Heraclitus and the Bath Water

Philosophical Review

Heraclitus and the Bath WaterAuthor(s): Helen Morris CartwrightReviewed work(s):Source: The Philosophical Review, Vol. 74, No. 4 (Oct., 1965), pp. 466-485Published by: Duke University Press on behalf of Philosophical ReviewStable URL: http://www.jstor.org/stable/2183124 .Accessed: 12/10/2012 10:48

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HERACLITUS AND THE BATH WATER'

I. N Reference and Generality, Geach directs the following argument against what he calls "the accepted treatment of

referring expressions like 'some A."'

According to this accepted view, we may treat the proposition:

(i) Heraclitus bathed in some river yesterday, and bathed in the same river today

as equivalent to:

(2) Something (or other) is a river, and Heraclitus bathed in it yesterday, and Heraclitus bathed in it today

or, using "bound variable" letters, as equivalent to:

(3) For some x, x is a river, and Heraclitus bathed in x yesterday, and Heraclitus bathed in x today.

Now by parity of reasoning we may analyze:

(4) Heraclitus bathed in some water yesterday and bathed in the same water today

as equivalent to:

(5) Something (or other) is water, and Heraclitus bathed in it yesterday, and Heraclitus bathed in it today

or again to:

(6) For some x, x is water, and Heraclitus bathed in x yesterday, and Heraclitus bathed in x today.

But, according to Geach, "We may assert the additional premise,"

I I owe a good deal of clarification and buttressing of arguments in this paper to discussion with members of the philosophy department of Wayne State University, especially Richard Cartwright and, in Sec. 9, Hector Castafteda; in this connection I am also grateful for David Sachs's criticism.

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HERACLITUS AND THE BATH WATER

(7) Whatever is a river is water

"or 'For any x, if x is a river, x is water," and

given this premise, (5) or (6) is inferable from (2) or (3). But clearly this premise would not warrant us in inferring (4) from (i): it is notorious that (i) could be true and (4) false. Hence the above analyses of (i) and (4), which stand or fall together, must both be rejected.2

The argument invites criticism at a number of points. But perhaps its most obvious weakness lies in the purported relevance of Geach's additional premise (7). It is tempting to say that if, as it is claimed, (7) warrants the inference of (5) or (6) from (2) or

(3), then "water" must be understood in such a way that we are warranted in inferring (4) from (i) and (7)-and that (4) is not, after all, a proposition Heraclitus would have denied in the circumstances. But no doubt it is more plausible to agree that (7) does not warrant the inference of (4) from (i), and argue that it does not, in this case, warrant the inference of (5) or (6) from (2)

or (3). This, in effect, is what Quine does in his review of Geach's book.

A mass term like "water" or "sugar" does not primarily admit ccsame" nor "an." When it is subjected to such particles, some special individuating standard is understood from the circumstances. Typi- cally, "same sugar" might allude to sameness of shipment. Now in the sense in which one resists saying that Heraclitus bathed in the same water twice, a water is an aggregate only of molecules that were near a man when he once bathed. But a river is not such a water.3

2. But although it is true that the river in which he bathes is not just a man's bath water, it is questionable whether, in his explanation of the fact, Quine has really ended the matter. If we agree that (7) is irrelevant, the fact remains that (5) and (6) can

2 P. T. Geach, Reference and Generality, An Examination of Some Medieval and Modern Theories (Ithaca, I962), pp. 150, I5I.

3 W. V. Quine, Philosophical Review, LXXIII (I964), 102.

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HELEN MORRIS CARTWRIGHT

quite naturally be understood in such a way that they are weaker than (4). All that follows on this reading from (5) or (6) is that Heraclitus bathed in a certain substance, namely water-rather than, for example, milk-yesterday and today. It does not follow that he bathed in the same water, as (4) requires. Furthermore, if, in order to guarantee that we have a pair of propositions equivalent to (4), we follow Quine's suggestion as to how "water" is to be here understood, it is not at all clear that we shall succeed. In fact, it is hard to see why Quine says that "water" does not ''primarily admit 'same' nor 'an,"' since normally it does not admit the indefinite article at all, while "the same water," as in (4), is quite unexceptionable. Someone who says that Heraclitus did not bathe in the same water twice is surely not thereby com- mitted to speaking of "a water," and if such a commitment is involved in the understanding of (5) and (6), their equivalence to (4) remains to be shown.

