Denying Ignorance about the Vague

The arguments in section 5.2.1 rested on a couple of assumptions. The first was a relatively fine-grained theory of propositions: one in which there are two kinds of proposition, the vague and the precise.

However, one could imagine denying that vagueness involves ignorance by denying the first assumption: opting for a coarse-grained theory of propositions in which the proposition that Harry is bald is just identical to a precise proposition you know. One could also directly deny that vagueness involves ignorance, even in the context of a fine-grained theory of vague propositions. If vagueness is not a distinctive source of ignorance then there is no special phenomenon that a linguistic theorist is hard pushed to explain.

5.3.1    The fine-grained no-ignorance view

Failures of principles like Ignorance should strike us as extremely surprising. To deny such principles would be to take seriously the idea that, for example, not only is there a nanosecond at which I stopped being a child, but that we in fact typically can, and often do, know which nanosecond this is! The view seems to be open to a simple refutation—as a matter of sociological fact, no one, not even those who reject the ignorance thesis, will ever go so far as to try to answer a question like ‘what is the length of my childhood in nanoseconds?’, even after I've filled them in with all the relevant details of my development.

While claims like Ignorance are widely acknowledged, their acceptance isn't universal. Both David Barnett [12] and Cian Dorr [34] argue that when people are knowledgeable about the relevant precise facts, people often do have knowledge of borderline cases.^11,12

Why is it, then, that we don't say ‘yes' or ‘no' to borderline questions whose answers we know? Let's consider a specific example: Suppose that Harry is a borderline case of the predicate ‘bald'. By the law of excluded middle, an assumption Barnett and Dorr accept, Harry is either bald or he isn't. Let us suppose for the sake of argument that Harry is bald; so by the no-ignorance view I know that Harry is bald, assuming I have the relevant precise facts to hand (hair number, distribution, and so on). Why is it, then, that I don't simply say ‘yes' when someone asks me the question ‘Is Harrybald?'?

There is an important choice to be made at this juncture regarding what one asserts when one utters a vague sentence. Let us begin by assuming a relatively fine-grained theory of propositions: one in which there is a vague proposition, the proposition that Harry is bald, distinct from the precise proposition that Harry has at most n hairs, for each choice of n. This view is naturally paired with the thesis that a vague sentence (uniquely) expresses a particular vague proposition. In particular:

The sentence ‘Harry is bald' expresses a unique vague proposition, namely the proposition that Harry is bald. The belief that Harry is bald can thus be communicated uniquely by uttering the sentence ‘Harry is bald'.

While Dorr's theory, to be discussed in section 5.3.2, is consistent with a coarse­grained theory of propositions, Barnett treats indeterminacy as an operator and is therefore committed to the existence of vague propositions.

Given this picture the problem of explaining why we are reluctant to assert ‘Harry is bald' or ‘Harry isn't bald' is extremely pressing. If I know that Harry is bald, and I can easily communicate that knowledge by simply uttering the sentence ‘Harry is bald', why don't I? In order to explain this, Barnett appeals to the principle:

One should aim to clearly satisfy the rule: (M) assert p only if p.

According to Barnett there will be cases where you know whether p, but you are not allowed to assert p (or —p) because it would result in its being unclear whether you've satisfied (M). That is, even though you know you’ve satisfied (M), i.e. even though you

¹¹             Barnett [12] argues for the mere possibility of knowing borderline truths, although doesn't conclude that we actually know any borderline truths. In Barnett [10] the more radical view that there are in fact borderline truths that we should believe is defended, although he doesn't talk about whether we would know them. It would, however, be puzzling to think that we're not in a position to know what we should believe in these cases. In what follows I shall continue to talk about knowledge although I think a similar discussion could be run if I replaced ‘knows' with ‘ought to believe'.

¹²             Crispin Wright [167] also denies principles like Ignorance, although his theory seems to require that one accept an intuitionistic logic.

There's a natural analogy to be drawn here between Barnett's rule and the claim that one shouldn't assert things that are rude, or mean, or that would break a promise to keep something secret. Even when you know that p, it can bad to assert p if it would violate these principles. But even so, one might think these types of norms can sometimes be trumped: maybe by other norms of communication, or by moral norms, or by something else. For example, if some knowledge you possess could somehow save someone's life, then one should provide it even if it would mean breaking a promise to keep it secret.

