Vagueness and Uncertainty Andrew Bacon June 17, 2009
ABSTRACT In this thesis I investigate the behaviour of uncertainty about vague matters. It is fairly common view that vagueness involves uncertainty of some sort. However there are many fundamental questions about this kind of uncertainty that are left open, questions I shall attempt to answer in this thesis. Could you be genuinely uncertain about p when there is no matter of fact whether p? Could you remain uncertain in a vague proposition, even if you knew exactly which possible world obtained? Should your degrees of belief be probabilistically coherent? 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 uncertain about matters one considers to be vague; uncertainty about vague matters is no different in this regard from uncertainty about the future, the deep sea or far away galaxies.
CONTENTS 1. Introduction................................. 5 1.1 Preliminaries............................. 6 2. The importance of uncertainty in theorising about vagueness..... 8 2.1 Two kinds of supervaluationism................... 8 2.2 Epistemicism and supervaluationism................ 10 3. Ersatz uncertainty............................. 17 3.1 New psychological vocabulary.................... 18 3.2 Mixed uncertainty part I....................... 19 3.3 Mixed uncertainty part II...................... 21 3.4 What is genuine uncertainty?.................... 23 4. Rejectionism................................ 25 4.1 Strange consequences......................... 25 4.2 Decision theory............................ 27 4.3 Deontic logic............................. 29 4.4 Two kinds of expectation...................... 30 5. Vagueness in epistemic attitudes..................... 31 5.1 Epistemic Sorites........................... 31 5.2 Indeterminacy and rational action................. 32 5.3 Indeterminacy and rational obligation............... 34 5.4 Dorr and Barnett........................... 36 6. Vagueness and uncertainty......................... 42 6.1 Behaviour............................... 42 6.2 Vague evidence............................ 44 6.3 Comparative judgements of uncertainty.............. 45 6.4 Probabilism.............................. 48 6.5 Genuine uncertainty......................... 50 7. Vagueness and desire............................ 53 7.1 Vagueness and probabilism..................... 53 7.2 Betting and vagueness........................ 54 7.3 Vagueness and desire......................... 57 7.4 Must we only care about the precise?................ 59
Contents 4 7.5 The supervenience of vague beliefs on precise beliefs....... 61 7.6 Appendix: decision theory and representation theorems..... 63
1. INTRODUCTION In a broad sense, this thesis is about the nature of uncertainty. A natural picture, a picture I want to deny, is one in which there are lots of different ways the world could be, different configurations of objects and events, and it is in the nature of uncertainty to be about these configurations. One might say that uncertainty is always about the world. I do not think this is how uncertainty works at a basic level, and this thesis is 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. I claim that sometimes we may be uncertain about the truth of a sentence, even when the truth of the sentence is not determined by the way the world is. I believe that vagueness is one such case. To bring out the contrast, imagine someone who had no worldly uncertainty whatsoever; that she knew the exact configuration the world was in down to the finest details - that there is, let us say, exactly one world compossible with the facts she is certain about. Would there be anything left for her to be uncertain about? A central claim of this thesis is that vagueness induces a kind of uneliminable uncertainty that would remain 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 need not be about the factual. Epistemicists accept the claim that vagueness involves genuine uncertainty, but maintain that all uncertainty is worldly uncertainty. They claim that one can only be uncertain whether p if one considers there to be a matter of fact whether p. There is thus a prima facie tension between the claim that vagueness involves genuine uncertainty and the view that if p is vague, 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 theory of vagueness. 2 One must explain what it means for there to be no matter of fact whether p, if one can (and must) be uncertain about whether p obtains. If I 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 to the configuration of the world, all the mental facts, facts about where you are located in the world, necessary a posteriori facts, and so on and so forth. The thesis can be strengthened in several 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 plays a central role in non-classical theories. To believe the negation of an instance of excluded middle - a contradiction even in non-classical theories - requires you to believe everything given very minimal assumptions about uncertainty. The non-classical theorist, then, needs an account of uncertainty if they want to defend the claim that it is ever rational to accept an excluded 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 these issues. I particular, I shall explicate how a sentence can be semantically indeterminate, and how one can be uncertain in it, without the view collapsing into epistemicism. Another way one could disagree with me is to deny that vagueness involves uncertainty. 