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First thoughts on relativity in the Peri Ideon

After seeing me talk about relatives (ta pros ti) in Aristotle’s Categories 7, James Warren suggested that I look again at the so-called ‘relativity argument’ in Alexander’s testimony on the lost treatise Peri Ideon. Many of you will know the argument from Owen’s classic 1957 paper A Proof in the Peri Ideon and the industrial-scale debate it generated. But here, I’m just going to offer one preliminary idea.

Alexander (In Met. 82, 11-83,16) reports Aristotle’s (type of) argument that reconstructs a Platonic argument for Forms. Most doubt Alexander quotes the Peri Ideon verbatim, but scholars take Alexander to be a good witness to Aristotle’s argument. But I wonder whether Alexander (and others) have misunderstood Aristotle’s intentions with the argument, even if Alexander’s report is accurate.

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Aristotle on Relatives in Categories 7 (Part 3): Opaque and Transparent Relatives

In the last post, I set up Quine’s distinction between transparent and opaque ways of reading a statement. What I need to do now, is connect this to Aristotle’s two conceptions of relative terms, as he sets them up in Cat. 7. I will first argue that D1 relatives are opaque, on Aristotle’s view, then that D2 relatives are transparent. In the next post, I will explain how this helps Aristotle avoid the conclusion that some substances are relatives. Roughly, when construed opaquely, a proposition involving a relative gives us less information, so the proposition is more ambiguous. It is this ambiguity that means some relatives appear to be substances. When construed transparently, a proposition gives us much more information, so is not ambiguous. At least, not ambiguous enough to allow some relatives to appear to be substances. But I’m getting ahead of myself. First I will argue that D1 relatives are relatives viewed opaquely, while D2 are viewed transparently.

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Aristotle on Relatives in Categories 7 (Part 2): Quine’s Safari

We saw in the first post in this series that  Aristotle gives two definitions of relatives in Categories 7, which I called D1 and D2. Aristotle worries that D1 will allow some substances to be relatives, so introduces D2. Specifically, Aristotle worries that parts of secondary substances, like hand, will turn out to be both substances and relatives. So, what is the difference, according to Aristotle, between D1 and D2?

One way we could understand the difference is as a difference of extension: D2 excludes some items that D1 does not. Hand would have to be one of the relatives excluded from D2, but included in D1. The existing suggestions in the literature are of this kind.[1]

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Aristotle on Relatives in Categories 7 (Part 1): Two Definitions

I got a taste of ancient conceptions of relativity pretty much in the first week of my PhD when I read David Sedley’s paper ‘Aristotelian Relativities’. I wrote my dissertation on the  scraps of evidence concerning relatives in Plato but stopped before I got to Aristotle’s feast of ideas. A very nice email from one of Dhananjay’s colleagues asking about relations in Aristotle prompted me to write down some ideas about Cat. 7 that have been cooking for a while. Bon appetit!

Aristotle’s class of relatives (ta pros ti), discussed in Categories 7, excludes some items that we consider relations and includes some that we do not. Aristotle excludes three or more place relations, such as between, but includes some monadic properties: e.g. large (6a36-b10); and virtue and vice (6b15). So what does Aristotle think relatives are? He defines them at Cat. 6a35 (D1) but gives a different definition later in the same chapter at 8a15, (D2).  Traditionally, scholars have thought that D2 is strictly narrower than D1: that is, at least one relative, that falls under D1 does not fall under D2. However, in this and the next few posts, I will argue, using a distinction formulated by Quine, that D1 and D2 give us two different ways to view relatives: the D1 relatives are relatives viewed opaquely, while the D2 relatives are relatives viewed transparently.

In this, the first post, I will discuss in more detail Aristotle’s definitions and explain Aristotle’s motivation for giving D2, roughly, that D1 may lead him into a contradiction. In the next post, I will introduce a distinction between two ways of understanding propositions involving relatives: transparently and opaquely. In the third post, I will argue that D1 relatives are relatives viewed opaquely, while D2 relatives are relatives viewed transparently. To prove my reading, I will show that the distinction I identify allows Aristotle to avoid the contradiction he worries about.

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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).

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Aristotle’s solution to Zeno’s paradox of the runner

This is a guest post by Matthew Duncombe, a fellow Cambridge ancient philosophy student. I hope it (and future posts!) will expand the range of topics on the blog. 

Aristotle’s answer to Zeno’s basic paradox of progress, the runner, is known to invoke a distinction between ‘actual’ and ‘potential’ infinities. Below, I ponder for your amusement, whether the solution based on the distinction of actual and potential infinities leads to incoherence.  The paradox, with which I imagine most people are familiar, is as follows. Suppose that in a finite period of time, T, a runner, Bolt, traverses a finite distance, AB. At some point within T, t1, Bolt will ‘touch’ (haptesthai in Aristotle’s presentation) the mid-point of AB, namely a1. At some time after t1, Bolt will ‘touch’ the midpoint between a1 and B, namely a2, and so on. In general, for any aN there is a point, aM, midway between aN and B which Bolt will ‘touch’ at some point in T. That is an infinite number of points. So (P1) if Bolt runs from A to B, then Bolt completes and infinite number of tasks. (P2) It is impossible to complete an infinite number of tasks. (C) Therefore, Bolt cannot run from A to B.[1]

Aristotle’s objection is that the argument trades on an ambiguity in the notion of ‘infinity’ in the argument (Phys. 263b3-9). There is an innocuous sense of infinity, potential infinity, which is the sense of ‘infinite’ in (P1), and a vicious sense of infinity, actual infinity, which is the sense operative in (P2) (Phys. 206a9-b2). The difference can be seen with the following case: Imagine another runner, Colt, who runs exactly the same course as Bolt, but who takes his hat off as he ‘touches’ point aN, and puts it back on as he touches point aM. Is he wearing his hat when he reaches B? It cannot be off, because each time he took it off, he put it back on again. But it cannot be on, because every time he put it on, he took it off. So it is neither on, nor off. So reaching B is impossible. This is because Colt has actualized each of the infinite number of potential points on the course AB. But Bolt did not actualize them, so there is no contradiction in Bolt’s case.[2]

To put the point another way: (AI) There is an actual infinity of tasks to complete running from AB just in case, between any two actual or potential tasks, there is a third actual task. (PI) There is potential infinity of tasks just in case between any two actual or potential tasks on AB there a third potential task. But (PI) does not entail (AI). Problem solved.

But: (M) If x is potentially F then it is possible that x is actually F. (PI) does entail that on course AB there is an infinite number of potential tasks to be completed to run from A to B. So the course AB is traversable only by completing an infinite number of potential tasks. But that is to say that AB is traversable only by it being possible to complete an infinite number of actual tasks. But Aristotle agreed that it is impossible to traverse AB if one must complete an infinite number of actual tasks.

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