Phenomenal Sorites Paradoxes and Looking the Same

Phenomenal Sorites Paradoxes and Looking the Samedltc_1267 327..344

Rosanna Keefe†

Abstract

Taking a series of colour patches, starting with one that clearly looks red, and making each sosimilar in colour to the previous one that it looks the same as it, we appear to be able to show thata yellow patch looks red. I ask whether phenomenal sorites paradoxes, such as this, are subject toa unique kind of solution that is unavailable in relation to other sorites paradoxes. I argue that theydo not need such a solution, nor do they succumb to one. In particular, I reject the claim made byFara and Raffman that looks the same is a transitive relation, which would allow us to solvephenomenal sorites paradoxes by denying the possibility of the required kind of sorites series.

Take a large vat of bright red paint and use it to paint a patch of colour on a whitewall: that patch will, of course, look red. Then, add to the vat of red paint smalldrops of yellow one at a time, mixing well, and paint a new patch on the wall eachtime. Each patch will look the same as the previous one, for we are unable to detectthe minute change in the composition of the paint. Consider now the compellingprinciple that if x looks red and x and y look the same colour, then y looks red.Using this principle, we could run through our patches of colour and reach theabsurd conclusion that the final patch looks red even if we have added five timesmore yellow paint than the original red. We can call this a phenomenal soritesparadox: it is a sorites paradox based on an observational predicate whose appli-cability depends on how things look and it involves a principle about when twothings appear the same in the relevant way. Other phenomenal sorites will involvepredicates such as ‘sounds loud’ and ‘smells sweet’, based on relations of sound-ing the same or smelling the same that hold between members of the series.

This paper will consider and reject a proposed solution to this important classof paradoxes. According to this solution (see e.g. Fara 2001), relations such aslooks the same as are transitive, and this means that there could never be the kindof series needed for the phenomenal sorites.

1. Phenomenal sorites

One general form for a sorites for the predicate F starts with the two premises:

(1) Fa1

† Department of Philosophy, University of Sheffield, 45 Victoria Street, Sheffield S37QB, UK; Email: [email protected]

dialecticadialectica Vol. 65, N° 3 (2011), pp. 327–344

DOI: 10.1111/j.1746-8361.2011.01267.x

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(2) If Ryx then if Fx, Fy

and a series a1 . . . an such that Rai+1ai for all i. Both premises appear true, but forsome suitably large n, the putative conclusion

(3) Fan

seems false. For example, if F is ‘is tall’ then R could be ‘is one-hundredth of aninch shorter than’, with the series starting with a 7-foot man and ending with a4-foot one. Or if F is ‘looks red’ then R can be ‘looks the same colour as’ and theseries can be one such as that described at the beginning of the paper.

Phenomenal sorites fit nicely into this form. Fara (Ib., 907) gives the followingtwo defining features of phenomenal sorites: “(i) the occurrence of ‘looks the sameas’ (or ‘smells the same as’ etc.) in the antecedent of the inductive premiss; and (ii)the occurrence of an observational predicate . . . in the other constituents of theargument [i.e. as ‘F’ in the above formulation]”, where “a predicate is observa-tional just in case its applicability to an object . . . depends only on the way thatobject appears”. ‘Looks red’ is, then, observational, but ‘is tall’ probably isn’t (forsomeone can be tall but not look tall). Regarding the first condition, we will notlimit ourselves to paradoxes involving the looks the same relation (or the equiva-lent for the other sensory modalities), but will be interested, for example, in serieswhere each member counts as ‘indiscriminable from’ or ‘indiscernible from’ thenext. I will talk of ‘matching relations’ to refer to all such relations, whoseapplicability to a pair of things depends on how those things look.

Do phenomenal sorites and non-phenomenal sorites call for distinct types ofsolution? On most theories of vagueness, the solution to the sorites paradoxinvolves denying that the inductive premise is true. So, for example, according toepistemicism and supervaluationism, it is false that ‘if x is tall and y is one-hundredth of an inch shorter, then y is also tall’, either because it has an unknow-able false instance (as according to epistemicism) or because it comes out false onevery way of making ‘tall’ precise (as according to supervaluationism). The denialof such a compelling principle for ‘tall’ is a prima facie implausible feature of atheory, and one that needs explanation: this is a task that the epistemic view andsupervaluationist tackle in different ways.1 But is it even less acceptable to denythe parallel inductive premise of a phenomenal sorites?

1 For defences of epistemicism and supervaluationism see Williamson (1994) and Keefe(2000) respectively. Williamson’s solution to the sorites paradox involves a detailed account ofwhy we are ignorant of the boundary of a vague predicate (i.e. ignorant of the false instance ofthe sorites premise). We cannot distinguish between our actual situation and a possible situationwith a different false instance, he argues. The supervaluationist’s theory of vagueness declares avague sentence true iff it is true on all ways of making it precise – an account that preserves thetruth of compelling ‘penumbral connections’ such as ‘this blob is red or pink’, said of a borderlinered–pink blob. Every way of making ‘tall’ precise renders some instance of the sorites paradox

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Let us see what Fara (Ib.) says about this, considering the following principle:

(I) For all x and y, if x and y look the same and x looks red, then y looks red.

Fara has described it as a truism that if two things look the same in respect ofcolour then if one looks red so does the other, and she defends the contrast with thecorresponding non-phenomenal principles. She writes,

If two men differ in height by even one-hundredth of an inch, then they differ in arespect that is relevant for the applicability of ‘tall’. But if two colour patches lookthe same (not just similar, but the same) in respect of colour, then they do not differ,on the face of it at least, in any respect relevant for the applicability of ‘looks red’(Ib., 908).

