be credible if Aristotle’s technical notion of epagōgē
were first developed only later in the Analytics. We could then believe that
Aristotle is here, in an endoxical tone, using the term with a loose or everyday
sense and will only later employ it as a technical contrast to syllogizing. What
makes this view untenable, however, is that epagōgē has already been used in
an incompatible sense in the very same chapter. It is contrasted with syllogistic
reasoning in the first paragraph (71a.6), where Aristotle explains that epagōgē,
rather than treating the student as one who already understands (ὡς ξυνιέντων,
71a.7), shows the universal (δεικνύντες τὸ καθόλου, 71a.8) by way of a clear
particular case of it (τοῦ δῆλον εἶναι τὸ καθ’ ἕκαστον 71a.8–9). The sentence
immediately preceding (71a.5–9) clarifies that the learning in question lacks a
characteristic feature of syllogistic learning (that it proceeds through a middle
term) and possesses a characteristic feature of learning via epagōgē (that it takes
place on the basis of particulars, not universal truths; cf. Pr. An. Β23, 68b.30–
32). Barnes’ reading therefore would require Aristotle to have switched from
using the vocabulary of epagōgē in its technical sense to using it in a looser
sense in the space of a few lines, which is not very plausible. It is even less
plausible that Aristotle would suddenly switch to using ἐπαχθῆναι simply to
mean συλλογίσασθαι, as Ross’s reading requires.
7
“the reasoning referred to is an ordinary syllogism” (Ross 1949, 506). Barnes (1993, 86)
has a similar reading.
5
A number of scholars have found this solution unsatisfying,8 and have proposed
ways to read ἐπαγόμενος at 71a.21 as having its usual meaning. McKirahan
(1983) has argued for the interpretative utility of thinking of epagōgē as “coming
to see individuals not simply as individuals or individuals of some sort or another,
but as individuals of a particular sort ” (McKirahan 1983, 10). LaBarge (2004)
has supplemented McKirahan’s account with a description of the psychological
process of retaining and recalling a phantasia that Aristotle might take to underwrite such recognition. Since recognizing an individual as a member of a kind is
plausibly a central part of learning a new universal fact, as in canonical usages
of epagōgē, and in perceiving the applicability of a universal fact already known,
as in Post. An. A1, this interpretation provides a way to give “ἐπαγόμενος”
at 71a.21 the same sense that verbs of epagōgē possess in other more canonical
passages.
But where this strategy may help with reading the episode of learning described
at 71a.19–24 as one of epagōgē rather than syllogismos, this only compounds the
difficulties associated with the fact that, at 71a.24, Aristotle seems to describe
the process explicitly as both one of epagōgē and syllogismos. For if the language
of epagōgē does have its technical meaning in this passage, it seems all but
impossible to understand how Aristotle can also describe the episode as a case of
λαβεῖν συλλογισμὸν.9 Furthermore, even on LaBarge and McKirahan’s reading,
8
9
[Ref LaBarge, McKirahan, others]
LaBarge (2004, 209), who is one of the few commentators to address this problem
directly, proposes reading “πρὶν δ’ ἐπαχθῆναι ἢ λαβεῖν συλλογισμὸν” at 71a.24–26 “not
epexegetically” – like Ross and Barnes – “but as a quotidian disjunctive.” He takes
Aristotle to be interrupting the discussion of simultaneous learning here so as to make
the point that something similar happens when we syllogize. But this is to read Aristotle
as making an obscure and highly cryptic digression. Up until this point, Aristotle has
been talking about the mechanics of a specific episode of simultaneous learning. After
this line, he again goes on to discuss a puzzle about the mental state of that person
before their learning: That, before they perceived the triangle in the semicircle, they
knew it had 2R in one sense but not in another (71a.26–27).
6
the episode of learning still seems to involve a syllogism in a central way.10
As McKirahan notes,iii even if the student is reasoning by epagōgē rather than
syllogismos, she is nevertheless in a position to form the following syllogism after
seeing the figure in the semicircle as a triangle:
1. Every triangle has 2R (premise)
2. This figure in the semicircle is a triangle (premise)
3. Therefore, this figure in the semicircle has 2R (conclusion)
This syllogism offers itself as a natural referent of συλλογισμὸν at 71a.24. Given
this, I don’t think LaBarge’s proposal that Aristotle is making a digression here
is plausible. A more natural way to read the phrase is, as McKirahan seems to
be suggesting, that the process of epagōgē puts the student in the position to
form this syllogism, or gives the student knowledge of this syllogism.