Thus, instead of warding off Geach's attack, these remarks of Quine's only suggest a further puzzle. We have not as yet been told why the analysis of (4) as (5) or (6) is not to be rejected. And if the analysis of (4) and of (i) "stand or fall together" as Geach claims, then there is reason to reject them both and with them the "accepted treatment of referring expressions like 'some A.'"

In what follows I propose first to take a somewhat different tack against this argument of Geach's and then to consider at some length a problem which remains. It may be conceded that Geach's analysis of (4) will not do; I mean to take issue with his claim to have derived it by "parity of reasoning" with his analysis of (i), and I shall discuss in this connection a certain rather neglected point of English grammar. This discussion will, however, lead to the formulation of a further question about the use of variables with words like "water," and it is this question which will be my principal concern.

3. Let us assume, then, that (5) and (6) are not equivalent to (4). For we need not go beyond the bounds of the "accepted view" to be guaranteed a pair of propositions which are equivalent to (4) . By "parity of reasoning" with the analysis of (i) as (2) or (3) we are no less justified in analyzing (4) as (5) or (6) than in analyzing (4) as

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(5') Something (or other) is some water, and Heraclitus bathed in it yesterday, and Heraclitus bathed in it today,

or

(6') For some x, x is some water, and Heraclitus bathed in x yesterday, and Heraclitus bathed in x today.

Whatever else may be said about them, (5') and (6') do not suffer from the defects of (5) and (6). The difference between the two pairs of propositions is just comparable to the difference between

(4a) Heraclitus bathed in some water yesterday and bathed in it (again) today,

which would normally be supposed to be equivalent to (4), and (4b) Heraclitus bathed in water yesterday and bathed in it

(again) today,

which would not. Thus it is false that Geach's analyses of (i) and (4) stand or fall together. We can retain the analysis of (i) and repair that of (4) by simply adopting (5') and (6').

Now if this point is not perfectly clear, it should be noticed that the equivalence of (4a), (i'), and (6') hangs on the function of "some") as an article; for the schema "some A" represents two quite different sorts of phrases, and it is possible to read these sentences in two ways. Indeed, it is worth emphasizing that "some" has a function distinct from that which it has in (i) or (3), since Geach and traditional grammar seem to have overlooked it.

We can suppose-not very plausibly-that "some water" in the formulation of (4a) is to be understood in a way analogous to that in which it is most naturally understood in, for example,

Some water tastes worse than Detroit water.

(4a) would then follow from (i); that is, it would follow from the fact that Heraclitus bathed in the water of a certain river and bathed in it again-and it would be a proposition weaker than (4). But plainly we would then also have a way of understanding the sentence in which (4) is formulated, which is no more im- plausible and which is such that (4) too is weaker than the notorious proposition Geach intends.

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It will be helpful to consider a sentence which is more obviously susceptible of alternative interpretations than is (4a); for example,

(9) Mary served some wine with dinner yesterday and served it (again) with dinner today.

(9) may formulate a proposition according to which Mary served a certain bottle or assortment of bottles of wine two days in succession; or, with equal plausibility, it can be supposed to formulate a proposition which requires only that Mary have served a certain wine-say, a certain rose-though she served different bottles of it, on the two days.4

Geach calls attention to the fact that the pronoun "it" in the formulation of a proposition like (4b) does not bear the burden of the phrase "the same water." Thus the trouble with (5) and (6) is that " 'Being the same water' cannot be analyzed as 'being the same (something-or-other) and being water.' "5 But if we suppose that "it" in (9) is replaced by "the same wine," the result on either interpretation of (g) is a proposition entailed by it. That is, given either way of understanding (9), it entails a proposition formulable as

(9a) Mary served some wine with dinner yesterday and served the same wine with dinner today.

The only question might be, in fact, whether, given one way of understanding "some wine," there is not a way of understanding

4 It is perhaps also possible to ignore "some" altogether or to take "it" to mean "some"; that is, it might be possible to take (g) to be equivalent to

Mary served wine with dinner yesterday and served it again with dinner today,

or to Mary served some wine with dinner yesterday and served some with dinner again today.

