vagueness and uncertainty

Vagueness and Uncertainty

Andrew Bacon

June 17, 2009

ABSTRACT

In this thesis I investigate the behaviour of uncertainty about vague matters. Itis fairly common view that vagueness involves uncertainty of some sort. How-ever there are many fundamental questions about this kind of uncertainty thatare left open, questions I shall attempt to answer in this thesis. Could you begenuinely uncertain about p when there is no matter of fact whether p? Couldyou remain uncertain in a vague proposition, even if you knew exactly whichpossible world obtained? Should your degrees of belief be probabilistically co-herent? Should you beliefs in the vague be fixed by your beliefs in the precise?Could one in principle tell what credences a person has in the vague?

This thesis defends the view that typically one ought to be genuinely uncer-tain about matters one considers to be vague; uncertainty about vague mattersis no different in this regard from uncertainty about the future, the deep sea orfar away galaxies.

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2. The importance of uncertainty in theorising about vagueness . . . . . 82.1 Two kinds of supervaluationism . . . . . . . . . . . . . . . . . . . 82.2 Epistemicism and supervaluationism . . . . . . . . . . . . . . . . 10

3. Ersatz uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1 New psychological vocabulary . . . . . . . . . . . . . . . . . . . . 183.2 Mixed uncertainty part I . . . . . . . . . . . . . . . . . . . . . . . 193.3 Mixed uncertainty part II . . . . . . . . . . . . . . . . . . . . . . 213.4 What is genuine uncertainty? . . . . . . . . . . . . . . . . . . . . 23

4. Rejectionism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1 Strange consequences . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Decision theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Deontic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 Two kinds of expectation . . . . . . . . . . . . . . . . . . . . . . 30

5. Vagueness in epistemic attitudes . . . . . . . . . . . . . . . . . . . . . 315.1 Epistemic Sorites . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Indeterminacy and rational action . . . . . . . . . . . . . . . . . 325.3 Indeterminacy and rational obligation . . . . . . . . . . . . . . . 345.4 Dorr and Barnett . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6. Vagueness and uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 426.1 Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.2 Vague evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.3 Comparative judgements of uncertainty . . . . . . . . . . . . . . 456.4 Probabilism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.5 Genuine uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 50

7. Vagueness and desire . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.1 Vagueness and probabilism . . . . . . . . . . . . . . . . . . . . . 537.2 Betting and vagueness . . . . . . . . . . . . . . . . . . . . . . . . 547.3 Vagueness and desire . . . . . . . . . . . . . . . . . . . . . . . . . 577.4 Must we only care about the precise? . . . . . . . . . . . . . . . . 59

Contents 4

7.5 The supervenience of vague beliefs on precise beliefs . . . . . . . 617.6 Appendix: decision theory and representation theorems . . . . . 63

1. INTRODUCTION

In a broad sense, this thesis is about the nature of uncertainty. A naturalpicture, a picture I want to deny, is one in which there are lots of different waysthe world could be, different configurations of objects and events, and it is inthe nature of uncertainty to be about these configurations. One might say thatuncertainty is always about the world.

I do not think this is how uncertainty works at a basic level, and this thesisis a sustained account of one way in which uncertainty fails to be like this.

In a more specific sense, this thesis is about vagueness and uncertainty. Iclaim that sometimes we may be uncertain about the truth of a sentence, evenwhen the truth of the sentence is not determined by the way the world is. Ibelieve that vagueness is one such case. To bring out the contrast, imaginesomeone who had no worldly uncertainty whatsoever; that she knew the exactconfiguration the world was in down to the finest details - that there is, let ussay, exactly one world compossible with the facts she is certain about. Wouldthere be anything left for her to be uncertain about? A central claim of thisthesis is that vagueness induces a kind of uneliminable uncertainty that wouldremain even if you came to know all the facts about the world you inhabit.1

There are two ways one could disagree with the claim that uncertainty neednot be about the factual. Epistemicists accept the claim that vagueness involvesgenuine uncertainty, but maintain that all uncertainty is worldly uncertainty.They claim that one can only be uncertain whether p if one considers there tobe a matter of fact whether p. There is thus a prima facie tension betweenthe claim that vagueness involves genuine uncertainty and the view that if p isvague, there is no matter of fact as to whether or not p.

For this reason I believe that uncertainty plays a central role in the theoryof vagueness.2 One must explain what it means for there to be no matter offact whether p, if one can (and must) be uncertain about whether p obtains. IfI say ‘either Princeton is Princeton Borough or it isn’t, but there’s no matter

1 I might also add that one could still be uncertain in the vague if one knew in addition tothe configuration of the world, all the mental facts, facts about where you are located in theworld, necessary a posteriori facts, and so on and so forth. The thesis can be strengthened inseveral ways, but the above description should be sufficient to convey the general idea.

2 I am mainly concerned with its role with respect to classical theories, however it also playsa central role in non-classical theories. To believe the negation of an instance of excludedmiddle - a contradiction even in non-classical theories - requires you to believe everythinggiven very minimal assumptions about uncertainty. The non-classical theorist, then, needs anaccount of uncertainty if they want to defend the claim that it is ever rational to accept anexcluded middle-denying non-classical logic.

1. Introduction 6

of fact which’, I have, on this view, expressed uncertainty as to which obtains.The epistemicist says that’s all I have done. In Chapter 2 I focus on theseissues. I particular, I shall explicate how a sentence can be semantically inde-terminate, and how one can be uncertain in it, without the view collapsing intoepistemicism.

Another way one could disagree with me is to deny that vagueness involvesuncertainty. Chapters 3, 4 and 5 are concerned with this type of response.One way to deny this is to maintain that vagueness does not involve genuineuncertainty. Uncertainty due to vagueness is sui generis, of a novel and funda-mentally different kind, and cannot be understood in terms of the psychologicalvocabulary used to describe worldly uncertainty. This is the topic of chapter3. Another way in which vagueness might not involve true uncertainty wouldbe to hold that whenever you are certain that p is vague, you should be an-ticertain (have credence 0) in p and in ¬p. This shall be the topic of chapter 4.Finally, one might argue that if you have no relevant worldly uncertainty, youjust shouldn’t be uncertain in p. You must be certain in p or certain in ¬p, andif p is vague, it will be indeterminate which you are certain in. This is the topicof chapter 5.

Chapters 6 and 7 are devoted to my preferred view. In chapter 6 I amconcerned with showing that vagueness-related uncertainty and ordinary uncer-tainty form a natural kind, and that even in the context of vagueness degreesof belief should obey the probability calculus. In 7 I turn to the functional roleof uncertainty. I argue that our desires and beliefs cannot be reduced purely todesires and beliefs in the precise, and caring about the vague is not esoteric andstrange, but commonplace. I end by arguing for some intuitively plausible prin-ciples governing rational preferences, principles governing how one should acteven when one knows the information relevant to your decision is vague. I ap-ply machinery from decision theory to show that we ought to be uncertain, andindeed, have probabilistically coherent degrees of belief, in vague propositions.

1.1 Preliminaries

Before we start I want to get clear on some things. Firstly, I am only going tobe considering uncertainty in the context of theories of vagueness that endorseclassical logic. I think that the role of uncertainty in non-classical logics isa central and fascinating topic, but it is well discussed, for example, in theworks of Hartry Field [9][8] and Graham Priest [16][15], and some of the topicsare tangential to the main thrust of this thesis. However, many of the issuesraised here have direct analogues in the non-classical setting. Each of the viewsdiscussed in chapters §3, §4 and §5 have non-classical counterparts, and theymostly fall afoul of the same criticisms I raise against their classical versions.The moral of chapters §6 and §7 mostly transfer to the non-classical setting.The thesis of probabilism is stated in a way that does not assume classicallogic. However the appeal to de Finetti’s theorem assumes classical logic, asdoes the decision theoretic framework, which assumes a Boolean algebra of

1. Introduction 7

propositions. I talk of ‘precisifications’ throughout the thesis. Hopefully anyonewho accepts the existence of penumbral connections can make sense of whatI mean here, for example, as ways of interpreting the language that respectpenumbral connections.3

Second. The word ‘proposition’ is often used for those things which arethe objects of peoples beliefs, the truth-conditions of sentences, obeyers of theT-schema, and so on and so forth. I do not believe there is one single thingthat fills all these roles. I shall thus use the word ‘proposition’ in two sensesthroughout this thesis; it should always be clear which I mean, but in casesof ambiguity I will mark which I mean. I shall treat the truth-conditions ofsentences as sets-of-worlds propositions. In this sense of the word, propositionsare always precise, and sentences are the true bearers of vagueness: sentencesexpress different propositions on different precisifications. On the other hand, Ithink that the possible objects of belief can be vague, so in that sense of the wordpropositions are not like sets of worlds. Nor are they sentences however; thepossible objects of belief are many in number, whereas there are countably manysentences of a language with a countable lexicon. There are vague concepts wecannot express, but nonetheless form propositions that are possibly believed. Ithus also use ‘proposition’ to mean a set of world/precisification pairs.

Lastly, what do I mean when I talk about ‘vagueness-related uncertainty’?I shall give a paradigm case: vagueness-related uncertainty regarding p is thecredential state one ought to be in if one knows all the relevant precise factsand is certain that p is vague.

To illustrate, consider the case of Hector the hairologist.4 Hector is a compe-tent speaker of English. Moreover, Hector knows pretty much everything thereis to know about hair, and keeps incredibly detailed records of each of his clientshair situation. For each client, he documents the number of hairs, whereaboutsthey are located on the head, the width of each strand of hair, the colour, thelength, and so on and so forth. One of these clients, Cedric, is on the verge ofbaldness. He still has tufts of hair here and there, but things don’t look good;he clearly is bordering on bald and Hector can see this. Despite knowing the sit-uation of Cedric’s head to incredible precision, including the number of hairs hehas, their length, colour, width and so on, Hector is neither willing to assert thatCedric is bald, nor to assert that he’s not bald. Such behaviour is characteristicof uncertainty, and in the case just described, people naturally will describeHector as being uncertain as to whether Cedric is bald. The credential stateHector is in, in the story just described, is a paradigm case of vagueness-relateduncertainty.

3 One need not insist that precisifications are classical; one might, for example, countenanceintuitionistic precisifications, paraconsistent precisifications, or what have you.

4 This is an example I shall refer back to throughout this thesis.

2. THE IMPORTANCE OF UNCERTAINTY IN THEORISINGABOUT VAGUENESS

The most important question concerning vagueness and uncertainty, in my mind,is whether uncertainty must always be about the factual. The purpose of thischapter is simply to get clear on what exactly this means, explain why it is notanalytically true, and generally clear the ground for further discussion.

Let us focus on a principle:

One may be uncertain whether p only if one considers there tobe a matter of fact whether p.

(2.1)

This principle is enticing. However - we must resist! For there is only a smallstep between accepting (2.1) and accepting epistemicism. For example: if onemaintains that considering p as vague requires being uncertain that p, then, by(2.1), one shouldn’t consider there to be no fact of the matter where vagueness isconcerned. Even non-classical logicians will typically accept classical logic wherethere is a fact of the matter, so there seems to be a quite general argument from(2.1) to the central tenets of epistemicism.

For the classical logician, the problem is even more acute, since the main(perhaps the only) point of contrast between supervaluationism and epistemi-cism, is that the former claims there is no matter of fact where vagueness isconcerned. What is needed, then, is an explanation of what non-factuality, or‘indeterminacy’, is, if it doesn’t obey (2.1). There is a fine line between ac-cepting the necessary co-extensiveness of indeterminacy and uncertainty andepistemicism about indeterminacy. One needs an explanation that doesn’t justidentify indeterminacy with uncertainty - that’s epistemicism - but leaves roomfor their coextensiveness.

The nature of uncertainty in vague contexts is an often overlooked, butcentral point in the dialectic between the various theories of vagueness. In thischapter I attempt to give an account of indeterminacy that allows for failuresof (2.1), i.e. allows for the co-extensiveness of uncertainty and indeterminacy,but does not permit a simple epistemicist reading of indeterminacy.

2.1 Two kinds of supervaluationism

Before we begin it is worth noting two different kinds of supervaluationism. Isee them coming as two packages of views - while the views don’t all have tocome together, it is very natural to group them as such. Throughout this section

2. The importance of uncertainty in theorising about vagueness 9

propositions shall be understood as whatever play the role of truth-conditions.The first package involves the following combination of views.

• Truth is identified with supertruth, and falsity with superfalsity.

• Vagueness is truth on some but not all precisifications and thus corre-sponds to a truth value gap.

• Validity is identified with preservation of supertruth.

• Propositions vary in truth value relative to a precisification.

• Propositions are the kinds of things which can be vague.

• ‘Determinately’ is treated as an operator on propositions.

The second package of views

• Truth obeys the T-schema, and is vague.

• Vagueness does not involve truth value gaps, rather, a sentence is vaguejust in case it is vague whether it is true.

• Validity is identified with preservation of truth at a precisification.

• Propositions do not vary truth value relative to the precisification.

• Sentences are the kinds of things that are vague. Propositions are alwaysprecise.

• ‘Determinately’ is treated as a predicate applying to sentences.

The second kind of supervaluationism is preferable for several reasons. (1) I takeit that the first theory is not a theory of semantic indeterminacy - each sentenceis determinately assigned exactly one semantic value, a proposition, specifically,a function from precisifications to sets of worlds. (2) The first theory appearsto be yet another version of a degree theory, with a partially ordered truthvalue set. (3) Given that ‘it’s true that’ has a trivial modal logic, and sincevagueness is identified with being neither true nor false, the first theory cannotaccount for higher order vagueness. (4) The first theory has a non-classicalconsequence relation. (5) If propositions are the kinds of things that obtain,then the first theory postulates an intolerable kind of ontic vagueness in theworld. (6) Compositionality fails for this kind of supervaluationist. (7) if onewere to maintain compositionality with respect to a precisification instead offull compositionality, every proposition would either be determinately true ordeterminately false (every proposition is precise w.r.t a precisification.) (8) Ifpropositions are additionally Russellian, one must posit ontic vagueness. Forexample, the proposition ‘Princeton is Princeton Borough’ is the Russellianproposition consisting of identity, Princeton, and Princeton Borough. Sincethis proposition is apparently vague, either identity is vague, or Princeton orPrinceton Borough are vague objects.


In what follows, the first kind of supervaluationism will often provide a wayout of the problems considered. I take the above to provide good reasons fornot taking this option.

2.2 Epistemicism and supervaluationism

Epistemicists openly accept the existence of ‘sharp cutoff points’, whatever theymay be, and it is typically thought to be this commitment that makes epis-temicism extremely counterintuitive. Supervaluationism, on the other hand,supposedly can retain classical logic while rejecting whatever it is that is socounterintuitive about epistemicism. But getting clear on what it is that is socounterintuitive, and seeing how supervaluationism does better is harder thanit seems.

The crux of the problem is that, if one accepts the second kind of super-valuationist theory espoused in the previous section in conjunction with thegenuine uncertainty view, there is not very much a supervaluationist can saythat distinguishes herself from epistemicism. For example, Field writes:

“the indeterminist needs to provide principles governing the notionthat are incompatible with its being given an epistemicist reading:incompatible, for instance, with reading ‘it is indeterminate whether’as ‘it would be impossible to find out whether.’ ” – [REF, p151,STfP]

Such a consequence should be worrying for supervaluationists if they are tomaintain that their theory is a genuine alternative to epistemicism. Distin-guishing it from epistemicism, however, is not the only problem: we must alsohave the means to state that the supervaluationist is not committed to the“sharp boundaries” of the kind that makes epistemicism so unattractive.

Sharp cutoff points. An initially compelling thought is that epistemicismis is committed to sharp cutoff points for vague predicates. A simple way tocash out the existence of sharp cutoff points for a predicate F is to say thatthere is some element in a Sorites sequence for F which is F while its successorin the sequence is not. It seems just outright implausible, for example, thatthere should be a last small number, or given that it’s vague whether Cedric isbald or not bald, that it should turn out that he is one or the other.

But this thought does not, and indeed, must not carry any weight for thesupervaluationist! For if this is what is so bizarre about epistemicism, superval-uationism is just as bizarre, as is any other view which retains classical logic.For the supervaluationist there is a last small number, and Cedric is either bald,or not bald - as dictated by the laws of classical logic.

The supervaluationist will typically have a more lenient notion of sharpcutoff point: ‘small’ has no sharp cutoff points because there is no n such thatdeterminately n is small and determinately n+1 is not small: ¬∃n(∆small(n)∧∆¬small(n + 1)).1 The supervaluationist then maintains that for a predicate

1 The operator, ∆p, is supposed to be read as ‘p and it’s not vague whether p’, and is


such as ‘small’ to have a sharp cutoff point is not for there to be a last smallnumber, but for there to be a number which is determinately last. There is alast small number, it’s just vague which one it is.

But if this is what we mean by a sharp cutoff point, the epistemicist isn’tcommitted to sharp cutoff points either! For the epistemicist, ‘it’s vague whetherp’ and ‘determinately p’ have epistemic readings, which when substituted into∃n(∆small(n) ∧∆¬small(n + 1)) entail there is an n that we know to be thelast small number - something which epistemicists reject. To say that thereis a last small number, but it is vague which one it is is just to say that weare ignorant as to which the last small number is. Lowering the standards ofcommitment to ‘sharp cutoff points’ does not allow one to escape whatever itis about epistemicism that seems so counterintuitive, unless we are to let theepistemicist off the hook too.

A final way of cashing out sharp cutoff talk might be to talk instead aboutthe state of the world determining cutoff points. For example, suppose Cedricis borderline bald. Let p be the conjunction all the (precise) facts relevant toCedric’s baldness (his hair number, hair colour, distribution et cetera) and letq be the use facts (which may or may not be precise.) For the epistemicist,this information is sufficient to determine whether or not Cedric is bald. Inparticular, necessarily if the use facts, and Cedric’s hair are as they actually are,then Cedric is bald, or necessarily if the use facts, and Cedric’s hair are as theyactually are, then Cedric is not bald: 2(p ∧ q → Cedric is bald) ∨ 2(p ∧ q →¬Cedric is bald). However, supervaluationists should accept this disjunctiontoo, since the truth of any proposition which precisifies ‘Cedric is bald’ followsfrom the complete description of Cedric’s head.2

Truth value gaps. Could indeterminacy be a lack of truth value? After all,epistemicists characteristically reject ¬True(ppq)∧¬True(p¬pq) as inconsistent.The distinction between supervaluationism and epistemicism on this view isgrounded in the characteristics of truth. Vagueness, in particular, is a kind oftruth status - being neither true nor false.

Does this explanation of vagueness meet Field’s challenge of giving a non-epistemicist reading of ∇? Certainly cashing out vagueness in terms of anantecedently understood notion of truth achieves this. But the notion of truthappealed to here is non-standard, and certainly is not pretheoretically available;for example it does not validate the equivalence between p and ‘p’ is true. Thenotion of truth at play here is closer to the notion of ‘determinate truth’ - anotion we cannot help ourselves to, if we are attempting to explain the notionof determinacy.

