## Transformative induction in Prior Analytics B21

Since this is my first post after the Blogistikon hiatus, I’d like to thank Dhananjay for asking me to contribute. I’m really excited to be able to get some ideas out and hopefully readers will enjoy it too. I expect I will post mainly on epistemology and logic in Plato and Aristotle because my current work looks at how logic gets used in different contexts: dialectical encounters, analysis, epistemology and so on. I got interested in the puzzle I address in this post because Prior Analytics B21 is a bit of a quirky text where Aristotle relates logic to epistemology.

Sometimes we fail to know logical consequences of our knowledge or to believe logical consequences of our beliefs. I can know the axioms of arithmetic and set-theory, for example, but not know whether the Goldbach conjecture is true. B21 asks why this is. Aristotle compares his answer to ‘the argument in the Meno’ (cf. A. Po. A 1 71a17-b8) then says something puzzling:

For it never turns out that someone knows the individual (to kath’hekaston) in advance, but she gets knowledge of the particular  (ton kata meros) at the same time, by induction, just like those who are reminded. For sometimes we know directly, for example that <such-and-such> has two right-angles if we see that <such-and-such> is a triangle (Pr. An. B 21 67a21-26).

Aristotle usually uses ‘epagoge’ to mean a move from cognitions of individuals to cognitions of universals [1]. But in this text, Aristotle apparently uses ‘epagoge’ to describe the following cognitive process:

1. a knows that two-right angles belong to all triangles;
2. a sees that triangle belongs to such-and-such;
3. a knows that two-right angles belongs to such-and-such.

This seems to be a cognitive move from the knowledge that all triangles are thus-and-so, to the knowledge that some triangle is thus-and-so. That is, a move from a universal to a particular. We would call this a deduction and Aristotle would call it a syllogismos. In fact, it looks rather like a syllogism in Darii.

I argue that if we understand ‘epagoge’ in the way Aristotle defines it just a few pages later (T2) Aristotle’s use of the expression in T1 makes perfect sense.

(T2) So, induction, i.e., the syllogism from an induction, is the syllogizing of one extreme to belong to the other, through the middle. For example, if B is the middle of AC, <an induction is> to demonstrate that A belongs to B through C. (Pr. An. B23 68b15-19)

Here, Aristotle tells us that an ‘epagoge’ transforms one syllogism into another. Specifically, a syllogism, (S) of the form ‘ArB, BrC, so ArC’ transforms into another syllogism, (E), of the from ‘ArC, CrB so ArB’ . We move to (E) from (S) by a simple operation that Aristotle gives. Convert one premise of S, using one of the usual Aristotelian conversion rules, and interchange the conclusion of S and the other premise of S. This results in E. The general scheme is given below, with S represented on the left, and E on the right:

 ArB BrC ArC ArC CrB ArB

Aristotle discusses the general structural characteristics of these transformations, which he calls ‘circular arguments’, in Prior Analytics B5-7. From that discussion, it is clear that you cannot transform any syllogism into any other. For example, to transform a syllogism (S) in Barbara to an epagogic syllogism (E) also in Barbara, the terms of the minor premise must be co-extensive. Otherwise, you can only move from Barbara to Darii. But why does Aristotle bring up transformation in B23? B23 is not exploring the mathematical structures that obtain in the syllogistic. B23 and B21 are concerned with epistemology.

It seems to me that the epistemic value of the transformation is that it ‘re-carves’ information we already have available. S tells us that that three terms are related thus-and-so; the transformation to E tells us that the terms are also related so-and-thus. We gain information in the sense that we have revealed relations we previously did not know obtained between terms. We could call this ‘transformative induction’.

