Aristotle, Eubulides and the Sorites

Mind Association

Aristotle, Eubulides and the SoritesAuthor(s): Jon MolineSource: Mind, New Series, Vol. 78, No. 311 (Jul., 1969), pp. 393-407Published by: Oxford University Press on behalf of the Mind AssociationStable URL: http://www.jstor.org/stable/2252254 .

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V.-ARISTOTLE, EUBULIDES AND THE SORITES

BY JON MOLINE

IN this paper I should like to inquire about the probable historical origin, target and solution to perhaps the most perplexing of the ancient paradoxes, the Sorites.1 The evidence which remains on these points is neither copious nor quite conclusive, but it does support reasonably compelling hypotheses which serve to illumin- ate further our picture of the period. And such hypotheses should be formulated, for, as W. and M. Kneale noted, it is in- credible that the ancient paradoxes were produced in the entirely pointless way the tradition appears to suggest.2

1. The Origin of the Sorites

It is traditionally and, I think, rightly taken that the Sorites was first formulated by Eubulides of the Megarian school.3 The

'That this argument is still perplexing is indicated by Max Black's article, " Reasoning With Loose Concepts ", Dialogue, II (1963), no. 1, pp. 1-12. An additional indication of the perplexity it generates is provided by Evert Beth, who saw the argument as leading to paradoxical results in set theory. Foundations of Mathematics (Amsterdam, 1959), p. 22.

2 W. and M. Kneale, The Development of Logic (Oxford, 1962), p. 114. This point has also been recognized by Beth (see n. 1). The traditional view is found in Kurt von Fritz's article, " Megariker ", in Pauly-Wissowa's Realencyclopaedia der Classischen Altertumswissenschaft (Stuttgart, 1931), Supplement V, p. 710. It also appears in E. Zeller's Outlines of the History of Greek Philosophy, 13th edition, transl. by L. R. Palmer (London, 1931), p. 107. Zeller viewed Eubulides' paradoxes as " clever but worthless fallacies ". This view dates back at least to Cicero, who considered the Sorites to be a " very vicious and captious style of arguing ", and the other Megarian paradoxes to be " far-fetched and pointed sophisms ". Academic Questions, transl. by C. D. Yonge (London, 1853), pp. 46, 58.

3 Scholars who hold this view are: P. Natorp (" Eubulides ", in Pauly- Wissowa); I. Bochenski, A History of Formal Logic (Notre Dame, 1961), p. 105; W. and M. Kneale (ibid.); Theodor Gomperz, Greek Thinkers, transl. by G. G. Berry (New York, 1905), II. 189-190, 203; Carl Prantl, Geschichte der Logik (Leipzig, 1855), pp. 54-56; Kurt von Fritz (ibid.). The paradox may have been suggested to Eubulides by Zeno's contention that if a bushel of wheat makes a noise in falling, every grain of wheat must make its proportionate noise. (Aristotle, Physics, 250a 19-25.) But Zeno cannot on this ground be credited with having invented the Sorites paradox, for his argument is merely an instance of the simpler fallacy of division. The Sorites consists not in the claim that each of the parts of a whole must have its characteristics, but in the claim that any modification of a whole having the same characteristics. And the modification may be increase as well as by decrease.

393



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evidence for his authorship of the Sorites and indeed most of what we know about him comes from Diogenes Laertius.1 According to Diogenes, Eubulides was a student of Euclides (himself a student of Socrates) and was the author of many dialectical arguments, of which the Sorites was one.2 He was also reputed to be famous enough to be commented on by a comic poet, important enough philosophically to carry on a controversy with Aristotle, and clever enough to get the better of Aristotle in the exchange.3

There is good reason to believe that two of these reported facts-Eubulides' authorship of the Sorites and his carrying on a controversy with Aristotle-are connected, though not, or at least not merely, in the ways that have hitherto been suggested.4 Surprisingly, there is not only good evidence that these facts are connected, but good evidence as to how they are connected.

2. The Target of the Sorites

We know nothing of Eubulides' work other than his paradoxes. He was remembered by the best ancient sources for no other

1 Diogenes Laertius, Vitae Philosophorum, II. 108. 2 Diogenes Laertius credits Eubulides with the Liar, the Horned One,

the Electra, the Veiled Figure (a variant of the Electra), the Disguised (another variant of the Electra), the Sorites, and the Bald man or Phalakros (a variant of the Sorites).