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 genuine uncertainty. Uncertainty due to vagueness is sui generis, of a novel and fundamentally different kind, and cannot be understood in terms of the psychological vocabulary used to describe worldly uncertainty. This is the topic of chapter 3. Another way in which vagueness might not involve true uncertainty would be to hold that whenever you are certain that p is vague, you should be anticertain (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, you just shouldn t be uncertain in p. You must be certain in p or certain in p, and if p is vague, it will be indeterminate which you are certain in. This is the topic of chapter 5. Chapters 6 and 7 are devoted to my preferred view. In chapter 6 I am concerned with showing that vagueness-related uncertainty and ordinary uncertainty form a natural kind, and that even in the context of vagueness degrees of belief should obey the probability calculus. In 7 I turn to the functional role of uncertainty. I argue that our desires and beliefs cannot be reduced purely to desires and beliefs in the precise, and caring about the vague is not esoteric and strange, but commonplace. I end by arguing for some intuitively plausible principles governing rational preferences, principles governing how one should act even when one knows the information relevant to your decision is vague. I apply machinery from decision theory to show that we ought to be uncertain, and indeed, 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 to be considering uncertainty in the context of theories of vagueness that endorse classical logic. I think that the role of uncertainty in non-classical logics is a central and fascinating topic, but it is well discussed, for example, in the works of Hartry Field [9][8] and Graham Priest [16][15], and some of the topics are tangential to the main thrust of this thesis. However, many of the issues raised here have direct analogues in the non-classical setting. Each of the views discussed in chapters 3, 4 and 5 have non-classical counterparts, and they mostly 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 classical logic. However the appeal to de Finetti s theorem assumes classical logic, as does the decision theoretic framework, which assumes a Boolean algebra of
1. Introduction 7 propositions. I talk of precisifications throughout the thesis. Hopefully anyone who accepts the existence of penumbral connections can make sense of what I mean here, for example, as ways of interpreting the language that respect penumbral connections. 3 Second. The word proposition is often used for those things which are the objects of peoples beliefs, the truth-conditions of sentences, obeyers of the T-schema, and so on and so forth. I do not believe there is one single thing that fills all these roles. I shall thus use the word proposition in two senses throughout this thesis; it should always be clear which I mean, but in cases of ambiguity I will mark which I mean. I shall treat the truth-conditions of sentences as sets-of-worlds propositions. In this sense of the word, propositions are always precise, and sentences are the true bearers of vagueness: sentences express different propositions on different precisifications. On the other hand, I think that the possible objects of belief can be vague, so in that sense of the word propositions are not like sets of worlds. Nor are they sentences however; the possible objects of belief are many in number, whereas there are countably many sentences of a language with a countable lexicon. There are vague concepts we cannot express, but nonetheless form propositions that are possibly believed. I thus 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 the credential state one ought to be in if one knows all the relevant precise facts and is certain that p is vague. To illustrate, consider the case of Hector the hairologist. 4 Hector is a competent speaker of English. Moreover, Hector knows pretty much everything there is to know about hair, and keeps incredibly detailed records of each of his clients hair situation. For each client, he documents the number of hairs, whereabouts they are located on the head, the width of each strand of hair, the colour, the length, and so on and so forth. One of these clients, Cedric, is on the verge of baldness. 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 situation of Cedric s head to incredible precision, including the number of hairs he has, their length, colour, width and so on, Hector is neither willing to assert that Cedric is bald, nor to assert that he s not bald. Such behaviour is characteristic of uncertainty, and in the case just described, people naturally will describe Hector as being uncertain as to whether Cedric is bald. The credential state Hector is in, in the story just described, is a paradigm case of vagueness-related uncertainty. 3 One need not insist that precisifications are classical; one might, for example, countenance intuitionistic 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 THEORISING ABOUT VAGUENESS The most important question concerning vagueness and uncertainty, in my mind, is whether uncertainty must always be about the factual. The purpose of this chapter is simply to get clear on what exactly this means, explain why it is not analytically 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 to be a matter of fact whether p. (2.1) This principle is enticing. However - we must resist! For there is only a small step between accepting (2.1) and accepting epistemicism. For example: if one maintains 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 is concerned. Even non-classical logicians will typically accept classical logic where there 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 epistemicism, is that the former claims there is no matter of fact where vagueness is concerned. 