I would deny the second half of the claim: the physical constitution of somethingis certainly relevant to the application of ‘looks red’ and this may vary slightly fora pair of things that nonetheless look the same. This may not seem enough of adifference to affect the applicability of ‘looks red’, but nor does one-hundredth ofan inch seem enough to affect the applicability of ‘is tall’: that is the force of thesorites paradox.

Next, take a second quotation from Fara: “someone who sincerely claimed thattwo colour patches looked the same and yet that one looked red and the othernot . . . would not merely seem to be plainly mistaken, but also to be in a state ofconfusion” (Ib., 909). Now, even if true, this does not threaten either supervalu-ationists or epistemicists – the two major theories of vagueness that deny the truthof the inductive premise of sorites paradoxes. For they can both maintain that evenif (I) is false, we could never be in a position to locate a false instance, eitherbecause there is no such unique instance (as according to supervaluationism) orbecause such an instance is unknowable (as according to the epistemic view). Boththeories can acknowledge that anyone who claimed to identify a sharp boundary to‘looks red’ that fell between two things that look the same to them would not countas a competent user of that phrase. There need be no contrast here with predicatessuch as ‘is bald’ or ‘is a tadpole’ that are, we are assuming, not observational: ifanyone claimed to identify a specific instance falsifying the inductive premise,they would generally appear to be either misunderstanding the expression orperhaps stipulating a boundary for present purposes.2

inductive premise false. So, since that generalized premise is false on all ways of making thelanguage precise, it comes out false simpliciter without commitment to a unique false instance.

2 Consider also Dummett’s objection (1975, 264) that if a looks red and b looks the sameas a, b cannot fail to look red, because I could not determine by looking that it does not look red,if it looks the same as a (based on the observationality of ‘looks red’). At best this shows that (I)cannot have a false instance. If, as the supervaluationist maintains, (I) is false but there is no falseinstance, then nothing will defy the principle that we can tell by looking whether the predicateapplies.

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Finally, can we show (I) to be true purely in virtue of the logical features of‘looks the same’? Suppose ‘looks the same’ amounts to ‘has the same look’, wherethis commits us to something which is ‘the look’ of the thing, which is the same forany two things that look the same and which may or may not be red. Then, theantecedent of (I) implies that there is something which is the look of x and thesame thing is the look of y, in which case the consequent of (I) follows trivially,amounting to the claim that if the look of x (= the look of y) is red then that samething is red.

So, there appears to be an interpretation of (I) on which it is guaranteed to betrue. But that interpretation may be problematic in various ways and may, forexample, commit us to an incoherent notion of ‘the look’.3 Additionally, it mightnot be a reading which is appropriate for a sorites paradox if, for example, it relieson a notion of ‘the same look’ for which there are no non-trivial instances (i.e.more than just cases where x has the same look as y because x = y).

On the reading in question here, then, ‘look’ functions as a noun in the scopeof ‘the same’ – as ‘has the same look’ – and logical properties of ‘looks the same’follow from logical properties of ‘the same’. On the alternative, more natural,reading, ‘the same’ is in the scope of ‘look’, so that ‘looking the same’ is a matterof appearance of sameness (with ‘look’ functioning as a verb). Saying that A lookssquare does not mean that there is a thing – the look of A – which is square;similarly, saying that A and B look the same needn’t mean that A and B have looksthat are the same.

I will return, in section 7, to interpretations of ‘looks the same’ that commit usto something which is ‘the look’ of the thing. First, I note that even if we haveconcerns about talk of ‘the look’, we can still talk freely about ‘how a thing looks’.‘How a thing looks’ can be a matter of whether it looks red or looks square orwhatever. ‘Judging how something looks’ is shorthand for judging that it looks F(for some F). We may want to distinguish between the way something looks andthe way it is, but we can do this by distinguishing apparent properties – ones itseems to have – from real properties of the thing. So we do not need to see theseas properties of two different subjects – the thing and the look. Talk of howsomething looks is talk of apparent properties of that thing.

Having asked whether we are obliged to regard (I) as true, I turn in the nextsection to responses to the phenomenal sorites paradoxes that do accept the truthof (I).

3 See e.g. Peacocke (1981, 131) on the incoherence of a notion of ‘the look’, sometimesdescribed as ‘the observational shade’. I am not committing myself to the claim that there is nocoherent notion of ‘the look’. Rather, we cannot assume that the notion required for thisinterpretation of (I) is a coherent one. See section 7 for further discussion, and see Martin (2010)for a recent detailed discussion of ‘What’s in a Look?’.

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2. A new solution to phenomenal sorites?

Fara argues that there is an alternative response to the sorites paradox, which isavailable in the case of phenomenal sorites, though not in other cases (Ib., 908–909). This response is to deny that there could be a series of items that would allowthe phenomenal sorites argument to yield its absurd conclusion. Now, in the caseon which we have focused, such a series would consist of patches where eachadjacent pair matches, but the endpoints don’t, and this will mean that matches, (orlooks the same as etc.) would have to be non-transitive. If we could show that suchrelations are in fact transitive, we could then deny that any such series is possible.This is Fara’s strategy in her (2001).4 Such a position implies a significant differ-ence between the treatment of phenomenal sorites and the treatment of othersorites: there is no option of denying that there could be a series of men eachone-hundredth of an inch shorter than the previous one, for example. My aim is toquestion Fara’s case for the transitivity of matching relations with a view toshowing that the attempted distinctive treatment of phenomenal sorites cannot bemaintained: we should not expect a different solution from that of other sorites.