3 Part of the exchange may have been slanderous, as Eusebius indicates in Praeparatio Evangelicca XV. 2.22. Eusebius, a fourth century bishop of Caesaria, claims to have been acquainted with Eubulides' book (there is no other evidence that he wrote a book). He says that it charged Aristotle with being a spy for Philip of Macedon and with disloyalty to Plato. Eusebius' report may be discounted somewhat, as he was probably not a learned man. But there is other evidence from the fourth century that Eubulides was thought to have made personal attacks on Aristotle; Themistius considered him a just judge of Aristotle, (Themis- tius, Opera Omnia, Venice, 1534.) Apparently the view that Eubulides made personal attacks on Aristotle dates back at least to the third century, for Athenaeus (Deipnosophists, 354c) takes the trouble to say that he knows that Epicurus, not Eubulides, was responsible for such attacks. In any case, personal attacks are not likely to have been the whole subject- matter of Eubulides' controversy with Aristotle, for he is represented in the most reliable sources as a logician, not a character-assassin. And it is quite unlikely that a writer as fair-minded as Diogenes Laertius could have thought it possible for Eubulides to get the better of Aristotle by such means.

4 Gomperz (ibid. II. 188)' conjectures that the Megarians wished to expose "the contradictions which ... traverse the whole fabric of our empirical concepts ". Beth (ibid.) suggests that the Sorites was an attack upon Aristotle's theory of potential infinity.



ARISTOTLE, EUBULIDES AND THE SORITES 395

reason.1 Hence it is likely that he did no other work of any consequence. Yet he was said to have carried on a controversy with Aristotle. It seems natural to ask if there is any reason to believe that the Sorites or any of the other paradoxes of Eubulides figured in this controversy. And, as we shall see, there is evidence that the Sorites, at least, was directed against Aristotle's views. But which of these views was the target ? No doubt there is a wide range of Aristotelian targets, both practical and theoretical, to which the Sorites might plausibly have been directed. Beth and Gomperz mention two candidates.2 One could name many more, for the Sorites is a notoriously poly- gamous paradox, ready to marry its way into almost any family of concepts. In this investigation I should like to point out the clearest and least speculative of the possible targets. I do not suggest that there may not have been other, perhaps even more basic points, to which the Sorites was directed.3 I suggest only that there are good historical reasons to believe that it was directed against at least one clear target-the doctrine of the mean.

In Book II of the Nicomachean Ethics, Aristotle defines virtue or excellence (arete) as

... a settled disposition, lying in the mean relative to us, deter- mined by measure (logos) and as a man of practical wisdom would determine. And it is a mean between two vices, the one of excess and the other of defect; it is a mean in that whereas these fall short of or exceed what is right in feelings (pathesi) and in actions (praksesi), virtue both finds and chooses the mean. Wherefore in relation to substance (ousia) and the account which states what it is, virtue is a mean, and in relation to the best and right, it is the extreme of perfection.4

The crucial points here for our purposes are

(a) that virtue is a mean, and (b) that it is so characterized in an account or logos which

states what it is, its essence.

What historical evidence is there to indicate that the Sorites was a criticism of this doctrine of the mean ? Two of the Greek commentators on the Nicomachean Ethics treat it as such.

1 Sextus Empiricus (Adversus Mathemcaticos vii. 13) maintains that Eubulides concerned himself solely with logical matters.

2 See n. 4, p. 394. 3 It seems quite possible, as Professor Julius Weinberg has suggested to

me, that the Sorites was also an attack upon the law of excluded middle. 4Aristotle, Nicomachean Ethics IT. vi. 15 (my translation).



396 J. MOLINE:

One, Aspasius, even says that Aristotle was aware of it or understood it in this way.' He says,