This would be especially tempting if (g) formulated some wildly unlikely possibility-as would be the case, for example, if "served" were replaced by "drank" and "some" read with weak stress. But I think that such an understanding only amounts to putting an interpretation on these words or taking something to be so "loosely speaking." Of course, it may be acknowledged that "we are not all (terribly or sufficiently) strictly brought up" (J. L. Austin, "Other Minds," Logic and Language, 2nd series, ed. by A. G. N. Flew [New York, I953], p. I24).

5 Geach, loc. cit.

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(9a) such that (9) does not follow-whether, that is, the phrase "the same wine" bears the burden of "it." What is important here, of course, is that phrases of the form "the same so-and-so" are not pronouns. In (9), as in (4a) and (4b), what the pronoun "it" can bear, since it is a pronoun, depends upon its antecedent.

The ambiguity of (g) and of (4a), in so far as (4a) is ambiguous, is one which can very often be resolved by paying attention to stress. If, in (9), "some" is read with weak stress, it will be understood in the former sort of way described above; if it is read with the same stress as "wine," then it will be understood in some such way as the latter. The implausibility of reading (4a) in any other than the obvious way is reflected in the oddity of stressing "some" and the first syllable of "water" alike-as it is natural to do in

Some water tastes worse than Detroit water.

Stress is, in fact, such a reliable mark of what is plainly a quite general distinction between uses of "some" that some modern grammarians distinguish two words "some," which, like objectt" and "object," have little more in common than their spelling.6

In (5') and (6') "some water" will, of course, be read with emphasis, but the use there of "some" is the same as its use in (4a) when (4a) is understood in the obvious way. So used, "some" is naturally called an "indefinite article," for it occupies a position which bears a widespread grammatical analogy to those positions occupied by "a" or "an.'7 When "some" is used in this way the

6 See, e.g., H. A. Gleason Jr., An Introduction to Descriptive Linguistics (New York, i96i), esp. pp. 224-226.

7 Thus in Did Heraclitus bathe in a river?

and Did Heraclitus bathe in some water?

"a" and "some" can alike be replaced by "any," "the," "his," "no," as well as "some" and "the same"; and they share with these words some requirements as to order in the noun phrases in which they respectively occur. For example, they precede but are not preceded by descriptive adjectives like "pretty" or "cool," they succeed but are not succeeded by words like "just" and "only," in their nondescriptive senses, and by phrases like "a picture of." On the other hand they alike differ from "any," "all," "s6me" in that they cannot precede "of" in sentences like

He gave me all of some water he had brought.

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schema "some A" like the schema "all A," is such that the result of replacing "A" by a noun is a singular form only if the noun there used is a "mass noun"-a noun which lacks a plural form; otherwise it is a "count noun." If the result of replacing "A" by a noun is singular, then the noun there used cannot, without a change in sense, be preceded by "a" or "an." We may speak of some rivers and some bathtubs as well as some water and some soap; but "some river" and "some bathtub," (/ s'm + river / and / sam + bdoetab /), are ungrammatical unless, in a special context, "river" or "bathtub" is given a sense in which it is a mass noun.8

Thus "some" preceding "water" in (4a), (5'), and (6') is meant to be the indefinite article common and peculiar to mass nouns and count nouns in the plural. It is there by analogy with the "a" preceding "river" in (2) and (3). Although grammar does not require it-as grammar does require the "a" in (2) and (3)- "some" surely is needed in (5') and (6'); and to suppose that the "accepted view" requires it be deleted is to make a straw man of that view.

