Resemblance, Universals and Sorites: Comments on March on Sorting out Sorites

Canadian Journal of Philosophy

Resemblance, Universals and Sorites: Comments on March on Sorting out SoritesAuthor(s): Fred WilsonSource: Canadian Journal of Philosophy, Vol. 17, No. 1 (Mar., 1987), pp. 175-184Published by: Canadian Journal of PhilosophyStable URL: http://www.jstor.org/stable/40231521 .

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CANADIAN JOURNAL OF PHILOSOPHY 175 Volume 17, Number 1, March 1987, pp. 175-184

Resemblance, Universals and Sorites: Comments on March on Sorting Out Sorites

FRED WILSON University College University of Toronto Toronto, ON Canada M5S 1A1

In a recent paper, Peter March proposes to sort out the traditional sorites paradox by distinguishing two senses of 'resemble/1 The paradox is generated in this way: we have the inference

(a) A is the same colour as B

(b) B is the same colour as C

Hence,

(c) A is the same colour as C

while also having

(d) A is not the same colour as C

The inference requires for its validity the transitivity of same colour as. March argues that in one sense of 'resembles,' (a), (b), and (d) can all be true, but we will not infer (c) since transitivity does not hold for a resemblance of this sort. In the other sense, 'resembles' is transitive, but in this case we will not have (d) true if (a) and (b) are true. The paradox arises because the two senses of 'resemble' are blurred together.

The two senses of 'resemblance' are defined by the evidence used to support their ascription:

Class I: ascription of (colour) resemblance to A and B is support- ed by evidence that the (colour) description of A and that

1 P. March, 'Sorting Out Sorites/ Canadian Journal of Philosophy 14 (1984), 445-54



176 Fred Wilson

of B (each description devised separately), jointly entail that A is the same colour as B.

Class II: ascription of (colour) resemblance to A and B is support- ed by evidence gained by placing A and B together, alongside each other, and observing directly whether they match (in respect of colour), that is, whether there is no notice- able (colour) difference between them.

Since matches is not transitive, Class II resemblance cannot be transitive. But Class I resemblance is, March holds, transitive.

March's discussion raises a number of interesting points, and in particular in its treatment of resemblance it raises some issues that touch on aspects of the traditional realism/nominalism controversy. In these comments I wish to take the discussion a few steps further through probing March's two senses of 'resembles' and exploring the connec- tions with the ontological problem of resemblance and universals. We shall come to the conclusion through in particular an examination of Class I resemblance that for the realist the question, What do two things that are the same shade of red have in common that makes them both red? should receive a very different sort of answer from the question, What do the several different shades of red have in common that make them all reds?

Consider two red spots - call them a and b - and assume that the same determinate shade of red is exemplified by each. The statement that

a is red

is true by virtue of the property, the red in a, that a exemplifies; and similarly for

b is red.

That is, the term 'red' correctly applies to a and b by virtue of the properties in a and b. But what is it about the red in a and the red in b that justifies applying the same term 'red' to a and b?2

Two answers have traditionally been given to this question. (1) The

2 Cf . F. Wilson, 'The Role of a Principle of Acquaintance in Ontology/ The Modern

Schoolman, 47 (1969), 37-56. Also 'Acquaintance, Ontology and Knowledge/ The New Scholasticism 54 (1970), 1-48; and Affability, Ontology and Method/ Philosophy Research Archives 9 (1983), 419-70.



Resemblance, Universals and Sorites: Comments on March on Sorting Out Sorites 177

realist answer: the red in a is literally the same as the red in b. On this view, the determinate property in a is a universal, and this same

property is also in b. (2) The nominalist answer: the red in a is different from the red in b, but the two reds (which, we must remember, are the same determinate shade) exactly resemble each other. On this view, the property in a is as particular as a itself; and what justifies saying that both it and the property in b are reds is the relation of exact resemblance.

March raises a problem that confronts both positions, provided one assumes that they both want to link the ontological notion of sameness or exact resemblance to the perceptual recognition that that relation obtains, after the fashion of anyone who adopts a Principle of

Acquaintance as a guiding principle in ontology. Let us say that the colour of a matches the colour of b just in case that, as shades of colour, they are not noticeably different. The realist assumes that sameness

among properties is transitive (since identity is transitive) and the nominalist assumes the same about exact resemblance (since that is what is meant by 'exact/ and since he aims to pick out as red exactly those

things that the realist picks out). But, March points out, Matches is not transitive.

