1 The version given here is my own (Plato Phaedo. Oxford 1975). except that ‘tall' and ‘short” have been substituted for ‘large’ and ‘small’ throughout for uniformity with the usage of this paper.

2 op. cit. 192-4.

3 Found, e.g., in Black, R.S., Plato’s, Phaedo (London 1955). 118-19.Google Scholar 124, n.3, and D. O'Brien, ‘Plato's Last Argument in the Phaedo’. Classical Quarterly 1967. 199- 200.

4 Hackforth, R., Plato’s, Phaedo (Cambridge 1955). p.155Google Scholar: ‘such a distinction would. in my opinion, be irrelevant both to [Plato’s] immediate purpose and to the whole final argument for immortality, of which this section forms a part'.

5 H.N. Castañeda. ‘Plato's Pheado Theory of Relations'. Journal of Phihhophical Logic 1972. 467-80. I am greatful to Professor castañeda for comments on an earlier version of this paper, for which his article was the original stimulus.

6 It should be noticed that ‘those words’ (τοις ὁήμασι) at 102b8 back to whenever you say that Simmias is taller than Socrates’ (ὅταν ∑ιμμίαο Zωϰράιους ωῇς μείζω ειυϰι) and not to ‘Simmias' overtopping of Socrates,(τὸυ∑ιμμο ὺτεεέχειυ ∑ωϰοάτοις) at 102bB-9. This latter expression stands for the truth which the words ‘Simmias is taller than Socrates’ are alleged to mispresent. Hackforth gives: ‘But of course you admit that the words “Simmias overtops Socrates “ do not express the truth of the matter'. This wrongly gives the impression that it is the sentence’ Simmias overtops Socrates’ that is supposed to be at fault. But the word; that are being held to misrepresent the truth are. surely. ‘Simmias is taller than Socrates'. and the reformulation in terms of Simmias’ Tallness and Socrates’ Shortness is a recasting of them. Cf. 102b5-6.

7 As Hackforth supposed. Op. Cit . 131. 135.n.1.

8 For the view of Socrates’ concerns advocated in this paragraph. See my phaedo. 172-3. I read his autobiography as apart . an ironical exposé of the inadequacy of ‘natural science’ to resolve conceptual problems about the possiblity of growth, change and and coming-to- be in general. These problems stem. ultimately . from eleatic logic. and left untouched by so called ‘natural science (περι φύσεως ίστορία) Socrates’ sardonic reference to the kind of wisdom which they call “natural science” ‘ (96a7-8) prepares us for his debunking of the scientists which follows. An ίσιορία περί ούσεως in the true sense would be a conceptual inquiry of the kind which Socrates claims to have conducted himself (99d1). an inquiry ἐν λόγοις rather than ἐν λόγοις (100a1-31. This latter contrast may be said to pre figure the modern antithesis between empirical and conceptual investigation.

9 op. Cit. 194.

10 102b2. c10. 103b7-c1. Ct. 78e2.

11 I am not wedded to this particular terminology. and nothing in my argument hangs upon it. But it is a convenient way of marking the distinction which I believe to be fundamental for our understanding of the whole passage. By contrast, Castañeda accuses earlier critics who have used it of ‘doing serious and irrelevant violence to the text’ (op. cit. 480.n.3).

12 For this reading of πεφυϰέυαι and τυγχάνει. see also O'Brien, op. cit. 199-200, 211-2. Hackforth (155) and Castañeda (480, n.3) both discount the evidence of these locutions. But neither offers any alternative account of their force. Castañeda's translation (op.cir.469) assumes an ellipsis of τυγχάυει three times at 102c2-4. But there seems no reason in grammar or logic to suppose this, nor does it lend his view of the passage any obvious support.

13 See note 5.

14 Castañeda. op. cit. 471.

15 ibid.

16 For a further development of this difficulty. see my note, op. cit. 194 (top). Although it seems not to have bothered earlier commentators, it seems to me a problem on any view of the passage, including the one defended in this paper. But it would be far more acute if Socrates’ primary purpose were to clarify the structure of relational facts. For he would be doing so only by reimporting the relevant relations on a new level. and by ascribing them to subjects that are, arguably, of the wrong logical type to stand in any such relations to one another.

17 op. cit. 469-70.

18 op. cit.470. I have quoted castañeda's translation as printed. But an additional ‘shorter’ need to be supplied after the penultimate word.

19 As might well be assumed from their occurrences earlier in the sentence at 100e9 and 101a2-4 in English ‘the taller’ and ‘the shorter (definite article of Comparative adjectives can only mean the taller and the shorter of two individuals previously Specitied. Hence it is natural for us to take ‘the shorter (man) (τολ ἐλάττα) at 100e9 and ‘the shorter (things)’ at 101a3-4 to be the Shorter of the two items specified (ἔτερο ἐτερου) at 100e8 and 101a2. and thus to think of two individuals who are taller and shorter than one another. This is certainly possible. But I think. necessary. For in Greek τὸυ ἐλάττω τὸ ἐλαττο may simply signity someone (or something) who (or which) is Shorter. We are thus not compelled to identity τὸυ ἐλάττω τὸ ἐλαττο at 100e8 and 101a2. Accordingly. The reading of 101a5-b2 preferred in the text seems both grammatically and logically tenable.