Furthermore, as noted in §1, supervaluationism is not committed to the ex-istence of truth value gaps. It’s open to the supervaluationist to say that everysentence is either true or false, but sometimes it’s indeterminate which. Thegappy view is rather an instance of the first type of supervaluationism discussed

usually pronounced ‘p is determinately true’.2 For simplicity, we can assume that the precisifications are the propositions that Cedric

has less than N hairs, for a certain range of N . Clearly the truth or falsity of each of thesefollow from the description of Cedric’s head.


in §1, and we have already argued that it is not particularly attractive. Thealternative, however, keeps the T-schema which brings supervaluationism andepistemicism into agreement so far as truth in its relation to vagueness is con-cerned. Given the T-schema and classical logic (specifically, the law of excludedmiddle) the following is a theorem: True(ppq) ∨ True(p¬pq). Furthermore,given a strengthened T-schema, ∆(True(ppq) ↔ p), the interaction betweenvagueness and truth is fully determined by the schema ∇p ↔ ∇True(ppq)). Ifa sentence is vague, then so is the statement that that sentence is true, andconversely. For this brand of supervaluationism, as for epistemicism, vaguenessisn’t to be identified with a kind of truth status.

Logic. Both supervaluationism and epistemicism retain classical validity inthe the sense that they agree with classical logic over whether each sentence isvalid or not. However, when it comes to the notion of consequence, the relationthat holds when a claim follows logically from some other claims, it is lessclear whether supervaluationism remains classical. On the view that identifiestruth with supertruth, and consequence with preservation of truth in all models,various classically valid inferences fail, such as contraposition and reductio adabsurdum, when the ∆ operator is in the language.

This clearly distinguishes the supervaluationist from the epistemicist. Fur-thermore, Field notes [REF, p165] we can go some way to answering the chal-lenge of giving the ∇ operator a non epistemicist reading. To say that p is vagueis to say that we should reject the instances of reasoning by cases on whether ornot p. I.e. to say that p is vague is to say that we shouldn’t reason from p |= qand ¬p |= q to |= q (for specific p and q.)

As noted already, this move is unnatural, unless one adopts the package ofviews that comes with identifying truth with supertruth that we rejected. Theother kind of supervaluationist might say that propositions are what fundamen-tally stand in consequence relations, and it is sentences, not propositions, thatare vague. Claims about inferences between vague sentences are to be treatedsupervaluationally, like any other claim involving vague sentences: ‘Γ entails p’is true at a precisification just in case the propositions expressed by Γ on thatprecisification entails (in the fundamental propositional sense) the propositionexpressed by p on that precisification. This is one way to restore a classical con-sequence relation, and is in full agreement with epistemicism over this matter.

Multiple precisifications I. A natural way for the supervaluationist todistinguish herself from an epistemicist is to engage the language of precisifi-cations. As a first try: supervaluationists believe there are multiple admissibleprecisifications of a vague language and epistemicists believe there is only one.

How are we to make this more specific? A precisification can be identifiedwith a bivalent model-theoretic interpretation of the language in question, thatmuch is clear. But when is a precisification admissible? Minimally a precisifi-cation is admissible just in case it assigns for each predicate of the language,extensions containing only things the predicate determinately applies to, andno things it determinately fails to apply to. Such an explanation of ‘admissible’clearly makes use of the notion of determinacy again. To adequately explainvagueness we need to eliminate this circularity - a topic we shall return to.


Notice that the minimal constraint has not yet distinguished supervalua-tionism from epistemicism. For as stated, the epistemicist should maintain thatthere are multiple admissible interpretations of the language. For given theminimal constraint on what an admissible interpretation is and the epistemi-cists reading of ‘determinately’, an interpretation is admissible just in case itmight be the correct interpretation for all we know, and indeed, the epistemicistsbelieves there are many of those.3

Notice also that there is considerable pressure on the supervaluationist inthe other direction: to maintain there is only one admissible interpretation. Calla precisification, v, ‘correct’ just in case the following disquotational schemataobtain:

‘φ’ is true when interpreted according to v iff φ. (2.2)The extension of the predicate ‘F ’ according to v, is just the setof F things.

(2.3)

According to v the name ‘a’ refers to to the object a. (2.4)

According to supervaluationism, there is exactly one correct precisification.Thus for our supervaluationist to maintain there are multiple admissible pre-cisifications, she must say there are admissible but incorrect precisifications- admissible precisifications where, for example, the extension of ‘red’ includesnon-red things, and other admissible precisifications where the extension of ‘red’exclude red things. This takes some getting used to.

The supervaluationist may try to lessen the blow by saying that althoughthere is only one correct interpretation, it is vague which one it is.4 But this ispresumably exactly what the epistemicist would say given her reading of ‘vague’:there is exactly one correct interpretation of the language, it’s just we cannotknow which it is.

Uncertainty and ignorance. The issue of uncertainty and ignorance isclearly quite central to this debate. A supervaluationist who believes that un-certainty and vagueness come hand in hand will find themselves agreeing withepistemicists about most things, as seen above. It is for this reason that somehave attempted to deny the connection between vagueness and uncertainty. Forexample Field, [9], suggests that to regard p as vague (or indeterminate) is to re-gard it as fundamentally misguided to speculate about whether p obtains. Whenproperly explained, this, Field claims, ensures we cannot give ‘indeterminate’an epistemicist reading. The sense in which an epistemicist finds it misguidedto speculate over p is much weaker - on Field’s preferred account, a person whois sure that p is indeterminate will have credence 0 in both p and in ¬p.

In [5] Dorr claims that vagueness does not involve ignorance or uncertaintyat all. Rather, in cases where one knows all the relevant facts, it vague whetheryou know p, when p itself is vague.

3 Of course, only a very crude epistemicis would identify vagueness with just any kind ofignorance, but this characterisation suffices to make my point.

4 Given that, according to this view, there is indeterminacy in the semantic vocabularysuch as ‘true’, ‘refers’ and ‘extension’, which precisification satisfies (2.2)-(2.4) relative to aprecisification, v, above will depend on (and indeed, will be identical to) v.


It should be noted that while both accounts succeed in distinguishing them-selves from epistemicism, neither view has anything positive to say about whytheir version of supervaluationism avoids the counterintuitive consquences ofepistemicism, whatever they may be. Indeed, if epistemicism accords with anyof our intuitions, it is the intuition that vagueness involves ignorance and in-termediate credence respectively. To say that not only is there a last smallnumber, but that I know which one it is, seems to invoke everything that is socounterintuitive about epistemicism, plus more.

Note that most epistemicists reject the thesis of uneliminable uncertainty.They will typically admit the possibility of knowing where the cutoff points ofvague terms lie. For example, Williamson [22] holds that the locations of thecutoff points in vague expressions supervene on the way the linguistic communityuses those expressions in the context of the language - it’s knowledge of how theysupervene on use that is so hard for us to achieve. However, no matter hard,surely it is at least possible that some being is able to see how the extension of avague predicate depends on the way it is used, and thus work out the extensionof that vague predicate.

This seems to be a point of departure with this brand of epistemicism. Forexample, according to the epistemicist above, it seems like one could knowthe facts about how English is used, and the facts about how the extension ofEnglish expressions gets fixed by how English is used, and thus could work outthe precise conditions for being in the extension of ‘bald’.

Of course, the supervaluationism I am defending holds that once you werecertain in the facts about how English is used and the use/meaning superve-nience facts, you wouldn’t be uncertain whether Cedric was bald. But there isno matter of fact about how the meaning of ‘bald’ gets fixed by the use facts -the supervenience hypotheses are themselves vague, so we could never be ratio-nally certain about these facts, in accordance with the uneliminable uncertaintythesis.

The kind of uncertainty I am postulating is not uncertainty over what theextension of ‘bald’ is - there is no matter of fact regarding this question. Ifthere were a matter of fact regarding this question - suppose bald means ‘hasless than 50,000’ hairs, and I knew Cedric had less than 50,000 hairs - then I’dbe inclined to say a I knew all along that Cedric was bald, I just didn’t knowwhich truth value the sentence ‘Cedric is bald’ had. By way of analogy, I wouldnot lose knowledge of a fact if it were rephrased into unfamiliar vocabulary ofwhich I did not know the meaning.

Finally, there are epistemicists who deny the supervenience of meaning onuse [14]. For this epistemicist nothing short of actually learning whether Cedricis bald will allow one to know whether Cedric is bald. For this epistemicistthere are physically duplicate worlds which differ over whether Cedric is bald,but not over the use facts.

One might object here that I am balancing a lot on a particular use of‘possible world.’ I, like the epistemicist who abandons supervenience, accepttwo physically identical epistemic possibilities, agreeing on the use facts, whichdiffer with regard to whether Cedric is bald, I just deny that there are also


metaphysical possibilities like this. I believe the distinction between epistemicand metaphysical possibility is at the root of the claim that uncertainty neednot be about the factual. Someone who denies that this distinction is a deep onewill not find much of interest in the thesis that uncertainy need not be aboutthe factual.

Multiple precisifications II. So far I have been distinguishing my viewfrom epistemicism on a case by case basis. What is it about each of the viewsI have considered that makes them epistemicist rather than supervaluation-ist? Intuitively, the supervaluationist claims that vagueness is fundamentallya matter of semantic indeterminacy, and that ignorance is merely a by prod-uct. Epistemicists on the other hand will outright deny there is any semanticindeterminacy, and maintain that vagueness is a purely epistemic notion.

So let us return to the question of providing a reductive explanation of‘determinately.’ Field [REF, p11] expresses skepticism that this could be donewithout appealing to the way in which indeterminacy interacts with uncertaintyand ignorance.5 I shall attempt to provide such an explanation.

The fundamental idea is that there are many different ways of interpretinga vague language which are compatible with the way we use that language. Tomake this rigorous we need to specify what ‘compatible’ means in a non-circularway that doesn’t merely mean compatible with our knowledge about the theway the language is used. In [13] Lewis provides such a non-circular account ofcompatibility with linguistic usage, which I shall adopt here. For our purposeswe may simplify: an interpretation of the language, I, is a function assigning‘truth-conditions’ to sentences of the language. An interpretation is admissiblefor a language L, just in case there is a convention among the speakers of L totry not to say sentences that would be false if interpreted according to I. Forsomething to be a convention requires a certain network of preferences, beliefsand desires to hold between the users of L.

For two distinct interpretations of a language, I and I ′, to be admissiblethere must be a convention among the speakers of L to try not to speak falselyaccording to I and I ′. Since they are distinct, there must be a sentence Sassigned truth-conditions with possibly different truth values by I and I ′ re-spectively. If the requisite conventions are in place, when a speaker knows thatI(S) obtains and that I ′(S) doesn’t, she should refuse to assert S and refuse toassert ¬S.

But how are we to distinguish the situation above from one in which there isexactly one admissible interpretation, J , that assigns finer-grained vague “truth-conditions” than either I or I ′ - and where a speaker would refuse to assert Sand ¬S because she is uncertain whether J(S) obtains?

What are ‘truth-conditions’ in this context? Could we, for example, assign‘Cedric is bald’ the truth-conditions that obtain just in case Cedric is bald,and that we should try not to assert ‘Cedric is bald’ unless Cedric is in factbald? A supervaluationist who believes that we should be uncertain in cases

5 We must be careful to distinguish indeterminacy, vagueness and ignorance here. For exam-ple, Williamson is not an epistemicist about indeterminacy: he believes there’s a determinatematter of fact about whether Cedric is bald, but we can’t know which.


of vagueness should not predict a difference in language use between a singlefine-grained interpretation that assigns ‘Cedric is bald’ the “truth-conditions”above, and several admissible interpretations assigning ‘Cedric is bald’ variouscoarse-grained truth-conditions.

Note that a similar issues arise concerning sentences like ‘Hesperus is Phos-phorus’. Are truth-conditions more fine grained than sets of worlds are? Is thedistinction between ‘Hesperus is Phosphorus’ and ‘Hesperus is Hesperus’ part ofsemantics proper? I think in the case in hand there are decisive reasons not tocut truth-conditions as fine as interpretations like J would. One could have Jassign each sentence itself as a “truth-condition”, which would certainly be finegrained enough to formulate homophonic conventions, but they would have nocontent. Truth-conditions are supposed to be tied to how the world is, ratherthan how our language represents it, and if it could be vague whether sometruth-conditions obtained that would involve an intolerable case of vaguenessin reality. It may be vague which truth-conditions a sentence has, but this isnot to say truth-conditions themselves are vague. While there may be some un-settled debates concerning the nature of truth-conditions, I take the distinctionbetween language and the world outlined above to be sacrosanct.

3. ERSATZ UNCERTAINTY

On first looks, uncertainty about the truth of vague propositions seems to beof a very different kind from uncertainty of the more commonplace kind. Un-certainty, for example, about the future, or about the goings on in far awaygalaxies, or about wildlife in the deep sea seems to have a very different nature.For one thing, the latter kind of uncertainty appears to be uncertainty aboutthe way the world is. The former kind cannot be like this, since the worldsimply does not give answers to questions that have no determinate answer.The world does not determine whether or not Cedric is bald, or which numberis the last small number. Secondly, it seems that once one is certain of all therelevant facts about the world - that is, one has no uncertainty of the latter kindwhatsoever - one may still be uncertain about the truth of vague propositions.Usually ignorance or uncertainty is curable; there are facts one could learn thatwould put one in a better position to judge whether or not p. It is difficult toeven imagine what it would be like to be in a better epistemic position withregards to p, if p were vague. If I knew all there was to know about Cedric’shead and I still didn’t know whether he was bald, it’s hard to see what elsecould I do to decide the question - there is nothing left to learn that is relevantto his baldness.

These characteristics are atypical in the latter kind of situation in whichone is uncertain about the world. An explanation for this disparity would bethat we are equivocating; two states that seemed to be of the same kind, arein fact different. Although both kinds of uncertainty play similar roles as faras assertion and action go, they actually correspond to different mental statesaltogether. Perhaps the kind of uncertainty that arises due to vagueness issui generis, and not to be assimilated to genuine uncertainty. For exampleSchiffer - a prominent defender of this kind of view - claims that vagueness-related uncertainty is ‘a new kind of propositional attitude, one that comes indegrees and that precludes standard partial beliefs,’ that it is ‘not a measure ofuncertainty’, and similar such things. I shall call this state ‘ersatz uncertainty’.If such uncertainty comes in degrees, we shall also talk about ‘ersatz credences’.

In this chapter I wish to explore the thesis that when we are sufficientlyinformed about the world, and we are considering whether or not p, where p isvague, we are not genuinely uncertain in p, and will typically find ourselves ina state of ersatz uncertainty.

I shall argue that ersatz uncertainty, if it were to exist, would give rise to

3. Ersatz uncertainty 18

‘mixture’ uncertainties. I give two examples:

p is second order vague. You are ersatz uncertain whether pis vague or precise. Your credence in p should be a mixturecredence: your conditional credence in p given p is vague (anersatz credence) times your ersatz credence that p is vague, plusyour conditional credence in p on p being precise (a genuinecredence) times your ersatz credence that p is precise.

(3.1)

You are genuinely uncertain whether p is vague or precise. Yourcredence in p should be a mixture credence: your conditionalcredence in p given p is vague (an ersatz credence) times yourgenuine credence that p is vague, plus your conditional credencein p on p being precise (a genuine credence) times your genuinecredence that p is precise.

(3.2)

The best way to account for mixture uncertainties is to treat vagueness-relateduncertainty and genuine uncertainty as mixable. I argue that the claim thatuncertainty due to vagueness is not ‘genuine’, that it is sui generis, loses its biteif it is mixable.

The overarching thesis of this chapter, however, is that ordinary uncertaintyand vagueness-related uncertainty form a natural kind; that it is genuine un-certainty.

3.1 New psychological vocabulary

Consider what happens to Hector, the hairologist considered in §1, when he ispresented with a Sorites sequence starting with clearly bald people and endingwith clearly non bald people. What is going on in Hectors brain as he movesalong the sequence and asks whether the current member is bald?

An initially plausible story is that he has high credence that the people at thebeginning of the sequence are bald, middling credence that the people aroundthe middle of the sequence are bald, and low credence that the people towardsthe end are bald. Is this picture compatible with the credences that he hasaround the middle of the sequence being of a radically different nature from thecredences he has at the beginning and end of the sequence?

The thesis that vagueness-related uncertainty is of a fundamentally differ-ent kind is, of course, terribly imprecise. A minimal characterisation, that willbe sufficiently precise for our purposes, I suggest, goes as follows: ‘to properlydescribe vagueness-related uncertainty, one must introduce new psychologicalvocabulary which is not explainable in terms of, nor reducible to the originalvocabulary of ordinary uncertainty.’1 This, for example, rules out the view inwhich ersatz uncertainty in p is simply intermediate credence in p accompaniedwith a positive credence in ∇p, since this was explained entirely using the orig-inal vocabulary of uncertainty. It should not be possible that my credence that

1 This looks like an empirical hypothesis, but I believe, even from the armchair, the pointsI am about to make are still relevant to this hypothesis.


this coin will land heads is the same credential attitude I have had towards theproposition that Cedric is bald (and vice versa.)

Could one be in both states? For example, could one be both uncertain in theordinary sense, and ersatz uncertain about the same proposition? Perhaps onehas both credential and non-credential attitudes (i.e. ersatz credences) towardseach proposition? To what extent would this corroborate the claim that whenit comes to propositions known to be vague, we are not genuinely uncertain? Iffor every x in the Sorites, Hector had credence 1 or 0 that x was bald (althoughperhaps he has intermediate ersatz credences), then there would be no sensein which Hector was genuinely uncertain. His intermediate credences wouldn’tbe genuine and his genuine credences wouldn’t be intermediate. This kind ofsituation, however, is greatly at odds with the intuitive picture described above,in which if we were to look in Hector’s brain, the attitudes he has to adjacentpropositions in the sequence would be fairly similar. Furthermore, if Hector isgenuinely certain that the nth member of the sequence is bald, presumably hemust be genuinely certain that earlier members are bald too. Given that weknow Hector ends up with 0 credence in the final members being bald, therewould be some point at which his genuine credence sharply drops from 1 to 0.Where this drop would occur, and why, is just too mysterious to take seriously.2

I conclude that if one has genuine credences at all in vague propositions,they will usually be intermediate. It thus seems that if we could be in bothstates, there will be cases of vagueness-related uncertainty in which we are bothgenuinely uncertain whilst having ersatz credences too. If the addition of ersatzcredence was supposed to account for the intuition that we are not genuinelyuncertain in cases of vagueness, it has not done so. Furthermore, the viewquickly comes upon embarrassing questions. Could one have credence 1 in p,but ersatz credence 0 in p? What should your ersatz credences look like whenyou are genuinely uncertain whether p is precise? What should your genuinecredences look like if you are ersatz uncertain that p is precise (if p has secondorder vagueness, for example)? Unless the compatibilist about these two statescan tell us what functional role ersatz credences play, or give a reason to positsuch things, I see no reason to think that there is anything more to our attitudestowards vague propositions than ordinary doxastic and credential attitudes.