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 accepting the necessary co-extensiveness of indeterminacy and uncertainty and epistemicism about indeterminacy. One needs an explanation that doesn t just identify indeterminacy with uncertainty - that s epistemicism - but leaves room for their coextensiveness. The nature of uncertainty in vague contexts is an often overlooked, but central point in the dialectic between the various theories of vagueness. In this chapter I attempt to give an account of indeterminacy that allows for failures of (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. I see them coming as two packages of views - while the views don t all have to come 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 corresponds 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 vague just 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 always precise. Determinately is treated as a predicate applying to sentences. The second kind of supervaluationism is preferable for several reasons. (1) I take it that the first theory is not a theory of semantic indeterminacy - each sentence is determinately assigned exactly one semantic value, a proposition, specifically, a function from precisifications to sets of worlds. (2) The first theory appears to be yet another version of a degree theory, with a partially ordered truth value set. (3) Given that it s true that has a trivial modal logic, and since vagueness is identified with being neither true nor false, the first theory cannot account for higher order vagueness. (4) The first theory has a non-classical consequence relation. (5) If propositions are the kinds of things that obtain, then the first theory postulates an intolerable kind of ontic vagueness in the world. (6) Compositionality fails for this kind of supervaluationist. (7) if one were to maintain compositionality with respect to a precisification instead of full compositionality, every proposition would either be determinately true or determinately false (every proposition is precise w.r.t a precisification.) (8) If propositions are additionally Russellian, one must posit ontic vagueness. For example, the proposition Princeton is Princeton Borough is the Russellian proposition consisting of identity, Princeton, and Princeton Borough. Since this proposition is apparently vague, either identity is vague, or Princeton or Princeton Borough are vague objects.
2. The importance of uncertainty in theorising about vagueness 10 In what follows, the first kind of supervaluationism will often provide a way out of the problems considered. I take the above to provide good reasons for not taking this option. 2.2 Epistemicism and supervaluationism Epistemicists openly accept the existence of sharp cutoff points, whatever they may be, and it is typically thought to be this commitment that makes epistemicism extremely counterintuitive. Supervaluationism, on the other hand, supposedly can retain classical logic while rejecting whatever it is that is so counterintuitive about epistemicism. But getting clear on what it is that is so counterintuitive, and seeing how supervaluationism does better is harder than it seems. The crux of the problem is that, if one accepts the second kind of supervaluationist theory espoused in the previous section in conjunction with the genuine uncertainty view, there is not very much a supervaluationist can say that distinguishes herself from epistemicism. For example, Field writes: the indeterminist needs to provide principles governing the notion that 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 to maintain that their theory is a genuine alternative to epistemicism. Distinguishing it from epistemicism, however, is not the only problem: we must also have 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 epistemicism is is committed to sharp cutoff points for vague predicates. A simple way to cash out the existence of sharp cutoff points for a predicate F is to say that there is some element in a Sorites sequence for F which is F while its successor in the sequence is not. It seems just outright implausible, for example, that there should be a last small number, or given that it s vague whether Cedric is bald 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 the supervaluationist! For if this is what is so bizarre about epistemicism, supervaluationism 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 sharp cutoff point: small has no sharp cutoff points because there is no n such that determinately 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