First, note one respect in which the range of sorites paradoxes that couldpossibly be solved by the suggested maneuver is narrow. As things are set up, it isonly directed at sorites built on observational predicates like ‘looks F’ (or ‘soundsF’ etc.), rather than ‘is F’ (e.g. ‘is red’ ‘is sour’, let alone ‘is tall’ or ‘is bald’). But,the strategy in question will not even help with all sorites involving these obser-vational predicates.

Take our original example of the vat of red paint to which yellow drops of paintare added one at a time. We formulated the sorites premise above by appealing tothe presumed fact that each patch will look the same as the previous one. But thesorites argument need not be formulated in this way: we can simply say, whenlooking at the series of patches, that if one patch looks red, then so will the next,or ‘if ai looks red then ai+1 looks red’, where the ai are the series of patches inquestion. This inductive premise is highly plausible in its own right, withoutexplicitly invoking the looks the same as relation. It might be claimed that actuallythe plausibility of this premise rests on the assumption that the consecutive pairsalways look the same. But I deny this. We might say of a pair ‘I don’t knowwhether they look (exactly) the same, but they are so similar that it is certainly thecase that if one looks red, so does the other’. We would accept the successiveconditionals corresponding to instances of the premise without first confirming

4 I focus, in what follows, on questions of whether patches look the same as each other.That may seem to ignore the disjunctive nature of Fara’s solution which leaves space for aresponse to a putative non-transitive series on which we deny that the patches are each homo-geneous. Since homogeneity is a matter of parts of the patch looking the same as each other, wecan, without loss, focus on features and behaviour of ‘looking the same’ and not explicitly discusshomogeneity.



that the relevant pairs look the same. It may be claimed that in such a series, somepair (or pairs) of patches may seem to look the same, but in fact do not; but thiswould not shake the conviction that the conditional ‘if ai looks red then ai+1 looksred’ holds of each pair. Denying the non-transitivity of looks the same as will thusnot help to solve this compelling sorites paradox on ‘looks red’, and the option ofdenying the possibility of the series is clearly unavailable since we have describedhow the series in constructed in terms of an unproblematic physical set-up.

This is just the first kind of case that cannot be solved by the suggestedmaneuver. As the paper progresses, we will see that it also fails for more and moreparadoxes seemingly of the right sort.

3. Types of matching relation

The alleged solution to the phenomenal sorites relies on a formulation of theparadox involving a particular kind of relation that holds between consecutivemembers of the relevant series. Let’s consider further what that relation must belike. As remarked above, various such relations give rise to a highly compellinginductive premise. For example, the premise may talk of ‘matching’, ‘looking thesame as’, ‘being indiscernible from’, ‘being indiscriminable from’ or ‘having thesame look as’ (interpreted in such a way as to commit us to something, perhaps aquale, that is the look of something). In this section, I explore some features theserelations may or may not have, with a view to clarifying which are suitable for aphenomenal sorites paradox and which, if any, are susceptible to Fara’s response.

Matching notions may be relativised to a person, amounting to ‘looks the sameto me’ or ‘is indiscriminable by Bob’, and use of this sort of relation might beencouraged given that, intuitively, what looks the same to one person may not lookthe same to another (perhaps even assuming something which is the look of a thing,we could acknowledge a different look for different people).5

We will also need to specify the respect in which the things do (or don’t) lookthe same, e.g. in respect of colour. For simplicity, we will focus on the example ofseries of colour patches of the same size and shape, so that there is no issue aboutlooking the same or different in other respects. Other aspects of context may alsoneed to be fixed; clearly, if something changes colour over time, we may want tosay that it is indiscriminable from another thing at one time but not at another. Therole of fixing context will be crucial to section 6 below.

5 Many philosophers have assumed (or argued) that ‘o looks F to S’ is semantically priorto ‘o looks F’; see e.g. Jackson (1977) and Tye (2002). Martin (2010), by contrast, offers anaccount which reverses this order of priority. I need not take a stance on this debate about thesemantics of ‘looks’: both sides accept the coherence of both the relativised and unrelativisednotions, though they differ on their account of the understanding of each and the relation betweenthem. Either assumption about the order of priority is compatible with my discussion.

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One way in which different matching relations may differ is in relation to theanswers to questions such as the following. If it seems to S that Rab (e.g. that a andb are indiscriminable), does it follow that Rab? And if it seems that not-Rab, doesit follow that not-Rab? There is a very natural notion of ‘looks the same as’ forwhich the answer to both these questions is ‘yes’: there is no gap between howthings look to someone and how they seem to look to them, so two things look thesame to someone iff they seem to look the same to that person. As we will see, thistype of notion is particularly important for phenomenal sorites paradoxes.

Some matching relations, on the other hand, may invalidate the first form ofreasoning. For example, if the look of something can outstrip what we know abouthow it looks, then perhaps it can seem to me that two things have the same look,when in fact they don’t. We’ll explore the idea of ‘the look’ further in section 7.

With ‘is indiscriminable from’, the answer to our second question must be ‘no’,for sometimes we make incorrect discriminations between physically identicalthings. If they are identical in all physical properties that ground colour, at most itcould merely seem as if I can discriminate between their colours. So, it doesn’tfollow from the fact that it seems that a and b are discriminable that they areactually discriminable.