20 I have assumed here. as in my note on 96c1-e5(op.cit.175), that εὰ ἐϰα and τωα όϰαώ at 96e2-3 mean number ten and eight. They could, however . mean ‘ten things’ and ‘eight things’ and it may be that Socrates is thinking of sets of ten and eight items. rather than of pure numbers. This issue may be important for other purpose. But it does not affect the analysis given here. which could easily be recast in terms of sets.

21 Or alternatively. that it would equally well answer the question ‘By what is eight fewer than ten?'. Here again. it is not clear whether the point should be (i) that (x > y by d) & (y < x by d), or (ii) that (x > y by d) & (x < z by d).See my discussion of Castañeda in section III above with note 19. and my note on 100e5-b8 (op.cit. 185). I prefer (ii). because it sharpens the paradox if both opposites in here in one and the same subject. and this is the situation to be considered at 102b-d. But the essential point can still be made on the alternative view.

22 In this example it is difficult to render the Greek literally without misleading the reader. For when. in English. A is said to be larger than B’ by half'. this is naturally taken to mean ‘by half the size of B'. not ‘by half the size of A'. If Goliath is taller than David ‘by half'. he is half as tall again a’ David. In the present example. however. Socrates is thinking of the double measure as larger by half of itself. It will. of course. always be possible to represent a quantity n times d given measure (m) as larger than that measure by some fraction of nm. since where n > 1 . ﹛nm > m by m(n) ﹜ . and m(n-1) is expressible as a fraction.n-1/n of nm.

23 The leg-pulling manner in which these examples are introduced should warn us against making too much of them. But at the risk of a pulled leg. it may be worth pursuing them further. The objection feared at 101b6-7 is. perhaps. the consequence that the double measure will be longer than the single one by half. For ‘double’ and ‘half’ may be thought of as opposites (cf. Republic 438. 479b). and any double measure exceeds a single measure (m) by half of itself: (m) (2m>m by 2m/2). It could thus be represented as paradoxical that what is double should be longer by what is half. If this is correct. the paradox in this case would be. as suggested in the previous note. that a longer length should be longer by something that is a fraction of. and therefore shorter than. itself. If the example of ten and eight is. parallel. the paradox in that case must be that ten is greater than eight by a quantity less than itself. Thus in both cases the paradox would be that a quantity x is greater than another quantity v by a difference d. where d is less than x. This interpretation would enable a similar paradox to be constructed for all positive values of x and y, since for all such values (x > y by d)ᑐ(d < x).On this view. ‘a head’ would have to be regarded as ‘short’ in virtue of its being shorter than the taller man, not as being shorter than the shorter one. (This differs from the view suggested in my note on 100e5-101b8, op. cit. 186). A possible alternative view. perhaps more in accord with the tone of the objection. is that the paradox consists simply in the idea that anything should be taller in virtue of something short, regardless of what the latter is shorter than. But this would be plausible, only if it were requisite that the ‘reason’ (αίτία) for something's being taller should not itself be shorter than anything else whatever.

24 cf., e.g.,104c5: ‘two is not opposite to three'. It may be worth pointing out an awkward problem faced by the Theory of Forms with respect to cardinal numbers. If each Form is unique, in accordance with the Theory as usually understood (d. Republic 597c, Timaeus 31a-b). how can a Form N be a ‘reason’ (αίτία) for a set's being n in number. where n is greater than 1? It is clear that cardinal number predicates applicable to sets of items taken together are not applicable to individual members of those sets. This point is made at length at Hippias I 301a-302b. 303b. and is one root of the puzzles at Phaedo 97a-b. Plato maintains this principle not only with respect to sensible particulars but also with respect to Forms. Thus at Republic 476a it is said that the Forms of Beautiful and Ugly are two. and consequently that each is one. Presumably. therefore. the Form N for any cardinal number n is in some sense one. But if the Form Twoness is one. how can it be a ‘reason’ (αίτία) for any set's being two? If, as argued in this paper. it is. requisite for any satisfactory ‘reason’ (αίτία) that it should not be characterised by any property incompatible with that of which it is the 'reason'. how can a Form which is single be a ‘reason’ for any set's being plural? This is. in essence. the much-debated problem of ‘self-predication', which — in relation to Forms for numbers — has perhaps received less attention than it deserves

25 ‘A Proof in the Peri Ideōn'. reprinted in R. E. Allen Pd. studies’ in Plato's metaphysics. ch.xv. See especially p. 102-9.

26 For a Platonic example. see parmenides', 129c4-d6.

27 op. cit. 303.