4. Again, however, there remain some puzzling questions about propositions like (6'); and the question I want to raise can be put by first considering an objection someone might make to supposing that this proposition is after all an improvement on (6). This is an objection on the part of someone who may well agree that (4) is equivalent to (4a) and that (5') is equivalent to both. For the sentence in which (5') is formulated is awkward, but in a familiar way; it is at any rate no worse than (2). And if we consider the schema,

8 It is plain that even where words are used normally, the classification is not exclusive. But if a word is both a count noun and a mass noun, there is a more or less obvious shift in sense from one use to the other. Thus, "wood" in "some wood" is quite different from "wood" in "a wood" (e.g., mahogany); and the shift from "some lamb" to "some lambs" serves to explain the ambiguity in "Mary had a little lamb," to borrow Quine's example (Word and Object [New York and London, i960], p. 9i). The shift is perhaps less obvious from "some rope" to "some ropes," since these two phrases might in some context be used interchangeably. But so might "some rope" and "a rope"; and since "some ropes" and "a rope" cannot, "some rope" cannot be supposed in general to share the use of either of these phrases. Further, two uses of "rope" are surely needed, as with "lamb," to explain the ambiguity in "a little rope" (the example is Norman Kretzmann's).

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x is some water, and Heraclitus bathed in x yesterday, and Heraclitus bathed in x today,

it is not at all hard to think of noun phrases which, substituted for "x," result in comparatively straightforward sentences of English: "this," "that," "that river water," "what Heraclitus bathed in." (6'), however, is formulated in a sentence and not in a schema for sentences; x is there a variable and not merely a schematic letter. And if (6') is to follow from a proposition formulated in a way that fits the schema above, we must have not only an appropriate nominal substituend of x, but a value which it has just in case that proposition is true.9

Suppose that (4) is false; suppose, that is, that Heraclitus bathed in some water yesterday and today, but not the same water. It will then be the case for some suitable substituends of x andy that

(a) x is some water, and Heraclitus bathed in x yesterday,

and

(b) y is some water, and Heraclitus bathed iny today,

and

(c) x :y.

This last conjunct might be, for example,

This water :A that water.

Now it is crucial to the notion of logical identity that if, in the formula (c), x andy are variables, their replacement results in the formulation of a true proposition just in case they are thereby assigned exactly two values, whatever these values might be- rivers, men, objects, and so forth. But in the case in question it is fair to ask what two values have been assigned; for, so far, this water and that water are simply some water and not anything of which we can say there are two. And, given that we cannot say what their values are, it may be doubted that x andy have been

9 On the importance of distinguishing the values of variables from their substituends, see, e.g., Quine's Mathematical Logic, ch. i, sec. 6.

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assigned values at all. By the same token, then, if (4) is true and we have the truth of some proposition of the form of (a) and (b) conjoined with

(d) x y,

it is again not clear that (6') must be true: that is, it is not clear that (6') is equivalent to (4).

5. It will be said in response to this objection that if it is true that

This water # that water,

then to say that the values of x and y for these substituends are two is just to put the matter another way. Indeed, it is just here that one is tempted to depart from normal usage in Quine's manner; for surely it is plausible to answer the question, "Two what?" in the sort of case in question, by speaking as Quine does of "a water" and "two waters"-simply inventing a count noun if one must, since the language does not provide one. Since variables do not occur in ordinary English, and specification of their values is not an ordinary procedure, it is perhaps not surprising that the language might sometimes lack a word which will serve. It is reasonable in any case to say that where x is replaced by "yesterday's bath water" andy is replaced by "today's bath water," their values are one or two things or objects or individuals.

But, given (4) alone, one would not have supposed that Heraclitus bathed in a thing-still less that he bathed in an object or an individual. If any of these words applies, it does not do so in any ordinary way; and simply to insist without any further explanation that some one of them does apply would be to beg the point at issue.

How is further explanation to proceed? This is the question I want to consider. We need to explain the use of some count noun which will serve to specify the values of x for which (6') is true. More generally, we need to explain the use of a count noun which will apply to whatever can properly be said to be some water.

6. Quine says that when words like "water" and "sugar" are preceded by "same" or "a," "some special individuating standard is understood from the circumstances," and this remark suggests

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an answer to the question. We can suppose that "some special individuating standard" is some standard according to which an ordinary count noun applies-or an ordinary phrase with a count noun as head-or, at any rate, some word or phrase which is no more extraordinary than the language of chemistry; a standard, then, which is "special" only in that it may differ from one set of circumstances to another. And Quine says, again, "Typically, 'same sugar' might allude to sameness of shipment," and "a water" in the case of Heraclitus is "an aggregate only of molecules that were near a man when he once bathed." We can suppose that "same sugar" might, under appropriate circumstances, allude to sameness of lump or grain, and "a water" might be a puddle or tubful or gallon. But if we can assume that in any set of circumstances some one or another of these words or phrases applies, we can explain the use of "a sugar" and "a water" by saying that they simply stand in for whatever such word or phrase will do in a particular case.