3.2 Mixed uncertainty part I

I submit, then, that if we are to countenance novel psychological vocabulary forvagueness-related uncertainty, it should not be compatible with the old vocab-ulary of genuine uncertainty. Thus, in the set up described, we should expectHector to be genuinely certain that the early members of the sequence are bald,and genuinely certain that the people towards the end of the sequence are notbald. In the middle, we may assume, he is not really uncertain, but in somefundamentally different state we have dubbed ersatz uncertainty.

2 I shall, for the moment, bracket the view that there is a sharp drop in Hector’s credences,but that it is vague where the drop occurs. This view is something I shall return to in §5.


So much for the beginning, middle and end; what about the transitions?Are we to suppose that Hector has genuine credential attitudes up until a cer-tain point in the series, and then, at some particular point, stops having theseattitudes altogether and starts adopting a fundamentally different attitude? Ifone thought there was a sharp transition in this Sorites sequence between thedeterminately true, vague and determinately false, then this would fit with apicture in which there were sharp transitions between the kinds of credentialstates Hector is in. One should have ordinary credences on the determinatelytrue and determinately false propositions, and ersatz credences on the vaguepropositions.

However, this is not at all how vagueness works. The transition from de-terminately true to vague, is not sudden - there are instances where it is vaguewhether the proposition is determinately true or vague. In these intermediatecases which attitude should one adopt? Given the current picture there are twofundamentally different kinds of uncertainty, one for the precise and one for thevague. It is tempting to say that whenever it is vague whether a propositionis precise or vague, it is vague whether Hector has the first or second kind ofuncertainty.3

I want to now consider the consequences this kind of vagueness would haveon such a view by focussing on a particular principle governing borderlineness:

If F and G admit a borderline case, then F and G are closeproperties.

(3.1)

Let me first of all clarify some issues. What I mean by ‘close’ is best demon-strated by an example. Red and orange are close, whereas navy blue and floures-cent pink are not: one could run a Sorites between red and orange to generateborderline cases of red/orange, but one could not do this for blue/pink. Thesense of closeness at work here is metaphysical. Someone can be borderlinebetween being bald and not bald because people who are bald aren’t fundamen-tally different from people who aren’t. On the other hand, an electron cannot beborderline between being positively and negatively charged, because these arefundamentally different properties. The principle could be defended by appealto the best fit theory of meaning: expressions referring to natural propertiescannot admit borderline cases, because there cannot be semantic indeterminacywhere there are reference magnets. I shall not pursue this thought here, but I’lljust assert that the principle seems to be independently motivated.

Secondly, F and G may have no borderline cases, even if there are truesentences of the form ‘it’s vague whether a is F or G’. If I were to stipulatethat the name ‘a’ referred to this clearly pink patch if the last small number iseven and this clearly blue patch if it’s odd, then it would be vague whether a ispink or blue. However, pink and blue do not admit any borderline cases.

According to the picture we are considering, someone could be in a statewhich was borderline between being a genuine and an ersatz credential state.

3 This would follow, for example, from the assumption that determinately, Hector has thefirst kind of uncertainty in p, just in case p is precise, and determinately Hector has the secondkind of uncertainty just in case p is vague.


This is not possible, however, if we are to simultaneously take (3.1) seriously,and maintain that genuine and ersatz uncertainty are fundamentally different.The existence of borderline cases between these two states suggests there’s notreally a deep distinction here, since the boundary between the applicabilityof ‘genuine’ and ‘ersatz’ is of the use determined sort, rather than the realitydetermined sort. If this is right, the claim that you are not genuinely uncertainin cases of vagueness loses its bite; it is only in a fairly stipulative sense thatthis isn’t genuine uncertainty, the uncertainty present due to vagueness is notfundamentally different from any other kind of uncertainty.

Before we discuss this view any further, I want look at another way in whichwe might want to posit mixture uncertainties.

3.3 Mixed uncertainty part II

In section 1 I argued that the best way to make sense of the claim that vagueness-related uncertainty is sui generis, is by thinking of there being two fundamen-tally different types of credential attitude which cannot be had simultaneously.In section 2 we saw that such a view has a hard time if there is higher ordervagueness. Let us now bracket that concern; I claim that even if one deniedthe existence of higher order vagueness, one would still have to make sense ofanother kind of mixed uncertainty: uncertainty that is partly induced by theworld, and partly induced by vagueness.

Here is a standard example of this kind of mixed uncertainty. Suppose youhave has a bag containing two small coloured balls. The first is clearly red, theother bordering on red/orange. You are about to take a ball out of the bag:what should your credence that you will pick a red ball be? A standard way toapproach this question would tell you to work out the probability that it’s redgiven that it’s the first ball and the probability that it’s red given it’s the secondball, then times each of these by the probability that it is in fact the first/secondball respectively, and add these together: Cr(red) = Cr(red|ball1)×Cr(ball1)+Cr(red|ball2)× Cr(ball2).

But what is the probability of the ball being red given it’s the second ball?Presumably that would be the credence you would have that it’s red if you learntthat you were holding the second ball - and this would be an ersatz credence.So the first puzzle is technical: we tried to stipulate that credences (notation:Cr(p)) and ersatz credences (notation: ECr(p)) were fundamentally different.A minimal constraint was that your genuine credence in p is undefined if you aregenuinely certain that p is vague (Cr(p) is undefined if Cr(∇p) = 1), and yourersatz credence in p is undefined if you’re genuinely certain that p is precise(ECr(p) is undefined if Cr(∆p ∨ ∆¬p) = 1).4 But what kind of function isthis conditional credence, call it Pr(·|ball2), defined loosely as ‘the credence Iwould have in p, if I learnt I was going to pick ball 2’? It can’t be a genuine

4 This is a minimal constraint. Of course we have been bracketing higher order vagueness- you might not even have a genuine credence in ∇p, which adds an additional layer ofcomplication.


conditional credence because Pr(∇red|ball2) = 1 but Pr(red|ball2) is defined,and no genuine credence has this property. It can’t correspond to an ersatzconditional credence because it seems to be defined on all sorts of propositionsyou can presumably have genuine credential attitudes towards, for example itis defined on the proposition that you are holding ball 2 (Pr(ball2|ball2) =1.) It seems there must be two notions of conditional probability for a givenantecedent. Roughly Cr(q|p) is ‘the genuine credence you would have in q ifyou learnt p’, and ECr(q|p) is ‘the ersatz credence you would have in q if youlearnt p’. This is the best sense I can make of conditional credences.

However, this raises a problem for the standard way of reasoning aboutthe puzzle above. The probability that you are holding a red ball should becalculated: Cr(red|ball1) × Cr(ball1) + ECr(red|ball2) × Cr(ball2).5 Now,certainly this will give us a number, but what are its ‘units’: what kind ofcredence does this number represent? The sum had terms representing bothersatz and genuine credences. Does the resulting number represent a genuinecredence or an ersatz credence? Call this kind of credence, whatever it is, a‘mixture credence’.

One could simply stipulate that you should have ersatz credence in anyproposition you have a positive genuine credence is vague. This would ensurethe sum above, and indeed all non-trivial mixture credences, corresponded toersatz credences. But this doesn’t seem like a plausible line to take in general.Suppose instead the bag contained a million determinately red balls, a milliondeterminately black balls, and 1 borderline red/orange ball. Are we still sup-posed to deny that we are genuinely uncertain that a randomly picked ball wouldbe red? The sense in which I am uncertain is intuitively very similar I kind tothe way I would be uncertain if there were a million determinately red ballsand a million determinately black balls and I didn’t know which was going tobe picked. The claim that this is a fundamentally different kind of uncertaintyloses some of its purchase. Alternatively, one could stipulate that you only haveersatz credences in propositions you’re genuinely certain are vague. This seemsimplausible for similar albeit symmetrical reasons.

To assimilate all mixture credences to ersatz credence, or genuine credencesseems to destroy the distinction we were going for. Perhaps your credence inp should be ersatz, if your genuine credence that p is vague is above somethreshold. On this view, some mixture credences are genuine, some aren’t.However, this proposal will inevitably involve situations where you ought tohave ersatz credences in p and in q, but not in p ∧ q, and similarly, that youought to have genuine credences in p and q but not p∨q. This prevents one fromdeveloping a serious theory of genuine and ersatz probabilities in the systematicway seen, for example, in the classical probability calculus, since the domain ofany such function will not form a σ-algrebra.

One could think of this problem and the previous one as being part of thesame difficulty of understanding how the two credences interact. In the last

5 The alternative way of calculating the answer with Cr(red|ball2) instead of ECr(red|ball2)would leave one of the terms in the sum undefined.


section, where it is vague whether p is vague or precise, Hector was ersatzuncertain whether p vague or precise. How should this affect your genuine andersatz credences in p? In this case, one is genuinely uncertain whether p is vagueor precise, and again, we may ask how this should affect our genuine and ersatzcredences in p.

3.4 What is genuine uncertainty?

The upshot of these observations seems to be that someone who insists vagueness-related uncertainty is of a different kind must have a story to tell about whichkind each mixture credence belongs to. And as we have seen, there is no suchstory to be told.

One of the most prominent defenders of the view that vagueness-relateduncertainty is sui generis is Stephen Schiffer in his recent book, [19], and withslightly more detail in [18]. Schiffer adopts the first option we considered, inwhich mixed uncertainties are always ersatz uncertainties, and so his particulartheory is subsumed by the discussion here. One thing worth noting, however,is that his view also suffers from the problems the threshold view has. Forexample, Schiffer stipulates that credences are calculated in such a way thatyour ersatz credence might be defined on a conjunction, but not on one of theconjuncts, making it very hard to see what kind of structure the propositionsyou can have ersatz credences in would have.

I propose a more elegant way to view vagueness-related credences; one thathas the benefit of simplicity and which does not deviate from our current un-derstanding of credence. The mixture of an ordinary credence and a vagueness-related credence, I claim, is exactly the same kind of credence as both. As such,vagueness-related credences and ordinary credences are one of a kind.

There is a worry that this issue is terminological. The initial target was aperson who thought vagueness didn’t really involve uncertainty, and talk aboutuncertainty in such cases would better be attributed to some other attitude.One might accuse me of using the term ‘genuine uncertainty’ more elasticallythan my opponents - after all, I certainly admit there’s something distinctiveabout vagueness-related uncertainty. Perhaps this is so; and if it is, I haveat best clarified how much mileage one can get from the claim that vaguenessdoesn’t involve genuine uncertainty.

But I think this is too modest a conclusion. We do have a reasonably goodaccount of when something counts as genuine uncertainty. For example, weall know that you do not need to be human to be genuinely uncertain aboutsomething. Presumably animals with a very different neurophysiology from uscould still be genuinely uncertain. We can recognise this, and there is no termi-nological question about whether they count as genuinely uncertain. The brainstate I am in when I’m genuinely uncertain about something could be radicallydifferent in configuration and constitution from the same state of genuine un-certainty in an alien. It is not the actual brain state, but the causal-functionalrole it plays with respect to the rest of the being that makes it a state of genuine


uncertainty.It could be that, physiologically, vagueness-related uncertainty is very dif-

ferent from ordinary uncertainty. But as with the case of the alien, this doesnot mean that vagueness-related uncertainty is not genuine uncertainty. Toestablish that vagueness-related uncertainty is not really genuine uncertainty,one must show that it plays a very different causal-functional role. On firstlooks, however, this does not seem to be the case. Someone who is in a state ofvagueness-related uncertainty in p because they’re certain p is vague, typicallywon’t assert p or ¬p, won’t stake things they desire on p, and will generallyact as if they where uncertain about p. Indeed, in chapter 7 it shall be arguedthat ordinary and vagueness-related uncertainty do play the same causal func-tional role with respect to our desires and actions, which is further explored ina decision theoretic framework in chapter 7.

If this state is playing the same causal-functional role as genuine uncertaintywith respect to our actions and desires, as is suggested above, there is littlesubstantial left to the claim that it is not genuine uncertainty. It is insubstantial,I claim, in the same way in which it is insubstantial to insist an alien can’t begenuinely uncertain - it is a verbal disagreement at best, since we do not needto introduce new psychological vocabulary to describe the alien.

4. REJECTIONISM

We have seen that vagueness-related uncertainty and ordinary uncertainty arepart of one continuous spectrum, neither of which occupy a privileged role. Ifwe are to maintain that vagueness does not involve genuine uncertainty, it isnot because there is ambiguity in uncertainty talk.

Hartry Field in recent work [7] has defended a classical view in which vague-ness does not involve uncertainty.1 However, his theory does not place anydivision between ordinary and vagueness-related credences, making it a seriouscontender. It does not posit different kinds of uncertainty for the vague and pre-cise, they are both included in a general theory of credence in which mixturesof ordinary and vagueness-related credences are dealt with uniformly. As such,classical probability theory drops out as a special case for precise languages.

We shall look at the details of Field’s theory later. For now we shall askwhat the characteristic features of the theory are. Does it deviate from classicalprobability theory? What impact would learning that p is vague have on youropinion about p? A characteristic feature of Field’s theory is that wheneverCr(∇p) = 1, Cr(p) = Cr(¬p) = 0, a thesis Williams [21] calls ‘rejectionism’.

I take it that in a salient sense, this is a view in which vagueness does notinvolve uncertainty, since it does not require intermediate credences. Despitethe absence of uncertainty, it is also not a view in which you can be certainwhere the cutoff points in vague terms lie. For to be certain of this, one mustpresumably be certain that F (a) for some a, and certain that ¬F (a′), for a’ssuccessor in a given Sorites sequence on F .

Field’s account of vagueness-related uncertainty is part of a more generalproject of explaining the conceptual role of indeterminacy in non-epistemicistclassical theories, (see chapter 2), and is extended to non-classical theories aswell in [8].2 We shall be concrentrating on Field’s theory purely as an accountof vagueness-related uncertainty.

4.1 Strange consequences

Probably the most important feature of Field’s account is the fact that youought to assign credence 0 to both p and ¬p if you’re certain p is vague. This is

1 He later considers a similar but non-classical view in [8] - my focus shall be on the classicalversion in keeping with purposes.

2 In the latter case, it plays an important role with respect to failures of excluded middle:one may reject excluded middle without accepting any counterinstances. The latter cannot bedone on pain of contradiction, if accepting p as a counterinstance involves accepting ¬(p∨¬p).

4. Rejectionism 26

an initially attractive feature. Anyone who is certain that p is vague will deny pand ¬p, will not accept bets on p or on ¬p even on good odds, and will displaybehaviour that is mostly in accordance with this assignment of credences. Soin many ways it has the benefit of according with our pretheoretic intuitionsabout degrees of belief.

The first thing I want to do is shed some doubt on some of the supposedlyintuitive judgements Field’s calculus delivers. In [4], Dietz presents a Dutchbook for Fieldian probabilities, based on the idea that people will reject anybet on a proposition they’re certain is vague. I suggest that normal people willusually bet ‘as if’ they had 0 credence in p because they don’t believe that ifthey accepted, they’d get their money if and only if p. However, this is no moremysterious than my rejecting bets arbitrarily in my favour that a coin will landheads, and similarly for bets that the same coin will land tails, because I knowmy bookie is too poor to pay up. It is clear in this case that we shouldn’t inferthat I have credence 0 that the coin will land tails and credence 0 that it willland heads. This simply because I don’t really believe I’m facing the decisionproblem originally advertised. When you’re certain that p - the propositionyou’re betting on - is vague, you should similarly be sure your bookie won’t payup iff p. Your bookie doesn’t have the means to check whether p any more thanyou do. For these reasons, I think the Dutch book arguments fail.

In the next section I shall argue that provided you can care about the vague,there are bets in which Field’s calculus delivers the wrong results. Right nowI want to briefly consider arguments against the claim that intuitively Field’scalculus delivers the right results. For one thing, if you are certain that p and qare vague, you cannot think that p is more likely than q, or that q is more likelythan p - they both receive credence 0. But this, I claim, does not accord withour intuitive judgements. One might be certain that it’s vague whether this60% full glass is pretty full, and vague whether that 66% full glass is pretty full,but you should be more certain that the latter is pretty full, since it is clearlymore full than the former.

Can one construct cases where one is certain p is vague, and is almost certainthat p? You could imagine a sequence of borderline cases of pretty full glasseswith larger and larger amounts of water, but one then runs the risk of confusingcertainty of vagueness with higher order vagueness. Imagine instead I’m aboutto roll a hundred-sided die, whose sides are labelled 0 to 99. Intuitively I shouldbe fairly sure that the number it’s going to land on won’t be the last two digitsof the largest small number. After all, there’s only one of them, and which everone it is, there a 99% chance it won’t land on that one. I should be 99% certainthat it’s not going to land on the last two digits of the largest small number.But I’m absolutely certain that whichever number it does land on, it’ll be vaguewhether it’s the last two digits of the largest small number, because I’m certainthere are at least 100 borderline cases of smallness. Prima facie, this seems tobe a case in which Cr(∇p) = 1 and Cr(p) = .99, contradicting Field.

Let me draw attention to one other bizarre feature of Field’s theory. In-tuitively, if one looks at the distribution of credences over a Sorites sequence,(pn)n∈ω, you imagine a smooth distribution with no sudden drops. This should

4. Rejectionism 27

be so even for someone who knows all the precise facts. There is nothing abouthaving non-classical credences which directly contradicts this: what one wouldexpect is that the credence in pn and ¬pn start off at 1 and 0 respectively for theearly n. As n increases, the credence in pn decreases until it meets the credenceof ¬pn at 0 for a little while. After that the credence of ¬pn starts to increase,leaving pn at 0, until ¬pn hits 1 as n reaches the end of the Sorites.

This is, however, not how it works on Field’s view. Field’s probability calcu-lus sits naturally in a context of an S5 modal logic for ∆.3 Despite being utterlyimplausible in that it appears to deny the existence of higher order vagueness, italso requires the sharp drops in credence of the kind just denounced. If a knowsall the precise facts, according to Field’s view, a’s credence in pn and ¬pn willbe 1 and 0 respectively until some N . At that point pN+1 will shoot from 1 to0, and remain there until some larger M , at which point ¬pM will shoot from 0to 1 at pM+1. As far as I can see, there is no way to fix this even if one droppedS5.4

4.2 Decision theory

In this section I shall look at how credences in Field’s framework are supposedto inform action. Since most of the points can be made without going intotoo much technical detail, I shall try to keep the formalism to a minimum,appealing mainly to feature that your credence in p and ¬p should be 0, whenyour credence in ∇p is 1.

I shall focus on one counterexample, and two refinements of the counterex-ample, to the natural way to formulate decision theory in this framework, andthen a puzzle about the deontic logic it would generate. Finally I shall dis-cuss another way to calculate expected utility which does better, but which isultimately unmotivated on philosophical grounds.

Let Bel(·) denote a Fieldian credence function. The standard way to calcu-late the expected utility of an act would be:

• Exp(p) :=∑

sBel(s)u(p ∧ s)

where the S form a partition of logical space into states ‘independent’ of theacts, and u is a assigns utilities to outcomes like p ∧ s.5 I have chosen todefine expectation like this so as not to make any assumptions about conditionalprobabilities in this framework, whereas the relevant notion of independenceshould be just as clear (or murky) in this context as elsewhere.

