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 . But in this text, Aristotle apparently uses ‘epagoge’ to describe the following cognitive process:
- a knows that two-right angles belong to all triangles;
- a sees that triangle belongs to such-and-such;
- 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:
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:
- a knows that two-right angles belong to all triangles;
- a sees that triangle belongs to isosceles;
- 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:
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?
 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.