Now, questions about variables aside, it is wholly plausible to say that in order to understand, for example,

Mary wiped up some water,

we rely upon the individuating standards of what, in addition to being some water, is a drop or puddle, or, in general, something that can be wiped up but not, say, poured or, as with (4a), bathed in. And if we agree that "same water" or "same sugar" is to be understood in such a way that

x is the same water (or sugar) as y if and only if x is some water (or sugar) and y is some water (or sugar) and x y,

then it is also plausible to say that if x andy are variables, their range of values has in it just such things as puddles, drops, and tubfuls of water; or shipments, grains, and lumps of sugar. Thus we have a view which, whether Quine actually holds it or not, is sufficiently tempting to merit consideration here.10 If we say that

10 Quine's discussion of mass terms in Word and Object suggests that he does hold such a view (see esp. p. 9i). But in light of his view that " 'water' and 'sugar' in the role of general terms, are true of each part of the world's water or sugar, down to single molecules but not to atoms" (p. 98), I am not sure.

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anything in the range of values of x and y is "a water" or "a sugar," we may say in explanation of these phrases that they are to be applied according to the individuating standards of all of these various things.

For a number of reasons, however, this view seems to me to be wrong. For one thing, it seems plain that in most circumstances more than one of the required sort of individuating standards will be applicable to what one would have supposed is the same thing. If, in a certain case, we have the same sugar only if the same shipment, then presumably we must also have at least a large subset of the same set of grains of sugar; and if what we have is lump sugar, we must have the same set of lumps as well-or, again, at least a large subset of that set. How, then, are we to decide which of these is, in this case, "a sugar"? It cannot be all of them, since the two sets must, at any rate, be distinguished; and there need be nothing in the circumstances which requires that we pick one rather than another. The sugar-that which is some sugar-can be said to be made up of or constituted by both the lumps and the grains, and it is in the shipment. But we cannot speak in this way of a sugar without beginning again to explain a use whereby we need not after all depend solely on the individuating standards of shipment, grains, or lumps.

This point can be seen in another way if we suppose that the sugar in question is melted-so that there no longer is a set of grains or lumps; or by consideration of the fact that the water Mary wiped up might have been spilled. A puddle of water is not the sort of thing which can be spilled; and what can be spilled- say, a glass of water-is not the sort of thing one can wipe up.

But this sort of consideration suggests that, in the second place, there might be circumstances in which no individuating standard of the required kind is applicable. It is obviously a contingent matter whether some sugar constitutes a shipment or is made up

Strawson seems to hold such a view in the later sections of Individuals and in "Particular and General" (Proceedings of the Aristotelian Society, LIV [I953-

19541), where he says, "The general question of the criteria of distinctness and identity of individual instances of snow or gold cannot be raised or, if raised, be satisfactorily answered. We have to wait until we know whether we are talking of veins, pieces, or quantities of gold, or offalls, drifts or expanses of snow."

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of grains, or whether some water is a puddle or glass of water. And although it is perhaps less obvious that bath water might not be made up of some aggregate of molecules or other, the molecular structure of water was, after all, a matter of empirical discovery. Furthermore, even if we agree that water cannot fail to be liquid H20, some of it could, on different occasions, be constituted by different aggregates of molecules. This would be very likely if it were instead sulfuric acid which had been dissolved and then reclaimed-since molecules of acid become dissociated in solution and presumably reassociate in new ways. It might be said that the aggregate in such a case is nonetheless the same; but if we are agreed that the molecules are different and it is an aggregate of molecules, then it is hard to see how "aggregate" is being used, unless according to that standard whereby we have, in any case, the same acid. To insist without further explanation that the word applies would be again to beg the question; for, so far, "an aggregate of molecules of acid" is just "an acid," and we have yet to be told what this means.

