Notes on G.E.L. Owen's 'Notes on Ryle's Plato'

The answer Owen and Ryle offer is that the Parmenides is not concerned with proving any particular conclusions. Rather, it is concerned with laying out a thick nest of problematic arguments which inform much of the 'late' Plato. That nest of problems has to do with the problems of the Theory of Forms, but more generally it has to do with problems for any theory of types in more modern terms. Russell too took it as such. The central problem here, for Russell, is that of self-reference:

• if we speak about the "class of all classes," it seems that that class must include itself.
• If we speak of "the class of all horses," however, that class is not a member of itself (because it is not a horse!).
• Now, what if we speak of "the class of all classes that are not members of themselves"?
• Does that class include itself?
• If we say "no," then it is a class that is not a member of itself, and so it must include itself, which makes no sense.
• If we say "yes," then it is included in the class of all classes that are not members of themselves, and so it cannot be a member of itself, which makes just as little sense.

1. Hypothesis: One is
   Subject of the antinomies: One
   1. 137c-142b
   2. 142b-155e
   3. Appendix on paradoxes of instantaneous change: 155e-157b
2. Hypothesis: One is not
   Subject of the antinomies: everything other than one
   1. 157b-159b
   2. 159b-160b
3. Hypothesis: One is
   Subject of the antinomies: One
   1. 160b-163b
   2. 163b-164b
4. Hypothesis: One is not
   Subject of the antinomies: everything other than one
   1. 164b-165e
   2. 165e-

More detail

1. Hypothesis: One is
   Subject of the antinomies (antinomy is a name for these opposed arguments): One
   1. 137c-142b
      1. The one is one and it is not many (137c).
         Therefore, The one cannot have parts or members or be a whole.
         Owen thinks 1Ai confuses the 'is' of identity (example: Jacques is Prof. Bailly) and the 'is' of predication (example: Jacques is confusing OR Jacques is a man). That may be one source of the self-predication that besets the theory of forms.
         This thesis conflicts with 1Bvi.
      2. The limits or extremities of anything are parts of that thing (137d).
         Together with 1Ai, this leads to the conclusion that the one cannot have limits, shape, or position.
      3. Having more than one character or attribute makes a thing a plurality (139e-140a and perhaps 138c, 139c)
         Perhaps this thesis together with 1ai proves that:
         1. One cannot change in quality or
         2. be identical with...
         3. differ from...
         4. be like...
         5. be unlike ...
            
         • ... itself or anything else,
         • because 1Aiiia-e all attribute an additional character/attribute to one, and so involve one in having >1 character or attribute
         But perhaps 1Aiiia stems directly from the Identity/Predication confusion in 1Ai. From 1Aiiia-e, it is argued that the One is not equal or unequal to anything and that it has no temporal attributes.
      4. If the statement that S is P differs in truth-value from the statement that S is Q, then P is different from Q (139d, 142b-c (with small variant: substitute "sense" for "truth-value").
         
