Explaining Ignorance about the Vague

Ignorance is an instance of one of a tightly knit cluster of phenomena that we associate closely with borderlineness. There are systematic connections between facts like Ignorance and borderline cases—facts like this just call out for some kind of general explanation.

According to the theory I ultimately defend, that explanation takes the form of a general principle, which in turn falls out of a general theory of vague propositions:

Epistemic: Necessarily, if it's borderline whether p, then it's not known whether p.

A linguistic theorist who has no paraphrase for Epistemic in her own ideology cannot accept it as an explanation. If she is to offer a general explanation of Ignhrance which still has something to do with vagueness she must instead seek to explain Ignhrance in terms of her favoured vocabulary: S is borderline in L relative to context c and other parameters. But if the fact that some sentence, S, is borderline as used in language L, by community C, at context c, at time t, and world w explains why no one in the international team of glass measurers knows whether the glass is pretty full (i.e. explains Ignhrance), we must ask:

(a)    Which language L and linguistic community?

(b)    Which sentence of this language is such that its borderlineness explains Ignhrance?

(c)     At what time must the sentence be borderline?

(d)    What context must the sentence be borderline in?

(e)    At what world must the sentence be borderline?

Before we move on let me stress that the demand is not just to explain why the English or Spanish or Mandarin speakers among our team don’t know whether the glass is pretty full.

they have inspected the glass, no matter what language they speak, or whether they even speak a language, no matter what the architecture of their brains and so on.

So let us begin with the first question, (a). Suppose, without loss of generality, that the sentence is simply the English sentence ‘this glass is pretty full', uttered in a context where ‘this’ refers to the 70% full glass in question. Could the fact that this particular sentence is borderline in English (relative to the relevant context and other parameters) explain why nobody knows that the glass in question is pretty full? Could it, for example, explain why none of the monolingual Spanish speakers who inspected the glass know that the glass is pretty full? The answer to this question seems to be obviously ‘no’—the linguistic practices of people in English-speaking countries can do nothing to prevent monolingual Spanish speakers from knowing whether the glass is pretty full.

The best we can do is explain my ignorance by appealing to the borderlineness of an English sentence, a monolingual Spanish speaker’s ignorance by appealing to the borderlineness of a particular Spanish sentence, and so on. Although it is unclear to me whether even these explanations are possible, it is natural to object that even if they were, they would be incomplete—they say nothing of intelligent creatures that do not speak a public language. Would it be easier for me to find out whether the glass is pretty full if I didn’t speak a language? And even if I do speak a language, why should my co-speakers’ linguistic habits bear on whether I can find out whether the glass is pretty full?

More importantly, it is not clear that we have an explanation of the fact that nobody who tried to determine whether the glass was pretty full succeeded.

Perhaps the general fact is that both Mary and Lucia speak a language containing a sentence which both expresses the proposition that the glass is pretty full, and is moreover used in whatever way suffices to make a sentence borderline in that language. This explanation is adequate only on the assumption that every language under consideration has a sentence which expresses the proposition that the glass is pretty full, and which is used in that special way that makes the sentence bor­derline. But some of the languages in question might fail to have a sentence which expresses that proposition; maybe there is no perfect translation of‘this glass is pretty full' into German, for example, even if there might be something pretty close. But this should not make it any easier for a German to find out what we cannot in this situation.

These remarks cast some doubt on the possibility of answering our second question, (b).

Finally, it seems we must specify a time, world, and context at which the sentence is borderline. Words often become more precise as language evolves. If the selected sentence S (‘this glass is pretty full' say) had once been determinate it would presum­ably still be impossible to know whether the glass is pretty full. Conversely, if‘electrons have negative charge' had once been borderline that shouldn't prevent me from finding out that electrons have negative charge. Maybe S has to be borderline at the time I'm trying to find out whether the glass is pretty full. If we have to pick a different sentence each time we want to explain why someone can't figure out whether the glass is pretty full then the explanation loses its generality.

Furthermore, had we used S in a precise way we wouldn't be in a better epistemic situation regarding Harry's head. We may therefore be able to explain why no one in fact knows whether the glass is pretty full by appealing to a selected sentence's borderlineness, but in order to explain why we couldn’t have known whether the glass is pretty full, even if S had been used in a precise way, we need also to relativize to worlds.