The first puzzle concerns what appears to be a very good set of bets, but onewhich the Fieldian decision theorist would recommend you rejected. Suppose

3 See Dietz [4] for a more detailed discussion.4 The characteristic principles would require you to have credence 0 that p was vague, if

you were certain p was second order vague, which doesn’t seem to put much constraint onyour credence in p. If one instead stipulated that your credence in p and ¬p ought to be 0whenever your credence that p is precise is 0, however, you would once again have steep dropsin credence.

5 And outcomes should be as complete and specific as they need to be to capture everythingyou could possible care about.

4. Rejectionism 28

we are looking at a glass that is filled two thirds the way up with water, andwe are considering whether or not it is pretty full. I offer you the following bet:you pay me £1, and if the glass is pretty full I’ll give you £3, otherwise you getnothing. Since you know that it’s vague whether the glass is pretty full, yourcredence should be 0 that it’s pretty full and 0 that it’s not pretty full, so theexpected utility (after paying the £1) is £-1 relative to the partition prettyfull, not pretty full. So you shouldn’t accept this bet. Symmetrical reasoningshows that the equivalent bet on the glass not being pretty full also shouldn’t beaccepted. However, if I were to offer you both bets simultaneously, surely youought to accept both? Since whether or not it’s pretty full, you’ll get £3 returnon one of the bets, minus the £2 that you paid to accept both bets: that’s £1profit guaranteed.

Perhaps even more strange, if one calculates the expected utility of the betthat pays out £3 if the glass is pretty full or not pretty full, and costs £2 to play,expected utility is £1 with respect to the partition pretty full or not prettyfull, despite the fact that this bet, and the set of bets in the last paragraphdeterminately and necessarily have the same outcomes no matter what youdo. This demonstrates a drastic failure of partition invariance in this setting.Admittedly the equation we chose is not invariant with respect to arbitrarypartitions, but it had better be invariant with regard independent partitions, orbetween partitions that are coarsenings of one another.

Now one might still argue that the decision theory recommends the correctaction in both cases, because in the two bet situation you can’t be sure you’ll get£3 iff the glass is [not] pretty full and you accept, whereas the payoff conditionsof the tautologous bet allow you to be certain you’ll get your payoff in thecorrect circumstances. As I have already argued, if one does not really believeone is in the decision problem described, one has no reason to perform the actrecommended for that decision problem anyway. Thus I do not see how thisresponse engages with the objection in the first place. Decision theory can andshould still make recommendations for decision problems that one could neverface because they are impossible to set up.

That said, I think that one can set up metaphysically possible decision prob-lems involving vagueness. For example, suppose you just cared intrinsicallyabout owning green things. I shall postpone discussion of these kinds of desiresuntil chapter §6, but for now it is enough to note that these kinds of desires seemto be coherent. Suppose I have a ball which is greeny blue, but it’s vague which.I offer you a bet: pay me £1, and if the the ball is green, I’ll give you somethinggreen for your collection, and if it’s not I won’t. The way to guarantee thatyou’ll get a green thing iff the ball is green and you accept is obvious: I’ll giveyou the borderline green/blue ball if you accept. If you also cared intrinsicallyabout owning blue things, I could offer the same bet on the ball being blue. Inthis case, you’re guaranteed to get something you want for £2 (a bargain, wemay suppose), because you know the ball is either green or blue. However, thisgenerates the same problem we discussed, since the decision theory recommendsyou reject both bets.

One could just insist that such a desire is incoherent: if you are certain

4. Rejectionism 29

p is vague, you should automatically assign p 0 utility - as prescribed by theexpected utility equation. Unfortunately, this suggestion turns out to be utterlyimplausible even if you do assume that you shouldn’t have vague desires of thekind described above. Suppose that the set up is as above, except that insteadof a ball I have a five pound note. Five pound notes, by the way, are borderingbetween green and blue in colour. The bet is as follows. You give me £1 toplay. If the five pound note is green, your reward will be £5 and a green pieceof paper, if it’s blue you’re reward will be £5 and a blue piece of paper. Thepenumbral connection is ensured by giving me the £5 note. As per usual, weought to reject this bet if we abide by the the expected utility equation, anddescribe the decision problem in the natural way.

Green BlueAccept £5 and a green thing £5 and a blue thingReject 0 0

Intuitively, however, this is absurd. You’ll end up having made £4 profit, soyou should definitely accept. The claim that you should assign p 0 utility ifyou’re certain p is vague is just implausible, for it entails you should also assign0 utility to ‘p∧ receive £1,000,000’ since you should be certain this conjunctionis vague also.

4.3 Deontic logic

We have seen that Field’s probability calculus gives strange prescriptions onhow we ought to value actions. Deontic logic gives us a more coarse grainedclassification of actions, allowing us to state things like ©p, read as ‘it ought tobe the case that p.’

The puzzles above translate into seemingly paradoxical statements aboutthe logic of ‘ought’. For example, failures of partition invariance could be takento demonstrate the existence of cases where you ought to accept a bet on pand a bet on ¬p simultaneously, but you ought to reject them separately. Thecombination ©(p ∧ q), ©¬p and ©¬q, however, is inconsistent in a minimaldeontic logic.6

In fact, if we apply deontic logic systematically, we can show that almostevery action is impermissible, unless you don’t care about anything. For everyprecise proposition p, perhaps an action that will guarantee you £100000, canbe partitioned into two vague propositions. For example, if q is vague, p ≡((p∧ q)∨ (p∧¬q)), both disjuncts of which are vague. Assuming©¬(p∧ q) and©¬(p∧¬q), due to their vagueness, it follows that©¬((p∧q)∨(p∧¬q)) ≡ ¬©p.

6 More precisely, it is inconsistent in D, which is K + 2p→ 3p.

4. Rejectionism 30

4.4 Two kinds of expectation

Field notes that his probability calculus appears to accord with Dempster-Shaferprobability theory. A Dempster-Shafer probability function (henceforth, a DS-probability function) is one that obeys the following conditions7:

1. Bel(⊥) = 0

2. Bel(>) = 1

3. Bel(⋃n

i=1 Ui) ≥∑n

i=1

∑I⊆n+1:|I|=i(−1)i+1Bel(

⋂j∈I Uj)

If P is a set of classical probability functions, then Bel(p) = inf(µ(p) | µ ∈ P)is a DS-probability function. However, not all DS-functions can be generatedin this way. Let us run with the assumption that all Fieldian probabilities aregenerated in this way. After all, we may assume an agent has classical credencesassigned unproblematically to each set-of-worlds proposition, and for each vagueproposition (set of world/precisification pairs) one can find a corresponding setof classical probability functions, based on the agents classical credence in eachprecisification of the vague proposition.

If the agents credential states are structured this way, we have more informa-tion to play with than with the original Bel(·) function. There are two naturalnotions of expected utility.

• Exp1(p) :=∑

s inf(µ(s) | µ ∈ P).u(p ∧ s)

• Exp2(p) := inf(∑

s µ(s).u(p ∧ s) | µ ∈ P)

Exp1 corresponds to our original definition of expectation. Exp2 is different, andit is easy to check that it does intuitively better in the cases we have considered.However, Exp2 makes use of information over and above the agents Fieldiancredence. It appeals to the information encoded in a set of probability func-tions, which were in turn characterised in terms of classical probabilities withrespect to different admissible precisifications of the language. Such an appealto admissibility completely undermines Field’s project of providing a reductiveexplanation of indeterminacy. Indeterminacy, and indirectly, admissibility, weresupposed to be defined functionally by their interaction with uncertainty.

7 The last scary-looking condition would just be a generalised statement of finite additivityif ≥ were replaced with =. In the = case the generalised version is usually stated, and isentailed by its n = 2 instance, but this entailment doesn’t hold for the ≥ version, which iswhy the complicated looking formula is necessary.

5. VAGUENESS IN EPISTEMIC ATTITUDES

In this chapter I am going to study vagueness in epistemic and doxastic attitudereports, by which I mean to include locutions such as ‘Hector believes that’,‘Hector is certain that’, ‘Hector knows that’, ‘Hector has a credence larger thanα that’, and so on. Given an ideally rational agent, could it ever be vaguewhether she is certain that p?

This is relevant to our investigation because it provides a plausible way forsomeone to hold on to the view that vagueness doesn’t involve uncertainty. Onecould maintain that, when you know the underlying facts and p is vague, you’redeterminately not uncertain, i.e., determinately, you’re certain in p, or you’recertain in ¬p, without being committed to an arbitrary sharp drop in credenceover a Sorites sequence of propositions. There will be a sharp drop in credence,but it’s indeterminate where that drop will be. Similarly, you are either certainin p or certain in ¬p, it’s just it’s indeterminate which. Let us call an attitudelike this transparent. That is to say, an agents attitude, E is transparent ifwhenever it’s vague whether p and the agent knows all the relevant precisefacts, Ep or E¬p obtains, but it’s indeterminate which.

The thesis that doxastic attitudes are transparent is a fairly strong one. Inthe first two sections of this chapter I shall concentrate on the much weakerclaim that it can ever be vague what an ideally rational agent believes, or vaguewhat their credences are. Surprisingly there are strong reasons to think thatthere couldn’t ever be vagueness in an ideally rational agents doxastic attitudes.In the latter part of this chapter I shall concentrate on the transparency thesis.

5.1 Epistemic Sorites

Surely, you might object, it is obvious that it may be vague whether a personbelieves or is certain in something - especially when the something is itself vague.Why should it be any different for an ideally rational agent? The considerationsin favour of such vagueness do not seem to depend on whether the agent inquestion is ideally rational.

Let us suppose our perfectly rational hairologist, Hector, is presented with aSorites sequence. The sequence begins with clearly bald people, and ends withclearly non bald people, and Hector knows the exact hair status of each personin the sequence. Consider, now, the corresponding epistemic Sorites:

• Hector is certain that the first person in the sequence is bald.

5. Vagueness in epistemic attitudes 32

• If Hector is certain that the nth person in the sequence is bald, he iscertain that the n+ 1th person in the sequence is bald.

• Hector is certain that the last person in the sequence is bald.

This argument seems to be Sorites on the predicate ‘Hector is certain that xis bald’. The conclusion of this Sorites is determinately false, and the firstpremise determinately true, and given that Hector is an expert whose credencesclosely track the truth, the middle premise seems just as compelling as in thethe corresponding Sorites, using the same line-up of people, on ‘bald’. As faras susceptibility to Sorites are good tests for vagueness, this establishes that‘Hector is certain that x is bald’ is vague.

To what extent does this show that the doxastic operator, ‘Hector is certainthat . . . ’, is vague? There is a parallel Sorites sequence of propositions: theproposition that x is bald, for each x in the original sequence. However, theterms of this Sorites sequence, unlike the terms of the original Sorites, are vaguewhich makes it hard to infer anything about the vagueness of ‘Hector is certainthat’.1 Indeed, the inference from it’s vague whether Hector is certain thatx is bald, to for some p, it’s vague whether Hector is certain that p is onlyvalid on a semantics where the objects of propositional attitudes are (at leastas fine-grained as) sets of world-precisification pairs, rather than sets-of-worldspropositions. This is interesting, because it is most natural for the transparencytheorist to go for the latter semantics.

Either way, I see no reason to suppose that vagueness in the things believedinduces vagueness in doxastic attitudes.2

5.2 Indeterminacy and rational action

I shall now consider a positive argument that it couldn’t ever be vague whatcredence an ideally rational agent has. I shall make essential use of the fact thatone ought to maximise expected utility.

©(MaxEU(A)→ doA) (5.1)

The basic idea is that in a case where one has indeterminate credences, therewill be decision problems where it is indeterminate which action maximisesutility. In such circumstances none of the available actions will be rationallypermissible. Since being rational requires the disposition to do only rationallypermissible actions, the mere possibility of such a decision problem is enoughto undermine perfect rationality.

1 For an analogy, “(a) the largest small number is less than 1,000,000, (b) if the largestsmall number + n is less than 1,000,000 then the largest small number + n+1 is smaller than1,000,000, (c) the largest small number + 1,000,000 is not less than 1,000,000” is a Soriteson ‘less than 1,000,000’, but no-one in their right mind would infer from this that ‘less than1,000,000’ is vague.

2 This is, of course, perfectly compatible with there being vagueness in the doxastic attitudereport.


Here’s an example. Suppose that action A would have highest expected util-ity if your credence were 0.7 in p and would not for any other credence, and thatit is indeterminate whether your credence in p is 0.7. It is thus indeterminatewhether or not A maximises utility. Should you do A? This question doesn’tseem to have a determinate answer.

Already we can see this is bordering on unsafe ground. For most decisionproblems, the available options simply won’t admit borderline cases. It’s simplynot possible to indeterminately place a bet, or indeterminately push a button,and even if it were, there would have to be the right penumbral connectionswith your beliefs. According to these views, however, this is exactly what youare required to do. The thought that ‘ought implies can’ thus seems to rule outthese views.

One might object at this point that to get this result, it has to be possiblefor it to be indeterminate which action has highest expected value. It seemsplausible, however, that any view in which beliefs are transparent should be onein which desires are too: if someone desires a glass of water that is pretty full,and they are given a glass that is two thirds full, it is indeterminate whethertheir desire has been satiated. If this were the picture, could it not be that it’snever the case that it’s indeterminate which action maximises utility? That is:do penumbral connections between desire and belief hold in such a way that theexpected utility of each action is never vague.

The answer is no, unless we place arbitrary restrictions on what one is ra-tionally allowed to care about. Consider a glass that is 65% full of water.

The glass is pretty full The glass is not pretty fullA 100 0B 0 100

In the example above you might ask how it could be possible to receive 100utils precisely if the glass is full. It would beg the question to insist that youcan (falsely) believe you are faced with such a decision problem, because, forexample, to do that you would involve believing: “if you chose A, you wouldreceive 100 utils iff the glass is full, and 0 otherwise”. But, in normal situations,this sentence is vague, so prima facie, it should at best be indeterminate whetheryou believe it.

The decision problem I have in mind is the following. Option A correspondsto giving the 65% full glass to person A, and B to giving person B the same65% full glass. You would really like it if A got a pretty full glass, and reallylike it if B got a glass that wasn’t pretty full. It is thus indeterminate whethergiving the 65% full glass to A (or B) satisfies the desire. On precisifications inwhich the glass is pretty full, option A gets 100 utils in the ‘full’ column, andyour credence is 1 that the glass is pretty full and 0 that it’s not.

The glass is pretty full The glass is not pretty fullA 100× 1 = 100 0× 0 = 0B 0× 1 = 0 100× 0 = 0


The expected utility of A is 100 + 0, and B is 0 + 0 on these precisifications.On precisifications on which the glass isn’t pretty full option B get’s 100 utilsin the ‘not full’ column, and your credence is 1 that the glass isn’t pretty full,and 0 otherwise:

The glass is pretty full The glass is not pretty fullA 100× 0 = 0 0× 1 = 0B 0× 0 = 0 100× 1 = 100

The expected utility of A is 0 + 0, and B is 100 + 0 on these precisifications.It’s thus indeterminate whether or not A or B maximises expected utility.

In short, there just does not seem to be any coherent prescriptions on howyou ought to act if it were indeterminate what your credences are. For assumingI can’t give the glass to both of them the relevant options are: (a) give the glassto neither, (b) give the glass to A, (c) give the glass to B, (d) perform anaction that is indeterminate between giving the glass to A and B. If I wereto do (a) I would determinately not have maximised utility. This seems bad.If I were to perform (b) or (c) it would be at best vague whether I’d donethe right action. Finally I could do (d). This would determinately maximiseexpected utility provided the action I performed was a giving to A on exactly theprecisifications that my credence that the glass is pretty full is 1, and a givingto B on the rest. But it is clear in this case that no such action is possible,unless one posited a radical and weird kind of ontic vagueness that one couldcontrol. Since (d) is impossible, it cannot be what I ought to do.

5.3 Indeterminacy and rational obligation

Let us return to the transparency thesis. I am mainly concerned here with theview that when an ideally rational agent believes/is certain/knows it is vaguewhether p, she won’t be agnostic/uncertain/ignorant in p. But how does thisrelate to what I, as an ordinary person, ought to do?

Let us consider an agent who has no relevant worldly uncertainty, but is notnecessarily perfectly rational. There seem to be at least two relevant principlesgoverning what the agent ought to believe, if she believes p is vague:

©B∇p→©∇Bp (5.2)©B∇p→ ∇©Bp (5.3)

Given that the agent ought to believe p is vague, then the first principle statesthat she ought to be such that it’s indeterminate whether she believes p, andthe second states that it’s indeterminate whether she ought to believe p. It isinteresting to see how these connect up to the truth norms on belief. For exampleWilliams, in a recent talk, suggests the truth norms may commit you to either(5.2) or (5.3), depending on how the truth norm is concieved: ∆© (Bp ↔ p)or ©∆(Bp ↔ p). However, one normally concieves of a truth norm as sayingsomething along the lines that one ought to believe p only if p. The biconditional


versions of the truth norm just alluded to are implausible, but are are essentialif we wish to get (5.2) or (5.3).3 Finally, it is not clear that the sense of ‘ought’being invoked here is relevant to our discussion. It is clear that in the senserelevant to decision theory, it would have been irrational for the cavemen, giventheir evidence, to have believed that spacetime was non-euclidean. For now, Ishall simply set aside the truth norms, and work directly with (5.2) and (5.3).

Should one hold both (5.2) and (5.3) together, or is it just one of the two weshould accept? The different combinations of the consequents hold, when youought to believe p is vague, gives rise to several views and a bunch of puzzlingquestions. For example, suppose you hold the first principle (5.2), but not thesecond (5.3), so you ought to indeterminately believe p whenever you ought tobelieve that p is indeterminate. It’s certainly not permissible to determinatelybelieve p, that would lead to inconsistency in a weak deontic logic, but is itpermissible to believe p? This would mean that it’s permissible that you believep and it’s indeterminate whether you believe p (P(Bp∧∇Bp)), so an essentiallyvague sentence is permissible.4 Suppose instead that it wasn’t permissible tobelieve p, so you ought not believe p. That would mean that its obligatory thatyou indeterminately believe p and not believe p. This is again necessarily vague,so it seems that a necessarily vague sentence can be obligatory. I have no knockdown objections to such combinations of views, but it’s certainly quite puzzlingwithout some further explanation of what’s going on.

What about accepting the second but not the first principle? Suppose it’sindeterminate whether you ought to believe p. Is it permissible to determinatelybelieve p and permissible to determinately believe ¬p? There is perhaps moreto be said for this view than the previous view. For example, if you believep is vague, there is simply no answer to the question of whether you ought tobelieve p. It is indeterminate. However, if I do believe in p, or its negation forthat matter, I have not done anything wrong.