Because it is difficult to grasp the mean, he cited as a paradigm not certain of the means relative to us but those relative to the thing itself. For not everyone grasps the mean (middle) of a circle, but only someone who knows. And if it is troublesome to grasp the mean in thinrgs such as this, in which the mean is one and the same for all, it is much more difficult in those actions in which the mean is accommodated to each (person). He said first that moral virtue is a mean and in what way, and also that these other things are means, but that the others are clear. For the mean relative to us involves choice, but not the mean relative to the thing. And at the end he said that as to quantity it is not easy to define until someone deviates either by excess or by blameworthy defect; he understands the things which are always said about it-that no action is accurately definable other than in outline. For it is no different with any other sensible thing from which they arrive at Sorites. For at what point is a man wealthy? When he has deposited ten talents, they inquire? And if one should take away a drachma, is he no longer wealthy? And if two? For there is no one of such things, since it is a sensible thing, which it is possible to define accurately, whether a poor man or a rich. And concerning the bald man they ask whether by one hair one can become bald, or by two, or three? Whence the arguments said that bald men were also Sorites. For concerning a heap (soros) they ask the same thing, whether by one grain of wheat the heap is made smaller, and then whether by two, and so on according to the pattern; and it is not possible to say where first there is no longer a heap because no sensible thing can be understood accurately, but only broadly and in outline. Thus this holds even of actions and of feelings. For it is not possible to say by how much anger one attains a mean in anger or surpasses it or fails to come up to it because of devia- tion towards too much or too little. Wherefore there is need of practical wisdom for discovering the mean in feelings and in actions.2

The other piece of evidence of Greek origin is taken from the anonymous commentator on the Nicomachean Ethics, Book II, who also regards the Sorites as directed against the doctrine of the mean and as answered by certain of Aristotle's remarks. He says,

1 Aspasius, a respected Peripatetic, flourished during the first half of the second century A.D. See Alfred Gercke's article, " Aspasius ", in Pauly- Wissowa.

2 Aspasius, In Ethica Nicomachea, in Commentaria in Aristotelem Graeca, ed. G. Heylbut (Berlin, 1889), XIX. 56-57 (my translation).




For it is not possible to determine in an account (logos) what additions make a heap, just as it is not possible concerning actions. And in fact in actions the one who diverges a little is not blamed, not even if it is on the side of excess; thus also in that case one who adds insufficiently does not make a heap. For when a heap comes into being out of the many things added it is surely not possible to determine in an account. For there is no account of perceivables, but perception is the judge. And such questions depend upon particular cirumstances. More- over, actions, he says, are that in which the mean is sought. And these are particulars. In perceivables, perception, not an account, is the judge. And concerning such things it is not possible to determine the mean in an account.'

Both commentators treat the Sorites as a criticism of the doctrine of the mean, and both find answers to it in certain views Aristotle expresses, although different views are cited by the two. Aspasius recommends practical wisdom as the antidote to the problem posed by the Sorites; the anonymous commentator recommends perception. This lends support to the hypothesis that the commentaries are independent, that the one commentator is not merely repeating what he has read in the other.2 Hence there seem to be solid historical reasons to believe that the Sorites was directed against the doctrine of the mean.

3. Was Aristotle Aware of the Sorites ?

What do these facts gleaned from the Greek commentators show ? In isolation they certainly do not show that Aristotle was aware of the Sorites as a criticism of the doctrine of the mean. Aspasius lived in the early part of the second century A.D., and the anonymous commentator's dates have not been established. Moreover, the term " Sorites " does not occur in the Corpus Aristotelicum. And there is no formulation of the paradox in the Nicomachean Ethics or Sophistic Refutations, the works in which

1 Anonymi, In Ethica Nicomachea, in Commentaria in Aristotelem Graeca, ed. G. Heylbut (Berlin, 1892), XX. 140 (my translation).

2 Other differences between the two commentaries tend to support the view that they are independent. Although the two treat of the same passage in Aristotle, the language is very different. Aspasius' style is more polished, that of the anonymous commentator more severe. Aspasius gives one form of the Sorites for the soros or heap, the anonymous commentator quite another. As"asius mentions but does not emphasize the contrast between perception and definition. The anonymous commentator finds this contrast so crucial that he alludes to it several times and finds the practical antidote to the Sorites in it.



398 J. MOLINE:

one would expect such a formulation to appear if Aristotle was indeed aware of the paradox. In fact, there is no formulation of the paradox anywhere in Aristotle's works. Hence one might be tempted to conjecture that Aspasius and the anonymous commentator are perhaps repeating a criticism advanced against the doctrine of the mean by some later philosopher, a criticism unknown to Aristotle himself. One would then conclude that Aspasius was incorrect in claiming that Aristotle was aware of the paradox.