2. The importance of uncertainty in theorising about vagueness 11 such as small to have a sharp cutoff point is not for there to be a last small number, but for there to be a number which is determinately last. There is a last 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 t committed to sharp cutoff points either! For the epistemicist, it s vague whether p 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 the last small number - something which epistemicists reject. To say that there is a last small number, but it is vague which one it is is just to say that we are ignorant as to which the last small number is. Lowering the standards of commitment to sharp cutoff points does not allow one to escape whatever it is about epistemicism that seems so counterintuitive, unless we are to let the epistemicist off the hook too. A final way of cashing out sharp cutoff talk might be to talk instead about the state of the world determining cutoff points. For example, suppose Cedric is borderline bald. Let p be the conjunction all the (precise) facts relevant to Cedric s baldness (his hair number, hair colour, distribution et cetera) and let q 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. In particular, 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 they actually are, then Cedric is not bald: (p q Cedric is bald) (p q Cedric is bald). However, supervaluationists should accept this disjunction too, since the truth of any proposition which precisifies Cedric is bald follows from the complete description of Cedric s head. 2 Truth value gaps. Could indeterminacy be a lack of truth value? After all, epistemicists characteristically reject T rue( p ) T rue( p ) as inconsistent. The distinction between supervaluationism and epistemicism on this view is grounded in the characteristics of truth. Vagueness, in particular, is a kind of truth status - being neither true nor false. Does this explanation of vagueness meet Field s challenge of giving a nonepistemicist reading of? Certainly cashing out vagueness in terms of an antecedently understood notion of truth achieves this. But the notion of truth appealed 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. The notion of truth at play here is closer to the notion of determinate truth - a notion we cannot help ourselves to, if we are attempting to explain the notion of determinacy. Furthermore, as noted in 1, supervaluationism is not committed to the existence of truth value gaps. It s open to the supervaluationist to say that every sentence is either true or false, but sometimes it s indeterminate which. The gappy 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 these follow from the description of Cedric s head.
2. The importance of uncertainty in theorising about vagueness 12 in 1, and we have already argued that it is not particularly attractive. The alternative, however, keeps the T-schema which brings supervaluationism and epistemicism into agreement so far as truth in its relation to vagueness is concerned. Given the T-schema and classical logic (specifically, the law of excluded middle) the following is a theorem: T rue( p ) T rue( p ). Furthermore, given a strengthened T-schema, (T rue( p ) p), the interaction between vagueness and truth is fully determined by the schema p T rue( p )). If a sentence is vague, then so is the statement that that sentence is true, and conversely. For this brand of supervaluationism, as for epistemicism, vagueness isn t to be identified with a kind of truth status. Logic. Both supervaluationism and epistemicism retain classical validity in the the sense that they agree with classical logic over whether each sentence is valid or not. However, when it comes to the notion of consequence, the relation that holds when a claim follows logically from some other claims, it is less clear whether supervaluationism remains classical. On the view that identifies truth with supertruth, and consequence with preservation of truth in all models, various classically valid inferences fail, such as contraposition and reductio ad absurdum, when the operator is in the language. This clearly distinguishes the supervaluationist from the epistemicist. Furthermore, Field notes [REF, p165] we can go some way to answering the challenge of giving the operator a non epistemicist reading. To say that p is vague is to say that we should reject the instances of reasoning by cases on whether or not p. I.e. to say that p is vague is to say that we shouldn t reason from p = q and p = q to = q (for specific p and q.) As noted already, this move is unnatural, unless one adopts the package of views that comes with identifying truth with supertruth that we rejected. The other kind of supervaluationist might say that propositions are what fundamentally stand in consequence relations, and it is sentences, not propositions, that are vague. Claims about inferences between vague sentences are to be treated supervaluationally, like any other claim involving vague sentences: Γ entails p is true at a precisification just in case the propositions expressed by Γ on that precisification entails (in the fundamental propositional sense) the proposition expressed by p on that precisification. This is one way to restore a classical consequence relation, and is in full agreement with epistemicism over this matter. Multiple precisifications I. A natural way for the supervaluationist to distinguish herself from an epistemicist is to engage the language of precisifications. As a first try: supervaluationists believe there are multiple admissible precisifications of a vague language and epistemicists believe there is only one. How are we to make this more specific? A precisification can be identified with a bivalent model-theoretic interpretation of the language in question, that much is clear. But when is a precisification admissible? Minimally a precisification is admissible just in case it assigns for each predicate of the language, extensions containing only things the predicate determinately applies to, and no things it determinately fails to apply to. Such an explanation of admissible clearly makes use of the notion of determinacy again. To adequately explain vagueness we need to eliminate this circularity - a topic we shall return to.