Are all matching relations equally effective in a phenomenal sorites paradox?There are good reasons, I suggest, to require a relation that validates at least thefirst of our inferences, from seeming to look the same to looking the same. For theidea with the phenomenal sorites paradox is that on the basis of how the subjectjudges a pair from the series, he/she is driven to classify the next item the sameway as the previous one and is thereby driven through the series. If we can’ttypically tell whether the matching relation involved in the principle actuallyapplies to the pairs – it can seem to apply when in fact the pair doesn’t match –then we will not be in a position to apply the principle and move through the series.The phenomenal sorites paradox is compelling because it seems that we would bedriven through the relevant series on the basis of how the items seem to us – thepairs seeming the same will warrant the application of the premise – so a matchingrelation reflecting this idea is particularly important. In section 5, we will considerhow this kind of thought undermines the alleged solution to phenomenal soritesparadoxes under consideration. First, I appeal to differences between differentmatching relations sketched in this section in asking how far other considerationsabout perception rule out the transitivity of matching relations.

4. The limitations on our powers of discrimination

Many philosophers have assumed that there clearly are, or at least could be,phenomenal continua, and this is incompatible with the solution to the phenomenalsorites under consideration. For example, Wright says it is “familiar that we may



construct a series of suitable, homogeneously colored patches, in such a way as togive the impression of a smooth transition from red to orange, where each patch isindiscriminable in colour from those immediately next to it; it is the non-transitivity of indiscriminability which generates this possibility” (1975, 338–339). Similarly, with something that changes very gradually over time, it may wellbe that it looks the same at some time as at a second earlier, but that severalseconds later it does not look the same as at the original time.

Fara considers the suggestion that non-transitivity might follow from consid-erations about the limitations on our powers of discrimination – the thought that“physics is finer than the eye” (Travis 1985, 350). She uncovers some argumentsto this effect and rejects them. She offers two formulations of the claim about thelimitations on our powers of discrimination, rejects one, (a), as unacceptable andshows that the other, (b), does not yield the non-transitivity of matching. I willargue that her two formulations do not exhaust the possibilities, and that analternative principle that does yield the non-transitivity of various matching rela-tions is not subject to her objections. Her formulations are as follows (they governchange through time rather than differences between two things, but they could beeasily adapted for the latter role):

(a) For some sufficiently slight amount of change [in colour, or in thephysical basis of colour], when we perceive an object for the entirety ofan interval during which it changes by less than that amount, we perceiveit as not having changed at all during that interval.

(b) For some sufficiently slight amount of change [in colour], we cannotperceive an object as having changed by less than that amount unless weperceive it as not having changed at all (as having changed by zeroamount) (Fara 2001, 917).

The intended difference between these principles is that with (a) the limitation ison the level of difference needed for us to perceive a difference – below a certainlevel of change we will perceive no change. With (b), the claim is that we can’thave experiences of a change of certain tiny amounts – amounts below the limit.So in (a) the threshold concerns physical differences between things (the thresholdbelow which no difference is perceived), whereas in (b) the threshold is on howsmall a difference we can represent or experience the difference between things asbeing.

(b) does not yield non-transitivity in the way that (a) does. But (b) is anywayan odd principle, appealing as it does to the amount of perceived difference. (Theidea of a scale of quantity of apparent difference does not seem relevant: surely itis only significant whether or not there is perceived to be a difference, not thedegree to which things are perceived as different.) The reason (a) is rejected,however, does not give us reason to move from considering thresholds concerning

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the difference in things to a threshold concerning represented differences. Fararejects (a) because it implies that if two things are physically the same, then wewill perceive them as such, when in fact we can mistakenly judge that two thingsare the same colour. As Fara summarises, (a) is incompatible with “the fact that wehave certain very mild hallucinations” (Ib., 932). But Fara’s objection leaves openthe possibility that we replace (a) with a related principle that avoids Fara’sproblem, yet still supports the inference to non-transitivity.

We need to adapt the principle so that it allows for things not looking the sameeven when they are the same. By contrast, it is reasonable to say we can neverdiscriminate between them when they are the same. This suggests that by using anotion with the logical properties of ‘indiscriminable‘ rather than those of ‘looksthe same’, we may be able to preserve the inference to non-transitivity while notsuccumbing to Fara’s objection.6 The following principle is better placed for thetask:

(a*) For some sufficiently slight amount of change in colour, or in the physicalbasis of colour, when we perceive an object for the entirety of an intervalduring which it changes by less than that amount, we cannot successfullydiscriminate between the way it is before the change and the way it isafterwards.

This principle still guarantees the non-transitivity of its central relation. For we canmirror Fara’s argument (Ib., 918) showing that (a) would support non-transitivity:

(A) Suppose, for example, that whenever we perceive an object that hasgrown in height by some amount less than e, we cannot discriminate withrespect to height at the end of the growth as at the start. As long as anobject can grow by an amount less than e, and as long as we caneventually discern a change in height after repeated growths of amountsless than e, the non-transitivity of indiscriminable with respect to heightclearly follows. (Directly based on Fara (Ib.), with ‘cannot discriminate’replacing ‘looks to us the same’.)