There is a prima facie puzzle with this view, in that if it’s indeterminatewhether you ought to believe p and it’s permissible to believe p, you have aFitch style sentence: ∇¬©¬Bp ∧ ¬©¬Bp (for reason, see footnote.5) So thetheory is a priori indeterminate. How could we ever rationally come to believea theory that we can be certain is at best vague? Of course, the theorist will say,“by the theories own lights the theory is acceptable, because it is permissible tobelieve things you believe to be vague. In this case it is perhaps even mandatory,given the strength of unspecified-philosophical-argument”. That is all very well,the theory is internally coherent, but it is hard to see how one could come tobelieve such a theory. One has to already believe it to believe it, as it were.

Rather than face these questions, it is more economic, I think, to just acceptboth (5.2) and (5.3).

3 That is, the single direction truth norms are satisfiable by remaining agnostic about p.4 Any sentence of the form∇p∧p is necessarily vague; the argument is completely analogous

to Fitch’s paradox, (see [3].)5 By assumption we have ©B∇p, which gives us the equivalent ©B∇¬p, so by the second

principle (5.3) ∇©¬Bp, which is equivalent to ∇¬©¬Bp. We may then conjoin this withthe permissibility of believing p.


5.4 Dorr and Barnett

In [5], Dorr defends a view on which a sufficiently informed person presentedwith a borderline case of baldness will typically be in a state which is indetermi-nate between knowing that they are bald and knowing that they are not bald.Similarly, Barnett [1] defends similar view, focusing on credences.

Since one typically finds oneself in such a state, should we infer that this isthe kind of state we ought to be in when we learn that p is indeterminate - or atleast, that it’s permissible? Is it indeterminate whether we ought to believe p,or is it just impermissible to believe p or believe ¬p, just as it is impermissibleto be agnostic?

Assertion

Which ever combination of (5.2) and (5.3) fits with this theory and the norms ofbelief and knowledge, there are some norms which definitely are not transparentwith respect to vagueness on Dorr’s account. For example ‘ought to assert p’ isnot vague when p is vague - indeed, you ought not assert p, or ¬p, when p isknown to be vague.

The requirement that you remain silent in cases where you know p is vagueis similar in many ways to the requirement that you remain silent in cases whereyour are ignorant about whether or not p obtains. However, Dorr stresses, weare not ignorant in such cases - and the explanation for why we ought neitherto assert p or ¬p is very different from the explanation for why we shouldn’tassert p or ¬p when we are ignorant about p.

An example he considers is a respondent who can see a glass of water justwell enough to see it is between 60% and 70% full. Questioner 1 asks whetherthe glass is pretty full, and questioner 2 asks whether the glass is at least 65%full. In both cases the respondent is reluctant to answer ‘yes’ or ‘no’. In the firstcase the reason respondent is reluctant to say yes or no is that such responsesindicate that the glass has a lot of water (roughly, more than 70%), or notvery much (roughly, less than 60%). Both the ‘yes’ and ‘no’ responses wouldsignificantly decrease the questioners credence in the propositions that the glassis x% full, for each 60 ≤ x ≤ 70. These are exactly the propositions respondentwants questioner 1 to have high credence in.

For questioner 2, respondent is reluctant to say yes or no because she hasa low credence, roughly a half, that the glass is 65% full or over. Saying yesor no is risky: it could well be over 65%, it could well not be, and answeringeither way will lower questioner 2’s credence in a proposition about the amountof water in the glass that could easily, for all she knows, be false, and wouldcertainly raise her credence in hypothesis that you can see the glass well enoughto tell.

Now it is somewhat worrying that the pattern of behaviour that silence dueto vagueness and ignorance produce are almost exactly the same. Dorr notesthat certain responses are more appropriate in the cases involving vagueness;for example one might say ‘it’s hard to say’ or ‘it’s borderline’. Undoubtedly


this distinguishes vagueness from cases of ignorance more generally, but it doesnothing to convince us that vagueness is not a special kind of ignorance, withthese responses reserved for expressing ignorance of that particular kind. Itseems to me that in some cases it can be appropriate in this situation to say ‘Idon’t know’. Although in many other circumstances you might be required tosay ‘it’s borderline’ - the maxim that you ought to be as informative as possiblerequires this, for example - this is not to say that saying ‘I don’t know’ in suchcases isn’t to speak truly but inappropriately.

The crucial difference, it seems, is that the explanation of why we oughtn’tsay ‘yes’ or ‘no’ in such cases is different. But it is still puzzling that knownvagueness and known ignorance in p are decisive reasons not to assert p or ¬pin exactly the same circumstances, and that despite the fact that they needn’talways come together, every other behavioural aspects of such situations are thesame.

Things would be much worse if the explanations in both cases were only su-perficially different. An initial disanalogy between questioner 1 and questioner2 is that questioner two at least thinks it is possible that the respondent (de-terminately) know the answer for certain if it were between 60% and 70% full.Suppose for some reason questioner 2 knew that the respondent couldn’t nar-row the amount of liquid in the glass to a range more precise than 10% width- i.e. the questioner knows that respondent’s eyesight has an error margin of+/- 5%, and that she knows respondent knows this. Once this knowledge is inplace, it is plausibly analogous to question 1 even if one assumes the ignorancetheory of vagueness. For on that theory questioner 1 may know for sure thatthe respondent wouldn’t know if the glass was pretty full if it were between 60%and 70% - it’s an unknowable fact.

Giving questioner 2 knowledge that the respondent’s perceptual capabilitiesare limited is like the knowledge questioner 1 has that the respondents abilitiesto determine the truth values of borderline cases is limited. The two exam-ples are now comparable. However, with the superficial difference removed, theexplanations become similar. When there is a certain degree of common knowl-edge in the fact that I couldn’t know whether the glass was at least 65% full if itwere between 60 and 70% full, then if I were to say ‘yes’ questioner’s credenceswould all be distributed over the >70% hypotheses, and if I were to say ‘no’questioners credences would all be distributed over the <60% hypotheses.

The difference in explanation for why you ought to refrain from assertingor denying that the glass is pretty full/more than 65% full collapses when thequestioner knows your limitations. If the ignorance theory were in fact true, themost substantial difference between these cases would be that one can usuallyknow, without knowing anything about the respondent, that the respondentcouldn’t know whether the glass was pretty full, if it were between 60 and 70%full. But one could imagine an exceptional case in which the questioner (mis-takenly) believed the respondent could tell what the truth values of borderlinesentences are. If this were common knowledge between the questioner and therespondent, the respondent should be unwilling to say ‘yes’ or ‘no’ if asked ifthe glass was pretty full, because, ex hypothesi, she doesn’t know whether it’s


pretty full, and it’s a risky answer. So it seems that one can modify both expla-nations of why one ought not affirm or deny each question so that it fits withthere being vagueness or ignorance in the answer.

Norms of assertion

It seems that on this view, ‘ought to believe’ is transparent with respect tovagueness. It is clear, however, that ‘ought to assert’ is not transparent - whenp is vague, you oughtn’t assert p, and you oughtn’t assert ¬p. Thus the followingsituation will be quite common: that it is indeterminate whether you ought tobelieve (know) p or ¬p, but you oughtn’t assert p and you oughtn’t assert ¬p.

On the other hand, it seems to be widely held that knowledge or belief is thenorm of assertion (and if not either of these, at least some similar propositionalattitude that falls under the transparency theory.) This commitment can beexpressed as follows:

You ought to: assert p (if and) only if you know/believe p (5.4)

I take it the biconditional version is plausible in circumstances where you areable and have a decisive reason to inform your audience about p. If so, (5.4)seems to be true and assertable (in particular, it isn’t “true but vague”.)

However, if p is vague, it’s indeterminate whether you ought to believe orknow p, but determinate that it’s not the case that you ought to assert p. (5.4)is thus indeterminate. Of course, this means that it’s at best indeterminatewhether you ought to accept (5.4) as a theory of assertion. This, however,seems to be too weak: we determinately accept (5.4) as determinate since as wenoted, it is typically taken to be assertable.

Action

We have seen that the requirement to perform certain kinds of actions, suchas assertion, fail to correlate penumbrally with requirements on our beliefs.Our beliefs and credences, however, are supposed to inform our actions moregenerally. We have seen already that positing these connections might requireus to do impossible things, such as indeterminately perform actions, if yourcredences are indeterminate.

So this raises a natural question for Dorr: how should decision theory lookon this view? If it’s vague whether A or B maximises expected utility, shouldyou refrain from performing A and B? This appears to be Dorr’s answer whenA and B correspond to assertions of p and ¬p respectively, and where conveyingcorrect information is all that is valued. It seems natural to extend that thoughtto this more general setting. This at least avoids the problems raised in §4.

In the case described above - where either A or B maximises expected utilitybut it’s indeterminate which - performing neither A nor B determinately doesnot maximise expected utility. Thus one seems to be required to violate the


central maxim of decision theory - that one ought to maximise expected util-ity.6 Even if you did perform one of A or B, it will be indeterminate whetheryou’ve violated the central maxim. It seems that if the central maxim is at bestindeterminate, there isn’t much of a decision theory to be had after all.7

This deficiency naturally leads to further worries. Credences, belief, and thelike, are characterised functionally by the role they play with respect to ourdesires, and in informing our actions. For credences and desirability, that roleis given by the expected utility equation. The significance of the representationtheorems is that for people with rational preferences, role of desirability andcredence are filled uniquely.

It is hard to see what it would mean for belief or credence not to fit this role;after all, if credence is not the thing that plays the expected utility role, what isit? One might certainly allow that there is a property, F (p), such that whenevera well informed person is confronted with a vague proposition, p, the agent goesinto a state which is indeterminate between F (p) and F (¬p). This property F ,however, cannot play the functional role of belief (or knowledge or certainty) -the fact that determinately F (p) ∨ F (¬p) usually holds does not vindicate theview that vagueness needn’t involve agnosticism, ignorance or uncertainty.

Epistemicism

Dorr offers the transparency of knowledge with respect to vagueness as a wayof distinguishing the role of indeterminacy in supervaluationist theories fromits role in epistemicist theories. Thus, in regard to the problem considered inthe last chapter, the supervaluationist may simply deny that vagueness involvesignorance. But the crux of the problem was not so much distinguishing theview from epistemicism, but explaining why supervaluationists who accept theT-schema aren’t committed to whatever it is that is so puzzling about epistemi-cism.

Dorr claims that it is a commitment to sentences like the following thatcapture the counterintuitive commitments of epistemicism:

This two-thirds-full glass is either pretty full or it isn’t, but nohuman being can know which it is.

(5.5)

One of the tall members of this department is shorter than allof the others, but we will never be able to find out who it is.

(5.6)

Each person has a mass in grams (to the nearest gram), but wecan never know what it is, no matter how accurately we weighthem.

(5.7)

It is not so much the commitment to sharp boundaries, as dictated by classicallogic, but the commitment to ignorance that is so counterintuitive.

6 Modulo ammendments for Newcomb problems.7 Given the story so far, we know that the central maxim can’t be asserted with the rest of

the theory.


I personally find the following sentence far worse than the ones above:

This two-thirds-full glass is either pretty full or it isn’t, and infact, I know which it is.

(5.8)

As far as counterintuitiveness goes, it is the first conjuncts in (5.5)-(5.7) thatseem to be doing the damage - the second conjuncts, if anything, lessen theblow.

One could imagine a conversation in which I claim, rightly according to theview, that I know whether the glass is pretty full. This naturally invites theresponse ‘Well, which is it? Is it pretty full, or isn’t it?’. Clearly, I can’t answerthis question. But why can’t I? Intuitively I want to say that this is because Idon’t know the answer. I can’t say this, because I do know the answer - andeven so, this response is, allegedly, never appropriate in borderline cases.

If asked why I can’t answer, even though I claimed to know whether ornot the glass is pretty full, I shouldn’t retract and concede that I didn’t in factknow, I should instead say something like: ‘I’m not going to answer, not becauseI don’t know, but because if I did, I’d raise your credence in false propositions.’This makes it sound as if you could answer if you wanted to, but you’re notgoing to for pragmatic reasons. In reality, you couldn’t even if you wanted to,and it is hard to see what the explanation for this fact is if it’s not because youdon’t know.

Moorean sentences

There are propositions which are consistent, but which one can’t coherentlybelieve. For any p, the proposition that p but I don’t believe that p seems toform one class of such examples. In the context of vagueness we could considersentences of the form ‘p but it’s indeterminate whether or not p’.8 Not only aresuch sentences unassertable, it seems like it would be incoherent to accept sucha sentence. One can’t simultaneously believe p, while at the same time admitthat p is indeterminate. Determinately, one shouldn’t believe sentences of theform ‘p but it’s indeterminate whether p.’

However, for transparency theorists, this is not true. When you are certainthat p is indeterminate, it is typically indeterminate whether you are certainin p. Thus it should be indeterminate whether you are certain that p and it’sindeterminate whether p.9 This seems wrong: whether or not you ought to becertain in this case isn’t borderline - it’s just plain false.

Belief in sentences

According to the view I have been espousing, you may be uncertain whetheror not the glass is pretty full, even if you know it’s exactly two thirds full, and

8 Supervaluationists such as Dorr take these sentences to be consistent. Others deny eventhat they are consistent.

9 One can get this by assuming certainty determinately distributes over conjunction, orsimply by noting that you should be certain that the conjunction itself is vague.


you know this is a borderline case. On this view the objects of belief are likesentences in many ways. For example, they are the sorts of things that canbe vague - perhaps they are even sentences of the language of thought. Theopposing view, in which beliefs are always in sets-of-worlds propositions (which,in turn, are precise) leads naturally to the transparency view, if one builds upbelief reports compositionally: ‘Hector believes that Cedric is bald’ is true iff theproposition expressed by ‘Cedric is bald’ belongs to Hector’s belief box. Whenit’s indeterminate which proposition ‘Cedric is bald’ expresses, and Hector onlybelieves some of these propositions, it will be indeterminate whether Hectorbelieves that Cedric is bald.

However, there is at least a derivative sense in which the transparency viewcan make sense of credences in sentences (or whatever the kinds of things thatcan be vague are.) At the beginning of his paper Dorr considers two creatureswho speak a very primitive vague language. A goes about searching for fruittrees. When he comes to a fruit bearing tree he lets out a hoot or a yelpdepending on whether there is more or less fruit on the tree, to communicate toB which. A feature of this situation is that A has a decisive reason to do oneor the other when he reaches a fruit tree, otherwise B wouldn’t even know if itwas a fruit bearing tree at all.

Interestingly, B’s credence distribution over the fruit number hypotheses issmoothly distributed when hearing a hoot or a yelp, since B might not knowhow A reckons B will update credences in response to hearing a hoot or yelp.This is interesting, since according to the view, B’s credence after updatingon the fact that there are hoot-many fruit should have a sharp distribution,although it is indeterminate where the sharp change is located. The answer tothis seeming tension is that B is not really updating on the proposition thatthere are hoot-many fruit, but rather on the proposition that A asserted thatthere are hoot-many fruit.

If one wanted to make sense of credence and belief in vague things, ratherthan as vagueness about which precise things are believed, it would be quitenatural to look at one’s credence distribution after hearing that there are hoot-many fruit on the tree. This produces B’s “ersatz-conditional-credence” onthere being hoot-many fruit. One could then develop two parallel theories ofcredence - Dorr-credence, and ersatz-credence. To what extent you think thatDorr-credence and ersatz-credence are important will depend on the details.But to the extent that one plays the role of genuine belief, I hope to have shownthat it is something more like ersatz credence that plays the correct functionalrole when it comes to informing our actions, interacting with our desires, andthe norms of assertion.

6. VAGUENESS AND UNCERTAINTY

We have studied several accounts of belief in vague contexts that entail that onecannot be genuinely uncertain once one knows sufficiently many facts. It is nowtime to consider the view that these scenarios involve genuine uncertainty.

[In section §?] We shall start off by looking at some evidence that someonein Hector’s epistemic position should be uncertain about whether Cedric is bald.Although I take these arguments to show that in general, we are inclined to talkabout Hector as if he were uncertain about something, it requires more argumentto show that the uncertainty in question is genuine. To answer this questionwe need to get a firmer grip of what it means to be uncertain, as opposed tobeing, perhaps, in some non-doxastic state, or simply behaving like you areuncertain. Getting clearer on the nature of genuine uncertainty will be the taskof the section 6.5. Finally, we will need to say something about how one shouldadjust the strengths of your beliefs, once you learn that something is vague. Onemight accept that in these cases, you may still be genuinely uncertain about p,yet maintain that the best way to represent the uncertainty would be in termsof a non-classical or indeterminate probability function. In §?++ I argue thatdegrees of belief in vague contexts ought to be classical probability functions.

6.1 Behaviour

Competent speakers of English will typically refuse to assert or deny a sentencewhen it is indeterminate whether it is true. A paradigm case is that of vagueness.Consider a sequence of people with successively larger amounts of hair on theirhead (a Sorites sequence.) Usually one will accept, and be willing to assert,that people with virtually no hair are bald, and will deny that people with afull mop of hair are bald. However, there will inevitably come a point alongthe sequence at which our speaker will find neither response appropriate. Suchbehaviour is characteristic of uncertainty, and in a case like the above it seemsnatural to say that one is uncertain whether the person in question is bald.

This is the behaviourist argument for uncertainty. When someone is certaina proposition, p, is vague, they will typically behave as if they were uncertainabout p. For example:

They will not assent to p (they will not assent to ¬p.) (6.1)They will neither be willing to assert nor deny p. (6.2)They will not accept bets with arbitrarily high odds on p (theywill not accept bets with arbitrarily high odds on ¬p.)

(6.3)

6. Vagueness and uncertainty 43

This behaviour, it seems, is characteristic of someone who is uncertain aboutp. But the jump from behaving as if one is uncertain about p, to actually beinguncertain about p is a big jump to make! After all, I do not wish to endorse fullblown behaviourism about mental states like belief and uncertainty.

That said, behaviour is a good way - and for those of us who lack the relevantscientific knowledge and equipment - the only way to discover the goings on inother peoples mental lives. Furthermore, to be told that someone behaving asdescribed is acting appropriately does tell us something about her mental life.For as far as we have intuitions about how someone would behave in this sce-nario we have intuitions about how they ought to behave - and those intuitionssuggest that the behaviour in (6.1)-(6.3) is only permissible accompanied withuncertainty about p. The inference that goes from the fact that someone is be-having appropriately, and that this kind of behaviour is only appropriate if oneis uncertain about p to that person being uncertain about p, does not assumebehaviourism, and its premisses have direct intuitive appeal.

Of course, these pretheoretic intuitions are only suggestive. After sometheorising, we see distinctions where there previously were none - being certainthat ¬p, being anticertain that p, being such that it is vague whether you arecertain that ¬p or even just being ‘ersatz’ certain, in some sense to be spelledout, that ¬p are all importantly different. We must tread with care.

Our starting point shall be a very crude piece of conceptual analysis:

Someone is uncertain about p just in case they’re neither certainthat p nor certain that ¬p.

(6.4)

The analysis is simple, and intuitive. To reject it, one must first of all providea compelling case where it fails, and show where the mistake is being made.