This is also what Aristotle’s wording suggests. Aristotle does not say that
you understand in one way and not another “before you perform an epagōgē
or syllogize” (which would be πρὶν δ’ ἐπαχθῆναι ἢ συλλογίσασθαι). He says “before you perform an induction or accept a syllogism” (πρὶν δ’ ἐπαχθῆναι ἢ λαβεῖν
συλλογισμὸν). Commentators have not, it seems to me, paid enough attention
to the difference between “syllogize” (συλλογίσασθαι) and “accept a syllogism”
(λαβεῖν συλλογισμόν).11 Gifford, for example, notes that “all [commentators]
agree that with λαμβάνειν συλλογισμόν Aristotle is signalling the process of
‘acquiring’ the simple syllogism under discussion and hence the process of performing the syllogistic inference itself.”iv
10
[McKirahan is more explicit about this than LaBarge.]
TODO: Reference needed.
11
Barnes translates this phrase as “given a syllogism,” using the passive meaning of λαβεῖν
as “receive”. My reading takes it to have the active sense of “accept”. The term’s
semantic range spans both this active and passive meaning: See Liddell and Scott
(1891, 405–6).
iv
TODO: Consider McKirahan as an exception
iii
7
Yet to “acquire” a syllogism is a very different thing from “the process of performing the syllogistic inference itself.” To say that someone performs an inference
(or makes an inference, or, simply, infers – I will use these interchangeably here)
is to describe a person’s occurrent thought process. Someone makes the inference from φ to Γ when they move from a thought of φ to a thought of Γ in the
understanding that the former commits or compels anyone who believes it to
also believe the latter. Such a process takes time, and may require significant attention. To say that someone comes to accept an argument, by contrast, leaves
open the question of how they came to accept that argument. Someone might
come to accept an argument by coming to better understand what the premises
mean, or by considering an analogy to another argument, or by learning that
someone they take to be an authority endorses it.12 To say that someone comes
to accept an argument is not to arbitrate between which of these psychological
processes leads to someone accepting an argument, but only to characterize the
result of this process: That the person, in some way or another, ends up in a
state of taking the argument from φ to Γ to be good. A similar distinction holds
in Aristotle’s Greek, I claim.
συλλογίσασθαι versus λαμβάνειν συλλογισμόν
Forms of the verb συλλογίσασθαι occur frequently in Aristotle, both within and
outside of the Analytics. In many cases it means, as we would expect, to deduce
via a syllogistic argument (although not necessarily one in the three figures).v
12
Of the many ways that someone could come to accept an argument, it is not clear
that making the inference is one of them. For we said above that someone only counts
as inferring Γ from φ if, in moving from the thought of φ to the thought of Γ, they
understand that the former necessitates the latter. But this is to already accept the
argument from φ to Γ. So it looks like making an inference requires one to already
accept the argument. On pain of Meno’s paradox, then, making the inference from φ
to Γ cannot be a way of coming to accept the argument from φ to Γ.