2. The importance of uncertainty in theorising about vagueness 13 Notice that the minimal constraint has not yet distinguished supervaluationism from epistemicism. For as stated, the epistemicist should maintain that there are multiple admissible interpretations of the language. For given the minimal constraint on what an admissible interpretation is and the epistemicists reading of determinately, an interpretation is admissible just in case it might be the correct interpretation for all we know, and indeed, the epistemicists believes there are many of those. 3 Notice also that there is considerable pressure on the supervaluationist in the other direction: to maintain there is only one admissible interpretation. Call a precisification, v, correct just in case the following disquotational schemata obtain: φ is true when interpreted according to v iff φ. (2.2) The extension of the predicate F according to v, is just the set (2.3) of F things. 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 precisifications, she must say there are admissible but incorrect precisifications - admissible precisifications where, for example, the extension of red includes non-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 although there is only one correct interpretation, it is vague which one it is. 4 But this is presumably exactly what the epistemicist would say given her reading of vague : there is exactly one correct interpretation of the language, it s just we cannot know which it is. Uncertainty and ignorance. The issue of uncertainty and ignorance is clearly quite central to this debate. A supervaluationist who believes that uncertainty and vagueness come hand in hand will find themselves agreeing with epistemicists about most things, as seen above. It is for this reason that some have attempted to deny the connection between vagueness and uncertainty. For example Field, [9], suggests that to regard p as vague (or indeterminate) is to regard it as fundamentally misguided to speculate about whether p obtains. When properly explained, this, Field claims, ensures we cannot give indeterminate an epistemicist reading. The sense in which an epistemicist finds it misguided to speculate over p is much weaker - on Field s preferred account, a person who is 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 uncertainty at all. Rather, in cases where one knows all the relevant facts, it vague whether you know p, when p itself is vague. 3 Of course, only a very crude epistemicis would identify vagueness with just any kind of ignorance, but this characterisation suffices to make my point. 4 Given that, according to this view, there is indeterminacy in the semantic vocabulary such as true, refers and extension, which precisification satisfies (2.2)-(2.4) relative to a precisification, v, above will depend on (and indeed, will be identical to) v.
2. The importance of uncertainty in theorising about vagueness 14 It should be noted that while both accounts succeed in distinguishing themselves from epistemicism, neither view has anything positive to say about why their version of supervaluationism avoids the counterintuitive consquences of epistemicism, whatever they may be. Indeed, if epistemicism accords with any of our intuitions, it is the intuition that vagueness involves ignorance and intermediate credence respectively. To say that not only is there a last small number, but that I know which one it is, seems to invoke everything that is so counterintuitive 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 of vague terms lie. For example, Williamson [22] holds that the locations of the cutoff points in vague expressions supervene on the way the linguistic community uses those expressions in the context of the language - it s knowledge of how they supervene 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 a vague predicate depends on the way it is used, and thus work out the extension of that vague predicate. This seems to be a point of departure with this brand of epistemicism. For example, according to the epistemicist above, it seems like one could know the facts about how English is used, and the facts about how the extension of English expressions gets fixed by how English is used, and thus could work out the precise conditions for being in the extension of bald. Of course, the supervaluationism I am defending holds that once you were certain in the facts about how English is used and the use/meaning supervenience facts, you wouldn t be uncertain whether Cedric was bald. But there is no 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 rationally certain about these facts, in accordance with the uneliminable uncertainty thesis. The kind of uncertainty I am postulating is not uncertainty over what the extension of bald is - there is no matter of fact regarding this question. If there were a matter of fact regarding this question - suppose bald means has less than 50,000 hairs, and I knew Cedric had less than 50,000 hairs - then I d be inclined to say a I knew all along that Cedric was bald, I just didn t know which truth value the sentence Cedric is bald had. By way of analogy, I would not lose knowledge of a fact if it were rephrased into unfamiliar vocabulary of which I did not know the meaning. Finally, there are epistemicists who deny the supervenience of meaning on use [14]. For this epistemicist nothing short of actually learning whether Cedric is bald will allow one to know whether Cedric is bald. For this epistemicist there 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, accept two physically identical epistemic possibilities, agreeing on the use facts, which differ with regard to whether Cedric is bald, I just deny that there are also
