At 101a5-b2 and 101b4-7 Socrates expresses 'fears' that throw light on what was wrong with the reason rejected earlier (96d8-el) for one man or horse being larger than another. There would be two things to fear in giving 'a head' as the reason.
(1) 'If you say that someone is larger and smaller by a head, then, first, the larger will be larger and the smaller smaller by the same thing' (101a6—8). The apodosis of this sentence could mean either
x is larger than y by a head and y is. smaller than * by a head, or
x is larger than y by a head and smaller than z by a head. With (i), the phrases 'the larger' and 'the smaller' will refer to two different items, x and y, with the converse relations of 'larger than' and 'smaller than' holding between them. On this view, the supposed 'contradiction' could be formally deduced from lx is larger than y by a head', given that:
(x, y) {(x >y by h) = (y < x by h)}. With (ii) 'the larger' and 'the smaller' will refer to the same item, x. which will be 'larger' and 'smaller' in relation to two different things, y and z. On this view, the 'contradiction' could not be formally deduced, since from (x >y by h) we cannot derive (3z) (x < z by h). But (ii) seems to follow more naturally from the antecedent 'if you say that someone is larger and smaller by a head', which mentions only a single subject, (ii) would seem a more typically Platonic way of making the point that particulars may have contrasted predicates in different relations. And it has the advantage of enabling the later discussion of 'Simmias is larger than Socrates and smaller than Phaedo' (102b—d) to be linked directly with the present example.
On either interpretation the essential point is that 'a head' could just as well be viewed as a reason for 'being smaller' as for 'being larger', and therefore cannot be what constitutes anything's being larger. Hence it cannot help to answer the conceptual question 'what is largerV Evidently, it is here being assumed that if any feature is constitutive of a characteristic F, that feature will be found in all and only those things that are F, and not in things that are G. These lines thus fulfil the promise of 100b3—4 to 'display' the sort of 'reason' with which Socrates is concerned. For they exhibit, without expressly stating, the requirement -lat the 'reason' in question should be both a necessary and sufficient condition of the concept that has to be explicated. Whatever is the 'reason' for a thing's being F, or more F, cannot also be the reason either for its or for anything else's being G, or more G.
Vlastos (loc.cit.) assumes interpretation (i), and objects that the fact that y is smaller than x by a head would be a spurious reason for rejecting 'a head' as what makes x larger. For, he urges, x and y are different items, and 'there is no contradiction in the same cause producing contrary effects on different things'. On interpretation (ii) this difficulty does not arise, since only a single item, x, will be involved. But even on interpretation (i) Vlastos' objection does not affect Socrates' point. For he is, in effect, here stipulating what shall count as a 'reason' for any given property, and he is excluding as a reason for F anything that features in cases of F and G alike. This point can be made whether the items concerned are the same or not.
See on 96e6-97b7.
(2) The other objection to 'a head' as the reason for one man's being larger than another is (a8-b2) that a head is itself a small thing, and 'it's surely monstrous that anyone should be large by something small'. This displays a further requirement for a 'reason'. Whatever is to be a reason for x's being F must not itself be characterized by F's opposite, G. No adequate 'reason' for a property, that is genuinely constitutive of it, can possess the opposite of that property. So a head, being characterized by smallness, cannot be what constitutes a thing's being larger. The examples at 101b4—7 can be understood similarly. Both the rejected answers, 'two' and 'half of two cubits', are to be thought of as something 'small', and therefore ineligible as 'reasons' for anything's being 'large', or 'larger'.
Here again Vlastos (op.cit. 316, n.64) finds the argument flawed. The fact that a head is a small thing would not, he objects, preclude it from making x larger, as distinct from large. For* may be a larger man than y, without being a large man. But this objection seems, once again, to miss Socrates' point. The difficulty is: how can something'small'or (in this case) 'smaller than y' be what constitutes x's being 'large' or (in this case) 'larger than y"> How could something 'small' or 'smaller' be the true 'reason' for anything's being 'large' or 'larger'? By calling this supposition 'monstrous' (bl), Socrates exhibits a further condition that a true 'reason' must meet.
His objections to the rejected reasons are here formulated in an eristical manner, no doubt in parody of the contradiction-mongers (101el-2). But the principles implicit in these objections have a serious role in the coming argument. Putting them together with the one noted earlier (see on 96e6—97b7) the requirements for a 'reason' may be summarized as follows:
No opposite, F, can count as the 'reason' for a thing's having a property, if its opposite, G, can also give rise to that property (97a7—b3).
Nothing can count as a 'reason' for a thing's having a property F, if it can also give rise to theoppositeproperty G(101a6—8).
A 'reason' for a thing's having a property F, cannot itself be characterized by the opposite of that property, G (101a8—b2).
Requirement (iii) is crucial for what follows. For when Socrates comes, later on, to improve upon the present 'safe* Form-Reason hypothesis with 'a different kind of safeness' (105b8), he does so with examples that are 'safe' from the objection here brought against 'a head'. Fire, fever, and oneness (or a unit) cannot be characterized by the opposites of the properties of which they are 'reasons'. Soul is a 'reason' of the same type. What qualifies it as a 'reason' for something's being alive is precisely what disqualifies 'a head' as a reason for something's being larger. Hence, in this trivial example a major principle underlying the final proof of immortality can be discerned.
The principle in (iii) is, however, highly questionable. It is well discussed by E. L. Burge, Phronesis 1971, 5. See also on 105b5—c8 (p.213), 105c9—dl2.
101b9—c9. The grounds for rejecting addition and division as reasons for things' coming to be two, or one, were given earlier (97a7—b3). The reason now recommended is that they participate in the Forms Twoness and Oneness. The assimilation of number concepts to others tends to mask the peculiar difficulties to which they give rise (see on 96e6—97b7). Adequate treatment of those problems would call for an inquiry into number concepts such as Plato envisages at Republic 525a—526c. But it is not to his purpose to pursue it here.
Plato uses two kinds of words for numbers: (1) the ordinary words for the series of cardinal numbers, and (2) a series of words ending in -as, (monas, duas, trias, etc.). It is uncertain whether any consistent distinction is intended between these, and in particular whether members of (2) refer exclusively to Forms, and whether members of (1) never do so. In the present passage members of (2) are used, and the Forms Oneness and Twoness are clearly meant. In the next few pages, however, the matter is often more debatable. For this issue, which becomes crucial for the interpretation of the whole argument, see J. Schiller, Phronesis 1967, 57—8, and D. O'Brien, C.Q. 1967, 217-19. The formal difference between the two series has been marked in translation by adding '-ness' for members of (2): 'oneness', 'twoness', 'threeness', etc., and omitting it for members of (1): 'one', 'two', 'three', etc. The English 'monad', 'dyad', and 'triad' are transliterations of the roots of words in series (2), but their special associations make them unsuitable as translations. The present renderings may, however, wrongly give the impression that the abstract character possessed by, e.g., sets of three things is meant. The Greek words contain no suggestion of this. The termination '-ness' has been used simply to mark the formal difference between these words and members of series (1). In this way no questions of interpretation are begged, or, of course, solved. See on 103e5-104c6, 104d5-e6, 104e7-105b4.