Now, if we are to believe the kinds of claims made in Barnett [10] and Barnett [12], Barnett knows, or at least is in a position to know, which the last small number is.^[68] Yet due to this primitive norm of assertion, he's not in normal circumstances permitted to tell us which it is. In the particular case at hand, however, I think the rule (M) is trumped by other considerations. If Barnett really does know which the last small number is, it would be of great value to the rest of the philosophical community if he were to tell us. After all, there has been a long-standing debate amongst classical and non-classical logicians about whether there even is one; if Barnett could settle the matter constructively, that would be significant progress. Therefore, given the great benefits, it seems like it would easily be worth the cost of flouting the rule and telling us which the last small number is.

5.3.2     The coarse-grained no-ignorance view

Unlike Barnett, Dorr's version of the No-Ignorance view is one in which all propos­itions are precise. Dorr tells a very different story about assertion:

When one asserts a vague sentence one asserts a large number of very similar precise propos­itions. When the sentence is borderline, some of those propositions are true and others false.

Thus, for example, when I utter the sentence ‘Harry is bald' I assert, for each n in an admissible range, the proposition that Harry has at most n hairs. When Harry has a borderline number of hairs—suppose that he has exactly k hairs—then some of these asserted propositions are true and some of them are false: the proposition that he has at most k + 1 hairs is asserted and true, and the proposition that he has at most k — 1 hairs is asserted but is false.

What happened to the proposition that Harry is bald? Since all propositions are precise on this view, the proposition that Harry is bald is just identical to the proposition that Harry has at most N hairs, where N is the cutoff point for baldness. Thus when one asserts ‘Harry is bald’ one asserts the proposition that Harry is bald (i.e. the proposition that Harryhas at most N hairs), along with further propositions.^[70]

Dorr hasastraightforward explanationfor whywedonot assert thesentence‘Harry is bald’: if I were to assertively utter this sentence I would increase my audience’s confidence in a number of false propositions.

This is not to say, however, that I can’t communicate my knowledge that Harry is bald: we have only demonstrated that I can’t communicate this knowledge using the sentence ‘Harry is bald’. In fact, I can communicate my knowledge that Harry is bald: I can do this by uttering the sentence ‘Harry has at most N hairs’, since this sentence expresses the proposition that Harry has at most N hairs, and the proposition that Harry is bald is identical to the proposition that Harry has at most N hairs. It follows that the sort of ad hominem charge we levelled against Barnett cannot be extended to someone theorizing in terms of coarse-grained contents.

It should be evident that disquotational principles for dissent and assent cannot be a part of this view. Take for example:

Dissent: Don’t dissent to ‘A’ if you know that A.

By ‘dissent’ I mean to include a range of linguistic responses that include flat out denial as well as a principled and persistent refusal to assent. According to Dorr, one should dissent from ‘Harry is bald’ in some cases, even when one knows that Harry is bald.

Now of course the scope of Dissent needs to be limited somewhat if it is to remain at all plausible. It doesn’t apply if verbally dissenting to ‘A’ will have other bad consequences—you might, for example, wake the baby (if you dissent too loudly), or perhaps you’re in a case where lying is morally justified, or ‘A’ has a presupposition that you want to avoid.^[71] However, it seems like a prima facie cost if one cannot accept some suitably limited principle of this form—such principles seem to be integral to the way that we learn what people believe on the basis of their linguistic behaviour: if I look all puzzled and refuse to say ‘yes’ or ‘no’ when you ask me a question like ‘is Harry bald?’ English speakers typically conclude, mistakenly according to Dorr, that I don’t know whether Harry is bald.

Here is another worry. Supposing that I know the number of hairs on Harry's head, then even though I know that Harry is bald I won't assert the sentence ‘Harry is bald’, for that would be to assert (or at least, raise my audiences confidence in) a number of similar but false propositions. Surely, one might think, if I know that Harry is bald it should be simple for me to introspect on that fact, and come to know that I know that Harry is bald. Why then couldn't I just assert the sentence ‘I know that Harry is bald' and allow my audience to conclude that Harry is bald from that? According to Dorr it is for exactly the same reasons I cannot assert the sentence ‘Harry is bald'—the knowledge ascription is also borderline. Generally, when S is borderline ‘I know that S’ is also borderline provided I'm in possession of the relevant precise facts.