2. The importance of uncertainty in theorising about vagueness 15 metaphysical possibilities like this. I believe the distinction between epistemic and metaphysical possibility is at the root of the claim that uncertainty need not be about the factual. Someone who denies that this distinction is a deep one will not find much of interest in the thesis that uncertainy need not be about the factual. Multiple precisifications II. So far I have been distinguishing my view from epistemicism on a case by case basis. What is it about each of the views I have considered that makes them epistemicist rather than supervaluationist? Intuitively, the supervaluationist claims that vagueness is fundamentally a matter of semantic indeterminacy, and that ignorance is merely a by product. Epistemicists on the other hand will outright deny there is any semantic indeterminacy, 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 done without appealing to the way in which indeterminacy interacts with uncertainty and ignorance. 5 I shall attempt to provide such an explanation. The fundamental idea is that there are many different ways of interpreting a vague language which are compatible with the way we use that language. To make this rigorous we need to specify what compatible means in a non-circular way that doesn t merely mean compatible with our knowledge about the the way the language is used. In [13] Lewis provides such a non-circular account of compatibility with linguistic usage, which I shall adopt here. For our purposes we may simplify: an interpretation of the language, I, is a function assigning truth-conditions to sentences of the language. An interpretation is admissible for a language L, just in case there is a convention among the speakers of L to try not to say sentences that would be false if interpreted according to I. For something to be a convention requires a certain network of preferences, beliefs and desires to hold between the users of L. For two distinct interpretations of a language, I and I, to be admissible there must be a convention among the speakers of L to try not to speak falsely according to I and I. Since they are distinct, there must be a sentence S assigned truth-conditions with possibly different truth values by I and I respectively. If the requisite conventions are in place, when a speaker knows that I(S) obtains and that I (S) doesn t, she should refuse to assert S and refuse to assert S. But how are we to distinguish the situation above from one in which there is exactly one admissible interpretation, J, that assigns finer-grained vague truthconditions than either I or I - and where a speaker would refuse to assert S and 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 fact bald? A supervaluationist who believes that we should be uncertain in cases 5 We must be careful to distinguish indeterminacy, vagueness and ignorance here. For example, Williamson is not an epistemicist about indeterminacy: he believes there s a determinate matter of fact about whether Cedric is bald, but we can t know which.
2. The importance of uncertainty in theorising about vagueness 16 of vagueness should not predict a difference in language use between a single fine-grained interpretation that assigns Cedric is bald the truth-conditions above, and several admissible interpretations assigning Cedric is bald various coarse-grained truth-conditions. Note that a similar issues arise concerning sentences like Hesperus is Phosphorus. Are truth-conditions more fine grained than sets of worlds are? Is the distinction between Hesperus is Phosphorus and Hesperus is Hesperus part of semantics proper? I think in the case in hand there are decisive reasons not to cut truth-conditions as fine as interpretations like J would. One could have J assign each sentence itself as a truth-condition, which would certainly be fine grained enough to formulate homophonic conventions, but they would have no content. Truth-conditions are supposed to be tied to how the world is, rather than how our language represents it, and if it could be vague whether some truth-conditions obtained that would involve an intolerable case of vagueness in reality. It may be vague which truth-conditions a sentence has, but this is not to say truth-conditions themselves are vague. While there may be some unsettled debates concerning the nature of truth-conditions, I take the distinction between language and the world outlined above to be sacrosanct.