This conjecture may be correct. But to rest satisfied with such an hypothesis would be to fragment our view of the evidence already cited, and would render incoherent our account of the period. First, if we accept such a conjecture we thereby give up the best (and perhaps the only) historical ground for answering the question " What was the subject of the dispute between Aristotle and Eubulides ? " Secondly, we thereby postulate an unknown critic when there is evidence of a known critic-Eubu- lides. Thirdly, Aspasius was probably in a far better positiolu to say what Aristotle was or was not aware of than we are; he was a member of the still-extant Peripatetic school, and was apparently respected.' Fourthly, if we adopt the view that the Sorites was unknown to Aristotle, we have difficulty in accounting for the fact that Aristotle was aware of other Eubulidean paradoxes.2 Theparadoxes are cited as a set byDiogenesLaertius. If Aristotle was aware of several of them it is likely that he was aware of the others. Fifthly, we must find some way of glossing over the fact that even though Aristotle does not use the term " Sorites " or formulate the paradox, he unmistakably alludes to it in a context in which he is discussing others of the Eubulidean paradoxes. In discussing an unsuccessful attempt to solve the Electra bv saying that in a sense one may know a person wearing a veil and in a sense one may not, Aristotle remarks,

1 Aspasius was noted solely for his exegesis of Aristotle's views. As Geroke noted (see n. 1, p. 396), one looks in vain for original thoughts in Aspasius' commentary. Aspasius is not likely, therefore, to have thought of the criticism himself. And he was not a passionate, philosophic partisan of Aristotle's views; he reports them faithfully, but more in the spirit of Andronicus the philologist than in that of one anxious to defend them from every attack. Such a man is not likely to have strayed far from what he had reason to believe was historical fact.

2 For the Hooded or Veiled man (by name), see Sophistic Refutations 179a30 Si. For the Electra (although not by that name) and points bearing on its solution, see Sophistic Refutations 179a35, 170b12 f, 171a18 if, 179b29 f, 175b28 f, 181a7 if, Posterior Analytics 71a25 Si. For the Liar (by name), see Sophistic Refutations 180a23-181b13, especially 180b2-7.




An error similar to that made by those we have mentioned is committed by those who solve the argument that every number is small, for if, when no conclusion is reached, they pass over the fact and say that a conclusion has been reached and is true because every number is both large and small, they commit an error.1

The argument that every number is small is a standard form of the Sorites, and is presented as such by Diogenes Laertius at VII. 82. I know of no other " argument that every number is small " in ancient literature. This appears to be an unmistak- able allusion to the Sorites. Hence it cannot be maintained that Aristotle was not aware of the paradox.

If we conjoin this conclusion with those argued for above, there would appear to be historically conclusive grounds for believing that Aristotle was aware of the Sorites and compelling, though perhaps not historically conclusive, grounds for believing that the reason he was aware of it was that Eubulides used it as a criticism of his doctrine of the mean.

4. The Probable Form of the Sorites as Known to Aristotle

But if we accept these conclusions, we must be prepared to acknowledge that Aristotle was aware of an important paradox which he did not explicitly answer. And this gives one pause. Not only is the Sorites more important than many paradoxes and fallacies he does deal with in the Sophistic Refutations, it is, as we shall see, prima facie damaging to his own theory of the mean. Failure to answer such a paradox is not characteristically Aris- totelian, and calls for an explanation.2 Why, if he was aware of the Sorites, as he now appears to have been, did he not answer it ? Anyone who has attempted to solve the Sorites can produce a ready reply-not even Aristotle could solve this paradox. Not being able to solve it, he ignored it, it might be said. And this is not entirely implausible, since Eubulides was said to have got the better of Aristotle in their controversy.

This answer to the question " Why did Aristotle not answer the Sorites ? " may be correct, but there are alternatives which deserve inspection. Aristotle answers two Eubulidean paradoxes explicitly. He has obviously appropriate machinery available for answering the rather unimportant third one, the Horned Man; probably he did not take up the Horned Man explicitly because he thought the "machinery for its solution was set out in

1 Aristotle, Sophistic Befutations, transl. by E. S. Forster (London, 1945), 179b33 ff. 2 See n. 2, p. 398.



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the Sophistic Refutations.1 Possibly he did not take up the fourth Eubulidean paradox, the Sorites, not because he was stymied by it, but because he thought it similarly covered by views already presented. I find it difficult to believe that he thought the solution obvious in his writings, b-ut I do not find it difficult to believe that he thought he had one implicitly. It cannot be maintained with any great plausibility that even so catholic a writer as Aristotle explicitly dealt with every philosophically interesting argument which came to his attention.