Note that there are two aspects of our linguistic behaviour that need explaining. One is our refusal to accept a sentence or its negation when it is a simple borderline sentence that does not involve attitude reports, such as ‘Harry is bald'. Dorr's explan­ation seems to account for this adequately. It is not clear, however, whether Dorr can accommodate our linguistic behaviour regarding borderline sentences involving knowledge and belief ascriptions like Alice knows that Harry is bald'. Unlike the simple borderline sentence ‘Harry is bald', where we are not inclined to assert it or its negation, we are inclined to outright deny things like ‘Alice knows that Harry is bald' and to even assert its negation. Yet according to Dorr, this sentence is also borderline, and we should not be asserting either it or its negation.

Not only are utterances of Alice knows that Harry is bald' utterances of borderline sentences, according to Dorr, but one would expect these utterances to be untrue in cases where the knower is in possession of the relevant precise facts. It seems, then, that this theory must endorse some kind of error theory regarding attitude reports: speakers frequently assent to the false sentence Alice doesn't believe/know that Harry is bald'.

Things get worse. The sentence Alice believes that Harry is bald', let's suppose, is borderline. Now, according to Dorr, if I'm in possession of the relevant precise facts (perhaps the fact that Alice believes that Harry has exactly n hairs) then I know whether Alice believes that Harry is bald. Thus I am in an even worse position than someone who goes about assertively uttering false sentences: I am going about asserting Alice does not believe that Harry is bald' when I in fact know that Alice believes that Harry is bald.

5.3.3     Non-Iinguistic behaviour

Setting aside the troubles with attitude reports, an adequate theory ought to do more than just account for linguistic behaviour that is associated with vagueness. It seems to be extremely hard to eradicate vagueness-related uncertainty from explanations of our non-linguistic behaviour as well. It is therefore hard to see, even in principle, how a view in which vagueness is a public language phenomenon could accommodate this behaviour.

For example: I may know exactly how much cheese I have, but still be unsure about whether I have enough to make a cheese sandwich that tastes reasonably good—that, for example, doesn't have too high a ratio of bread to cheese. It is natural to attribute this uncertainty to the fact that I have a borderline amount of cheese. There is no precise fact that I am unsure of, but yet I still hesitate: it would be a waste of bread if I don't have enough cheese, but it would satisfy my need for food if I do. How is this to be explained if all ignorance is ignorance about the precise? You might try to attribute the hesitation to uncertainty about some precise fact about what the resulting sandwich would taste like. Let me clarify then: I know exactly what it would taste like, I just don't know whether that taste is tasty—with that cheese to bread ratio, the result would be borderline. The hesitation is not like the refusal to assert or deny a sentence—I needn't even speak a language to be in this situation. The problem is completely internal to me: I don't know whether the sandwich would be tasty, and so my actions will depend on how confident I am about its tastiness.^[72] Phenomenal sorites show that these kinds of scenarios are ubiquitous—the states you are in when something looks red to you, when you feel cold or feel hungry, and so on, are all soritesable, and these states all seem to play an important role in guiding our non- linguistic behaviour in ways that couldn't be explained if we were always certain about whether we were cold or hungry.

Another example in which vagueness makes its way into our thought, although plausibly not via a public language, is when we acquire evidence through imperfect perceptual faculties. IfI see a tree in the distance I learn some things about its height. It seems implausible that my total evidence, after seeing the tree, is that the tree is between x and y centimetres tall, or any precise proposition of this type—my credences will presumably fit some kind of smooth curve over the possible precise heights, but no credence that is gotten by conditioning on a precise proposition would have this smooth shape. It is natural to think that in this case my total evidence is vague; yet the visual experience and my resulting epistemic state had nothing to do with my ability to speak a language. Of course, there is much more to be said about this argument, and we'll return to that in chapter 6, but on the face of it, vagueness is pervasive in our non-linguistic mental lives.