How does this help us make sense of ‘epagoge’ in B21? First a general point. B21 is all about how we extract ‘new’ knowledge from things we already know and how this process goes wrong. Transformative induction is one way in which this extraction happens. Second, I think that the ‘such-and-such’ in T1 is an individual sort of thing. Reading the passage this way ‘such and such’ should refer to a sort of triangle, e.g. isosceles. This is how ‘to kath’hekaston’ functions in B23 and, I suggest, T1 as well. We can now read the problematic example as a syllogism in Darii, with the relevant cognitive operators:

1. a knows that two-right angles belong to all triangles;
2. a sees that triangle belongs to isosceles;
3. a knows that two-right angles belongs isosceles.

Why would Aristotle call this process an epagoge? Given how he explains that term in B23, it is because this Darii is the result of re-carving information already available to a. Where ‘A’ is ‘two-right angled figure’, ‘B’ is ‘isosceles’ and ‘C’ is ‘triangle’, when a sees that some triangle is an isosceles, a re-carves the information contained in the Barbara on the left hand side, to the Darii on the right:

 AaB BaC AaC AaC CiB AiB

B23 68b30-37 says that the result of the transformation, presented on the right, is clearer to us, while the corresponding syllogism is prior. This precisely reflects my construal of this epagoge in B21. We see that triangle belongs to some isosceles, perhaps from a diagram, and, when we also know that two-right angles belong to all triangles, we instantly know that two-right angles belongs to some isosceles. In B21 Aristotle confuses us because he describes the result of this transformation as an ‘epagoge’. But, as we have seen the result of such a transformation is a syllogism, which is, of course, a deduction.

Thoughts? I’m a bit worried that in T1 Aristotle switches between ‘individuals’ and ‘particulars’. Maybe he intends an important difference with the shift in vocabulary? Also, the ‘transformative’ sense of ‘epagoge’ in B23 seems officially introduced as an example of using the syllogistic in the practise of ‘rhetoric and absolutely any form of persuasion and any kind of method’. Does it seem like transformative induction can be so used?

[1] This is a simplification. Engberg-Pedersen distinguishes six different sorts of epagoge in Aristotle, although he concludes that they all, somehow or other, move from particular to universal, usually in a multi-agent context. See More on Aristotelian Epagoge T. Engberg-Pedersen Phronesis  Vol. 24, No. 3 (1979) , pp. 301-319.

• djr
• December 23rd, 2013

Logic (Aristotelian or otherwise) is certainly not my forte, but I’m both intrigued and worried by a few things here. What intrigues me is the attempt to read epagōgē in Pr. An. B21 in its technical sense rather than some attenuated way. The parallel passage in Post. An. A1 likewise describes coming to know that a certain figure has two right angles by coming to know that it was a triangle hama epagomenon. So far as I know, epagomenon there is usually taken in a non-technical sense (Barnes takes it that way, but cites McKirahan JHP 21 (1983) as reading it in the technical sense). I’m often troubled by that sort of move, and the view you end up with — that epagōgē ‘re-carves’ information already available to the subject — is interesting in its own right. But the Post. An. A1 passage leads me to worry about your interpretation of the Pr. An. B21 bit. Here’s two reasons why.

First, it seems plain that the Post. An. passage presents us with an example in which we are dealing with a particular figure, so that one of our terms is a particular, and not a type like ‘isosceles.’ Looking back at Pr. An. B21, it gives the same impression: “for we know some things directly, e.g., that the angles are equal to two right angles, if we see that the figure is a triangle” (trans. Jenkinson rev. Oxford). Admittedly, the Greek here is compressed, but Jenkinson’s rendering seems more accurate than yours. By supplying ‘such-and-such,’ your translation gives the impression that we’re talking about a type; but the omission of any subject in the Greek suggests that we’re talking about a particular figure, as in Post. An. A1. Does to kath’ hekaston give us reason to think otherwise? I can’t see that it does; just above it contrasts knowing universally that every triangle has two right angles with knowing this of every particular triangle, and while that might apply to types like isosceles, surely it needs to extend to particulars like this here Δ. So if your view depends on to kath’ hekaston referring to types like isosceles &c. but not to particulars, then I’m skeptical.

