Unknowable self-truths and the KK principle∗ November 25, 2014

Abstract Recent philosophical literature has persuasively argued that there are unknowable truths. In this article, I argue for the related claim that there are many truths about oneself that one can never know. My argument is more conclusive if the KK principle is false, and so I defend the popular margin-for-error argument for its falsity from recent objections.

Some truths are unknowable. While this claim is still controversial, in my opinion it has been conclusively shown to be true, and I believe that this is a rare and therefore important example of philosophers conclusively answering a fairly big philosophical question (cf. Chalmers forthcoming). Here I argue for the related claim that there are many truths about oneself that one can never know. In other words, there are many unknowable self -truths. Section 1 spells out my claim more precisely. Section 2 presents the argument that there are certain propositions about oneself that one can never know. Roughly, one can never know the ‘level’ at which one knows any particular proposition. In Sections 3 and 4, I argue that almost all of our knowledge is at some level or other, and so there are many truths about one’s levels of propositional knowledge. If the KK principle (that if one knows something, then one knows that one knows it) and several related principles are all collectively true, then some of the assumptions in my argument are false; so in Section 4 I present the standard margin-forerror argument against the KK principle and reply to several recent objections. Section 5 defends my claim from an objection based on cases of self-ignorance, and also briefly mentions that Elga’s Dr. Evil case may contain another way of constructing unknowable self-truths.

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Unknowable self-truths

In my opinion, it has been proven that some truths are, in the broadest sense, unknowable (Fitch 1963), though the proof is still somewhat controversial (Brogaard and Salerno 2013). Here I set out a version of the Church-Fitch proof, since I use a similar proof to argue for my claim. ∗ Acknowledgements

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We assume that there is some true proposition that is actually never known (though perhaps knowable).1 To take a common example, no one will ever know how many socks were in my drawer on New Year’s 2014, but either (i) the number of socks in my drawer on New Year’s 2014 is even or (ii) the number of socks in my drawer on New Year’s 2014 is odd. It would have been easy for me to know one of these propositions, but as it turns out neither will be known, and since one of them must be true, one of them is an unknown truth. Let ‘EVEN’ abbreviate ‘the number of socks in my drawer on New Year’s 2014 is even’. Now consider this more complex proposition: EVEN, and no one knows that EVEN It is impossible to know this proposition. For say some person knows that (EVEN, and no one knows that EVEN). Because knowledge distributes over conjunction, they know that EVEN.2 But because knowledge is factive, it is true that (EVEN, and no one knows that EVEN), and so it is true that no one knows that EVEN. So they do not know that EVEN. This is a contradiction, and so it is not possible for some person to know (EVEN, and no one knows that EVEN). Similarly, it is not possible for some person to know (ODD, and no one knows that ODD). One of these propositions must be true, however. So since both are unknowable, there is some unknowable truth.3 While there is a sizable literature discussing this proof (Brogaard and Salerno 2013), I take it to be sound. Its soundness immediately suggests an interesting research project, one proposed by the last paragraphs of (Williamson 2000): if there are indeed necessary limits to knowledge, then it seems worthwhile to find out exactly what these limits are. My goal here is to show that there are certain facts about oneself that one can never know. The notion of a fact being ‘about oneself’ is a bit vague, so here is one way of trying to make my claim more precise. The Church-Fitch proof above showed that there is some way of filling out the ‘. . . ’ where it is true that: It is not possible that there is some x that knows that . . . 1 If an omniscient divine being exists, then we assume there is some true proposition that is never known by any non-divine being, and the conclusions of this paper only concern nondivine beings. 2 The assumption that knowledge distributes over conjunction is somewhat controversial. I find the arguments for this assumption in (Williamson 2000, c. 12) persuasive; Williamson also presents plausible ways of strengthening other assumptions to avoid the need for this one. Also note how plausible the needed instance of this assumption is here. It seems very intuitive that if one knows that BLAH is an unknown truth, then one knows that BLAH is a truth. 3 For ease of exposition here, I am assuming the logical equivalence of phrases like ‘I know that Ed left’ and ‘I know the proposition that Ed left’. Having noun phrases such as ‘the proposition that Ed left’ with similar meanings to sentence phrases such as ‘Ed left’ is useful in this topic. We want to quantify over these meanings, and English grammar makes this easier when one is using noun phrases. For example, ‘something’ can replace a noun phrase, but there is no corresponding English expression that can replace a sentence. The claims I want to make and the arguments I want to present could be given without assuming this logical equivalence, but it seems to me this would require either being quite long-winded or using another language, for example English supplemented with propositional quantification.