If this is the case, it is worthwhile trying to find in Aristotle's writings machinery which he might have thought adequate for handling so formidable a paradox.2 And in this attempt it will be helpful to have before us a formulation of the paradox not unlike that probably used by Eubulides against the doctrine of the mean. The argument was probably a double application of the well- known schema.3 I suspect that it ran something like this: " You Aristotle, say that virtue is essentially a mean. And you claim that this applies to all of the various virtues as well. Very well. Consider a case in which I am called upon to give a generous gift. Let the man of practical wisdom consider the case and my means, and say what would count as a generous gift. It does not matter what he says or why, for whatever amount he

'The Horned Man is not mentioned by name in Aristotle's writings, but similar confusions are treated in the discussions of ambiguity cited in n. 2, p. 398, in reference to the Electra and in the treatment of the fallacy of complex question at Sophistic Refutations 167a38 ff, 181a36 ff.

2 An alternative explanation of why Aristotle did not answer the Sorites explicitly might be this: Aristotle did not need to construct any machinery for solving the paradox, for, like Max Black, he recognized the looseness of at least our ordinary-language ethical concepts. This view might be thought to gain some support from Aristotle's repeated warnings against the inevitable lack of precision in ethical matters. (These warnings appear at EN I. iii. 1-3, vii. 18; ii. II. 8; IX. ii. 6.) This explanation is untenable, however. The imprecision to which Aristotle draws attention in these passages is not in our ethical concepts, but in the subject matter of ethics. This is especially clear in II. ii. 3. And Aristotle can still demand precision despite this admission that the subject matter of ethics is not amenable to precise treatment, for in Topics 146b he remarks, " Similarly, too, we must state what quantity of money which he desires makes a man avaricious and what quality of pleasures which he desires makes a man incontinent.... And similarly, in all cases of this kind; for the omission of any differentia whatever involves a failure to state the essence."

3 This schema appears in Diogenes Laertius at VII. 82, in Horace's epistle to Augustus (Epistulae, Book II, no. 1), in Cicero's Academic Questions II, 49, in his De'Finibus Bonorum IV. xviii. 50 (although here he extends the schema to the addition and subtraction or intensification and lessening of qualities, not merely quantities), and also in Sextus Empiricus' Outlines of Pyrrhonism III, 80-81.




seizes upon as the generous amount for me to give will lead to a paradox. For suppose the man of practical wisdom in my circumstances would give n drachmas. Giving just so much as the man of practical wisdom would give and would declare to be the mean is, you would say, generous. But it is not. For suppose I give just one obol less. Is my gift not generous ? Surely you must concede that giving just one obol less than the generous amount is generous, for an obol is a trifle. Yet if we apply the principle you concede a sufficient number of times, it follows that it is generous to give nothing. But this, clearly, is not generosity, but the extreme of meanness. And suppose that I give one obol more than the mean as specified by the man of practical wisdom. I am giving one obol more than the generous amount, but surely my gift is still generous, for again, an obol is a trifle. But if we apply a sufficient number of times the principle that giving one obol more than what is generous is generous, it follows that it is generous to give not n obols, but one's entire fortune. And this is not generosity, but the extreme of prodigality. You define virtue as a mean. But it follows that virtue is both of the extremes. You see, Aristotle, what trouble there is in trying to define sensibles-it always leads to paradox."

This fable, I suggest, is not unlike what Eubulides said if we can trust the evidence cited and if we are at liberty to put two and two together. As for the paradoxical, Zenonian turn in this formulation of the Sorites argument, we saw in his allusion to the Sorites as quoted above that someone with whom Aristotle was acquainted did give the argument just such a turn. Additional evidence to support the historical appropriateness of the double application of the Sorites is provided by the fact that Aspasius takes away grains of wheat in his formulation of the argument, while the anonymous commnentator adds them. One would need only to conjoin the addition and subtraction forms to get the paradoxical view that a heap can be diminished to nothing and yet remain a heap and that one can pile up as many grains as one likes without making a heap. One could conclude that any given pile of grain both was and was not a heap. As for the Megarians having had sufficient reason to give the argument such a Zenonian turn, we have the testimony of Diogenes Laertius that they were under Eleatic influence,1 the statement of Cicero, who considers the Eleatics paleo-Megarians,2 and hints by Aristotle

1 Diogenes Laertius, II. 106, II. 109. 2 M. T. Cicero, Academic Questions, transl. by C. D. Yonge (London,

1853), p. 84.



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himself.' Hence it seems reasonable to maintain that the Sorites was probably advanced against the doctrine of the mean in a form not unlike that suggested.