v
TODO: Ref Barnes
8
Aristotle also uses the term to mean “infer” or “reason” more broadly, where the
reasoning is not deductive. In all cases, however, he seems to intend the active
meaning of performing the inference or engaging in the reasoning. The locution
λαμβάνειν συλλογισμόν is far less frequent in the Aristotelian corpus, but those
usages which we do have in Aristotle, and in the commentators, confirm that
it means something like “acquire a syllogism”. Later in Post. An. A1, Aristotle uses the phrase “acquire a demonstration” (λαμβάνειν ἀπόδειξιν), and since a
demonstration is a kind of syllogism (71b.18), we can take this as another example of the locution. Aristotle is rejecting the view of a group of philosophers
who claim that knowing a universal consists in knowing that it applies to those
particulars which you are acquainted with. He objects: “What they know is
what they possess a demonstration of and what they acquire [a demonstration]
of, but they acquire [a demonstration] not of everything which they know to be
a triangle or a number, but of every number or triangle simpliciter.”13 Aristotle
presumably has in mind his distinction between knowledge in an active and a
passive sense; in the passive sense, we have knowledge in virtue of possessing a
demonstration, in the active sense, we come to know something when we acquire
a demonstration of it. On Galen’s reading, the locution λαμβάνειν συλλογισμόν
occurs in the Sophistical Refutations. After listing that there are six species of
a certain subdivision of elenchus, Aristotle claims that there are arguments by
syllogism and epagōgē for the correctness of this division:
It can be verified by induction and by syllogism (and another argument might be taken [ληφθῇ]), that this is the number14
13
καίτοι ἴσασι μὲν οὗπερ τὴν ἀπόδειξιν ἔχουσι καὶ οὗ ἔλαβον, ἔλαβον δ’ οὐχὶ παντὸς οὗ
ἂν εἰδῶσιν ὅτι τρίγωνον ἢ ὅτι ἀριθμός, ἀλλ’ ἁπλῶς κατὰ παντὸς ἀριθμοῦ καὶ τριγώνου
(71a.34–71b.1)
14
τούτου δὲ πίστις ἥ τε διὰ τῆς ἐπαγωγῆς καὶ συλλογισμός, ἄν τε ληφθῇ τις ἄλλος καὶ ὅτι
τοσαυταχῶς (165b27–30). vi
9
Galen, in his commentary De sophismatis, complains:
Now the inductive procedure is familiar. […] But what Aristotle says
next is wholly unclear; namely, what he means by ‘and a syllogism,
(and perhaps there is some other [syllogism] that may be taken), that
this is the number of ways in which we may fail to mean the same
things by the same words and sentences.’ For it has not been stated
how we could take some other syllogism, and the sentence, ‘this is
the number of ways in which we may fail to mean the same things
by the same words and sentences,’ resembles a syllogistic conclusion
rather than a syllogism.15
On Galen’s reading, the implied object of Aristotle’s ληφθῇ at 165b28 is
συλλογισμὸν. Aristotle, on this reading, is saying that there are two arguments
we might take as proof for the claim that his sixfold division is correct, one
through epagōgē, one through syllogismos, and perhaps we might accept [λαμβάνειν]
some other argument. Galen complains that Aristotle, on his reading, claims
that there is some other syllogistic argument that might be given without even
making clear what the syllogistic argument he has in mind. What interests
us here is not whether Galen’s criticism is fair, but the way that he reads the
passage, and his usage of λαμβάνειν. Clearly, here, ληφθῇ συλλογισμὸν, as
Galen uses it – and, if we follow his reading, as Aristotle uses it, too – refers to
an attitude one might hold towards an argument: Accepting it as proof for a
claim. Neither Galen nor Aristotle are speaking here about the psychological
process of drawing an inference, but rather the grounds for credence (πίστις) in
15
τὸ μὲν οὖν τῆς ἐπαγωγῆς γνώριμον […] τὸ δ’ ἐφεξῆς παντάπασιν ἀσαφές ἐστι, τί ποτε
βούλεται λέγειν ἐν τῷ ‘καὶ συλλογισμός, ἄν τε ληφθῇ τις ἕτερος, τοῖς αὐτοῖς ὀνόμασι
καὶ λόγοις οὐ ταὐτὰ δηλώσαιμεν’· οὐδὲ γὰρ ὅπως ἂν ἕτερόν τινα λάβοιμεν συλλογισμὸν
εἴρηται, τὸ δὲ ‘καὶ ὅτι τοσαυταχῶς ἂν τοῖς αὐτοῖς ὀνόμασι καὶ λόγοις οὐ ταὐτὰ δηλώσαιμεν’
συλλογισμοῦ συμπεράσματι μᾶλλον ἔοικεν ἢ συλλογισμῷ.