A class of cases where (6.4) seems to fail are cases of belief in vague propo-sitions of the kind we have been considering here. Hector, recall, knows everyfact there is to know about the status of Cedric’s head - he has ruled out all butone maximally specific state for Cedric’s head to be in. However, given Hector’sreluctance to commit to any position on Cedric’s baldness, we naturally say thatHector is not certain that Fred is bald, nor is he certain that Fred is not bald.

This view, I take it, is compatible with a view in which Hector is not gen-uinely uncertain, such as Hartry Field’s view, or Schiffers. For Field Hector,given he is a rational believer, is neither certain that Cedric is bald, nor certainthat he’s not bald. But he is not genuinely uncertain either, because he is an-ticertain in both.1 For Field we must reject the analysis; the mistake was toidentify anticertainty in a proposition with certainty in its negation. A betteranalysis, perhaps, would identify uncertainty in p with being neither certainnor anticertain that p. Schiffer, on the other hand, holds that Hector is neithercertain that Cedric is bald, nor certain he’s not bald, but maintains that Hectoris not genuinely uncertain: Hector is in some sui generis state which is not thesame as real uncertainty.

1 The claim that someone is anticertain that p is to the claim that she has credence 0 in p,as the claim that someone is certain that p is to the claim that she has credence 1 in p.


Yet other views accept (or may accept) the analysis, but reject the claimthat Hector, when he is acting rationally, is neither certain he’s bald nor certainhe’s not. For example, it may be supertrue that Hector has certainty in p or inits negation (that is, it is supertrue that Hector is not uncertain, assuming theanalysis) but indeterminate which he is certain in. The reason Hector typicallyacts like he is uncertain is because he is not acting rationally; to act rationally,would involve performing actions which are indeterminate between being goodwhen p is true and good when ¬p is true. If it’s not vague that doing A isgood when p, and bad when ¬p, then this would require one to indeterminatelyperform A - a difficult feat indeed!

It is important that we bear these thoughts in mind when we consider thebehaviourist argument for uncertainty: we must make sure that when someonebehaves as if they were uncertain about p that (a) this is not because theycannot behave as they ought to if they were certain about the truth value of p,(b) that this behaviour could not also be explained by anticertainty in both pand ¬p, or some other sui generis kind of state.

Taking these intuitions at face value, we can put (a) to one side. I claim wedo have direct intuitive evidence that Hector is behaving as he ought to givenhis epistemic position regarding Cedric’s hair. Make of this what you will; for amore detailed discussion of this view I shall simply refer you back to chapter 5.

As for (b), it certainly does seems a live possibility that being in somenovel doxastic state, or being anticertain in both p and ¬p might explain thebehaviour displayed towards p. However the views under consideration, for allwe’ve said about them so far, are completely neutral on how one should act insuch circumstances. A theme we shall return to in the following sections is howthese views can generate a sensible decision theory. We shall see that eitherthese decision theories will deliver the wrong prescriptions on how to act, or, byconsidering the causal functional connections that desires and beliefs stand toeach other in, that one is actually just describing genuine uncertainty after all.

6.2 Vague evidence

It is commonly claimed that sentences known to be are vague are unassertable.There is a sense in which this is not true: you may know the exactly how faraway the nearest post office is, 0.973 miles lets say, but if asked where the nearestpost office is it would be perfectly acceptable to say that it’s about a mile away,despite knowing that this is a vague proposition. Clearly, a sentence is onlyunassertable when it is in fact vague, and not when it is merely possibly vague.

The fact that you can permissibly assert vague sentences in the circumstancesdescribed above, allows me to acquire vague evidence by your testimony. I canupdate my credences on the vague information that the nearest post office isabout a mile away; which you were perfectly within your rights to assert eventhough it is possibly vague. In doing so I will get a smooth distribution ofcredences over the various hypotheses about the precise number of miles awaythe nearest post office is, with a peak around the 1 mile mark. If my credence


that the post office is 0.67 miles away, after hearing this information, is 12 then

my credence that it’s about a mile away and is 0.67 miles away (a propositionthat is at best vague) is going to be strictly between 0 and 1 - it’ll be half of myoriginal credence that it’s about a mile away.

Of course, this distribution of credences could equally well be explained bymy having updated on the precise proposition that you said that it was abouta mile away. The smooth distribution would then be explained in terms of thevarious hypotheses I have about how you think I would update my credences ifyou said that, and how you think I think you think I would update my credencesif you said that, and so on and so forth (for more detailed explanation of how thiswould work, see section one of Dorr [5].) However, this response is sometimesjust not available. For example, what if you hear someone hesitantly utteredthe sentence ‘it’s a mile away’, in such a way that it was vague whether theyhad said it. Or perhaps you vaguely remember it being a mile away. It’s vaguewhether you remember it’s a mile away, but this is still constitutes some kindof evidence. Similarly, after hearing you make the hesitant utterance I shouldcertainly rule out the possibilities in which it is 10 miles away. So it seems Ihave acquired evidence, but it is very difficult to see what precise evidence thiscould possibly be assimilated to.

6.3 Comparative judgements of uncertainty

Another observation that I take to suggest an uncertainty account of belief invague sentences, is that we can make comparative judgements of uncertainty,even when we know all the facts. That is to say, uncertainty in vague sentencescomes in different degrees very much like ordinary uncertainty. Furthermore,degrees of uncertainty in vague sentences interact with ordinary degrees of un-certainty in a systematic way.

The first couple of examples we shall consider involve scenarios where youare absolutely certain about all the facts.

First case: suppose Hector is presented with a Sorites sequence of his clients.The first person in the sequence is clearly bald, the last one is clearly not bald.He knows everything about each person’s hair situation to the finest detail.For convenience, also assume that hair number is the only factor relevant tobaldness. Now suppose we come to a stretch along the line-up where Hectoris no longer willing to commit either way over the baldness of people in thisstretch. These people are all clearly borderline bald, and Hector is certain ofthis.

Now consider two successive people in this stretch, A and B: A has n hairs,B has n + 1 hairs. It seems quite natural at this point to say that it is morelikely that A is bald than B - for after all - A doesn’t have as many hairs asB, and, ex hypothesi, hair number is all that matters to baldness. If offeredto bet on A or B over who’s bald, it would be sensible to go for A over B.You might reason as follows: provided A and B’s hair status remains the same,it’s an analytic impossibility that B is bald and A isn’t, since it’s an analytic


necessity that if someone is bald, then anyone with fewer hairs is also bald [sic.]You know you can’t have gone wrong by going for A over B.

Second case: you’re presented with a coloured patch that is borderlineblue/green, and your show a picture of Cedric, who strikes you as clearly bor-derline bald. Which should you be more certain in: ‘Cedric is bald’ or ‘Cedric isbald or the patch is green’. If offered to make a bet, again, it seems you oughtto go for the disjunction - after all, it’s an analytic necessity that if the firstsentence is true, so is the second, so it just does not seem to be analyticallypossible that you’ll regret your decision (given that all you care about is thepayoff from the bet, the patches don’t change colour, etc etc...)

What does this show? I claim that, at the very least, it shows you need notbe anticertain that p whenever you are certain that it’s vague whether p. Forthere would never be strict comparisons of uncertainty between propositions youare certain are vague - they would all be equally likely for you. But I also claimthat these kinds of comparative uncertainty claims in vague cases interact withordinary comparisons of uncertainty. This suggests that the kind of uncertaintyinvolved is of a similar nature to ordinary uncertainty about the world. Let usturn to mixed cases - cases where you are also uncertain about how things willturn out.

Third case: Richard Dietz [4] gives us the following example of how uncer-tainty about the world, and uncertainty due to vagueness might interact:

Consider a coin, where the heads side is clearly green and the tailsflip side a clear borderline case of being green. The coin is goingto be tossed, and our reliable informant tells us that the coin ismanipulated to fall always heads. However as we cannot rule outthat our informant is this time wrong, we cannot rule out that thecoin is in fact not manipulated and that it will fall next time tails.In this case, the hypothesis

(#) The outcome of the next coin toss will be a green side

is highly likely to be definitely true, but owing to a small chance -say 0.1 - that the informant is wrong after all, we should considerthe hypothesis as potentially indefinite after all. Suppose that giventhe coin is manipulated we are 100% confident that it will fall tails;and furthermore that given that the coin is not manipulated, we are50% confident that it will fall tails. Then we should be 95% confidentthat (#) is definitely true, and (as definite truth entails truth) atleast to the same degree confident that (#) is true. In that case,it would be unreasonable to accept a bet against (#) at a bettingquotient higher than 5%; that is a betting quotient lower than 95%regarding (#) could not be considered as fair.

Views that postulate two different kinds of credences - ordinary credences andvague credences (see, for example, Schiffer [19]) - have difficulties explaining howthe two kinds of probabilities interact. However, it still seems to be compatible


with the above argument that one should have credence 0 in both p and ¬pwhen one is certain p is vague. In the example above you’re not certain thatit’s indeterminate whether the coin will land green side up - in fact, you’re 99%certain its determinately true.

The next example we shall consider hopefully demonstrates that you can benearly certain in p, say 99% certain, even if you’re 100% certain that p is vague.

Fourth case: let n denote the largest small number modulo 100 (that is, nis the last two digits of the largest small number.) Thus 0 ≤ n ≤ 99 but it isvague which one n is. Indeed, assuming there are at least 100 borderline smallnumbers, for each 0 ≤ k ≤ 99 it is vague whether k = n. Suppose you also have100 balls in a bag numbered from 0 to 99, and you have just picked one out ofthe bag but have not yet seen which you’ve picked.

Now ask how much credence you should have that you are not holding thenth ball. Intuitively it should be pretty high; it’s determinately true that onlyone out of the 100 numbers is n, and that 99 of them aren’t identical to n. Ifyou could put a number to it at all, your credence should be 0.99 that you arenot holding the nth ball.

Furthermore, it’s clear to you that there are at least 100 borderline casesof smallness. So in this example you are certain that whatever ball you arecurrently holding right now, it is indeterminate whether it is the nth ball.

This example can be modified to show that one can be certain that p, evenwhen one is certain that p is indeterminate. For example, suppose that a uni-formly coloured patch is borderline between green and blue. Say that the colourof the patch is on the boundary between blue and green iff it is either the green-est blue colour, or the bluest green colour.2 Since you know that the patchis borderline, and you know that the set of borderline cases of green/blue aredense (any colour between two borderline green/blue colours, is a borderlinegreen/blue colour) you know there are precisifications on which the patch is aboundary colour. Thus you should be certain that it is vague whether the patchis on the boundary between green and blue. However, intuitively chances thatit is on the boundary seem to be incredibly slim - they are comparable to thechance a dart thrown at the real line would land on π. In the latter case, Iclaim, one should be certain (or, at least, have credence 1) that the dart won’tland on π. Similarly, it seems, one should be certain that the patch is not onthe boundary (you may find your intuitions are strengthened if you assume thatyou have not yet seen the patch and all you know about it is that it is border-line between green and blue. Indifference reasoning in this situations suggestsassigning credence 0 to every colour hypothesis, including the hypothesis thatit is the boundary colour.)

2 Suppose that ‘bluer than’ and ‘greener than’ correspond to positions on a given rainbow,so that they are linear orders. Technically one must also assume that colours are ordered likethe reals, in that every bounded set has suprema (if, for example, they were ordered like therationals there might be no boundary colours.)


6.4 Probabilism

In the previous sections we found some concrete cases where we could makecomparative judgements about how uncertain one should be in a proposition,even when it is known to be vague. It is time to see how much mileage we canget out these judgements.

Let us encode a given person’s strength of beliefs by an ordering, ≤, overa set of propositions Ω.3 Define p < q := (p ≤ q) ∧ ¬(q ≤ p), and p ≈ q :=(p ≤ q) ∧ (q ≤ p). Roughly p < q says that a’s belief in q is stronger than herbelief in p and p ≤ q says that a believes q at least as strongly as she believesp. We have seen that, even when a is certain that p and q are vague, she maystill believe one more strongly than the other.

Let us now consider what constraints must be imposed on ≤. Say a proposi-tion p is null just in case p ≤ ⊥. We shall concentrate on the following principlesin turn:

1. ⊥ < >

2. For every p ∈ Ω, ⊥ ≤ p ≤ >

3. ≤ is a total (pre)order.

(a) p ≤ p for every p ∈ Ω

(b) If p ≤ q ≤ r then p ≤ r for every p, q, r ∈ Ω

(c) p ≤ q or q ≤ p for evey p, q ∈ Ω

4. Suppose p ∧ q = ⊥ and p ∧ r = ⊥. Then q ≤ r iff (q ∨ p) ≤ (r ∨ p).

5. If q < p then there is a finite collection r1, . . . , rn of mutually incompatiblenon-null propositions such that (q ∨ ri) < p for each i.

The first three principles seem to be pretty firm. One should always find atautology more likely than a contradiction, and every other proposition mustbe somewhere in between. Everything is at least as likely as itself, if p is morelikely than q and q more likely than r according to a then p is more likely thanr according to a. The only controversial thesis in (3) is the requirement thatevery proposition be comparable to every other propositions. This is going toessential to anyone who thinks that ones judgements of likelihood are going to berepresentable by a probability function - for after all, there is no getting aroundthe fact that [0, 1] is linearly ordered. But does vagueness give us particularcause for concern in this regard? I suggest not. We have seen that we canmake comparative judgements of uncertainty, even when vagueness is involved.Examples where it is hard to make comparative judgements usually can be madejust as well using precise sentences.

3 We will not make many assumptions about Ω but we do assume it is a complete Booleanalgebra - thus ruling out non-classical approaches to vagueness. One could think of Ω as thepowerset of the set of precisification/world pairs.


Someone whose likelihood relations are not linearly ordered because they fail(3c) but obey the rest of the axioms (1)-(5) can have their credences representedby sets of probability functions, with each function, Pr, quasi-representing it,in the sense that p ≤ q only if Pr(p) ≤ Pr(q) for every p and q. We havealready tacitly criticized similar views when we discussed Field’s calculus. Thatsaid, there are independent reasons to think that credences should not be setsof probability functions (and hence, given the rest of (1)-(5), should be linearlyordered). For two separate and convincing criticisms of the sets of probabilitiesview see White [20] and Elga [6].

Principle (5) might seem a little mysterious, but it is really only there toensure that Ω is infinite, and that any proposition you’re not certain is false,can be split up into a disjunction of propositions you are not certain are false.This axiom precludes us considering agents who know exactly what the worldis like, but this restriction does not seem to be relevant to the issues concerningvagueness.

The remaining axiom, and by far the most crucial one is axiom (4). To givean example of (4) at work in vague contexts let us consider some examples.Recall the example we considered in §6.3 concerning two patches that are bor-derline between being green and blue. Suppose that p is the proposition thatboth patches are blue, q is the proposition that the first patch is green, and r isthe proposition that the second patch (the greener patch) is green. How likelyshould we find p ∨ q compared to p ∨ r? Intuitively it seemed that it was morelikely the second patch was green than the first - that is r > q. It seems morelikely that, if at least one of the patches is green (i.e. ¬p), the second patch isgreen than that if at least one of the patches is green the first patch is green.But this is just to say that we are more confident that the second patch is greenor neither is, than we are that the first patch is green or neither is (p∨r > p∨q).

With (4) we are in a position to refute Field’s claim that one may be justas confident in p as you are in ¬p and just as sure of both of these as you arein ⊥. For once we are just as sure of p as of ⊥, (p ≈ ⊥), we may deduce thatp∨⊥ < p∨¬p (since by (1) and the laws of a Boolean algebra p∨⊥ = p ≈ ⊥ <> = p ∨ ¬p.) But since ¬p is incompatible both with p and with ⊥ it followsthat ¬p > ⊥ by (4). Thus, pace Field, whenever one is just as certain in p as in⊥, one must be less certain in both than in ¬p.

What we have here is a qualitative theory of comparative likelihood. Theview I want to defend is that credence’s, even in the presence of vagueness,should be represented by probability functions. Although it seems impossibleto have direct intuitions about quantitative theories, such as probability theory,that postulate numerical values, it is possible reconstruct such a theory purelyfrom the principles (1)-(5).

Theorem 6.4.1. de Finetti: Supposing that ≤ is a comparative likelihoodover a complete Boolean algebra, Ω. Then there is a unique finitely additiveprobability function, Cr, such that:

• p ≤ q if and only if Cr(p) ≤ Cr(q)


6.5 Genuine uncertainty

One could imagine a theorist of the kind considered in chapter §3 agreeing witheverything that has been said so far. They would agree that vagueness-relatedcredences are probability functions, that vagueness usually involves having in-termediate credences, but would just insist that this didn’t constitute genuineuncertainty.

But what exactly does it mean for a state to constitute genuine uncertainty?There are weak ways this can be understood on which I would agree. I believethat there is something peculiar about vagueness-related uncertainty that de-rives loosely from the fact that it is not about the world. If that is all that ismeant by ‘genuine’ then I agree; I just do not think this constitutes a deep orfundamental difference in kind.

To dispel the intuition that “genuine” uncertainty must, in some way, beabout the world, let me take an example of which I think many people wouldagree constitutes real uncertainty, but is clearly not about the world. Theexample I am thinking of is what has come to be called ‘self locating uncertainty’;uncertainty, not about the way the world is, but uncertainty about where youare located in that world. I might know I have a lecture at 12.00pm but notrealise that the lecture is now, or I might know the layout of a maze particularlywell, but not know where I am in it because it all looks the same from the inside.If the world is completely symmetrical, containing two people receiving the sameperceptual experiences, I might even have no worldly uncertainty at all, I mightknow every objective fact about the world, but still be genuinely uncertain asto which of the two people is me.

I think the mental state one is in when one has vagueness-related uncertainty,self locating uncertainty or worldly uncertainty form a natural kind. I can thinkof two ways to make this more precise: one from a functionalist understanding ofmental states, the other from a representationalist theory of intentional mentalstates. I do not take these two approaches to intentional mental states to beincompatible, but I shall not spend time here defending that view.

Representationalism

A representationalist about belief will typically say something like the following(see for example, Field [10]):

a believes S iff there is a sentence in Mentalese, M , such thatM belongs to a’s ‘belief box’, and M means the same as S.

(6.5)

There are several things worth flagging here. Firstly, the true objects of belief,on this account, are sentences in a’s language of thought - not (precise) propo-sitions. This leaves room for the view that objects of belief are genuinely vague.Secondly, genuine agnosticism about S seems to be nothing more and nothingless than the absence of some representor of S, and of some representor of ¬Sin a’s belief box. Let me also remark that ‘means the same as’ is supposed to


be understood so as to involve penumbral equivalence: ∆(M ↔ S).4

There is a propositional variant of (6.5), analysing the relation ‘believes∗’ be-tween a person and a set-of-worlds proposition, in which believing∗ a propositionis having a Mentalese sentence in your head which expresses that proposition.However, since it might be vague which proposition a Mentalese sentence ex-presses, I have gone for the sentential version to avoid comlexity. That said, onewould ideally like to describe the general relation between people and possiblecontents of belief. However, the possible contents of beliefs vastly outnumberthe sentences, for there is presumably at least one possible content of a belieffor every set of world/precisification pairs. To do this would be to invoke aspecialised notion of vague proposition which I shall not go into here.