Let me end by mentioning one more point which, although not a fully fleshed out objection, is something I find worrisome. In many ways explaining Ignorance is one of the easier jobs for a linguistic theorist. Other facts are harder to explain. Suppose that Harry is as before a borderline case of baldness, and that we rationally believe this. Then it seems that:

Agnosticism: It would be irrational to believe (given what we know) that the glass is pretty full, and it would be irrational to believe that the glass isn't pretty full.

This fact is prima facie quite puzzling. After all, you know that either Harry is bald or he isn't, so at least one of the two beliefs above is true. Furthermore you have all the evidence you could possibly have available. It feels like there should be some general fact about vagueness-related phenomena that explains this.

Since there appear to be plenty of things we do not know which are rational to believe, it seems, therefore, that there is a separate and harder problem of explaining what feature of linguistically vague sentences prevents us from rationally believing certain propositions. This seems a lot harder to do—for example, in a discussion of Williamson’s epistemicism, Horwich writes: ‘the ignorance due to vagueness is attributed to a special form of unreliability—an external failure—and not, as it should be, to the internal difficulty in making a judgement’ (Horwich [71]). But surely any explanation of our ignorance that appeals to the use of a term in a public language is going to be an external one.^[63] Even if we bracket the problems surrounding the explanation of Ignorance, I am much less confident that an explanation of Agnosticism in terms of language use can be given.

5.2.1    Explaining ignorance via metalinguistic safety principles

Let me now turn to a specific attempt to explain Ignorance within a linguistic theory.

The crux of Williamson’s explanation is a controversial principle that has come to be known as the ‘metalinguistic safety principle’. In order to understand the principle we firstly need to introduce some definitions. Say that As belief that P is safe iff it couldn’t easily have been the case that (i) A believes that P and (ii) it is not the case that P. A’s belief that P is metalinguistically safe iff it couldn’t easily have been the case that (i) A produces the belief token that actually resulted in a belief that P and (ii) that belief token expresses a false proposition (let us say that a belief token ‘expresses’ a proposition P if it constitutes a belief that P). The expression ‘it couldn’t easily have been the case that P^, is a term of art, and means something roughly like P isn't true in any nearby world, where what is ‘nearby' is determined by some epistemically significant measure of similarity. It is unclear whether one can get an understanding of the relevant notion of similarity without already having a grip on the concept of knowledge, but however this question turns out, the notion of a safe belief is still of interest and can be used to put some important structural constraints on knowledge that would be hard to motivate without invoking the notion.

Although it is by no means uncontroversial, a sizeable number of philosophers take the notion of a safe belief to have some epistemic force. In particular, these philosophers subscribe to something like the following safety principle:^[64]

Safety: One knows that P only if one does so via a belief that is safe.

Roughly, the thought is if you could easily have falsely believed that P, then even if you were in fact correct about P you were lucky to be correct, and so your belief could not constitute knowledge. It is important to contrast this principle with its metalinguistic variant:

Metalinguistic Safety: One knows that P only if one does so via a belief that is metalinguistically safe.^[65]

The safety principle says that if you know that P then you couldn't easily have been wrong about P. The metalinguistic safety principle entails no such thing: being easily wrong about P is neither necessary nor sufficient for having a metalinguistically unsafe belief—all you need to do to be metalinguistically unsafe is to have a false belief in some proposition in a nearby world; although not necessarily a false belief that P, it need only belong to a belief-type whose tokens are beliefs that P in the actual world.

It should be clear that the ordinary safety principle cannot explain Ignorance. For example, in the present example, given my knowledge, the glass is 70% full in all worlds that count as nearby for me, and since whether the glass is pretty full supervenes on how full it is, it is either pretty full in all nearby worlds or not pretty full in all nearby worlds. Thus for all the safety principle says, a belief that the glass was pretty full could constitute knowledge provided it was in fact a true belief. Metalinguistic safety does better in this regard. One could make a reasonable case that the semantic properties of belief tokens are correlated to the semantic properties of corresponding public language sentence tokens. On that hypothesis, we can attribute a similar degree of semantic plasticity to vague beliefs corresponding to vague public language sentences, and we can also attribute to them the feature Williamson identifies with borderlineness: in some nearby worlds they express a true belief and in others they express a false belief. If I form a belief that the glass is pretty full by forming a belief that is borderline in this sense, then this belief is not metalinguistically safe: although I couldn't easily have been wrong about whether the glass is pretty full, I could easily have been wrong about another proposition—one that, in that same nearby world, would have been expressed by the same belief token.