Second, I wonder whether your translation of Pr. An. B21 misconstrues the dative of epagōgē by taking it as instrumental / causal. It seems rather to be the object of the preposition hama, so that the claim would not be that we get knowledge of the particular (whether that’s isosceles or this here Δ) by induction, but that we get it at the same time as the induction that leads us to see, e.g., that this here Δ is a triangle or that isosceles figures are triangles. The Post. An. A1 parallel — hama epagomenon — doesn’t rule out a causal interpretation, but seems even more clearly to be temporal. On this reading, it would seem hardly mysterious at all why Aristotle thinks of this as an epagōgē, provided we understand epagōgē as coming to see a particular as an instantiation of a universal: I see that this Δ here is a triangle, and at the same time (hama) I immediately (euthus) understand that it has two right angles, because I already know universally that all triangles have two right angles.

So, whether or not that counts as an instance of the technical or the loose sense of epagōgē, it seems simpler than the reading you propose, cool as that reading is.

Yes? No? Maybe?

• mbduncombe
• December 24th, 2013

Hello! Thanks for your reply which has given me a lot to think about. I’ll tell you where I’m up to, but it may not yet be satisfactory.

The first question. Does my reading rely on taking the subject of they key sentence, for which I used the stand-in ‘such-and-such’, as type? Short answer: now that you make me think about it, no. I intended ‘such and such’ to be neutral between those two readings. Sorry if that was not clear. And in fact, my reading does not rely on taking ‘such and such’ to be a type. I want to take those sentences as a sketch of a Darii, and, as long as Aristotle accepts that (i) ‘A belongs to some B’ is true if ‘A belongs to some individual B’ is true, I can take it that way.

While were at it, your quoted translation made me realise that I mistranslated the ‘enia’ as ‘sometimes’, when really it should be ‘some things’. That would give a translation: ‘For we know somethings directly, for example that this has two right-angles if we see that it is a triangle.’ This is a big point in favour of taking the subject here to be a token.

As an aside, the APr B23 passage does unambiguously refer to types, and uses ‘kath’hekaston’ to mean an ‘individual type’. But as long as you think (i) holds, the passages can still be seen as parallel in the way I want.

On your second point about taking the epagōgē with the hama. That is a really interesting suggestion, which I don’t think I have seen before. Taken your way, epagoge here amounts to realising that this ^ is a triangle. The text says that we get knowledge ‘ton kata meros’ at the same time as the epagoge. But in the context of the Prior Analytics ‘ton kata meros’, which I translated as ‘particulars’ usually means ‘particular propositions’, e.g. a proposition which says ‘A belongs to some B’. In a Darii, both of the minor premise and the conclusion are of this form. So we get knowledge of both the minor premise and conclusion of the Darii at the same time as the epagoge, that is realising this ^ is a triangle.

(This also makes nice sense of the switch between ‘kath’ hekaston’ and ‘ton kata meros’ which I was worrying about.)

• djr
• December 23rd, 2013

To clarify, on the reading I’m proposing of hama tēi epagōgēi in An. Pr. B21, what I come to know by epagōgē is just that this figure is a triangle, or that isosceles figures are triangles; coming to know that this figure has two right angles, or that isosceles figures have two right angles, is not something that I come to know by the epagōgē, but immediately with it, as I apply what I have learned via epagōgē to my prior universal knowledge that triangles have two right angles. Hence we avoid supposing that epagōgē there describes a move from cognition of universals to cognition of particulars rather than vice versa.

• DavidB
• February 23rd, 2015

I am also working on an interpretation of epagoge that makes it a type of circular proof. I thought that this was a new interpretation, so I find it galling that this has already been noticed. However, although there can be a circular proof in Darii, this is because the terms of the major are convertible, whereas in the inductive syllogism it is the terms of the minor that are convertible, which means that the syllogism must be Barbara or Celarent. I don’t think something in Darii can be epagoge.