But it would seem to be a contingent matter whether, given any ordinary word or phrase of the required kind, its individuating standard will apply where what we have is some acid or water or sugar. And although it does not, of course, follow from this that there might be a case in which no such word or phrase is applicable, it is not at all obvious that there could not be such a case.

7. Finally, there is, in the suggestion that we rely upon various individuating standards, the further suggestion that some water or sugar might, in various circumstances, fail to be the same water or sugar-that there is no way of telling, apart from a particular context, whether some water or sugar is or is not to be counted the same. Such is the case, for example, with words; and we might compare the question whether "some" as it occurs in the formulation of (i) and as it occurs in that of (4a) is the same (ambiguous) word or two different words. There is no general way of saying; we need to know in a particular case whether the phonemic difference is to be taken into account-and of course there might be other considerations as well.

If we consider the mass noun "work" there is a similar flexibility in the way in which questions of identity are to be decided. If

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people are to get equal pay for equal work, it may be that time is to be taken into account; but it may be that the same work can be done by one man in less time than by another. The same work may have to be equally hard; but it might also be the case that the same work can be done by one man with greater ease than by another. It may be that the accomplishment must be the same; but it is also possible for someone to do the same work and accomplish less.

Other mass nouns are even more loosely tied to any general questions of identity; and in some cases there is-as Quine suggests in the case of "water" and "sugar"-something suspicious involved in the use of the word "same." Thus,

He showed some intelligence on that occasion (he did not show the same intelligence today).

When the drug wears off you may have some pain (the same pain Mrs. Jones had after her operation).

Let us hear some opinion on the subject (the same opinion that was heard last night).

From where I sit, I can see some blue (can you see the same blue ?).

It will not do, however, to suppose that mass nouns generally are distinguished from count nouns with respect to this sort of com- plication. We have the word "word" to compare with "work," and there are other cases which, like those just cited, are even worse. Compare, for example, any of the count nouns which would appear in the above sentences if an indefinite article were put in place of "some," or compare those which occur in

There is a virtue in necessity, or

From where I sit I can see a number of things.

It seems altogether likely that there are as many complicated cases among count nouns as there are among mass nouns, and equally likely that mass nouns differ from one another in this respect as much as they do from any count noun.

Thus, in particular, "water" and "sugar" are very different

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from "work" in the generality and precision with which it is possible to deal with questions of identity. Indeed, the concept of water is among the most stable and precise that we have; it is hard to see how otherwise water-or sugar, for that matter- could be supposed to have a molecular or any other sort of dis- coverable constitution. There could be no such discovery about work unless the word "work" were given a special use. But neither could there be about words. To say that words (of English) are made up of letters is not to claim a discovery about their constitution, but to settle on a use of the word "word." And discovery of the phonemes or morphemes of which words are constituted would be impossible without first agreeing to use the word "word" in such a way that words are either phonemic or morphemic in their constitution. "Words are made up of morphemes" might therefore be a statement of policy about the word "word." But "water is made up of molecules" could hardly be such a statement.

Within the bounds of normal usage, some water could no more fail to be the same water than could, say, some people fail to be the same people. And if we suppose that a count noun is applicable wherever we have some water, we cannot suppose that it is applied according to various individuating standards.

8. But now what is perhaps most important in all of this is a certain logical priority which the question whether what is some water is the same water has over any question as to its identity as a tubful or aggregate of molecules or any other such thing that comes to mind. For in this there is the suggestion that it is in- appropriate to try to rely on the individuating standards according to which these words apply. Indeed, it is more plausible to return to the apparent equivalence of (4) and (6') and to add to this the hosts of other cases in which, without appearing to need any ordinary count noun, analyses involving variables seem quite unexceptionable. Thus, given

All water is liquid H20.

No water is left in the tub.

All the water in the tub is river water.

There isn't any water left in the tub.

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we may set out the (apparently) equivalent propositions:

Given any x, if x is some water, then x is liquid H20.

For no x is it the case that x is some water and x is left in the tub.

Given any x, if x is some water and x is in the tub, then x is river water.

It is not the case that there is an x such that x is some water and x is left in the tub.