Belief, described in these terms is blind to the distinction between vague andprecise sentences. Agnosticism in a vague sentence, on this picture, involvesexactly the same mechanism as agnosticism in a precise sentence. I take thisto support the thesis that vagueness-related belief is fundamentally no differentfrom ordinary beliefs.

Functionalism

A functionalist explanation of belief (which is not necessarily at odds with theaccount in the previous section) would tell us that what counts as a belief isdetermined by the causal-functional role it plays with respect to desire andaction. For example, typically the desire that p, and a belief that doing Awill bring about p, causes one to intend to do A. Beliefs, desires, intentionsand actions stand in a complex network of causal relations to one another. Thefunctionalist says that belief just is whatever stands in the right causal relationsto some other states.5 This is compatible with a large range of physiologicalprocesses playing the role of genuine belief.

I propose a similar functionalist account of credence, very much analogous tothe account for belief, desire and action. Credences are more fine grained thanbeliefs; I similarly posit finer-grained versions of desires: desirabilities. Usually,if an agent has a high credence that doing A will bring about p, and they have ahigh desirability in p they will intend to do A. They will weight the desirabilityof each consequence to the credence they have that it’ll happen if they do A,and that will be the desirability of A. That is roughly the causal role role thatdesires, beliefs, intentions and and actions play with respect to each other. LetA ≺ B represent the relation between actions that says the agent would intendto perform B over A if given a strict choice between the two. Des(p), a functiononto real numbers, represents the agents desirability in p, and Cr(p), also afunction onto real numbers, her credence in p. Let S range over a partition of

4 This biconditional belongs to neither English nor Mentalese, but it should be clear whatI mean. Cross linguistic penumbral connections are widespread - the German word ‘rot’ andthe English word ‘red’, for example, are penumbrally connected - so there presumably is someway to make this fact precise in terms of the speakers linguistic practices of both languages.

5 Such definitions can be made rigourous and non-circular using the Ramsey-Lewis methodfor defining theoretical terms.


logical space.

A ≺ B iff∑

S Cr(S/A)·Des(A∧S) <∑

S Cr(S/B)·Des(B∧S) (6.6)

I have here written the ‘degree to which the agent thinks S will happen if shedoes A’ as Cr(S/A).6

What is particularly interesting in this regard are a class of theorems whichstate that if ≺ represents the preferences of a rational person, then there arefunctions Des and Cr which satisfy the functional role of credence and desir-ability. In some versions of the theorem such a representation is unique.7 Thisshows that we can ascribe credences and desirabilities over any vague proposi-tion (set of world-precisification pairs) to anyone who has rational preferences.To have a certain credence in a vague proposition p, just is to have a certaincombination of preferences. For the functionalist, it makes little sense to saythat a credence in a vague proposition isn’t ‘genuine’, if it simply falls out ofyour network of preferences. If one were to list all my preferences over all propo-sitions (vague and precise), for example my preference to have lots of hair overbeing bald, to have not too little and not too much salt in my food, and so onand so forth, one can in principle calculate my credence in any such proposition,vague or precise.

6 There is some controversy over how this should be defined. For example, should it be thedegree to which you think A will cause the outcome, or the degree to which doing A would

give you good evidence for the outcome, e.g., Cr(A S) (Stalnaker) orCr(A∧S)

Cr(A)(Jeffrey).

7 For evidential and causal decision theories (Jeffrey, [11], Joyce [12]) they are not quiteunique unless the agent has an unbounded utility function. In a Savage style decision theory[17] such a representation is unique up to a choice of additive constant. Savage style expectedutility is expressed using unconditional credences, and thus does not take sides on the eviden-tial/causal debate (which is left up to ones choice of how to carve up the states.) Despite this,the conditional expected utility equation above is generally considered to be more satisfactory.

7. VAGUENESS AND DESIRE

People appear to have vague desires all the time. I might, for example, want alarge house, but I could not put a precise figure on the volume it would have tohave for it to satiate my desire.

In this chapter I shall be examining vague desires. This raises some highlyrelevant questions for our investigation. How should one act upon vague desires?Must we only care about the precise? Are your beliefs in the vague fixed byyour beliefs in the precise? Is it possible in principle to find out what a personsbeliefs and desires about the vague are? Understanding how desire interactswith uncertainty allows us to get a better grip on the role vagueness plays inboth.

7.1 Vagueness and probabilism

In section §6.4 we appealed to intuitions directly about comparative credence.Notice, however, that unless we have a concrete theoretical interpretation ofcredence, we cannot expect these intuitions to hold too much weight. A standardway to get a fix on the nature of credence’s is to consider rational bettingbehaviour. That is, one trades in intuitions about how one’s credence’s oughtto be, with intuitions about how one ought to act. These latter intuitions, it issuggested, are then free of dubious uninterpreted theoretical vocabulary.

This is not, of course, to say that we cannot have direct intuitions aboutcredence’s. Talk of credence is normally grounded in intuitions about strengthsof beliefs - for example - my belief that this coin will land heads is not as strongas my belief that the sun will rise tomorrow. As we saw, given some minimalassumptions about the ordering of strength between beliefs, we can prove thatstrength of belief can be measured by a unique probability function.

But this is not the whole story. Credence talk does more than just encodeour judgements about comparisons in strength of belief. Credence’s are intendedto do important theoretical work: they are meant to encode the fine grainedaspect of our beliefs that is supposed to inform our decisions. When someonehas a very high credence in p, they should not take actions that would resultin bad consequences if p were to hold. You should act in a way that maximisesgood consequences, weighting each consequence proportionally to the degree towhich you believe that consequence is likely to occur (as a result of your action.)

This interpretation of credence falls out naturally from the functionalistaccount of uncertainty given in the last section. In what follows I intend to

7. Vagueness and desire 54

argue for a claim complementary to the view that one ought to be genuinelyuncertain in the presence of vagueness. It a thesis about the degrees to whichwe are uncertain in vague propositions. The claim goes that one’s degrees ofbelief in vague propositions behave much like they would in ordinary cases.

To be more explicit, I shall argue for probabilism: the view that ones cre-dences ought to be (or at least ought to be representable by) probability func-tions:

• Cr(φ) = 1 whenever ` φ.

• Cr(φ) = 0 whenever φ `.

• Cr(φ) + Cr(ψ) = Cr(φ ∧ ψ) + Cr(φ ∨ ψ).

• Cr(φ) ≤ Cr(ψ) whenever ` (φ→ ψ).†

The characterization of probabalism here does not make assumptions aboutwhat the logic of ` is.1 In this chapter, however, I shall be concentrating on thespecial case where ` is classical.

7.2 Betting and vagueness

A standard kind of argument employed among philosophers in defence of theclassical probability calculus are the so called Dutch book arguments. The ideais to get a quantitative measure of what your credences in a proposition, p,ought to be, by looking at the bet’s you you ought accept. Let an x-bet onp be a bet that you believe costs £x to participate in, and pays back £1 if pis true, and nothing if it is false. Assuming that you only care about money,and the degree to which you care is linear in pounds, then, the argument goes,your credence in p is the supremum of x | you ought to accept an x-bet on p.Finally, if you ought not accept an x-bet on p, you ought to be willing to sellan x-bet on p. This last principle is perhaps the most controversial, in that itrequires your credence’s to be linearly ordered.

With this in place, we are in a position to place constraints on what rationalcredence’s should look like. For example, if your credence in p was greater than1, then I could offer you a (1 + ε)-bet on p, where 1 + ε < Cr(p), which youought to accept, even though you’ll make a guaranteed loss. If p is true thenyou’ll lose £ε and if it’s false you’ll lose £1 + ε. Similar arguments can be madefor finite additivity, and various other probabilistic claims.

While the Dutch book argument might be compelling in ordinary contexts,it is not so clear that the argument can carry over to contexts where the betsin question are bets on propositions you are certain are indeterminate. Thereseems to be something deeply wrong about participating in a bet concerning

1 The status of the last principle † is controversial. It is redundant if ` is classical, but sincemost non-classical logics have special conditionals it is required. There is a stronger thesis,call it probabilism+, which replaces the last claim with: Cr(φ) ≤ Cr(ψ) whenever φ ` ψ. Forintuitionistic ` this is not a strict strengthening, but in the context of logics for vagueness,the deduction theorem usually fails, making this a strict strengthening.


Cedric’s baldness. Not in the least because it seems hard to see how you mightget paid in the event that Cedric is bald, given that this is indeterminate.

The problem seems to be that bets on vague propositions are somehow defec-tive. To get clearer on why, exactly, this is problematic, we must get clearer onhow and in what sense these bets are defective. Are they defective in the sameway that a bet on a question is defective? This seems like a category mistake.To be offer a bet you must specify conditions under which the payoff is received- a question provides no such conditions. Bets in vague propositions do not seemto be like this - there are conditions, albeit vague ones, under which you receivea payoff. Indeed, suppose I have just flipped Dietz’s coin (one side of which,recall, is painted definitely green, the other borderline green) and suppose thatin fact it’s going to land on the borderline green side. A bet at even odds thatthe coin will land green side up makes perfect sense, and given that the coin isbiased towards the definitely green side, it is a bet I ought to accept. However,the proposition we are betting on - that the coin will land green side up - isindeterminate (although I don’t yet know this.)

Perhaps bets on vague propositions are defective in the same way a bet onwhether the king of France is bald is defective. If I were under the illusion thatFrance had a king, and he was bald, there would be a large class of bets on thisproposition it would be rational for me to accept. This case shares with vaguebets the property that the things we are betting on are possibly truth-evaluable,and in cases where you have a high credence that they are truth evaluable, youmay even be required to accept these bets. The reason the bet is defective isthat once we realise that France has no king, we do not know how to fix thepayoffs. Should the bet be called off? Or should it be treated as if it were false?One problem with the latter option is that a bet on whether the king of Franceis not bald appears to be equally defective, for presuppositions project out ofnegations.

If presupposition failure is sufficient for a bet to be defective, a bet onwhether the kind of France is bald and the moon is made of green cheese shouldalso be defective. But intuitively the payoff is well-defined here: anyone ac-cepting a bet on this proposition will lose, since the moon is not made of greencheese!

There are much more mundane ways a bet could be defective. For exam-ple: an extremely shifty looking stranger approaches you in an alleyway with aproposition. He describes to you a plausible looking method he has for cheatingcasinos out of money, and offers to double any money you give him by playing itin the casino. The cheating method either works or it doesn’t, so we may thinkof this as a bet on whether or not the method will work. Should you give him£100? You’re pretty sure the method would work if tried, but you feel uneasyabout giving this stranger £100. How do you know he won’t do a runner? Inthis kind of case, it seems less plausible to attribute my unwillingness to acceptthe bet to my having low credence that the method will work. I just don’t thinkthe self described conman will pay me back if he wins.

In this case, I could tell the payoff wouldn’t correlate with the outcomebecause there were a number of clues: the circumstances of our meeting, he


looked suspicious, the giveaway cackle, the peg leg, - you can fill in your ownstory. Can we tell whether a bet on p would be defective in this sense withoutthese signs? Can there be something about p itself, which could make youbelieve the bet would be defective even if everything else looked respectable?It seems so. Let p a proposition we could never hope to verify. A bet on thestatus of some event far outside our light cone: for example, whether the galaxyTN J0924-2201 is currently in collision with another galaxy. The result of thiswe will not find out for another 11 billion years. You should immediately besuspicious of anyone who offers you a bet on this proposition. You can’t be surethat your bookie will pay up if and only if the galaxy in question is in collision.The fact that you oughtn’t accept any bet on p, or on ¬p, if p is hard to verify,does not indicate that we ought to have 0 credence in both.

In a way these bets are not defective in any troubling sense. They are, inprinciple, good bets since supposing, per impossible, the bookie could accuratelyguarantee that if you accepted, you’d get the money iff the proposition you’rebetting on is true, they are perfectly acceptable bets to take. Decision theorycan still tell you what you ought to do in a decision problem, even if it would bedifficult or impossible for that decision problem ever to arise. Indeed, someonewho had been somehow misled to believe that her bookie really could guaranteethat she’d get paid precisely if the proposition in question is true, would acceptnon-trivial bets. This is unlike, for example, a bet on a question, for whichpayoff conditions are not even well defined.

I want to argue that bets in vague propositions are defective, if defective atall, only in as much as this latter kind of bet is. What I want to argue is thatreasons to think that bets on vague propositions are not well posed boil downto the same kind of reasons we think that bets on unverifiable propositionsare defective. In both cases, the fact that the proposition being bet upon isunverifiable means you can’t be certain your payoff will be correlated with theoutcome if you accept.

The reason bets in vague propositions seem to be defective in the sense out-lined is that receiving a payoff is usually a precise matter - you either end upwith the money in your pocket or you don’t. Let p be the proposition you arebetting on, and let q be the proposition that you will receive a certain payoff.Given the set up we have ∇p and ¬∇q. A standard logic of indeterminacy(namely, one in which ∆ obeys K) implies that ∇(p ↔ q). That is, it is inde-terminate whether you will receive your payoff iff p. Given that we are certainthis is the setup, and given the entailments I outlined, it is uncontroversial thatCr(∇(p↔ q)) = 1. It is at this point that the theories begin to disagree over thevalue of Cr(p ↔ q). The consensus is, however, that it is not (determinately)true that Cr(p↔ q) = 1.2 Thus you are not certain that you will receive yourpayoff just in case p is true, which puts one in a situation similar to the onedescribed in the preceding paragraph.

2 According to the view I espouse, there may be cases where you are certain that ∇p andcertain that p, but these are esoteric. The other views we have considered do not even havethis as a possibility.


One might think the problem with these bets is that they are essentially likethis. Not even God could settle the question of whether Cedric is bald.

However in suitably constructed cases, I claim it is possible to be certainthat your payoff will be correlated with the truth of the proposition you arebetting on, even if you’re certain the proposition you are betting on is vague.Consider the following example. Over a period of 4 years Fred revises his will17 times at regular intervals. At the beginning of the 4 years he is perfectlysane, and by the end of the 4 years he is clearly insane. At the time of Fred’sdeath, the most recent valid will determines the transfer of money and propertyto any beneficiaries. A will written by a mad man is not legally valid, and wemay assume this is the only reason any of the wills might fail to be valid.

Now suppose at some point at which he is bordering on madness this manoffers you a bet over whether he is currently mad. The bet costs £50 to par-ticipate in: if he is mad you’ll get £100 and if not you’ll get nothing. How canyou possibly be certain that you’ll receive the £100 just in case he’s mad? Itis simple - he writes it into his will. We may assume that when he dies it isindeterminate whether the £100 belongs to you, or to whomever it is promisedin the other versions of his will. However, due to the nature of the situation, itis determinately true that you will legally own the £100 if and only if he wasmad. Thus you may be certain that the bet is not defective; you are certainthat you’ll get the money if and only if he’s mad.

Another real life case is one in which two partners, both of whom had sellingrights to a property, unaware of each other, simultaneously sold the propertytwo different people. It should be clear now how to generate the kinds of betsrequired to make these examples work.

7.3 Vagueness and desire

The examples above made essential use of an assumption usually taken forgranted in the context of Dutch book arguments: that the agent in questioncares intrinsically about legally owning money (and that this is all she caresabout.) The assumption, in ordinary cases, seems innocuous, since one canalways replace the monetary payoffs with whatever it is the agent really caresabout. However it is not so clear in this case that this assumption is innocuous- most people don’t care intrinsically about legally owning money, they careabout what they can spend it on.

To see how this might make trouble, consider how things would look if thehedonists were correct. Suppose personal utility is proportional to some measureof your pleasurable subjective experiences. Suppose now that you can have avague bet that is not defective: if you are betting on p, and q is the propositionthat you receive a given pleasurable experience (the payoff), then if you acceptthe bet, ∆(p ↔ q) must hold. The resulting scenario is one in which it isindeterminate whether you are having a pleasurable experience, a result whichseems intolerable (and at least, very hard to set up.)

Fortunately the hedonistic account of utility is about as implausible as the


view that one should only care about owning money. For example one mightreasonably care about the environment, despite the status of the environmenthaving no direct influence over your subjective experiences. It doesn’t evenseem to be irrational to want to be a philosophical zombie - a consequence thatwould receive zero utility in a hedonistic model.3 A more plausible model ofutility could be spelled out in terms of desire satisfaction. Indeed, some versionof this account is suggested by the functionalist theory of uncertainty and desireespoused in §6.5.

One point worth flagging at this juncture is that utilities on an accountlike this will presumably end up being assigned to propositions or sentencesof mentalese (or whatever the objects of desire turn out to be.) Crucially,this opens up the possibility that the kinds of things that we assign utility toare the very same as the sorts of things that can be vague. If bets on vaguepropositions are somehow anomalous, then this is not because we cannot makesense of someone caring about a vague outcome.

It seems that no-uncertainty views about vagueness cannot simply dismissthe Dutch book arguments for probabilism even as regards vague propositionsfor the reasons just alluded to. One can construct coherent bets with well-defined payoffs. For example, what if you just cared intrinsically about whetherp held? p’s obtaining is payoff enough - and there is seemingly nothing incoher-ent about caring about something vague. This is to be expected if the objectsof our desires are the very same kinds of things that are vague. To demonstratethis consider the following example. Suppose I have the desire to be bald. It’sa simplistic desire: I look in the mirror and think I have too much hair, I wishI were bald. This is not to say that I have any specific look in mind, or that Iwish to have less than any given number of hairs. My desire is a non-specificdesire to be bald: if I want to have less than 687 hairs, say, this is only becauseI think this will make me bald - I don’t really have any independent reason tocare how many hairs I have numerically. This is brought out in scenarios inwhich I am definitely bald (or definitely not bald.) I am completely indifferentbetween such scenarios, even though I may have different hair numbers in thesescenarios - it is not hair number, per se, that I care about.

Although my desire may be eccentric, it does not seem to be confused -it’s not like my desire to be bald has somehow misfired or that I’m simplymisdescribing what I want.

Now how does this help us with the Dutch book arguments for probabilism?While it is clear that, if p is vague, it is hard to ensure a monetary payoffdepending on whether or not p obtains. However what happens if you just careintrinsically about whether p obtains? Suppose I now wish to satiate my desire,so I find a hairdresser that can make me bald. I want to be bald so badly thatI’ll pay £100 to be bald, but any haircut that left me non-bald would be of novalue to me at all. But my hairdresser is a perverse fellow: he wants to find outwhat my credence is that one is less than 894-haired iff one is bald.4 Thus he

3 It also seems to make sense to talk about zombies wanting things, and acting so as tomaximise the satisfaction of their desires, despite having no subjective experiences.

4 I am making the simplifying assumption that hair number is the only contributing factor


says he’ll make me 894-haired for some amount of money. If £x is the maximumamount of money I’d be willing to pay for this haircut then my credence thatone is 894-haired iff one is bald is x%.