More specifically, many properties come in families - colours, pitches, etc. - that can be ordered by a betweenness relation, or, what is logically equivalent, by a relation of less than which is asymmetrical and transitive. For some at least of such classes, the matching relation is not transitive. This means that properties on the continuum cannot be sorted perceptually into neat equality classes. The classes will be

jointly exhaustive of the field of matches but they will not be mutually exclusive. Thus, if we take the class of properties that match the colour of a and the class of properties that match the colour of c, where the colour of a does not match the colour of c, so that the two classes are not identical, nonetheless the colour of b might match both the colour of a and that of c, so that the classes are not mutually exclusive. The

'equivalence' classes defined by matching thus as it were merge into one another. As N.R. Campbell pointed out long ago,3 it is precisely this breakdown of transitivity of perceptual matching that is charac- teristic of the physical continuum. Be that as it may, however, it is clear that there is a problem for either the realist or the nominalist commit- ted to a Principle of Acquaintance, namely, how to generate, respec- tively, either a sameness or an exact resemblance relation which satisfies the principles of symmetry and transitivity, from matching, which does not satisfy the latter.

3 Cf. N.R. Campbell, The Foundations of Science (New York: Dover 1957), Ch. XVI.



178 Fred Wilson

There is also the problem of defining an order on the physical continuum. If less than or between are to satisfy the usual axioms then that presupposes that there is an equality relation that satisfies the axioms of symmetry and transitivity - which matches does not. As we shall see, this latter is crucial if March's 'nominal sameness' is to become a legitimate case of sameness, one satisfying the principles of symmetry and transitivity.

The way out of this difficulty was pointed out by Nelson Goodman,4 and it is available to both the realist and the nominalist. If we let 'S' stand for either sameness or exact resemblance and let /M/ stand for matches, then we can define:

xSy = Df(z) (xMz^yMz)

In particular, we will have

fS the colour of a = (g) (gMf sgM the colour of a)

It is easy to deduce that the relation S is not only symmetrical but also transitive, therefore satisfying the axioms of equality, so that the classes into which it sorts properties are genuinely equivalence classes in the sense of being not only jointly exhaustive of their field, which is also the field of matches, but also mutually exclusive.

March argues that a sorites argument based on resemblance can proceed from true premisses to a false conclusion. This is so if the resemblance in question is matches. But then the argument is unsound because it assumes as an implicit premiss the falsehood that resemblance = matches is transitive. This is a correct point, and one well worth making. One should also note, however, that if the Goodman device is used to restore transitivity, then one will not be able to move from true premisses to a false conclusion concerning what resembles what, that is, resembles exactly.

March suggests that there is a way other than Goodman's for introducing a transitive sense of 'resemblance' or 'sameness.' This is his 'Class F resemblance. This way begins by noting that, as we ordinari- ly speak, the term 'red' applies to several shades of colour, not just one determinate shade. 'Red' in this sense is so defined, March claims, that resemblance with respect to red is transitive. If the colour of a resembles

4 Cf. N. Goodman, The Structure of Appearance (Cambridge, MA: Harvard Univer-

sity Press 1951), 222ff. For a discussion of Goodman on matching, see A. Haus- man, 'Goodman's Ontology/ Ch. VI, in A. Hausman and F. Wilson, Carnap and Goodman: Two Formalists (The Hague: Nijhoff 1967).




that of b in this sense, and b that of c in this sense, then the colour of a resembles that of c: the transitivity that is built in assures us of this. For this reason, one can never have a sorites argument which will lead from true premisses to a false conclusion, since transitivity is guaranteed. On the other hand, it does not follow that two reds must match. The appearance of invalidity in the sorites arises by illegitimately shifting from 'resemblance with respect to red' to 'matching.'

But is March's second way of introducing a transitive resemblance or sameness relation - what he calls 'nominal' resemblance - really legitimate? This alternative proceeds by dividing colours into classes; the members of these classes are to be reckoned as the same or as exactly resembling. This is a feasible approach. An equality relation is symmetrical and transitive in its field. Any such equality relation can be used to define a set of equivalence classes that are jointly exhaustive of the field and mutually exclusive, where two things are members of the same equivalence class just in case that they stand in the equality relation to each other.5 One can also proceed conversely. If one has a set of classes that are jointly exhaustive and mutually exclusive, then these can be used to define an equality relation with the usual properties of symmetry and transitivity.

Let the field be F, and let the set C of subclasses Cv C2, ..., Cn of F be jointly exhaustive and mutually exclusive. Now define for a, beF

aSb = Df (3f) (fcC-acf-bef)

It is easy to prove that S is an equality relation, that is, symmetric and transitive, given that every member of F is in exactly one of the classes Cir that is, as we said, given that the classes C{ are mutually exclusive, and that the classes are jointly exhaustive of F. March's Class I or 'nominal' resemblance or sameness proposes to use this device. If successful then it will indeed be transitive, as March claims.