At the least, then, this argument shows that the transitive matching relations cannotinclude ‘indiscriminable’, which further restricts the class of sorites that can besolved by Fara’s maneuver. But can we also argue from the non-transitivity ofindiscriminable to the non-transitivity of looks the same? Typically when twothings are indiscriminable for S, they do look the same to S. So, when the

6 In fact, it is more than the logical properties of ‘indiscriminable’ that are needed, since(a*) also rules out discrimination when the difference between x and y is below the threshold.This is accountable for by the fact that ‘discriminable’ is a success term – below the threshold anydifference cannot be detected.



difference between two things is below the crucial threshold of discrimination,they will typically look the same. We can thereby chain together pairs that look thesame and get end points that do not look the same. So, a non-transitive sequencefor indiscriminable will often also be a non-transitive sequence for looks the same.We can acknowledge that in some cases S is unable to discriminate between twothings although they do not look the same to S, so some non-transitive sequencesfor indiscriminable will not demonstrate the non-transitivity of looks the same; butas long as some sequences serve both purposes, we reach the desired conclusionfor looks the same.

This, however, still leaves open the possibility that there are no such sequencesfor looks the same as. There may be no a priori guarantee that successive stagesof our series look the same, because there are no circumstances under which wecan guarantee that two things look the same. But first, this should make ussuspicious about whether there can be an argument for transitivity either. Andsecond, if we can know by looking that pairs in a series do in fact look the same,as you might expect, then we will be home and dry. The defender of transitivitywill have to deny that we can know this: this feature of the required notion of looksthe same will feature prominently in what follows.

In this section I have argued that appealing to limitations on our powers ofdiscrimination can show the non-transitivity of certain matching relations andgives us good reasons to expect the non-transitivity of others.7 Fara’s response tophenomenal sorites paradoxes won’t work for the former, and I will go on to arguethat maintaining, against the above arguments, that ‘looks the same’ is transitivecomes at too high a price.

5. Maintaining transitivity

It seems to us as if there are series of things that demonstrate the non-transitivityof matching relations, and indeed that the series involved in phenomenal soritesare among such series. Take the example from the beginning of this paper in whichpatches are painted from a vat of paint as drops of yellow paint are added one ata time. The patches start out looking red and finish off looking a yellowy orange,and each patch looks the same as the previous one. What can the defender oftransitivity say about these apparently non-transitive series?8 In section 6, I willconsider responses that appeal to a change of context for the different compari-sons. In this section I focus on options that do not rely on claims of context change.

7 Hellie offered a detailed account of perceptual indiscriminability that accommodatesits non-transitivity by appeal to the “unavoidable presence of noise in perception” (2005, 489),which ensures inexactness in our colour perception.

8 The argument that follows depends on the assumption that there are series that at leastappear to demonstrate non-transitivity. I cannot show this a priori, but my claim is none theworse for that.

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The most obvious response to restore transitivity would be to say that, in everyapparently non-transitive series, there is in fact a non-matching consecutive pair.This saddles us with considerable error in our judgment of how things look to us:I think that consecutive pairs look the same to me, but in fact not all of them do.It drives a wedge between what seems to look the same to me and what in fact doeslook the same. And there will be a non-matching pair that I judge to matchwhenever there is apparent non-transitivity, which will occur very frequently alongthe sorites series. So, it is not simply that each sorites series from somethinglooking F to something looking not-F commits us to a pair about which we arewrong (as the epistemicist is committed to a single false instance of the soritespremise). Transitivity fails over and over again through small sections of the series,so on this view there will be many pairs about which we are wrong and we will noteven count as generally reliable when it comes to our judgments of when twothings look the same to us.9 I return in section 7 to this widespread fallibility aboutour judgments of when things look the same.

The matching relation most crucial to phenomenal sorites and the most naturalinterpretation of ‘looks the same’ is one for which looking the same cannot comeapart from seeming to look the same. But could someone object that there is in factno such coherent notion? I think this line is implausible, but anyway the problemwould surface again if we appealed instead to the idea of ‘seems to look the same’:when we judge of each consecutive pair of the series that they look the same aseach other, they surely at least seem to look the same. But if we allow thatconsecutive members of the series seem to look the same, then we can simply runthe phenomenal sorites with ‘seems to look the same’ instead of ‘looks the same’.The reformulated phenomenal sorites premise may then read (S) if x seems to lookred and x seems to look the same as y, then y seems to look red. Clearly thispremise is highly plausible. Again, it appears that you can have series which, giventhis premise, lead us through from the indubitable to the absurd. Taking the linedescribed above would require saying that there are pairs that apparently seem to

9 Fara says that she is focusing on “a sense of ‘looks’ that carries no explicit epistemo-logical implications, so that to hold that a person does or could know everything about the waythings look to her, or even to hold that a person could have no false beliefs about the way thingslook to her, is to hold a substantive thesis” (2001, 910). Resisting the wedge between looking thesame and seeming to look the same does not amount to such a substantive thesis in full generality.It may be, for example, that we are not altogether reliable informants on whether something looksthe same to us as it did an hour ago: our memory of how things looked to us is far from infallible.So, Fara’s case – where she only notices that her friend’s hair is lighter than the previous day afterit is pointed out to her (Ib., 927–928) – is significantly different from the kind of judgmentinvolved in the sorites series. We can, of course, present the members of a sorites series so thatconsecutive members are adjacent and seen together. The proponent of the described view iscommitted to rampant error over our judgments about whether things look the same to us evenwhen things are presented in ideal conditions.



look the same but actually don’t. But this does not seem to be taking seriously theintended notion of seeming to look the same.

Of course, standard solutions to the sorites paradox will, as in other cases, haveto deny the truth of (S). But, take the supervaluationist solution: there is noinstance that is a counter-example to this principle. There is no a and b of whichit is true that a seems to look red, a seems to look the same as b, but b does notseem to look red. Rather, the vagueness of ‘seems to look red’ accounts for thefalsity of the premise: there is no unique point at which it suddenly becomes truethat the patch seems to look red and so we have to consider all the possible patchescorresponding to ways of making that notion precise.