5. Possible Aristotelian Answers to the Sorites

Now let us consider possible Aristotelian answers to the paradox. Naturally, one wonders if the brief answers given by the Greek commentators are adequate or complete versions of an answer Axistotle himself is likely to have given. Although I shall conclude that they are versions of an answer Aristotle is likely to have given, unfortunately they do not seem to be adequate or complete versions of that answer. The answer suggested by the anonymous commentator does suggest how, on the commentator's view, the Sorites bore on the doctrine of the mean by representing it as hopelessly imprecise. Hence it lends some support to the formulation of the paradox given above. But the commentator merely shrugs off the imprecision as unavoidable. And in doing so he distorts Aristotle's remarks.2 For Aristotle does not say that it is impossible to define the mean relative to us. Rather, he says

These then, to sum the matter up, are the precautions that will best enable us to hit the mean. But no doubt it is a difficult thing to do, and especially in particular cases . . . yet to what degree and how seriously a man must err in order to be blamed is not easy to determine in an account. In fact this is not easy for any object of perception; moreover, such questions of degree depend upon particular circumstances, and the decision lies with perception.3

Aristotle's remarks do suggest the practical solution favoured by the anonymous commentator (" Look and see ".) But one is led to suspect that Aristotle perhaps had a more incisive answer than this. For this is not a theoretic solution or lysis; it does not expose that on which the argument turns. In dealing with others of the Eubulidean paradoxes, Aristotle remarks,

... it is not the exposure of every fault that forms a solution (lysis), for it is possible for a man to show that a false conclusion has been reached without showing on what point it turns, as,

'In Metaphysics 1046b29-1047a22, Aristotle explains how the Megarians " do away with both motion and generation ". This, of course, was a paradigm Eleatic activity, one accomplished by Zenonian argument.

2 There is, however, some, support for the line taken by the anonymous commentator. It is found in Metaphysics Z, 1039b27 f, 1035bl-20, 1040a28 f, not in the Nicomachean Ethics itself.

3 Nicomcahean Ethics II. ix. 7-8. (My translation. Emphasis added.)



ARISTOTLE. EUBULIDES AND THE SORITES 403

e.g. ill Zeno's argument that motion is impossible. Even, therefore, if one were to attempt to infer the impossibility of this view, he is wrong, even though he has given countless proofs; for this procedure does not constitute a solution, for a solution is, as we saw, an exposure of false reasoning, showing on what the falsity depends.1

Aspasius' suggestion, unfortunately, is no more a solution than is that of the anonymous commentator. To say that there is need of practical wisdom for determining the mean in particular cases is merely to repeat a corollary of the view under attack, not to fend off the attack by showing in what respect it depends upon faulty reasoninig.

Again, one suspects that Aristotle had a more perspicuous answer, a solution more in keeping with his practice of treating fallacies and paradoxes as involving structural, linguistic or conceptual defects in argument.2 But the reconstruction of such a solution is both hazardous and difficult. It is hazardous because Aristotle himself never formulated it in any one place. It is difficult because several alternative, though untenable, answers suggest themselves as well.3 But I submit that it is nevertheless possible.

The view I believe to have been Aristotle's own best means of dealing with the Sorites is implicit in his theory of measure. This can be discerned by attending to three claims:

(a) Aristotle's claim at EN III. iv. 5 that the good man (spoud- aios) is the standard (kanon) and measure (metron) in everything, a claim reinforced and generalized at EN I. iii. 5, IX. xi. 4, X. v-vi. 6, Topics 116a22, 145a25, 163b, Rhetoric 1364b, and Metaphysics 1053b35.

(b) Aristotle's characterization of measure at Metaphysics 1052b20 as that by which quantity is known, and

(c) Aristotle's general discussion of measure and calibration in units at Metaphysics 1052b33 if.

1 Aristotle, Sophistic Refutations, transl. by E. S. Forster (London, 1955), 179bl8-24. A similar view is echoed in the Nicomachean Ethics itself, where at VII. xiii. 3 Aristotle says, " We ought however not only to state the true view but also to account for the false one, since to do so helps to confirm the true; for when we have found a probable explanation why something appears to be true though it is not true, this increases our belief in the truth ". (Rackham's translation.)

2 This suspicion is reinforced by Aristotle's claim at Sophistic Refutations 179bl2 that " The method of correcting arguments which turn on the same principle ought to be identical ". None of his solutions to other paradoxes or fallacies turn on appeals to perception or to practical wisdom.