5.3.4     The Contextualist no-ignorance view

One of the main problems with the no-ignorance account described above is its treatment of attitude reports. Indeed, one can see this as an instance of a more general problem of trying to account for propositional attitude ascriptions in a theory where the objects of attitudes are reasonably coarse-grained. If the proposition that Harry is bald just is the proposition that Harry has less than n hairs, for some n, then the fact that we are disinclined to say that someone believes that Harry is bald even if we are inclined to say that they believe that Harry has less than n hairs might be subsumed by a more general theory for dealing with Frege puzzles.

Of course, one solution is to adopt a fine-grained account of the things appearing in the complements of attitude ascriptions (i.e. ‘propositions', in my terminology). This would allow you to accept the conjunction of Ignorance and Knowledge, and falls under the fine-grained views we have already considered in earlier sections. If we want to maintain a coarse-grained account of propositions it seems as though we'd either have to bite the bullet and accept the problematic attitude reports, or adopt a contextualist view in which the word ‘knows' expresses a different relation between people and propositions in different contexts.^[73] On this view it is consistent to say that the expressions ‘that Harry has less than n hairs' and ‘that Harry is bald' denote the same proposition, p, but still maintain that utterances of‘Jane knows that Harry has less than n hairs' and ‘Jane knows that Harry is bald' can have different truth values provided that they are made in different contexts. In one context ‘knows' must express a relation that Jane bears to p, whilst in the other context it must express a relation which she doesn't bear to p.

Let me suggest one way to cash this idea out a bit further. When a person has a belief-like relation to some proposition p they typically do so by being in a particular kind of mental state. Naturally lots of different mental states can result in the same proposition being believed: I might mentally token something like the sentence ‘Hesperus is Hesperus' in believing the necessary proposition, or I might mentally token something like the sentence ‘Hesperus is Phosphorus’. Presumably it would be fairly easy to come to know the necessary proposition by believing it the first way, and not so easy to come to know it by believing it the second way. For convenience, call these different ways of believing a proposition ‘modes of presentation'. Presumably before the discovery that Hesperus was Phosphorus, nobody knew the necessary proposition via the second mode of presentation, although everyone knew it via the first mode of presentation.

So far I have been using locutions like ‘S knows/believes p via m’ Let us grant the empirical assumption that this is really what is going on psychologically when we believe things—that this expression picks out a natural three-place relation between propositions, people, and modes of presentation. How does this technical ternary relation relate to the binary ordinary language expressions ‘knows' and ‘believes' and so on? These are the terms that we have been theorizing with, and are the terms relevant to the puzzles we have been developing. According to contextualism, there is no one relation that these expressions pick out: in contexts where a mode of presentation m is salient, ‘knows’ picks out the relation of knowing via m, and in other contexts knowing via m'. A context, then, can be thought of as providing a way of matching up coarse-grained propositions with modes of presentation.

It’s natural to think that the that-clause we use when making an attitude ascription brings to salience certain modes of presentations and not others. Thus when I say Alice doesn’t know whether Harry is bald’ I bring to salience a vague mode of presentation corresponding to the vague sentence ‘Harry is bald’, and when I say Alice does know whether Harry has less than n hairs’ I bring to salience a precise mode of presentation, even if, in both cases, I am just ascribing some relation between Alice and one and the same proposition. Note that the view in question is importantly different from the view which treats propositions, the denotations of that-clauses, to be ordered pairs of sets of worlds (or some other coarse-grained entity) and modes of presentations. There are affinities, but this view is a version of the fine-graining strat­egy: that view can straightforwardly make sense of the conjunction of Ignorance and Knowledge, and is therefore covered by the criticisms in section 5.3.3.

The contextualist cannot accept the conjunction of Ignorance and Knowledge— at least, not unless the context changes mid-sentence.^[74] For whatever mode of presen­tation is salient, since we are relating the agent to one and the same propositions, it cannot be both known and not known relative to that mode of presentation.