Let me firstly note, along with many others, my reservations about the metalinguis­tic safety principle.^[66] Suppose that, unbeknownst to me, my fellow English speakers had decided to start using the numeral ‘1' to mean 100. Grant also the assumption, needed in the explanation above, that the semantic properties of beliefs and their cor­responding public language sentences are correlated. Since the actual world certainly counts as nearby, it appears as though there are nearby worlds where a belief token corresponding to ‘1+1=2' expresses the false proposition that 100+100=2; in other words my belief that 1+1=2 is metalinguistically unsafe. Yet, I hope, it should be clear that this form of unsafety does not in any way undermine my knowledge that 1+1=2, a fact which I can verify by performing a simple calculation.

Although the metalinguistic safety principle seems somewhat suspect, notice that the ordinary safety principle, which is on much firmer footing, does allow us to explain a related fact: that we cannot know whether the sentence ‘this glass is pretty full' is true or not in English relative to the context and other parameters described. If that sentence is borderline in that context then it expresses a false proposition in a nearby world, and so, we are to suppose, the belief corresponding to this sentence is not metalinguistically safe.

Recall, however, that I took pains to distinguish Ignorance from Linguistic IGNORANCE—the latter claim only states that it is impossible to know whether the sentence ‘this glass is pretty full' is true in English relative to the relevant parameters. The fact that there are close worlds where we use ‘this glass is pretty full' differently may explain, by ordinary safety principles, why we cannot know that this sentence is true in English. But whether the glass is pretty full or not is not unstable in the same way: it is just as full as it actually is at all the relevantly close worlds. So there is also the further fact that, given the status of the glass, we cannot know whether it is pretty full, and there is no obvious way to infer ignorance of the latter fact from ignorance of the former linguistic fact.

One way to bridge this gap would be to invoke knowledge of the disquotational schema. Suppose I know that if this glass is pretty full then the sentence ‘this glass is pretty full’ is true in English, at the present context, etc. However, since we have just shown that I do not know the consequent on the basis of an orthodox safety principle, one can infer that I do not know that the glass is pretty full, assuming a small amount of closure. A parallel argument could be made to show that I do not know that the glass is not pretty full either. Of course, this argument still suffers from the defect that it is not completely general. To explain why a monolingual Spanish speaker does not know whether the glass is pretty full we’d appeal to knowledge of a principle that does not take the form of a disquotational principle when it is stated in English: if Lucia knows that this glass is pretty full then she knows that ‘este vaso esta bastante lleno’ is a true sentence of Spanish in this context. The fact that neither I nor Lucia know whether the glass is full seems to call out for a general explanation—something holding both the cases together—which the explanation I have just given does not seem to provide. The explanation we gave relied on specific knowledge I had about English, and Lucia had about Spanish—knowledge relating the truth of sentences in these languages to a fact about the glass—which is not present in all cases where there is ignorance about whether the glass is pretty full.

Generality aside, there are some important limits to when knowledge of disquo- tational reasoning can be appealed to, even among people who speak the relevant languages. When words change their meanings, or we are no longer sure of a word’s meaning, we are generally not in a position to know instances of disquotational principles involving those words. To demonstrate the general point, let us suppose that Alice is looking at Madagascar for the first time, and correctly concludes that it is an island. However, Alice is living during the time in which the word ‘Madagascar’ was being used to refer to a portion of mainland Somalia. So although Alice does in fact have a correct belief that Madagascar is an island, and in fact she knows it is an island, she also knows that the sentence ‘Madagascar is an island’ is not a true sentence of English. Similarly, cases where a word’s meaning is unknown give rise to cases where the relevant disquotational principles are unknown. If Alice is observing a particular cow chewing grass, then it’s plausible that she knows that the cow is masticating, although if she doesn’t know that ‘masticating’ means chewing, she might not know that the sentence ‘the cow is masticating’ is true in English in her context.