3. ERSATZ UNCERTAINTY On first looks, uncertainty about the truth of vague propositions seems to be of a very different kind from uncertainty of the more commonplace kind. Uncertainty, for example, about the future, or about the goings on in far away galaxies, 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 about the way the world is. The former kind cannot be like this, since the world simply does not give answers to questions that have no determinate answer. The world does not determine whether or not Cedric is bald, or which number is the last small number. Secondly, it seems that once one is certain of all the relevant facts about the world - that is, one has no uncertainty of the latter kind whatsoever - one may still be uncertain about the truth of vague propositions. Usually ignorance or uncertainty is curable; there are facts one could learn that would put one in a better position to judge whether or not p. It is difficult to even imagine what it would be like to be in a better epistemic position with regards to p, if p were vague. If I knew all there was to know about Cedric s head and I still didn t know whether he was bald, it s hard to see what else could I do to decide the question - there is nothing left to learn that is relevant to his baldness. These characteristics are atypical in the latter kind of situation in which one is uncertain about the world. An explanation for this disparity would be that we are equivocating; two states that seemed to be of the same kind, are in fact different. Although both kinds of uncertainty play similar roles as far as assertion and action go, they actually correspond to different mental states altogether. Perhaps the kind of uncertainty that arises due to vagueness is sui generis, and not to be assimilated to genuine uncertainty. For example Schiffer - a prominent defender of this kind of view - claims that vaguenessrelated uncertainty is a new kind of propositional attitude, one that comes in degrees and that precludes standard partial beliefs, that it is not a measure of uncertainty, 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 sufficiently informed about the world, and we are considering whether or not p, where p is vague, we are not genuinely uncertain in p, and will typically find ourselves in a 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 p is vague or precise. Your credence in p should be a mixture credence: your conditional credence in p given p is vague (an ersatz credence) times your ersatz credence that p is vague, plus your conditional credence in p on p being precise (a genuine credence) times your ersatz credence that p is precise. You are genuinely uncertain whether p is vague or precise. Your credence in p should be a mixture credence: your conditional credence in p given p is vague (an ersatz credence) times your genuine credence that p is vague, plus your conditional credence in p on p being precise (a genuine credence) times your genuine credence that p is precise. (3.1) (3.2) The best way to account for mixture uncertainties is to treat vagueness-related uncertainty and genuine uncertainty as mixable. I argue that the claim that uncertainty due to vagueness is not genuine, that it is sui generis, loses its bite if it is mixable. The overarching thesis of this chapter, however, is that ordinary uncertainty and vagueness-related uncertainty form a natural kind; that it is genuine uncertainty. 3.1 New psychological vocabulary Consider what happens to Hector, the hairologist considered in 1, when he is presented with a Sorites sequence starting with clearly bald people and ending with clearly non bald people. What is going on in Hectors brain as he moves along the sequence and asks whether the current member is bald? An initially plausible story is that he has high credence that the people at the beginning of the sequence are bald, middling credence that the people around the middle of the sequence are bald, and low credence that the people towards the end are bald. Is this picture compatible with the credences that he has around the middle of the sequence being of a radically different nature from the credences he has at the beginning and end of the sequence? The thesis that vagueness-related uncertainty is of a fundamentally different kind is, of course, terribly imprecise. A minimal characterisation, that will be sufficiently precise for our purposes, I suggest, goes as follows: to properly describe vagueness-related uncertainty, one must introduce new psychological vocabulary which is not explainable in terms of, nor reducible to the original vocabulary of ordinary uncertainty. 1 This, for example, rules out the view in which ersatz uncertainty in p is simply intermediate credence in p accompanied with a positive credence in p, since this was explained entirely using the original 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 points I am about to make are still relevant to this hypothesis.