These seem no more complicated than similar cases involving count nouns; and although they are all liable to the objection which was brought against supposing (6') to be equivalent to (4), the sheer number of otherwise innocent cases in which that objection is to serve seems more convincing than the suggestion that for every such case some special individuating standard is applicable.

9. In any case, however, there is an alternative to the view so far considered. Because of the number of cases like those just cited in which variables do seem to work, it is plausible to say that there must be objects which will serve as their values whether any ordinary count noun is applicable to them or not. But in addition it is possible to give an explanation of a count noun which will serve to specify those objects; in fact, it is possible to give something like a definition of such a word. For, given anything which can properly be said to be some water, there is something to which an ordinary count noun applies-namely, an amount of water-and the phrase "an amount of water" applies with all the stability and precision of the phrase "some water." Indeed, the relation which some water bears to some other water, if they are the same amount of water, looks for all the world like an equivalence relation whose field is all of those things which can appropriately be said to be some water or-by abstraction-an amount of water.

Let us return to the formulas (a), (b), and (d) of Section 4. It was there claimed that if the proposition (4) is true, then for some suitable substituends of x and y we shall have:

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(a) x is some water, and Heraclitus bathed in x yesterday,

(b) y is some water, and Heraclitus bathed iny today,

(d) x =y.

Suppose that for x we put "the water Heraclitus bathed in yesterday" and for y "the water Heraclitus bathed in today." Then

(d) will be

(e) The water Heraclitus bathed in yesterday the water Heraclitus bathed in today.

But consider (e). It seems to require that

(f) There is exactly one x such that x is some water, and Heraclitus bathed in x yesterday;

yet (e) might be true where (f) is surely false. Even if he took only one bath yesterday, Heraclitus bathed in most of what he bathed in as well as all of it; he bathed in all but a quart and all but a pint; and these things are surely distinct-all of the water he bathed in could not be less water than it is. What (e) requires is not (f) but

(g) There is exactly one x such that x is all of the water Heraclitus bathed in yesterday, and exactly oney such thaty is all of the water Heraclitus bathed in today, and x =y.

The values of the variables x andy in the formulas (a), (b), and (d) can only be all of that to which any appropriate substituend refers: (e) is equivalent to (g). And although (4) entails neither (e) nor (g)-given (4), after all, we have not even been told how many baths Heraclitus took-(e) and (g) entail (4); and (4) entails and indeed is equivalent to the proposition that there is (or at any rate was) something which is some water all of which Heraclitus bathed in yesterday and today. Herein lies the equivalence of (4) and (6').

But now it is just such things as the values of x andy for which (g) is true that can properly be supposed to be the same amount of water or not. If (g) is true, then all of the water Heraclitus bathed in yesterday is neither more nor less water than all of that

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in which he bathed today. All of today's bath water could not have been more or less water than it is. And today's bath water could not be the same amount of water as anything unless all of it were. More or less water than all of it would be a larger or smaller-hence different-amount. But such is the case for anything which can be supposed to be an amount of water. The relation which one thing bears to another if they are the same amount of water is thus reflexive in its field11; it seems obviously to be transitive and symmetric and, further, to include in its field everything which can be said to be some water.

By a sort of "principle of concretion," then, we may charac- terize the values of the variable x for which it is true that

x is some water

by saying that they are those things, each of which is capable of being the same amount of water as something distinct from it, and each of which must be the same amount of water as anything with which it is identical.12

io. Now this definition depends upon the fact that "an amount of water" is related to "some water" in much the way that "a number of people" is related to "some people." Although amounts are very different from numbers, the similarity involved can be seen if we again contrast such things as water and sugar with such things as work or words. Thus water and sugar are measurable. If, in a recipe, one is told to use a cup of sugar, it is clear enough

11 That is, it is such that if something bears that relation to anything, or anything bears that relation to it, then it bears that relation to itself. Of course this follows from the assumptions that the relation is transitive and symmetric, but I am supposing that any objection someone might have to these assumptions is dealt with in the foregoing argument.