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In this article I will argue that there are also interesting ways of filling out the ‘. . . ’ that include a bound ‘x’.4 In a familiar theory from (Lewis 1979), knowledge is a relation between a knower and a property that the knower self-ascribes. For any proposition, there is a corresponding property that one has iff one is in a world where the proposition is true. So Lewis treats propositional knowledge as knowledge of the corresponding property. There is some ‘irreducibly de se’ knowledge however—e.g., knowledge of one’s location in the world—that cannot be treated as propositional knowledge and must be treated as knowledge that a certain property is true of one. In the terminology of this picture, I aim to show that, in an interesting way, many of these properties that we instantiate are unknowable.5 Self-knowledge is often seen as something relatively easy to acquire (Gertler 2011). If this article is right, sometimes the opposite is true. There is interesting self-knowledge that is impossible to acquire.

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One cannot know the level of one’s knowledge

Here I argue that one cannot know the ‘level’ of one’s knowledge. First, I need to explain what I mean by ‘level’. It seems somewhat plausible that one could know some proposition without knowing that one knows it. For example, my dog knows that when I pick up my keys I’m leaving the house, but she does not know that she knows this. My dog’s knowledge of the proposition is at level one: she knows it but does not know that she knows it. We can define higher levels similarly: if one knows that one knows a proposition but does not know that one knows that one knows it, then one’s knowledge is at level two, and so on for higher levels, all the way up the natural numbers. Recent epistemological work on the KK principle suggests that much of human knowledge only reaches some finite level. In this section, I argue that one can never know the finite level at which one knows a proposition. In the next sections, I turn to the KK principle and discuss whether indeed human 4 The qualification ‘interesting’ is because, given the Church-Fitch proof above, there are various uninteresting, ‘cheesy’ ways of filling out the ‘. . . ’ that include a bound ‘x’. For example, if knowledge distributes over conjunction, then you can always add conjuncts to an unknowable proposition while preserving its unknowability. So it is not possible that there is some x that knows that (EVEN, and no one knows that EVEN, and x is hungry). There are other unknowable self-propositions. For example, one can never know that one knows nothing. These are less interesting because they are almost never true of one. In contrast, this paper argues that there are many unknowable self-propositions that are true of one. 5 One subtlety: many people draw a distinction between Lewis knowing that he himself is a philosopher, and Lewis knowing that David Lewis is a philosopher. On their way of thinking, the two different sentences are trying to get at two different knowledge states that normally accompany each other, but that can come apart if, for example, Lewis has a sudden brain injury and cannot recall who he is. I do not know what the right account is of these cases and this language, but if you do think there is this distinction, I am making the claim that there are many interesting properties P true of David Lewis, such that it is impossible for David Lewis to know that he himself has P . I am not making what you hold to be the distinct claim that it is impossible for David Lewis to know that David Lewis has P . I discuss these cases and distinctions more fully in section 5.