At this point it is worth drawing the analogy between this type of view and a similar view in the philosophy of modality. For example, when railing against the third grade of modal involvement, Quine writes ‘being necessarily or possibly thus and so is in general not a trait of the object concerned, but depends on the manner of referring to the object’ [114]. According to Quine an object is only necessarily F relative to some linguistic ‘mode of presentation’ of that object: the number nine is necessarily composite relative to the mode of presentation ‘the square of three’ but not ‘the number of planets’. It’s natural for the contextualist to make an analogous move here concerning determinacy operators. Rather than taking the relation ‘S is borderline in L relative to parametersp’ as primitive (as suggested in section 4.2), this theorist might theorize instead with a slightly more complex expression ‘It’s borderline whether p relative to the mode of presentation S in L and parameters p’

The view does not completely solve all the problems. It predicts, for example, the existence of a puzzling context in which it is okay to assert Alice knows whether Harry is bald’ Moreover, there will be no context in which it is okay to assert ‘John knows that Jane’s height is less than xcm but does not know whether she’s tall’: whatever mapping from propositions to modes of presentation the context provides, if the proposition that Jane is less than.vcm is identical to the proposition that she's tall, it will be assigned the same mode of presentation by the context and John must either know the proposition or not know it relative to that mode of presentation.

5.3.5     More on non-linguistic behaviour

The puzzles raised about the role of vague beliefs in guiding our non-linguistic behaviour in section 5.3.3 seem to be just as pertinent here. If it is modes of presentations that are vague, and these are vague in virtue of their relation to public language sentences (perhaps they are in some sense synonymous with vague public language sentences), it is hard to see how they play a role in behaviour that doesn't seem to involve language.

Moreover, most of our vague beliefs aren't articulable, either in a public language or even in a private language of thought. If I've been blindfolded and rolled down a hill, I have lots of beliefs about which direction is roughly up, which I acquire through some form of proprioception. I certainly have these beliefs, since they affect the actions that I make, but I couldn't articulate these beliefs in English, or even mentally. They are just beliefs that I have, they do not appear to be beliefs about any precise fact, and they do not appear to be linguistic. Such facts are puzzling for a view on which it is only linguistic items that are vague.

Consider the following scenario: Alice ran because she believed that Jack the Ripper was following her—not because she believed that Harmless Harry was following her, even though Jack the Ripper and Harmless Harry are the very same person. In order to explain why Alice acted in the way that she did we must appeal to a potentially hyperintensional distinction in her attitudes. The way we behave depends on what we believe and desire, and any decent theory of propositional attitudes ought to be able to accommodate explanations like this one. It is crucial to note that hyperintensional differences in a person's beliefs do not just generate differences in linguistic behaviour—the contextualist would not be hard pressed to accommodate differences in linguistic behaviour within her theory of modes of presentation— they also generate differences in non-linguistic behaviour as well. Alice's behavioural profile could be instantiated by someone who didn't speak any languages whatsoever.

Decision theory is an extremely general and powerful framework for representing a rational agent's decisions, beliefs, and desires. The intuition behind decision theory is very straightforward: to find out how good things are conditional on some sup­position, A, first of all assume A, and calculate a weighted average of how good the remaining epistemic possibilities are, with each possibility weighted by its probability given A.

A standard way to formalize this intuitive idea, due to Richard Jeffrey [73], rep­resents propositions by sets of indices. In Jeffrey's presentation the indices are in fact possible worlds, but nothing turns on this—they could be something more fine­grained like epistemically possible worlds. A utility function is a function mapping each index to a real value, representing how good that index represents things to be, and one's degrees of belief are presented by a probability function on the set of propositions (i.e. the sets of indices) satisfying the usual axioms of probability.^[75] The expected utility, or news value, of a proposition is then given by summing (or integrating if necessary), over each index, the result of multiplying the utility of that index with the probability of that index conditional on that proposition. According to the orthodox interpretation of this formalism, the indices correspond to epistemic possibilities, sets of them to propositions, and the probability and utility function correspond to graded attitudes equivalent in nature to the ungraded attitudes of belief and desire. (For a more thorough presentation of decision theory, see chapter 9.)

Decision theory is well established. There should, I think, be a default obligation on those who wish to abandon it to provide some reasonable alternative. Now, a crucial feature of this framework is that it is a theory of rational action that operates directly on the contents of the agent's beliefs and desires. The different ways of having beliefs with those contents, the modes of presentation, play no role whatsoever in its formulation—perhaps such things are needed in the correct semantics of belief reports, but for orthodox decision theory they remain completely idle.^[76] Conclusions that can be drawn about fineness of grain in this framework, then, cannot be explained away by standard contextualist manoeuvres invoking modes of presentation.