We are left with an even narrower range of matching notions that may betransitive. Correspondingly, at best, denying the existence of the appropriatesorites series has provided a reply to a smaller class of sorites arguments and notthe paradoxes it was particularly important to solve (as we saw at the end of section3). In section 7, I will ask just how narrow the category of solved paradoxes is. Butfirst I will consider another option for the defender of transitivity – an appeal tochanges in context that occur between different judgments that different pairs lookthe same.

6. Transitivity and context

‘Taller than’ is a transitive relation. But Arnie can be taller than Barney in 1990,Barney taller than Carlie in 2000, while Arnie is not taller than Carlie in 2000.Intuitively, this fails to show that ‘taller than’ isn’t transitive, because the threecomparisons do not all take place in the same context. Similarly, Raffman con-siders the case where “A is indiscriminable from B in infrared light, B is indis-criminable from C in incandescent light, and A is discriminable from C in thenoonday sun” (2000, 161) and claims, plausibly, that this won’t work as a counter-example to the transitivity of indiscriminability.

A counter-example to the transitivity of R will be of the form

(C) Rab & Rbc & ¬Rac

The above cases suggest that there must be a single context in which all three ofthese conjuncts hold together.

Now, the way that something looks to us with respect to colour can change ifits surroundings change. There are striking cases where the background againstwhich a coloured square is set can make the square look darker or lighter orperhaps more green or more blue (see e.g. Hardin 1988, plate 2). Apparent colourmay even change as the subject focuses on something different within a scene.

So, even if we find a triplet of things where the subject judges the first andsecond to look the same, the second and third to look the same, but the first and

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third to look different, this does not serve to establish that looks the same isnon-transitive unless we can be confident that context hasn’t changed between thecomparisons.

This point could then be used in response to the phenomenal sorites as follows.Imagine going through the sorites series for ‘looks red’, and judging each con-secutive pair to look the same. Transitivity and the phenomenal sorites inductivepremise would compel you to say that everything looks red given that the first thingdoes. But, suppose you judge c to look the same as b, when b, after its comparisonwith a, has been judged to look red. This need not force you to say that b and c bothlook red; you can say that neither of them does. This isn’t to contradict yourassertion that b looked red when compared with a, for the look of b can changewith the context – change induced by comparing it with something different. Thisis the basic idea of Raffman’s defense of transitivity, and is also central to Fara’sdefence; see e.g. her (2001, 934). It can be traced back to Jackson and Pinkerton(1973), who sought to defend ‘sensory items’ (i.e. visual qualia) against attacks vianon-transitivity.

Is the simple fact that the middle term, b, is compared with different things inthe two comparisons enough to yield a different context? If so, this would meanthat there could never be a single context in which Rab & Rbc & ¬Rac. But then,nor could there be a context obeying transitivity in which Rab & Rbc & Rac buta � c (for when a � c, Rab and Rbc will always hold in different contexts) andthis renders transitivity a trivial matter. This cannot be the intended interpretation.

What matters for putative counterexamples to transitivity is not what we calldifferent contexts, but whether there is a relevant difference between the contextsin which the two comparisons occur. So the putative counterexamples are only metif the contexts in which Rab and Rbc are relevantly different. If the middle term,b, looks the same in the two comparisons, then why think that the change in whatit is compared to is significant?10

And I suggest that b will frequently seem to look the same, even though it iscompared with two different (though very similar) things; that, for one, is myexperience. Suppose the subject does judge that b looks the same when beingcompared with the patch on the left as it looks when it is compared with that onthe right. Can’t we trust them on this? Raffman leaves the burden of proof with thedefender of non-transitivity, pointing out the problems with guaranteeing that thepatches look the same throughout (2000, 163). But, as long as the subject thinksthey look the same throughout, this again would mean that their judgment of

10 Raffman says “If the argument for nontransitivity is to succeed . . . each patch mustlook the same in both of its comparisons” (2000, 162). In Fara’s terms, if b looks the same in thetwo comparisons, we have “license to carry over the ‘middle term’”, (2001, 913). These quotesboth strongly suggest that there is a coherent possibility of b looking the same in the two contextsof comparison.



whether two things look the same was not a reliable guide to whether they do lookthe same to that subject. That this is an unacceptable consequence for Raffmanshould be apparent from the following quote: “discriminatory judgment and phe-nomenology go hand in hand: stimuli look the same (different) to a subject in acontext just in case the subject would make a judgment of sameness (difference)were he to compare the stimuli in that context” (Ib., 158).11

Might Raffman reply that the fallibility I have diagnosed involves looking thesame across contexts, not within a single context? Maybe such fallibility is easierto swallow. But is that distinction sustainable? If you can compare two things (orone thing in two different circumstances), then there must be a context in whichyou are comparing them. (For example, for the cases where one stage is after theother, you will be comparing them in the context of the later stage.) And thatcomparison is the one that is problematic for Raffman. She may then say that thisis a new context, so it does not amount to comparing across contexts, so nor canit serve to challenge transitivity. But this is then to return to the trivial version ofdenying non-transitivity, by ruling that anything of the required form is impossibleregardless of how the things look in the various contexts.12

Certain substantial changes in environment can change how something looks,and we can notice that change, but if the small change made by comparing b withc rather than the (barely different) a has any effect on the colour of the patch, wecan (often) honestly say that it is an effect that we cannot see. Thus, as with thenon-contextualist way of maintaining transitivity discussed in the previous section,the contextualist response commits us to widespread fallibility about our ownjudgments of whether things look the same, or are indiscriminable, to us.