The critical point here is that if the argument from semantic plasticity is to work, then there are nearby worlds in which the borderline sentence does not mean what it actually means. Assuming the ordinary safety principle, it follows that unless our beliefs about meanings are extraordinarily sensitive to very slight differences in meaning due to slight differences in use, we simply do not know what borderline sentences mean. Thus it follows that the above argument for Ignorance appeals to exactly the kind of knowledge of disquotational principles that we have seen to be suspect.^[67] Indeed we can effectively prove that this instance of the disquotational principle is unknown from Safety. That is, we can argue that although the sentence ‘“Harry is bald” is true if and only if Harry is bald' expresses a truth at all nearby worlds (even if it is a different truth at each world), the regular safety principle predicts that, were there nearby worlds where ‘Harry is bald' is true even though Harry isn't bald (for example), we would not count as knowing that ‘Harry is bald' is true iff Harry is bald. Let us suppose for the sake of argument that Harry is not bald (a symmetrical argument can be made if he is) and that he has the same number of hairs at every nearby world (let's suppose I know how many hairs he has). So there are no nearby worlds at which Harry is bald, since baldness supervenes on hair number. Yet by hypothesis the sentence is borderline, so there are nearby worlds where ‘Harry is bald' expresses a truth and nearby worlds where it expresses a falsehood. So there are nearby worlds where ‘Harry is bald' is true even though Harry is not bald, so by the safety principle it follows that I do not know that ‘Harry is bald' is true if and only if Harry is bald.

The epistemicist might insist at this juncture that although there are nearby worlds where Harry is bald, even though ‘Harry is bald' isn't true in English (because it means something else), our beliefs are surprisingly sensitive to the differences in meaning of ‘Harry is bald' between these different worlds. Sensitive enough that we are able to have the belief that Harry is bald iff‘Harry is bald' is true in English only at the worlds where ‘Harry is bald' does indeed mean something that's true iff Harry is bald. On its face this suggestion looks absurd: according to the picture described, the meaning of ‘Harry is bald' in English depends on very particular features of usage, and can change on the basis of slight differences of use that are clearly far beyond the knowledge of ordinary humans. On appearances this sounds like the type of view where we cannot know what vague sentences mean—the best we can hope for is to know the range of meanings a word could have.

The epistemicist might try to defend the position by noting that at the nearby worlds where ‘Harry is bald' means something else, so does the corresponding disquotational principle: ‘“Harry is bald” means that Harry is bald in English'. Indeed, whatever ‘Harry is bald' means at this world—p, let's say—the instance of the disquotational principle will mean something true, namely that ‘Harry is bald' means that p, and not the false claim that ‘Harry is bald' means that Harry is bald. The thought, then, is that we get to know what ‘Harry is bald' means simply by tokening some mentalese equivalent of the sentence ‘“Harry is bald” means that Harry is bald', which is guaranteed to express a truth whatever it means.

It is hard not to think there is something extremely fishy about this way of acquiring knowledge of something. Consider again Alice, who does not know what ‘masticate' appearing in ‘p’. However, this is still not sufficient to guarantee that one knows that ‘p’ is true if and only if p, unless one has knowledge of the more complicated instance of the T-schema: ‘“p” is true if and only if p’ is true if and only if ‘p’ is true if and only if p. But knowledge of this is suspect for the same reason that knowledge of the initial instance is suspect.

means, but can see, and therefore knows, that a particular cow is masticating, even though she has no idea whether the sentence ‘the cow is masticating' is true in English. I take it that there is something obviously wrong about inferring from her non-linguistic knowledge about the cows eating behaviour, that a particular English sentence, whose meaning she is completely unsure about, is true. To conclude that ‘the cow is masticating' is true in English in Alice's position seems just as obviously wrong even once we have pointed out that if she were to form a belief of the form ‘if the cow is masticating then “the cow is masticating” is true in English' her belief would most likely express a truth, even if she does not know which truth it is.

5.3