According to orthodox decision theory, then, whether an agent's action is rational or not supervenes on the contents of her beliefs and desires (construed broadly to include assignments of credence and utilities):

Content to Action: If Alice and Bob's beliefs and desires have the same con­tents, then they will act the same way if they are rational and have the same actions available to them.

Content to Action allows us to draw conclusions about how fine-grained contents are in a way that are impervious to the standard contextualist responses. For example, it entails that if two rational people can behave differently whilst having only beliefs and desires with necessarily equivalent contents then contents are more fine-grained than sets of worlds.

Let us demonstrate this strategy with an utterly trivial example. If we were willing to relativize all operators to guises, in the way the contextualist does to attitudinal operators, then it is a live option that there are only two propositions: the true and the false. However, even if we can account for attitude reports and other non-truth functional operators using the Contextualist apparatus there is a distinct problem deriving from decision theory. Given Content to Action there are only sixteen types of people, even including probabilistically incoherent people, differing only in the combinations of beliefs and desires they hold towards the two propositions. It is clear, however, that there are more than sixteen different rational ways of behaving.

This is, of course, a very boring version of the argument; however, you might think a similar argument can also be applied to show that propositions are more fine-grained than sets of worlds too. The basic point here is that the argument from attitude reports and the argument from decision theory are two distinct problems that both need addressing.

One can see how these considerations might also be relevant to the case of vagueness: almost all of our beliefs and desires, involving ‘medium sized dry goods, are vague. Although I concede that one could in principle have a perfectly precise belief or desire, perhaps about some aspect of logic or mathematics, we would not be equipped to deal with ordinary day-to-day life if all our beliefs and desires were like this. I will not pursue the application of these ideas to vagueness here, however, as I will resume discussion of these issues in chapter 10 which is devoted to this topic.

Returning to our initial line of thought, it should be clear, I think, that a contextual- ist possible worlds theorist can at least in principle accommodate our intuitions about the two sentences beginning with ‘Alice ran because... ’by postulating a change in context. The problem is not that the contextualist cannot explain particular utterances that purport to explain actions in terms of beliefs and desires. The problem is the rather more theoretical one of integrating particular explanations of this sort into a general theory of rational action. To my knowledge, no theorist of this stripe has ever developed anything looking like a half decent decision theory involving modes of presentations that avoids Content to Action.^[77] To do this, one would have to replace orthodox decision theory with a theory where the rationality of an action depends not only on the contents of your beliefs and desires, but on how you have them, or in other words, that depends on the modes of presentation under which you assign credences and utilities.

I think the technical obstacles to this project would be immense. Here are just few problems I can foresee, although I doubt these will exhaust the difficulties. Firstly, in order for credence to depend on modes of presentation one needs to relativize credences to modes of presentations. I can see what it might mean to have a credence of 0.6 in the necessary proposition via a mode of presentation corresponding to ‘Hesperus is Phosphorus', but then I presumably won’t have a credence of any defined value in the proposition that snow is white relative to this mode of presentation. This makes it unclear how to even formulate probabilistic axioms for mode-of- presentation-relative-credences so understood. Secondly, recall that the basic thesis of decision theory is that the expected value of a proposition p is the sum, weighted according to probabilities conditional on p, of the utilities of a partition of epistemic possibilities. On the present view people only assign probabilities and utilities to epistemic possibilities relative to modes of presentation, and there is no canonical way of pairing epistemic possibilities with modes of presentations of the possibilities. There is therefore no reason to expect that one can assign a unique expected utility to p relative to a mode of presentation for p, since one has a choice about how to pair the epistemic possibilities with modes of presentation. Even worse, I think, is that there is no guarantee that the probabilities defined relative to some pairing will sum to one, leaving it open whether the resulting notion of expected utility will conform to the ordinary constraints on rational preferences.

Until basic problems like these are solved, it's totally unclear how to go about formulating a decision theory that takes into account modes of presentation. The simplest way to avoid these troubles would be to treat propositions as ordered pairs of sets of worlds and modes of presentation, and to treat negation, disjunction, and so on as operations on both coordinates of these things simultaneously. This is, of course, just a version of the fine-graining strategy I have been urging for: a view in which the objects of our attitudes are more fine-grained than sets of worlds.