7. What a transitive matching relation would have to be like

We have reached some conclusions about what a transitive matching relationmust be like. First, it cannot be a notion of indiscriminability according to whichtwo things can never be discriminated when they are the same in the relevantrespects; otherwise we could show that the relation must be non-transitive (seesection 4 above). Second, the fact that someone, in ideal conditions and aftercareful consideration, always judges that a looks the same as b does not implythat a and b do look the same to her. Whether things look the same to me canbe (and often is) beyond my grasp. This feature was needed to ensure that series

11 Mills also argues that “perceptual inconstancy does not lie at the heart of the paradox”(2002, 388).

12 Fara (2000) and Raffman (1994) both, more generally, defend theories of vaguenessthat could broadly be called ‘contextualist’ and that seek to accommodate vagueness by appealto changes in context between different judgments; see e.g. Fara (2000) and Raffman (1994). Forsome objections to such accounts, see e.g. Stanley (2003) and Keefe (2007).

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in which we make judgments that appear to commit us to non-transitivity do notshow that there is non-transitivity. Let us consider this last feature in a littlemore detail.13

Fara’s ‘phenomenal sense of looks’ is supposed to be the sense used inobservational reports (2001, 910); but if two things seem to look the same, that willbe the content of our observational report, leaving no room for those two things infact to fail to look the same. If there is a gap between how things look and how theyseem to look, observational reports will line up with the latter and so the requirednotion of matching would need to line up with it too. The fallibilist notion neededfor the view in question is thus not what Fara meant. It is, as I argued earlier, alsonot the notion that best draws us down the sorites series, since our judgments ofconsecutive pairs looking the same does not ensure the applicability of the soritespremise or its corresponding inference about the next item in the series. Moreover,the assumption that we cannot be wrong about how things look to us madeparticularly striking the contrast between non-phenomenal sorites and othersorites. For example, you may allow surprising ignorance regarding an expressionsuch as ‘is tall’ – we are not quite the masters of such expressions that we thought– but deny it for expressions of which we cannot fail to be master since theirapplicability depends entirely on our judgment. If we turn out to be ignorant aboutapplicability of the phenomenal notions in question, then the standard solutionsmay be just as viable as in the non-phenomenal cases.

Returning to the notion required for transitivity: when do two things look thesame to me on this understanding, given that it is not simply when they seem tolook the same? Perhaps the answer is that whether they look the same depends onthe look of those things. Looking the same can amount to having the same look,where we needn’t assume that all aspects of the look of something are accessibleto us. But what does having the same look amount to? Is there a coherent notionof ‘the look’ to play the required role?

Recall that there is no guarantee that two things have the same look justbecause they are the same in the relevant physical respects: Fara used this in herrejection of a proposed argument for non-transitivity. So, for Fara at least, it cannotbe that the look of a thing is entirely determined by its physical properties and itssurroundings (e.g. viewing conditions).

Could the look be an apprehended qualitative property, i.e. the quale perceived?On this understanding, two things present the same qualia iff they look the same.Fara and Raffman both cite a particular argument against qualia or sense-data asanother reason why we should be interested in whether matching relations arenon-transitive. The argument appeals to the alleged non-transitivity of matching

13 See also Mills (2002) on the fallibility required in relation to the notions involved ina phenomenal sorites.



relations as follows (see e.g. Armstrong 1968). Suppose a, b and c constitute acounter-example to the transitivity of looks the same. Then if a and b look thesame, they present the same qualia, and if b also looks the same as c then all threeof a, b and c must present the same qualia, which is incompatible with a and c notlooking the same.

Some have thought that the above argument demonstrates that there can be noqualia. Fara and Raffman can respond that no such non-transitive triple of a, b andc is in fact possible because looks the same is transitive. They appeal to the aboveargument as a reason to place the burden of proof on those arguing againsttransitivity. But, with a clearer idea of what a transitive matching relation must belike, we can see that it is not the kind of relation to which the typical defender ofqualia is likely to appeal.

For example, typical characterizations of qualia have it that we are generallyright in our beliefs about our own qualia. This is in tension with our fallibilityabout how things look to us to which, I have argued, the defender of transitivity iscommitted. On the other hand, there are physicalist and/or functionalist charac-terizations that are compatible with the non-transitivity of indiscriminability (seee.g. Clark 1985), so those will not help our defender of transitivity.

One way to explain what it is to have the same look would be to adoptsomething like Goodman’s criterion of sameness of phenomenal shade: a and bhave the same phenomenal shade/look iff they match all the same things(Goodman 1951). This would explain why you cannot always tell whether a andb have the same look just by looking at a and b, for it may be that they appear tomatch, but that only one of them matches a third thing, c. For Goodman, then,looking the same (‘matching’ in his terminology) is not the same as having thesame look (or presenting the same quale). Although having the same look implieslooking the same, it is not implied by it. So, we could consider a sorites premiseinvolving a Goodmanian notion of ‘having the same look’ and then deny theexistence of the corresponding series. But this is, at best, not the most naturalformulation of phenomenal sorites paradox. A phenomenal sorites paradox willnot generally appeal to this notion but rather to the notion of matching on whichthis notion is built; no response has been given to that central case. And it cannotbe maintained that the non-transitive notion of matching is not coherent, for that isexactly the notion required to comprehend Goodman’s transitive notion (which isa matter of matching all the same things).