This is pertinent, since it demonstrates it is possible to have a non-trivialcredence in a proposition which you are certain is vague. What is more, it canbe demonstrated that my credences obey various conditions. For example, itis then clear that my credence cannot be larger than 100%. For suppose mycredence is 110% - if it were I should be willing to pay £110 for this haircut,guaranteeing that I have paid more than I would have for a haircut that wouldmake me clearly bald. Similarly, if I’d pay £x for a 894-haircut and y-for an895-haircut, then I shouldn’t pay more that £x+ y for both.5

7.4 Must we only care about the precise?

Bets in vague propositions are not defective in the same way that bets in, say,a question would be - caring about and believing in the vague does not involvesome kind of category mistake. One might respond by conceding that this maybe so, but maintaining that nonetheless, one should only care about the precise.I shall consider a couple of ways of cashing this out.

The first way in which we might cash out the claim that we should onlycare about the precise, is to say that as soon as you know that p is vague, youshouldn’t care about p at all: you should be indifferent between p obtaining and> obtaining - their news value should be 0.

If you are certain that p is vague then p ≈ >. (7.1)

This principle extends, in a natural sense, the discussion of Field’s non-classical credences in Chapter 4. It was left an open question whether thedesirability in a proposition known to be vague is its expected utility.6 Field’sprobability calculus equipped with an affirmative answer to this question entailsprinciple (7.1). Conversely, it seems like one could construct a convincing ar-gument starting from (7.1), that you ought to have credence’s that accord withField’s probability calculus. Roughly, suppose you are offered a bet on p withpayoff q if you accept (and it costs a positive amount to accept.) If you are cer-tain p is vague, and certain the bet is not defective, in that p and q are suitablypenumbrally connected, then you must be certain the payoff proposition, q, isvague. (7.1) tells us we shouldn’t care at all about q, which in turn means weshould reject bets on p. Symmetrical reasoning holds for bets on ¬p. Assumingcredence is the maximum amount of money you ought to pay to accept the bet,

concerning whether someone is bald.5 To get both, I have one and then the other in succession. Since I only care about being

bald for some period of time, if either 894 or 895 counts as being bald, my desire will besatisfied.

6 The standard equation for expected utility isn’t partition invariant according to this view,so by expected utility I shall by default mean the expected utility with respect to the mostfine grained partition: the set of singletons of world-precisification pairs.


your credence ought to be zero in both p and ¬p - the characteristic feature ofField’s calculus.7

Unfortunately (7.1) is demonstrably false. It is quite simple to constructvague propositions you assign high utility to, by simply conjoining things youcare about with things you know to be vague. For example, I really don’t careone way or another whether Cedric is bald. With regard to this proposition, Ihappen to obey (7.1). On the other hand, it matters a lot to me that I do wellin my BPhil. This second fact seems to guarantee that I should assign a veryhigh news value to the conjunction ‘I passed my BPhil exam and Cedric is bald’.After all, if I learnt that conjunction I would be extatic, and not irrationallyso. That said, I am just as certain that this second fact is vague as I am that‘Cedric is bald’ is vague. If I were to follow (7.1) I’d be rationally required toassign the conjunction no utility. This seems plain wrong.

Perhaps it is not that one must only care about the precise, but that itshould only be the precise content of a proposition that contributes to yourdesires? The second way to cash out the idea that we ought only care aboutthe precise is as follows:8

〈w, ν〉 ≈ 〈w, ν′〉 for every ν and ν′. (7.2)

In other words, once you are certain of all the precise facts, you should beindifferent between any further vague outcome. Once I know exactly how manyhairs I have, and, if it matters, any other conjunction of precise facts, I shouldnot care whether I am bald or not.

(7.2) tells us that given all the precise facts, we should be indifferent be-tween the vague outcomes. This is not particularly restrictive. In particular,it is compatible with (7.2) that the degree to which I care about being bald isdifferent from the degree to which I care about any precise proposition, and itis compatible with me having non trivial degrees of belief in vague propositions.

In short, (7.2) is compatible with probabilism. Furthermore, supplementing(7.2) with other natural principles governing rational preference guarantees thatthe agent can be thought of as maximising expected utility with respect to aprobabilistically coherent credence function over precise and vague propositions.(For more details see the appendix to this chapter.) As far as the central claimsof this thesis are concerned, nothing too much rests on (7.2). For now let mejust express my scepticism that about such a thesis. From where I see it, (7.2)is no more plausible than the thesis that one ought only care about Christmasornaments.

7 This is of course just an argument for the characteristic feature. A full representationtheorem would take us too far afield. See Dietz [4], however, for a Dutch book argumentalong similar lines.

8 Here p ≈ q means that the agent would be ‘equally happy’ if they learnt that p as theywould if they learnt that q.


7.5 The supervenience of vague beliefs on precise beliefs

The view I have defended here is one in which it is permissible to have a widespectrum of different doxastic attitudes towards the vague. A natural questionto ask is how our credences in precise propositions should affect our credencesin vague propositions? For example, could two rational agents agree on theprecise, in the sense that they assign the same credence and utility to eachprecise proposition, but disagree about the vague?

I’m inclined to think that people could permissibly disagree about the vaguewhile agreeing about the precise. To see why this is plausible, consider thefollowing scenario. Suppose there is this ‘chocolate machine’ that dispensesarbitrary finite numbers of pieces of chocolate to two people, a and b, at random,the same amount to both. Let us suppose the two people in question speak avery simple language in which they can describe different numbers of chocolatesthat they could receive from the chocolate machine. For any n, the propositionthat they both receive exactly n chocolates can be expressed in this language, ascan negations and arbitrary disjunctions of expressible propositions.9 We maysuppose each enjoys chocolate just as much as the other, and for each n, agreeabout how likely it is that the chocolate machine dispense n chocolates.

Suppose we now increase their vocabulary: they can now express the propo-sition that they will both receive a small number of chocolates, and anythingthat can be got from this by negation and arbitrary disjunction. Should theyagree about the newly generated propositions? It seems bizarre to say theyshould. By way of analogy, suppose instead they augmented their vocabularyso they could express the proposition that the machine will dispense alien-manychocolates, which is true iff the machine dispenses at most as many chocolatesas there are planets supporting intelligent non-human life-forms in the MilkyWay. There’s absolutely no reason they should agree in their degree of beliefabout the number of planets supporting intelligent life forms in the Milky Way,why should they agree about the largest number it takes to be small?

The possibility of this kind of disagreement raises some deeply puzzling ques-tions for the view proposed here. Could we tell if two people disagreed about thevague? Could they articulate their disagreement? The idea that it would be im-possible to detect from a persons behaviour and dispositions their credence in agiven vague proposition is puzzling, and from a functionalist perspective, prob-lematic. What’s more, if any probability function compatible with my credencesin the precise could adequately describe my beliefs, as far as my behaviour goes,then to my mind the thesis that one generally has intermediate credences in thevague ceases to be as interesting as it purports to be. It seems it would even becompatible with my credences having sharp drops over Sorites sequences.

So how would we go about testing whether a and b disagreed about thevague? It is simple to test whether they disagree over the number of life sup-porting planets. Let us adorn the chocolate machine with various buttons, la-belled with descriptions of what they do, like ‘dispenses at most 100 chocolates’,

9 Thus the proposition that they both receive more than ten, a prime number of, or finitelymany chocolates are all expressible, as are many many more.


‘dispenses a small number of chocolates’ and ‘dispenses alien many chocolates’.For example, a might choose 200 over alien many chocolates, whereas b mightdo the opposite, because a is certain, where b is not, that there are more than200 intelligent-life-supporting planets in the Milky Way.

Distinguishing behaviourally between a and b’s beliefs about the vague, youmight have thought, is thus not a problem: a could choose to have, say, at most88 chocolates to a small number of chocolates, whereas b could have made theopposite choice. The problem is that such a preference might disrupt our initialconstraint that a and b agree over the precise. Once we fill out the details,it seems like a and b disagree over the precise proposition that they press thisbutton, as opposed to that one. Could a and b agree over the precise, but displaydifferent behaviour because they disagree about the vague?

To put it another way, the problem is that any action a might performthat would distinguish herself from b can be redescribed in precise terms. Forexample, ‘pressing the middle button’, or ‘outstretching your finger and movingit forward 10 centimetres and back again’, rather than ‘choosing a small numberof chocolates.’ On first looks this seems to constitute a disagreement over howthey value the precise things, since, even if only instrumentally, they assigndifferent news value to propositions about pressing the middle button.

However, even if we brought a and b’s preferences about precise matters,like pressing buttons, into line, what makes us suppose that a and b are be-haviourally identical. More generally, why should we assume that behaviour isonly describable in precise terms? Suppose that a and b’s precise actions are thesame. In fact, let us suppose that the physical processes underlying a and b’smotions are exactly the same. Does it then follow that a and b are displayingexactly the same behaviour? I suggest that it does not. For example, a intendsto make it the case that she receives a small number of chocolates, and indeed,she presses the middle button, whereas b does not, he just likes to press buttons.

That said, I think that disagreement about the vague might even be madeapparent by a and b’s physical movements. Suppose, for example, that a prefersit to be the case that p, which is vague between zero and two utils, than for itto be the case that q, which determinately has one util, and b has the reversepreference. So far I have not specified what action would need to be performedto make it the case that p. It could be that a and b agree in their preferencesabout the precise, including every possible way to make it the case that p, andevery possible way to make it the case that q. Now, as we noted, a and bdiffer behaviourally, since in certain situations, a will intend to bring it aboutthat p whereas b will intend to bring it about that q, even though they mayend up performing the same actions. However, I don’t see why even this isnecessary. a might follow through with his intention to bring about p, and indoing so incur unwanted precise consequences, since a was uncertain about howthe vague supervened on the precise.

For a more concrete example, suppose there are two buttons on the chocolatemachine. Both a and b think that the first button will dispense two chocolates ifthe last small number is even and nothing otherwise, and that the second buttonwill definitely dispense one chocolate. a presses the first button, b presses the


second button. In this circumstance they certainly disagree about the precise,since they assign different news values to pressing the two different buttons.However, we may ask what their preferences are before they even know thereare two buttons. a prefers to wager two chocolates on the last small numberbeing even, than to receive one chocolate for certain. a may even form theintention to make the former true, which, if left unchecked, may result herpressing the first button when she sees what it does.

7.6 Appendix: decision theory and representation theorems

In chapter 6 we looked at the role credences played in psychological explanation,and concluded that credences are fixed by the functional role they play withrespect to action and desire. In particular, credences and desirabilities canbe thought of as whatever those things are that represent preferences betweenactions in the following sense:

a ≺ b iff∑

s∈S Cr(s | a)u(s ∧ a) <∑

s∈S Cr(s | b)u(s ∧ b) (7.3)

In the last section we saw that certain rational seeming dispositions to behavein a given way can force us to assign to rational agents intermediate credences invague propositions. Note that probabilism is an inherently normative claim, it isthe thesis that if Cr represents my credences in the sense above, then Cr oughtto obey the probability axioms. Functionalism about credence and desirabilityalone won’t give you probabilism.

In this section I shall briefly outline and discuss a class of theorems thatpurport to show, given that a class of constraints on how you ought to act aresatisfied, i.e. how your preferences ought to be ordered, that your credencefunction Cr is a probability function. The theorems say that if your preferencesare thus and so, then your credence is representable by a probability functionin line with (7.3). What is a theorem (vacuously) ought to be the case, thus ifyour preferences ought to be thus and so, your credence ought to be a proba-bility function. The question of probabilism appears to have been reduced to aquestion about what your preferences ought to be like.

Savage’s decision theory

Probabilism, even in the presence of vagueness, is guaranteed by qualitativenormative constraints on preferences such as reflexivity, transitivity, dominanceand so on. What are these purely qualitative constraints on preferences, andcould they fail if one brought vagueness into the mix? It is hard to see how,for example, someone could prefer A to B and B to C and not prefer A toC, even if there was vagueness involved. What about dominance, or the ‘surething principle’? Something like these two principles are at the heart of manydecision theories based on Savages [17]. The sure thing principle says that if adecision maker would take a certain action if she knew that p obtained, and alsoif she knew that ¬p, then she should take that action even if she knows nothing


about p. Dominance says that if the decision maker would prefer an action nomatter which state of the world she learnt obtained, then she should prefer thataction without knowing which state obtains.

Like transitivity, dominance and the sure thing principle seem to be justincontrovertible, whether you bring vagueness into the mix or not. Let’s consideran example that is explicitly about vagueness, and not at all about the world.Let’s suppose you have no relevant worldly ignorance, and that you have one ofthese vague desires we have been discussing: you want lots of sweets. No amountof sweets would be too much for you, but one sweet would be clearly too little.In short, ‘being a number of sweets that would satisfy you’ is Soritesable, andit is vague what number of sweets will leave your sweet hunger satisfied. I nowoffer you a choice between taking N sweets, or N + 10 sweets, where N is suchthat it’s vague whether N,N + 1, . . . , N + 10 sweets will satisfy your desire.Whether you choose N or N + 10 sweets depends purely on whether they areenough to satisfy you or not. In this context dominance seems to offer prettystraightforward argument that you should choose to have N + 10 sweets: anyprecisification of ‘number of sweets that would satiate your desire’ that includesN + 10, includes N as well. On precisifications where neither option satisfiesyour desire you should be indifferent and thus should weakly prefer N + 10sweets, similarly for precisifications in which both satisfy your desire, and forthe ten precisifications in which N + 10 but N sweets don’t satiate your desireyou should strictly prefer N+10 sweets. So whatever ‘state’ (i.e. precisification)were to obtain you would weakly prefer N + 10, so you should weakly preferN + 10 simpliciter. More generally, if the utility of an action is precisificationdependent, dominance should apply just as forcefully.

Jeffrey’s decision theory

Throughout this chapter we have not concentrated on Savage style decisiontheory, but on Jeffrey’s decision theory, because it gave us a unified treatmentof acts, states and outcomes which allowed us to uniformly model vagueness ineach. The qualitative constraints on preferences take a slightly different form,but, apart from (4), appear to be fairly intuitive:

1. ≺ is transitive and connected.

2. Ω is a complete atomless Boolean algebra

3. If p ∧ q = ⊥ then

(a) If p ≺ q then p ≺ (p ∨ q) ≺ q(b) If p ≈ q then p ≈ (p ∨ q) ≈ q

4. If (p ∧ q) = ⊥, p ≈ q, and (p ∨ r) ≈ (q ∨ r) for some r inconsistent withboth p and q then

(a) (p ∨ r) ≈ (q ∨ r) for every such r.


5. If pn is a chain in Ω and q ≺∨

n pn ≺ r then for some N , q ≺ pn ≺ r forall n > N .

(1) we have discussed, and seems uncontroversial. (2) can be satisfied by choos-ing some atomless subalgebra of the powerset of world/precisification pairs. Thisis an awkward constraint, but one that has to be put up with, even in the precisecase. (5) is just a continuity condition that ensures ≺ behaves like an orderingover the reals, rather than, say, the rationals. The core of the theory is (3) and(4). The intuition behind (3) seems to be just as strong in vagueness relatedcases as in precise cases. For example, suppose you collect marbles and you hateorange marbles and love red marbles. If O stands for ‘you get an orange marble’,and R for ‘you get a red marble’, we have: O ≺ R. I offer you a choice betweenthree marbles: an clearly orange marble, a marble that is borderline betweenred and orange, and a clearly red marble. This is a case, we may suppose, whereyou don’t have any relevant precise ignorance. It seems clear that you shouldprefer the red to the borderline marble, and the borderline to the orange mar-ble: O ≺ (O ∨ R) ≺ R. This is just one example, but it demonstrates how thepreference constraints hold across the board, whether in vague contexts or not.Admittedly, (4) is harder to intuitively justify without antecedently appealingto expected utility. This is a problem in the precise case as well, but see Bolker[2] and Joyce [12] for attempts to explain it intuitively.

The representation theorem for Jeffreys theory was proven in Bolker [2]. Itstates:

Theorem 7.6.1. ( Bolker.) Given a decision problem 〈Ω,A,S,O〉, where Ω isa complete atomless Boolean algebra, and given a preference relation satisfying(1)-(5) over Ω, ≺, there exists a countably additive probability function, Cr overΩ, and a utility function, u, over the outcomes, O, such that

• a ≺ b iff∑

s∈S Cr(s | a)u(s ∧ a) <∑

s∈S Cr(s | b)u(s ∧ b) for a, b ∈ Ω

Furthermore, if u is unbounded for any such representation, this representationis unique up to affine transformations of u.

Although this representation is not always unique, if we furthermore requirethat the ordering of comparative judgements of belief are rational, in the senseof de Finetti (see chapter 6 §4) we obtain a unique representation

Theorem 7.6.2. ( Joyce.) If one also has a coherent likelihood ranking over Ω,≤, in addition to ≺, there is a unique-up-to-affine-transformation representationthat also satisfies:

• a ≤ b iff Cr(a) ≤ Cr(b) for a, b ∈ Ω

BIBLIOGRAPHY

[1] D. Barnett. Vagueness, knowledge, and rationality (unpublished, online).

[2] E.D. Bolker. A simultaneous axiomatization of utility and subjective prob-ability. Philosophy of Science, pages 333–340, 1967.

[3] B. Brogaard and J. Salerno. Fitch’s paradox of knowability. StanfordEncyclopedia of Philosophy, 2008.

[4] R. Dietz. Betting on borderline cases. Philosophical Perspectives, 22(1):47–88, 2008.

[5] C. Dorr. Vagueness without ignorance. Philosophical Perspectives, 2003,page 83, 2003.

[6] A. Elga. Subjective probabilities should be sharp.

[7] H. Field. Indeterminacy, degree of belief, and excluded middle. Nous,34(1):1–30, 2000.

[8] H. Field. The semantic paradoxes and the paradoxes of vagueness. Liarsand Heaps: New Essays on Paradox, pages 262–311, 2003.

[9] H. Field. Saving truth from paradox. Oxford University Press, USA, 2008.

[10] H.H. Field. Mental representation. Erkenntnis, 13(1):9–61, 1978.

[11] R.C. Jeffrey. The logic of decision. University of Chicago Press, 1990.

[12] J.M. Joyce. The foundations of causal decision theory. Cambridge Univer-sity Press, 1999.

[13] D.K. Lewis. Convention: A philosophical study. Blackwell Publishers, 1969.

[14] O Magidor and W Breckenrigdge. Arbitrary reference (unpublished).

[15] G. Priest. Doubt Truth to be a Liar. Oxford University Press, USA, 2006.

[16] G. Priest. In contradiction. 2006.

[17] L.J. Savage. The foundations of statistics. Dover publications, 1972.

[18] S. Schiffer. Vagueness and Partial Belief. Philosophical Issues, page 220,2002.

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[19] S. Schiffer. The things we mean. Oxford University Press, USA, 2003.

[20] R. White. Evidential Symmetry and Mushy Credence. Oxford Studies inEpistemology. Oxford University Press, 2009.

[21] JRG Williams. Aristotelian indeterminacy and partial belief (unpublished,online).

[22] T. Williamson. Vagueness. Burns & Oates, 1994.

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