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knowledge typically only reaches certain levels. The form of the argument is similar to the Church-Fitch proof above. I will use superscripts over ‘knows’ to indicate iterations of knowledge. ‘Sancho knows2 that the donkey is grey’, for example, means that Sancho knows that Sancho knows that the donkey is grey. If someone’s knowledge of a proposition is at level 2, this means that they know that they know it, but they don’t know that they know that they know it. In other words, they know2 it but they don’t know3 it. So, given a proposition p, to know that one’s knowledge of p is at level n, is to know that both (i) one knowsn it, and (ii) one does not known+1 it. Say that there is some person and some proposition p where the person knows that their knowledge of p is at level n. So they know that both (i) they known p and (ii) they do not known+1 p. Since knowledge distributes over conjunction, they know that they known p. In other words, they known+1 p. Since knowledge is factive, however, their knowledge of (i) and (ii) entails that (ii) is true: they do not known+1 p. This is a contradiction, and so, for any proposition p and level n, it is impossible for there to be someone who knows that their knowledge of p is at level n. One cannot know the level of one’s knowledge.6,7 It is essential here that we are discussing self-knowledge: you cannot know the level at which you know something, but others can know the level at which you know it.8 There is thus a strong asymmetry here between your knowledge 6 This style of argument also shows that it is impossible to know the level at which you V a certain proposition, where V -ing is any state entailed by knowing. For example, perhaps you are justified in believing p, but you are not justified in believing that you are justified in believing it. It is impossible for you to know this fact about yourself. If you did, then since knowledge distributes over conjunction, you would know that you are justified in believing p. Since knowing something entails that you are justified in believing it, this entails that you are justified in believing that you are justified in believing p. But since knowledge is factive, it would also be the case that you are not justified in believing that you are justified in believing p. This is a contradiction, and so the assumption that you know this fact about yourself is impossible. Using this style of argument, it can also be shown that certain facts about common knowledge are unknowable. KK deniers often also deny that there are many examples of perfect common knowledge, where (everyone knows that)n p, for all n. This would mean that common knowledge often only reaches a certain finite ‘level’, and one can show that those within a group can never know that both (i) (someone within the group knows that)n p and (ii) it’s not the case that (someone within the group knows that)n+1 p. One instance is where you take the group to be all of the knowers in the universe. For any n and p, it is impossible to know both that (someone knows that)n p and that it is not the case that (someone knows that)n+1 p. Note that this is not an unknowable self-truth, but just an ordinary unknowable truth. 7 Williamson deserves some priority for the argument here over this article, for in his (1992, p.221 fn.3) he in passing uses the Church-Fitch proof to show that one cannot know that both (i) one knows p and (ii) one does not know that one knows p. 8 One can construct very simple models in epistemic logic to show the possibility of others knowing the level at which you know something. Epistemic logic makes strong simplifying assumptions about the closure of knowledge under logical entailment, but in this case these assumptions improve our argument: they show that there is no proof of the impossibility of this scenario even given the incredibly strong assumption that knowledge is closed under logical entailment. For a simple model, say there are three worlds x, y, and z, and say that p is true in x and y but false in z. Say that person a’s knowledge accessibility relation is {< x, y >, < y, z >, < x, x >, < y, y >, < z, z >}, and person b’s accessibility relation is

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of yourself and others’ knowledge of you. It is not just harder for you to achieve the knowledge they can have about you, it is impossible.

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The KK principle and transfinite levels

The previous section shows that one cannot know the level at which one knows any particular proposition. This way of phrasing the conclusion, however, tacitly assumes that there actually is a level at which one knows particular propositions. Arguing for this assumption requires looking at the KK principle—the principle that if one knows something, then one knows that one knows it. The KK principle is logically equivalent to the denial of the claim that some person knows some proposition at level 1. If there is no instance of knowledge at level 1, however, then there is also no instance of knowledge at any finite level. For knowledge at level n of proposition p is knowingn p and not knowingn+1 p. But, where q is the proposition that you known−1 p, this level n knowledge of p is the same as knowing q, and not knowing that you know q. In other words, level n knowledge of p is the same as level 1 knowledge of the proposition that you known−1 p. So if there is no case of the latter, there is no case of the former. Consequently if you accept the KK principle, you will not accept that our knowledge is ever at a certain level. Or, at least, you will not except that our knowledge is ever at a certain finite level, which is the only kind we have defined so far. We can define the transfinite limit level ω as the level of knowledge one has in a proposition p when both (i) for every natural number n, one knowsn p and (ii) one does not know that for every natural number n one knowsn p. (And we could keep defining further transfinite levels of knowledge.) Now, the argument from the previous section can clearly be used to show that it is impossible to know that one is at level ω. For if one did, then since knowledge distributes over conjunction, one would know (i). But since knowledge is factive, (ii) would be true, and so one would not know (i), which is a contradiction. And it is similarly impossible to know that one is at any of the other transfinite levels. Even if the KK principle is true, perhaps much of our knowledge is at ω or some other transfinite limit level. For example, since I do not believe the KK principle, I do not believe that (i) holds of me, for many p. If the KK principle is true, then for any p that I know, (i) does indeed hold of me. Since I do not believe that (i) holds of me, I do not know that (i) holds of me, and so (ii) also holds of me, and I am at level ω for many p. Regrettably, this fact about my level of knowledge of any of these p is impossible for me to know, though if you know the KK principle is true, perhaps you know it of me. So even if the KK principle is true, the thoughts in the previous section might show that there are interesting truths about many of us that are impossible for us to know. {< x, x >, < y, y >, < z, z >}. In this model, at world x, person b knows that a’s knowledge of p is at level 1. One could construct a more complicated model where b is not omniscient and where all the relevant non-logically-true propositions are only known by every knower at some finite level.