To define this concept of colour resemblance or sameness, one di- vides (chromatic) colours into reds, oranges, yellows, greens, blues, and

purples. Members of each class are reckoned to be equal, that is, the same - nominally the same, though, of course, things that are in this sense the same need not match. If 'nominal sameness' is to be any sense of 'sameness' then it must be, and March admits this, symmetrical and transitive. It will be symmetrical and transitive only if the classes used to define it are mutually exclusive and jointly exhaustive. However, one must recognize the existence of borderline cases or, as March puts it, the existence of 'odd couples.' Consider reds and oranges. Arrange

5 Cf. R. Carnap, Introduction to Symbolic Logic (New York: Dover 1956), 136ff.



180 Fred Wilson

them according to the matching relation. Not all reds will match, of course, nor all oranges. Yet we may proceed along steps of matching through the reds to the oranges and then on through the oranges. The crucial point is that there are samples on the borderline which one wants to reckon as both red and orange, i.e., as both red and not-red. This is not only paradoxical, but it means that the classes defining nominal sameness are not mutually exclusive. Hence, 'nominal sameness' is not a case of sameness. To the extent that 'nominal sameness' is not a case of sameness, it is not a legitimate alternative to the earlier (Class II) concept of sameness. But it can be made legitimate; that is, it can be made into a device that actually does succeed in defining an equality relation. The problem, in fact, is essentially the same as that which arose for matches and is to be resolved in the same way, by invoking Goodman's device.

Let us see. We assume that various shades of red are all reds. It is difficult to

say what they all have in common. For the realist, it is not difficult to say that all shades of colour have in common the generic property of being a colour. But unlike colour, the 'property' red, if it be a property, that gathers together the various shades of red, is hazy at least at the

edges, with shades of colour that one wants to say are reds but are not reds, too. If the property red that gathers together the various reds is a universal, then it is not a neat one. For this reason, it has often seemed that at this level, at least, the nominalist has an easier time. After all, it is possible for him to introduce over and above exact resemblance a relation of inexact resemblance: what all shades of red have in common is that they inexactly resemble one another. But in fact, this won't do, as Butchvarox has pointed out.6

The problem is that of degrees. There is a variety of inexact resem- blances. The reds inexactly resemble each other, but the reds also in-

exactly resemble oranges, more at least than they inexactly resemble blues. So there is not a single dyadic relation of inexact resemblance. If we stick with resemblance, we need a tetradic relation: 'x resembles

y more than v resembles w.' But this won't be either. The relation as- serted by the statement that 'x resembles y more than v resembles w' can only be understood as asserting a dyadic relation of difference in degree the terms of which are two separate instances of a dyadic relation of resemblance, the first instance having x and y as its terms and the second instance having v and x. Thus, inexact resemblance can't be a dyadic relation, and neither can it be triadic or tetradic. The nominalist's ap-

6 P. Butch varov, Resemblance and Identity (Bloomington, IN: University of Indiana Press 1966), 115ff.




peal to inexact resemblance is therefore not a real possibility. Thus, nominalism cannot provide a reasonable answer to the question, What do the several different shades of red have in common that make them all reds? Can the realist?

To answer this question we must recognize that the crucial point is that there are in fact degrees of resemblance. Or rather, there is an order among colours, and it is this order that permits arrangement of colours into classes of specific shades, where the members of such a class of shades are all between certain extremes. It is this being between certain extremes that all the shades of, say, red have in common, while the oranges and the blues have in common the feature of being between certain other extremes. It is this relation that is used to define the classes that are in turn used to define the notion of Class I or 'nominal' resemblance or sameness.

In greater detail, if Sj is a standard example of the lower extreme of shades of red and Sg is a standard example of the upper extreme of shades of red, then to say of a shade f that

f is a red

is to say that

f is inclusively between the colour of Sj and the colour of Sg

and to say of an individual x that

x is red

is to say that

there is an f such that x is f and f is a red.

What is common to all shades of red, then, is not a property or universal but rather that each is a constituent of a fact, and these facts share a common structure, to use Reinhardt Grossmann's term,7 which is not present in the corresponding fact for any other shade of colour. Thus, the answer to the question, What do two things that are the same shade of red have in common that makes them both red? has a very different answer from the question, What do the several different shades of red have in common that makes them all reds? For the former the answer

7 R. Grossman, Ontological Reduction (Bloomington, IN: University of Indiana Press 1973), 186ff.



182 Fred Wilson

is: they both exemplify one and the same common property; while for the latter question the anwer is: they are constituents in facts that share a common structure.