3 With the exception presented in n. 4, p. 404, I shall not present even the least implausible of these.



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In citinig the choice of the man of practical wisdom in the account of what virtue (arete) is, Aristotle is following a methodo- logical principle of great consequence in both his theoretic and practical philosophy. The good man or man of practical wisdom, it would appear, is not merely good at discerning the mean; rather, the mean is what the good man says it is. He is the standard or measure.

A measure is that by which quantity is known, Aristotle maintains.' If one wishes to know the quantity of money the contri- bution of which would constitute generosity in a particular case, not meanness or prodigality, one must discern that quantity picked by the man of practical wisdom in such a case. The measure of generosity is not the obol, drachma, or talent, but the size of the gift given by the man of practical wisdom. It is the only appropriate measure. And in all cases the appropriate unit of measure is taken as indivisible.2 It need not be the smallest unit available.3 The furlong is divisible into feet and inches, but it is still treated as the only appropriate unit for measuring the length of a race-track.

This machinery allows for a solution to the Sorites in the following way: If the only appropriate unit by which generosity can be measured is the size of the gift of the man of practical wisdom, and if this unit is taken as indivisible, then the Sorites cannot be generated. No one would be inclined to claim that taking away one such unit would still leave the gift a generous one. Rather, it would leave nothing at all-and would do it at one fell swoop. And no one would be very strongly inclined to claim that adding another such unit would still leave the gift a generous one and not an ostentatious or prodigal one, for the gift would be double the right and generous size, not merely larger by some trifling amount.

The reasonableness if not the correctness of such an ans'wer to the Sorites is apparent if we consider additional examples.4 Consider the example of the soros or heap.5 If we take a clear

'Metaphysics 1052b20. 2 Ibid. 1052b32. 3 Ibid. 4 I do not suggest that this answer to the Sorites is an entirely adequate

one philosophically. It does have the merit of recognizing a crucial factor on which the Sorites turns-the size of the unit of measurement used. And to this extent it is not unreasonable. But this factor is only one of several on which the argument turns, and Aristotle's view that the unit of measurement is taken as indivisible apparently prevented him from recognizing a second factor-the number of times one adds or subtracts such a unit. A third factor would appear to be the amenability of loose ordinary- language predicates to exploitation in a process akin to mathematical induction. This third factor had been pointed out by Max Black (ibid.).

5 This example occurs both in Aspasius and in the anonymous commentator.




case of a heap of grain and regard the number of grains in the heap as a single unit for purposes of measuring heaps, it is obvious that subtracting one such unit from the heap would at one fell swoop leave nothing at all. No one would maintain that there was still a heap.

The machinery applies in a similar way to means between extremes. Consider Aspasius' examples of the mean in anger. If the mean in anger is that degree of anger displayed by the man of practical wisdom, and if, perhaps metaphorically, we regard this degree as an indivisible measure of anger, then to " subtract " one such unit would render one not angry at all. To " add " one such unit would be to leave one in so obviously exaggerated a state of anger as to make clear in one stroke the transition from the right degree of anger to a terribly excessive degree.

At least one obvious objection arises, however. The per- petrator of the Sorites might well ask, " If the appropriate unit for the measurement of, say, generosity, is the gift of the man of practical wisdom in given circumstances, then why would we call a man 'generous ' who gave one obol less ? " Aristotle's proper reply here is clear. He would appeal to his ubiquitous doctrine of analogical predication.' We call one who gives an obol less than the man of practical wisdom " generous " because he is more like the generous man than the ungenerous. But in doing so we are not predicating generosity of him essentially. Rather, he is called "generous " by analogy.2 The term " generous " is not entirely univocal, nor is it entirely equivocal.3 It is used in relation to one central, essential meaning-identity with, or failing that, likeness to the gift of the man of practical wisdom in given circumstances. If one wishes to know the point at which the resemblance becomes so slight that we would no longer, even by analogy, call the giver " generous ", then one must notice when the man of practical wisdom ceases to praise the giver, and begins to blame him for niggardliness, for here too the man of practical wisdom is the measure.4

1 Metaphysics IV; ii, V. iv, Topics 108b7, Generation of Animals 761b32 ff, Nicomachean Ethics VII. ix. 6-x. 4.

2 Because it does not bear in an important way on the Sorites or its solution, I have ignored throughout this discussion a distinction Aristotle would make between a generous gift and a generous giver. The basis for this distinction is found at EN V. ix. 10-16.