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Nevertheless, I think some KK acceptors might argue that we are never at a finite or a transfinite level of knowledge. If we assume that the KK principle is not only true, but also always known to be true, and we also assume some strong closure principle of knowledge in this domain, then we can prove that we are never at a transfinite level of knowledge. The key step is that the KK principle and one’s knowledge of p entail that, for all n, one knowsn p. So, given a suitable closure principle of knowledge here, if one knows the two premises in this entailment, one knows its conclusion. (Note however that this requires the assumption that the KK principle is always known, which is a significantly stronger assumption than the KK principle itself.) Consequently it seems worth discussing the truth or falsity of the KK principle itself.

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The KK principle and finite levels

Denying the KK principle used to surprise philosophers, but now the principle is widely held to be false, and accepting the principle is what surprises philosophers (Greco Forthcoming). To deny the KK principle is to hold that there are instances of knowers only knowing something at a certain finite level. One could deny the KK principle, and yet hold that these instances of finite-level knowledge are exceptional and rare. As it turns out, however, some of the arguments that have led to the widespread denial of the KK principle are also arguments that this finite-level knowledge is not rare but rather a normal part of our cognitive life. In this section I present a version of the influential marginfor-error argument from (Williamson 1992) against the KK principle. I defend it from various recent objections, and then argue that it shows that much of our knowledge only reaches a certain level. Scenario: G¨ odel looks at a tree somewhat far away. The tree is actually 666 cm tall. G¨ odel does not know this of course, because of the limitations of human vision, but he does know that the tree is, for example, taller than 1 cm. Let ‘p1 ’ stand for the sentence ‘The tree is taller than 1 cm’, and likewise for other numerals. So G¨odel does not know p666 , since of course the tree is not taller than 666 cm. But G¨odel does know p1 . We stipulate that G¨odel is a competent logician, and that in the scenario he is constantly working out what follows from his knowledge concerning the tree. We also stipulate that he knows something about his limitations in this scenario. In particular, he knows the following: (Margin)

For all numbers n ≥ 1, if G¨odel knows pn , then pn+1 .

Here is why (Margin) is true in this scenario. If the tree is 666 cm tall or less, then there is no way G¨odel can know in this scenario that it’s taller than 665 cm. Either the tree isn’t taller than 665, and so by factivity one can’t know it’s taller than 665, or it is just barely taller than 665. But because of the imperfections of human vision and the limitations of G¨odel’s ways of learning 6

about the tree’s height in this scenario, if the tree is 666cm tall or in between 666 and 665 cm tall, there’s no way for G¨odel to know in this scenario that it’s taller than 665—his vision just isn’t that accurate, and no amount of thinking from that far away will allow him to know that the tree is taller than 665. More generally, for all numbers n ≥ 1, if the tree is n + 1 cm tall or less, then, G¨ odel does not know that it’s taller than n cm. In other words (taking the contrapositive of the previous sentence), if G¨odel does know that the tree is taller than n cm, then the tree is not n + 1 cm tall or less; or equivalently, since the tree has a height, if G¨ odel does know that the tree is taller than n cm, then the tree is taller than n + 1 cm. In other words, if G¨odel knows pn , then pn+1 . We assume therefore that (Margin) is true, and then we stipulate that in this scenario G¨ odel knows that (Margin) is true. We assume that the above scenario is possible. We then show that the KK principle is false by assuming that the principle is true and showing that this leads to a contradiction. In this scenario G¨ odel knows p1 . So by the KK principle, in this possible scenario, G¨ odel knows that he knows p1 .9 By (Margin), if G¨odel knows p1 , then p2 . Since G¨ odel knows (Margin) and is competently doing logic, G¨odel comes to know this conditional. Since G¨odel knows this conditional, knows the antecedent of this conditional, and is competently doing logic, G¨odel comes to know the consequent, p2 . But by the KK principle, G¨odel knows that he knows p2 . . . If we keep following this reasoning all the way up the tree-height numbers, the result is that G¨ odel knows that the tree is taller than 666 cm, but since knowledge is factive this entails that the tree is indeed taller than 666 cm, which is a contradiction, since 666 cm is in fact the tree’s height. So the KK principle must not be true. There must be some height where G¨odel knows that the tree is taller than that height, but he doesn’t know that he knows that the tree is taller than that height. Furthermore, even with the proposition that the tree is taller than 1 cm, there must come an iteration of ‘he knows that’ where G¨odel does not know that he knows that . . . he knows that the tree is taller than 1 cm. For if G¨odel knows that he knows that . . . he knows that the tree is taller than 1 cm, no matter how many ‘he knows that’ phrases we put in place of the ‘. . . ’, then we never have to use the KK principle in the above argument, and can keep stripping away ‘he knows that’ phrases using G¨odel’s knowledge of (Margin) to eventually arrive at the contradictory conclusion that he knows that the tree is taller than 666 cm. So the argument shows that there is some number n, where G¨ odel knowsn p1 , but Godel does not known+1 p1 . The argument shows that there is a finite level at which G¨odel knows even this incredibly obvious truth. 9 The