10
a certain division. Hence, λαμβάνειν συλλογισμὸν rather than συλλογίσασθαι.16
Post. An. A1 again
Now consider Posterior Analytics A1 again. Commentators have noted, rightly,
that there would be an incompatibility between describing the geometer both
as performing epagōgē and as performing a syllogistic inference. That would
be to attribute two opposed thought processes to the learner: One, from the
particular truth that this figure in the semicircle is a triangle or that it has 2R,
to the universal truth that all triangles have 2R, and another from the universal
truth to the particular. It is hard to see how this could be an apt description.
But where there is an incompatibility between performing a syllogistic inference and performing an epagogic inference, there is a tight connection between
ἐπαχθῆναι (performing an epagogic inference) and λαβεῖν συλλογισμὸν (coming to
acquire a syllogism). In this case, we only have a single description of the
thought process: The person’s thoughts are described as performing epagōgē.
But, as a result of this thought process, the person comes to accept a syllogism.
This in fact seems to be just what happens in the scenario Aristotle describes
in A1. Call the triangle in the semicircle T. The learner does initially have at
her disposal the deduction “All triangles have 2R; T is a triangle; therefore, T
has 2R.” She does have this deduction because she does not know that T exists.
Consequently, she does not know anything about it.17 In particular, she does
not yet know that it is a triangle, and so she has no reason to think the minor
premise is true.
16
We find the same usage in other commentators. See e.g. Philoponus, In An. Post.,
13,3.5.5 and Pseudo-Alexander 2, In Soph. El. 18.24.vii
17
Aristotle, at least, seems to assume that we need to know the existence of something
before we can know anything about it (cf. A1, 71a.26–27, also Β1).
11
Now imagine she comes to learn of the triangle in the semicircle. Upon constructing the third side, the figure jumps out at her from the tangle of lines and
it at once becomes clear to her (δῆλον) that it is a triangle. At the very same
time, it becomes clear to her that it has 2R. She recognizes these things not by
making two simultaneous observations but by making a single one. She sees a
triangle having 2R, and in doing so, she recognizes that there is nothing special
about this triangle in particular – any triangle has 2R; and so this figure, just
because it is a triangle, has 2R. That is, she sees at once that it has this angle
sum because and in so far as it is a triangle.viii
What can we say about her epistemic condition at this point? At a minimum, we
can say that she knows three things: “All triangles have 2R,” “this is a triangle”,
“this has 2R”. She knows these, however, not as three disconnected facts which
she happens to have gotten to know at the same time. Because of how she
has learned them, she understands their logical connection. The fact that she
knows them as a logically connected triad is marked by the fact that she would
accept the following piece of reasoning: “All triangles have 2R; this is a triangle;
therefore, this has 2R.” But this piece of reasoning is not something which she
come to accept by inferring that the triangle has 2R. All that was required for
her to accept it was seeing a figure in a certain way.18 Hence she comes to accept
a syllogism involving a particular triangle, a syllogism which she didn’t accept
before, not by engaging in syllogistic inference but by engaging in epagōgē.
viii
18
TODO: Ref I.5?
Keep in mind here the point I attempted to hammer home above: Accepting a syllogism
does not describe an occurrent mental state but rather a propensity or an attitude.
Compare McKirahan (1983, 9–11). My view is broadly sympathetic to his, at least in
the centrality it accords to seeing the triangle as a triangle. Unlike McKirahan (1983,
10), however, I do not think that these cases involve “reflecting on or reasoning about
one or more perceived instances,” at least if this means that some additional mental
effort is required over and above the act of seeing the triangle as such. Instead, I am
arguing that the learner’s seeing the triangle as such can be viewed as a process of
accepting a syllogism without “reflecting on or reasoning about one or more perceived
instances.”