12 Given some water, together with all of the things which can be said to be some or all of that water, and the relation which one of these bears to another just in case all of the former is the latter, we shall have either a model of Lesniewski's mereology (A. Tarski, "Foundations of the Geometry of Solids," Logic, Semantics, Metamathematics, trans. by J. H. Woodger [London, 1956]); or the calculus of individuals of Henry Leonard and Nelson Goodman, ("The Calculus of Individuals and Its Uses," Journal of Symbolic Logic, V [1940]); or a Boolean algebra, depending on whether one can suppose there is, among those things which are some of the water in question, something with the features of a null element.

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what is meant, and even if one does not know much about recipes, the direction can be followed with a good deal of precision-a precision which is obviously impossible if, without explanation, one is told to do, say, fifty cents' worth of work (if, as it has in effect already been noticed, people are to get equal pay for equal amounts of work). But without explanation it is similarly impossible to be precise about the number of words on this page. We need to be told how to count words in order to say whether there is the same number of words on this page as on the next; and we need to be told how to measure work in order to say whether, in a particular case, people have done the same amount of work. But we need no more be told how we are to measure sugar or water, in order to say whether one amount is the same as another, than we need to be told how to count people in order to compare numbers of them. If it is objected that the measurement of water is, after all, a more sophisticated procedure than the counting of people, we may compare instead the counting of, say, animal cells-which requires the use of a complicated machine-or the counting of pennies in penny rolls, which is done by weight.

It may be objected that the results of measurement vary in a way that the results of counting do not-according to the method employed and the conditions under which it is carried out; in different circumstances different results can with equal propriety be regarded as specifying a certain amount of water. Some water can be purified and thereby give us varying results as to its amount. But from none of this does it follow that what under one set of circumstances is the same amount of water might, under other circumstances, fail to be the same. Specification of it may be more or less precise, and it may vary in other ways. Sixteen ounces of water might, under certain conditions, measure seventeen ounces; some recipe books need correction for high altitudes. But none of this could be said if we did not suppose that something-specified differently according to conditions and procedures-remained the same.

Differences between the procedures of counting and of measuring are in fact beside the point. And the fact that such things as water or sugar can be more or less pure is only further testimony to the point at issue. To purify some water is to purify a certain amount

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of it; we have here only one more or less accurate measure of the precision with which that amount is to be specified. But we should not have such a measure at all if an amount of water did not retain its identity in the process of purification. Pure water is water and nothing else; if water is not pure, it has something in it which is not water. We have no such univocal sense of the words in which we may speak of "pure work" or "pure words"; and we have in neither case the corresponding measure of precision. It does not help our precision in specifying an amount of work to be told without explanation to measure work and nothing else; and neither does it help in the case of words to be told to count nothing but words.

i i. It follows from the fact that a word or some work might, under various circumstances, fail to be the same word or the same work, that the identity of numbers of words and amounts of work is subject to similar shifts from one set of circumstances to another. The identity of numbers of people does not shift in this way, and neither does the identity of amounts of water. But just as the same number of people might not be the same people, so the same amount of water might not be the same water; just as in the former case two things then bear a certain relation to one another, so in the latter case two things bear that relation to one another in virtue of which they may be said to be one amount of water. If there were nothing which could be said to be a value of x when- ever a proposition of the form

x is some water

is true, it would be very mysterious how, by abstraction, we nonetheless speak of amounts of water. It would seem, in fact, that we should have to suppose that a certain two-term relation could hold in a case where it lacks terms-or, worse yet, where it lacks one term.

Thus, finally, we may say that if proposition (4) is true, there is in the field of that relation which one thing bears to another, just in case they are the same amount of water, something which is a value of the variable x for which (6') is true. If (4) is false, and Heraclitus bathed in some water, but not the same water, twice, then there are two such things. Given that no ordinary

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count noun will do, it is perhaps a matter of indifference what we

call them. We might call them "waters"; but given the way in

which they are defined, it is appropriate to follow Russell in

calling them quantities of water-for although a good deal more

would be needed to explain the use of this word in general, it is a

defining characteristic of quantities of water at any rate, that "A

quantity is something which is capable of quantitive equality to

something else."''3 HELEN MORRIS CARTWRIGHT

Wayne State University

13 Bertrand Russell, The Principles of Mathematics, 2nd ed. (New York, I937),

p. 159.

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Heraclitus and the Bath Water