3. Ersatz uncertainty 19 this coin will land heads is the same credential attitude I have had towards the proposition that Cedric is bald (and vice versa.) Could one be in both states? For example, could one be both uncertain in the ordinary sense, and ersatz uncertain about the same proposition? Perhaps one has both credential and non-credential attitudes (i.e. ersatz credences) towards each proposition? To what extent would this corroborate the claim that when it comes to propositions known to be vague, we are not genuinely uncertain? If for every x in the Sorites, Hector had credence 1 or 0 that x was bald (although perhaps he has intermediate ersatz credences), then there would be no sense in which Hector was genuinely uncertain. His intermediate credences wouldn t be genuine and his genuine credences wouldn t be intermediate. This kind of situation, 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 adjacent propositions in the sequence would be fairly similar. Furthermore, if Hector is genuinely certain that the nth member of the sequence is bald, presumably he must be genuinely certain that earlier members are bald too. Given that we know Hector ends up with 0 credence in the final members being bald, there would 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 both states, there will be cases of vagueness-related uncertainty in which we are both genuinely uncertain whilst having ersatz credences too. If the addition of ersatz credence was supposed to account for the intuition that we are not genuinely uncertain in cases of vagueness, it has not done so. Furthermore, the view quickly comes upon embarrassing questions. Could one have credence 1 in p, but ersatz credence 0 in p? What should your ersatz credences look like when you are genuinely uncertain whether p is precise? What should your genuine credences look like if you are ersatz uncertain that p is precise (if p has second order vagueness, for example)? Unless the compatibilist about these two states can tell us what functional role ersatz credences play, or give a reason to posit such things, I see no reason to think that there is anything more to our attitudes towards 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 for vagueness-related uncertainty, it should not be compatible with the old vocabulary of genuine uncertainty. Thus, in the set up described, we should expect Hector 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 not bald. In the middle, we may assume, he is not really uncertain, but in some fundamentally 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.
3. Ersatz uncertainty 20 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 certain point in the series, and then, at some particular point, stops having these attitudes altogether and starts adopting a fundamentally different attitude? If one thought there was a sharp transition in this Sorites sequence between the determinately true, vague and determinately false, then this would fit with a picture in which there were sharp transitions between the kinds of credential states Hector is in. One should have ordinary credences on the determinately true and determinately false propositions, and ersatz credences on the vague propositions. However, this is not at all how vagueness works. The transition from determinately true to vague, is not sudden - there are instances where it is vague whether the proposition is determinately true or vague. In these intermediate cases which attitude should one adopt? Given the current picture there are two fundamentally different kinds of uncertainty, one for the precise and one for the vague. It is tempting to say that whenever it is vague whether a proposition is precise or vague, it is vague whether Hector has the first or second kind of uncertainty. 3 I want to now consider the consequences this kind of vagueness would have on such a view by focussing on a particular principle governing borderlineness: If F and G admit a borderline case, then F and G are close properties. (3.1) Let me first of all clarify some issues. What I mean by close is best demonstrated by an example. Red and orange are close, whereas navy blue and flourescent pink are not: one could run a Sorites between red and orange to generate borderline cases of red/orange, but one could not do this for blue/pink. The sense of closeness at work here is metaphysical. Someone can be borderline between being bald and not bald because people who are bald aren t fundamentally different from people who aren t. On the other hand, an electron cannot be borderline between being positively and negatively charged, because these are fundamentally different properties. The principle could be defended by appeal to the best fit theory of meaning: expressions referring to natural properties cannot admit borderline cases, because there cannot be semantic indeterminacy where there are reference magnets. I shall not pursue this thought here, but I ll just assert that the principle seems to be independently motivated. Secondly, F and G may have no borderline cases, even if there are true sentences of the form it s vague whether a is F or G. If I were to stipulate that the name a referred to this clearly pink patch if the last small number is even and this clearly blue patch if it s odd, then it would be vague whether a is pink or blue. However, pink and blue do not admit any borderline cases. According to the picture we are considering, someone could be in a state which was borderline between being a genuine and an ersatz credential state. 3 This would follow, for example, from the assumption that determinately, Hector has the first kind of uncertainty in p, just in case p is precise, and determinately Hector has the second kind of uncertainty just in case p is vague.