12
There is nothing contradictory, then, in saying that the geometer comes to
“accept a syllogism or [equivalently] engage in epagōgē ” (71a.25). These are, as
Ross says, “both ways of referring to the same thing”.ix Yet that same thing is,
as Aristotle says, a process of engaging in epagōgē, not a process of deduction
as commentators have assumed.19 It is an argument by epagōgē in just the
sense Aristotle defines in the first paragraph of A1. The universal (that all
triangles have 2R) is shown by the clarity of the particular triangle (δεικνύντες
τὸ καθόλου διὰ τοῦ δῆλον εἶναι τὸ καθ’ ἕκαστον, 71a.8). The universal is shown
by the particular not in the sense that it is proven, but in the sense that the
particular triangle is what calls to mind this theorem for the student.x Were
the student ignorant of this fact, such an experience might form a part of the
empeiria by which she could eventually come to grasp the universal. But since
she knows the universal already, the same experience instead results in the
learner being reminded20 of what she already knows.21
Another way we can put the point is as follows. We must distinguish between the
temporal order in which the person thinks the things that which she does, and
the relations of explanatory priority that the reasoner takes to pertain between
the propositions she thinks of. In many cases, these two relations of priority
may coincide. This is the case, for instance, for someone who first knows the
theorems that all triangles have 2R and that all figures with three equal angles
are triangles, and then, reflecting on these facts, goes on to conclude that all
figures with three equal angles have 2R. In this case, the premises are both
what she knows first and what she takes to logically imply the fact that she
ix
TODO: Reference needed.
Indeed, it is not clear how one even could come to accept a syllogism by syllogizing it.
For presumably one cannot carry out a deduction if one does not already acknowledge
the inference as sound. Issues of Meno’s paradox lurk here.
x
TODO: Cite Byrn?
20
Note about anamemnesis in B19.
21
Heath here?
19
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concludes. In this case, a single argument is representing two different aspects
of the subject’s reasoning. That is, the argument from premises “all triangles
have 2R” and “everything with three equal angles is a triangle” to the conclusion
that “everything with three equal angles has 2R” represents both the fact that
the person takes the premises to be what logically entail the conclusion, and
the fact that premises are what precipitates the learner to go on to have the
thought of the conclusion.
But that someone knows something first does not in general imply that they will
take it to be logically prior to what they learn subsequently. Someone may learn
what they then take to be the logical grounds for what they already know. In
this case, temporal and the logical orders come apart. Consequently, a different
argument is needed to represent the temporal order of her reasoning, on the one
hand, and the logical order that she takes to hold between the propositions she
is thinking about, on the other. This is just what happens in the case of the
geometer in A1. The temporal order of her reasoning is indeed an ascent from
particulars to universals, since it is the recognition of a particular as belonging
to a certain kind that precipitates her to think of the universal that al triangles
have 2R in the first instance. Nevertheless, the person does not take the fact
that the figure is a triangle to be, on its own, what necessitates the fact that
it has 2R. The student understands that this only follows because all triangles
have 2R. And so if we wish to represent the logical order among the propositions
the student is reasoning about, we find ourselves back at the original argument
by syllogism:
1. Every triangle has 2R (premise)
2. This figure in the semicircle is a triangle (premise)
3. Therefore, this figure in the semicircle has 2R (conclusion)
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It makes sense, then, that Aristotle would describe this event as ἐπαχθῆναι ἢ
λαβεῖν συλλογισμὸν. ἐπαχθῆναι describes what the learner does: She learns
by reasoning through this inductive argument. She does not learn by reasoning
through this syllogism (not δι’ συλλογισμοῦ, in Aristotle’s language): She cannot,
because she doesn’t yet know both of its premises (she only learns the minor
at the very same time as the major). But she nevertheless comes to acquire
(λαβεῖν) this syllogism in the act of her epagōgē. Both arguments describe one
and the same event of learning: The inductive argument describes the order of
her grasping; the syllogism, the order which she thereby grasps.
References
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Byrne, Patrick H. 1997. Analysis in Aristotle. Albany: State University of New
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Gifford, Mark. 2000. “Lexical Anomolies in the Introduction to the Posterior
Analytics, Part I.” In Oxford Studies in Ancient Philosophy, edited by Brad In-
wood, XIX:163–223. Oxford: Oxford University Press.
Heath, Thomas. 1949. Mathematics in Aristotle. Oxford: Clarendon Press.
LaBarge, Scott. 2004. “Aristotle on ‘Simultaneous Learning’ in Posterior Analytics 1.1 and Prior Analytics 2.21.” Oxford Studies in Ancient Philosophy 27:
177–215.
Liddell, Henry George, and Robert Scott. 1891. A Lexicon. Abridged from
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