The problem of 'odd couples' or borderline cases is this. Suppose the shade is such that

f is a red

while the shade h is such that

h is an orange.

We assume that the colour of the sample is of the lower extreme of orange. Hence,

- (f matches h).

But it is entirely possible, given what we know about matches, that there is a shade of colour g such that

f matches g

g matches h.

Shall we say that

g is a red

or that

g is an orange

i.e., that

~ (g is a red)?

March is correct that one does not generate a sorites paradox here: one does not move through steps of nominal sameness from reds to oranges. But that is only because the shade g is not clearly placed by the structures defining reds and oranges in one class or the other. But it is, surely, the same problem as arose in the case of matching.

The purpose is to define a sense of 'sameness.' In order to do this one forms a set of classes that are mutually exclusive and jointly exhaustive. The case of the 'odd couples' shows that the color classes




we have defined are either not mutually exclusive or not jointly exhaustive. We use the relation of betweenness to define the classes. But we can have an order relation only if we presuppose an equality relation; for, only with an equality relation satisfying the standard axioms of transitivity, symmetry and reflexivity in the field can one assign members of the field to a set of classes that are mutually exclusive. That is, we can introduce March's 'nominal sameness' in terms of classes of individuals only if we have a deeper, underlying notion of sameness. What the 'odd couples' show is that matches cannot be this underlying equality relation. But we already know why this is so: matches is not transitive. The problem with nominal sameness is thus the same problem that permitted matches to generate a sorites paradox. And the solution is the same: Goodman's device.

Once that is done we will reckon the shade g to be the same as neither f nor h. So it is neither a red nor an orange. To make the classes we use to define 'nominal sameness' jointly exhaustive, we must now re- define either red or orange to include g - or, perhaps, introduce another class in between reds and oranges, say the reddish oranges. The point is, that once we use the Goodman device to determine basic sameness, we can then use betweenness to define equality classes that can in turn be used to define 'nominal sameness.'

Finally, let me comment on March's point that 'non-transitive resemblance' is comparative since it is based on what he calls 'Class II Evi- dence,' judgments in which the two relata are directly compared in a single perceptual experience, while 'nominal resemblance' is not comparative since it is determined by the application of predicates based on direct inspection of separately perceived instances. Now, it is cer- tainly true that in acts of perceiving the intention is often of the sort

This is red

which is to say,

There is a colour f such that this is f and f is a red.

Such complex facts are indeed given in perceptual experiences where at the same time the only particular presented is the this and the only colour presented is the colour of the this. In particular, the colours defining the extremes of red in the nominal sense are not given in those perceptual experiences. One can speculate that this is possible by virtue of information processing mechanisms built into the physiological ba- sis of perception. The this which is a shade of red emits light of a certain wave length which strikes the eye. If this radiation is between



184 Fred Wilson

certain extremes then the organism will perceive the complex fact that the this is red.

But now let us let 'red' denote a specific shade, say, the shade of sample Sr:

red = the (shade of) colour of Sr Then, to say that

This is red

is to say that There is a colour f such that this is f and f matches the colour of Sr.

It seems to me that there is no reason to suppose, as March does with his Class II resemblance, that both the this and Sr must be presented for one to judge that the colour of the this that one is perceiving matches the colour of Sr. To judge that this is red one need not have another patch present, the two 'alongside each other/ and 'observe directly' that they match. One can in fact 'directly observe' without any cons- cious comparison that this is red. All one requires is that the organism respond to the fact that the wave lengths emitted by this are the same as or sufficiently close to the wave lengths emitted by Sr. It would seem, then, that judgments of comparative or Class I sameness do not need to involve the perceptual presentation of more than one individual.

March has neatly analyzed a sorites argument involving sameness of colours. His two senses of 'sameness' bring out some important points. What I have argued is that the two senses can be cases of genuine sameness only if one invokes Goodman's device. I have also suggested that March's two accounts of sameness have interesting im- plications for the realism/nominalism issue. About some other points March raises issues on which I have not commented. Thus, for example, March proposes to extend his analysis of sorites to the case of con- cepts like human and like bald. Human involves several dimensions, and so similarity judgments are more complicated that in the case of colours. Bald brings in quantity, also absent from the case of colours. Whether the nice things that March does with the colour sorites can be extend- ed to these more complex cases is itself a complex issue. I will merely say that the shift from the simple to the complex cases is not as easy as March suggests it is. But be that as it may, March's discussion is important not only for the light it sheds on the sorites paradox but for the implication it forces one to recognize for the realism/nominalism set of issues.

Received July, 1985



Documents

Resemblance, Universals and Sorites: Comments on March on Sorting out Sorites