3 See Categories la ff, Metaphysics 1030a21-b4, 1003a33-bl9, 1047a30-33, 1048b36 ff. for the background on this point.

4 It might be thought that a simpler way of answering the Sorites could be found in Aristotle's claim at Topics 128a that essences do not admit of degrees. Aristotle does say at EN II. vi. 15 that virtue or arete has an



406 J. MOLINE:

Another point remains to be clarified, however. Aristotle was aware of the Sorites in the form in which it was used to argue that every number is both large and small. Yet it is not immedi- ately clear how this proposed solution bears on the paradox in that form. There are no experts on the detection of large and small numbers as such. It would be absurd for Aristotle to maintain that there are experts who, are entitled to say " Ten thousand is a large number. And we will regard that as an indivisible unit for purposes of measuring large numbers ".

But the proposed Aristotelian solution to the Sorites does apply to this form of the argument. No one would be inclined to maintain in abstraction that numbers are large or small. One would have to consider numbers.,of this or that. "Large " and "small" are terms of relation, not of quantity. A large number of human offspring is perhaps six or seven; a large number of bacteria on a slide might be in the millions. The notion that numbers are large or small, and certainly the notion that they are both large and small, is either senseless or dependent upon specification of numbers of this or that. But this indicates that Aristotle's solution to the Sorites does apply in this case. Suppose we ask the Fire Marshall of a given city what a large number of fires on a day in mid-February would be. He might say, " Fifteen ". And clearly he can do so with authority. He is the expert, the man of " practical wisdom " in such matters. And as such his measure is amenable to the same kind of treatment as that outlined above for the man who measures generosity.

But was Aristotle aware that a number (or indeed, anything at all) is large or small only relatively, in a given context ? He plainly was. He says,

But someone might possibly say, ' great ' and ' small', 'nmuch' and 'little' are contraries. These are, however, more properly regarded as terms of relation: as suck, things are not great or small. They are so by comparison only. Thus a hill is called small, a grain large; but we really mean greater or smaller than

essence, as we saw above. Hence it would follow that there are no degrees of virtue. And the fact that we talk as if there were could be explained by appealing to the doctrine of analogical predication. But I am inclined to think that this simpler way does not really meet the Sorites. The Sorites is not, e.g., an argument to the effect that adding insignificant sums to a generous gift will make the gift more generous. It is merely an argument that the gift is still generous (not more or less so) no matter how many individually insignificant quantitative changes it undergoes. " Admitting of degrees " (mallon kai hetton) for Aristotle means " being subject to qualification as more or less such and such ". And no extant form of the Sorites employs the notions of " more " or "less " in this way.



ARISTOTLE, FTEUBULIDSF AND TETE SORITTE 407

similar things of the kind, for we look to some external standard. If such terms were used absolutely, we should never call a hill small, as we should never call a grain large. So, again, we may very well say that a village has many inhabitants, a city like Athens but few, although the latter are many times more; or we say that a house contains many, while those in the theatre are few, though they greatly outnumber the others. While ' two cubits ', ' three cubits long ', and the like, therefore, signify quantity, 'great', 'small' and the like signify not a quantity but rather a relation, implying some external standard or something above and beyond them. The latter, then, are plainly relative.'

It is clear who this external standard would be-the man of practical wisdom. It seems clear, then, that the solution proposed earlier applies also to the argument that every number is large and small.2

6. The Historical Status of this Answer to the Sorites

This solution to the Sorites can be recommended as historically probable on several grounds. Unlike the view that Aristotle did not answer the Sorites because he could not, it illuminates further our picture of the period. It is suggested by Aristotle's theory of measure. Moreover, it is not only compatible with the practical answers given by the Greek commentators, it provides a link between them.3 Finally, it counters the Sorites in a way con- sistent with Aristotle's usual way of treating fallacies and paradoxes.4

The University of Wisconsin

1 Aristotle, Categories, transl. by H. P. Cooke and H. Tredennick (London, 1962), 5b.

2 It is perhaps significant that Aristotle chooses to speak of calling a grain large here, since the paradigm of the Sorites deals with heaps (soroi) of grain.

3 The man of practical wisdom is the measure. He measures by per- ceiving and choosing. His practical wisdom is his qualification for being the measure.

4I am indebted to Friedrich Solmsen and Julius Weinberg for a number of helpful remarks on this paper.



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Aristotle, Eubulides and the Sorites