In short, if there is a coherent notion of the look or qualia, it may indeedsupport the transitive notion of ‘having the same look’. But, it is not the relationcentral to phenomenal sorites paradoxes. Fara herself, however, rejects Good-man’s distinction between ‘looks the same’ and ‘has the same look’, claiming tofind it ‘barely coherent’. But we should doubt whether there is any coherent notionof ‘the look’ that plays all the roles she needs.

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Conclusion

Denying non-transitivity and so the availability of a sorites series does not providea satisfactory solution to phenomenal sorites paradoxes. It is, I suggest, likesolving the paradox of the heap by considering a formulation of the inductivepremise such as ‘if we take one grain away from a heap without altering thearrangement of grains, then we are left with a heap’. This principle is highlycompelling, and is true because we cannot remove a grain without changing thearrangement of grains. No series can be constructed that chains together instancesof the premise and results in absurdity. But how significant is this when we still geta paradox of the heap using the highly compelling premise ‘if we take one grainaway from a heap (with minimal alteration of the arrangement of grains), then weare left with a heap’?

The transitivity solution to phenomenal sorites paradoxes that we have dis-cussed here implies that our proper response to a phenomenal sorites paradoxwould be to refuse to acknowledge that pairs exemplify the premise: ‘ah, theydon’t actually look the same even though they seem to (or even though I judge thatthey do, or some such), so it may be that one looks red and the other doesn’t’. Canthat really be an adequate response to the force of the argument that pulls youdown the series? You may feel as compelled to carry on down the series even afteracknowledging that some pairs do not literally fulfill some such demanding rela-tion. That suggests that the problem has to be pinned on the vagueness of ‘looksred’ and that one of the standard solutions to the paradox is required to get to theheart of the matter.14

The category of phenomenal sorites paradoxes is of particular interest becauseof the observationality of the terms involved. But, I have argued, there is no goodreason to think that phenomenal sorites and non-phenomenal sorites call fordifferent solutions.*

14 Such a solution will not be the end of the story in relation to the paradox for ‘looksred’, as further explanation may be needed of the vagueness of such a term. But that gap couldbe filled in a variety of ways. For example, suppose you adopt Martin’s account according towhich “looks statements are made true just by properties of objects that we need to appeal to inorder to explain the truth of sentences that are not explicitly looks sentences” (2010, 197) andsuch statements are to be understood in terms of comparisons with other things that we take tobe paradigm examples. Vagueness in ‘looks red’ can then naturally be attributed to vagueness inthe level of similarity required or the choice of paradigms; and, for example, someone seeking totreat vagueness as a species of ignorance can locate our ignorance in such matters withoutdenying our authoritative knowledge of our own psychological states.

* I’m very grateful to several anonymous referees for helpful comments and suggestionsand to Dominic Gregory for discussion of a draft of the paper. I am also very grateful to theAHRC, who funded a period of leave in which I worked on material for this paper and Iacknowledge the ‘Borderlineness and Tolerance’ project, of which I am a part (ref. FFI2010-16984, MICINN, funded by the Government of Spain).



References

Armstrong, D. M. 1968, A Materialist Theory of the Mind, London: Routledge.Clark, A. 1985, ‘A Physicalist Theory of Qualia’, The Monist 68, pp. 491–506.Dummett, M. 1975, ‘Wang’s Paradox’, Synthese 30, pp. 301–324.Fara, D. Graff 2000, ‘Shifting Sands: an Interest-Relative Theory of Vagueness’, Philosophical

Topics 28, pp. 45–81. (Originally published under the name ‘Delia Graff’.)Fara, D. Graff 2001, ‘Phenomenal Continua and the Sorites’, Mind 110, pp. 905–935. (Originally

published under the name ‘Delia Graff’.)Goodman, N. 1951, The Structure of Appearance, Cambridge, MA: Harvard University Press.Hardin, C. L. 1988, Color for Philosophers, Indianapolis: Hackett.Hellie, B. 2005, ‘Noise and Perceptual Indiscriminability’, Mind 114, pp. 481–508.Jackson, F. 1977, Perception: A Representative Theory, Cambridge: Cambridge University Press.Jackson, F. and Pinkerton, R. J. 1973, ‘On an Argument Against Sensory Items’, Mind 82, pp.

269–272.Keefe, R. 2000, Theories of Vagueness, Cambridge: Cambridge University Press.Keefe, R. 2007, ‘Vagueness Without Context Change’, Mind 116, pp. 275–292.Martin, M. G. F. 2010, ‘What’s in a Look?’ in: B. Nanay, ed., Perceiving the World, New York:

Oxford University Press, pp. 160–225.Mills, E. 2002, ‘Fallibility and the Phenomenal Sorites’, Nous 36, pp. 384–407.Peacocke, C. 1981, ‘Are Vague Predicates Incoherent?’, Synthese 46, pp. 121–141.Raffman, D. 1994, ‘Vagueness Without Paradox’, Philosophical Review 103, pp. 41–74.Raffman, D. 2000, ‘Is Perceptual Indiscriminability Nontransitive?’, Philosophical Topics 28, pp.

153–175.Stanley, J. 2003, ‘Context, interest-relativity and the sorites’, Analysis 63, pp. 269–280.Travis, C. 1985, ‘Vagueness, Observation and the Sorites’, Mind 94, pp. 345–366.Tye, M. 2002, Consciousness, Color and Content, Cambridge, MA: MIT Press.Williamson, T. 1994, Vagueness, London: Routledge.Wright, C. 1975, ‘On the Coherence of Vague Predicates’, Synthese 30, pp. 325–365.

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Phenomenal Sorites Paradoxes and Looking the Same