      5. Changes (strictly speaking, movements to a place) take time: to describe S as becoming P is to describe something temporally intermediate between an initial and a final state (138d)
         From 1Aii, the One cannot move in one place.
         From 1Ai and this thesis (and probably 1Aiv as well), the One cannot change place either.
         This thesis conflicts with the argument at 1Cii.
      6. If (when) X is becoming different from Y, it cannot be the case that Y is different from X: otherwise X would already be different from Y and not merely be becoming so (141b).
         Hence, if X is becoming older than itself, it is also becoming younger than itself.
         But from 1Aiii and 1Aiv, the Once cannot have temporal properties nor can it even remain the same age as itself. Therefore, it does not exist in time.
         "If S is becoming P it cannot yet be P" is the general premise here.
      7. What exists exists in time (141d-e).
         From this and 1Avi, the One is not in any way and is not anything: i.e. the One does not exist.
      8. If S is P, then S is or exists (141e)
         From this and 1Avii, it follows that the One is not one (or not the One).
      9. What does not exist can have nothing related to it.
         From this and 1Avii's conclusion, it follows that the One cannot be named and that there can be neither speech nor knowledge, perception, nor idea of it at all.
         This thesis conflicts with 3A.
   2. 142b-155e
      This movement uses some of the steps from the last movement (1A), but it also challenges some of the same steps as it uses (namely 1Ai and 1Avi).
      • By 1Aiv, if the One exists, its existence is not the same as its unity.
      • Hence, by 1Aiii, unity and existence are pluralizing parts of the One (this conflicts with 1Ai).
      • These pluralizing parts have pluralizing parts (oneness and existence).
      • These pluralizing parts have pluralizing parts . . .
      • Therefore the One is a whole with infinite parts and unity and existence are infinitely distributed.
      • What is more, if the One and its existence are differentiable, then they exhibit difference, which is distinct from either of them (143a-144a).
      1. What is not one is nothing at all (144c).
         From this it is argued that any part of a plurality is one, that anything divisible is divisible into some number of parts, and generally that any number is a number of units.
         This points out something interesting about unity and existence: we cannot abstract them away from a subject as we can things like 'largeness' or 'redness.' The reason is that if you abstract existence from a thing, you are left with something that nonetheless must exist. So you have to immediately reimport existence. The same holds for unity. (2Ai will try to challenge this).
      2. A whole contains and so limits its parts (144e-145a).
         This amounts to a claim that the limit or limiting factor is external to, not part of, what it limits.
         This conflicts with 1Aii and also 1Biv!
      3. If X is limited, X has spatial extremities (145b).
         Hence the One has shape.
         See 1Bv.
      4. If X has parts, X is identical with the aggregate of those parts.
         This says, in effect, that X is nothing more than its parts. 1Bii said that the whole is something external to its parts. In the Theaetetus 203e-205a, these two positions are given as an exhaustive disjunction. Here they are a conjunction. It is hard to see how they both could be the case. Sophist 252e-253c has an argument that supersedes this analysis.
      5. What exists must exist somewhere (145e).
         In Greek philosophy, to be somewhere is to be in something.
         From this and 1Bii and 1Biv, it is argued that the one must be both in itself and in another. Insofar as it is in itself, it is always 'in the same,' but insofar as it is in another, it is always 'in something different.' From that, the argument goes, it follows that the One is static and in motion and that it is different from itself. That argument is a fallacy. The problem is that 'the same' and 'different' above do not relate to one thing, but rather to distinct things.
         In the Sophist 259c-d, Plato chastises those who make that mistake by failing to complete predicates such as 'same' and 'different.'
      6. Anything is related to anything in one of three ways (146b):
         1. by identity
         2. by difference
         3. as part to whole or whole to part
         From the treatment of part-whole relationships in what follows, this distinction seems to say that 'partaking in unity' or 'being in a way one' is a part-whole relationship rather than one of identity or predication. That might serve to clear up the confusion in 1Ai (and elsewhere).
      7. What is not one is related to one by b and not by a or c (147a)
         Therefore things that are not one have no unity at all of any sort. And therefore, they have no number at all. They are neither identical nor predicated by one or any number.
         That supposedly conflicts with 1Bi.
      8. Words are names, and to repeat a word is to name the same thing twice (147d).
         Thus when one claims that the one is different from everything else and everything else is different from the one, one attributes to the one and everything else the same property, namely difference. So they share one property at least. And the One is like the others too! (this is all fallacious, and obviously so: why does Plato include it?)
      9. Smallness is small, largeness is large (150a-d).
         This is apparently used to claim that nothing but smallness can exhibit smallness, nothing but largeness can exhibit largeness (150a-b).
         This is a blatant claim of self-predication. By this point in the dialogue, Plato surely recognizes the problems with self-predication. Why does he nonetheless include it in his schema?
      10. If X is becoming Y, then at any 'present time' in the process X is (and not 'is becoming') Y (152c-d).
          Hence, if the one is growing older, at any present time it is older.
          Also the one is becoming younger and neither older nor younger than the others.
          Owen points out that this step is plausible if you use something like 'older' or 'younger,' but not plausible with things like 'an octogenarian' or 'six feet tall.' Step 1Avi, however, is plausible with things like 'an octogenarian' or 'six feet tall.'
          That is important, because it kills the assumption of the Theory of Forms that there are only two alternatives: either becoming or being. It depends on the question "Becoming what?"
          This conflicts with 1Avi.
      11. What exists has (or can have) other things related to it (155d).
          This is the converse of 1Aix.
          By this step, the one can be named and spoken of and there can be knowledge, perception, and thought of it.
   3. Appendix on paradoxes of instantaneous change: 155e-157b

2. Hypothesis: One is not
   Subject of the antinomies: everything other than one
   1. 157b-159b
   2. 159b-160b
3. Hypothesis: One is
   Subject of the antinomies: One
   1. 160b-163b
   2. 163b-164b
4. Hypothesis: One is not
   Subject of the antinomies: everything other than one
   1. 164b-165e
   2. 165e-