KK principle above has ‘knows’ followed by a noun phrase (‘something’), rather than followed by a sentence. So, strictly speaking, it may only entitle one to move from ‘G¨ odel knows the proposition that p1 ’ to ‘G¨ odel knows that he knows the proposition that p1 ’. As I mention in fn. 3, I assume for ease of exposition that, where ‘S’ can be replaced by any sentence, ‘G¨ odel knows S’ is logically equivalent to ‘G¨ odel knows the proposition that S’, and so do not distinguish between these sentence forms here.

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´ e 2009) have an objection (Sharon and Spectre 2008) and (Dokic and Egr´ to the margin-for-error argument, which I think can be answered fairly quickly. They both make the useful point that, while (Margin) may be true for scenarios where G¨ odel’s only knowledge about the tree height is directly from far-off visual glances, it’s clearly not true for other scenarios. For example, if G¨odel talks to someone that has gone up to the tree and measured it, then G¨odel can know via testimony that the tree is taller than 665 cm, even if the tree is actually 666 cm. They point out that in the G¨odel scenario, G¨odel does not necessarily just have knowledge about the tree’s height directly from his visual glance, but may also have some relevant knowledge about the tree’s height by logically deducing new facts from his old knowledge obtained by his visual glance. Because of this, they insist that we should replace (Margin) with a premise that applies in a wider variety of scenarios, (for example, by replacing ‘knows’ with ‘knows directly by his vision’), and then they show that certain new premises that could stand in for (Margin) would not result in a valid argument against the KK principle. I think the proper response to this objection is just to insist that (Margin) is true in this scenario, even though it is certainly false in other scenarios. Even if G¨ odel can also logically deduce new facts about the tree height here that he didn’t know solely through direct vision, there is still no way in this scenario for G¨ odel to know through just vision and logic that the tree height is taller than 665 cm, if its height is 666 cm. (And the same goes for the other integers above 0.) This claim is exceedingly plausible, and until we have a direct argument against it, there is no need for us to also come up with adjusted margin-forerror arguments containing premises other than (Margin) that apply in a wider range of scenarios. The original margin-for-error argument above is enough to show the falsity of the KK principle. Stalnaker (2009) and Mott (1998) have a different objection to the marginfor-error argument. As we said, G¨odel has some knowledge that the tree height must be within a certain range. For example, G¨odel knows that the tree is taller than 1 cm. Since the tree is not infinitely tall, there should be a largest integer n, such that G¨ odel knows the tree is taller than n cm. (Here I’m thinking of G¨ odel’s knowledge that he would recognize when picking out n by its canonical numeral, and not by some description like ‘the largest integer less than the actual height of the tree in cm’.) So for this number n, G¨odel knows that the tree is taller than n cm, but he does not know that the tree is taller than n + 1 cm. If we assume (and this is a crucial assumption!) that G¨odel also knows that he knows that the tree is taller than n cm, then we can deduce that G¨odel does not in fact know (Margin), for if he did there would be a contradiction. (Margin) and the fact that G¨odel knows the tree is taller than n cm collectively entail that the tree is taller than n + 1 cm. So if G¨odel does indeed know the first two facts, he will have done some logic and thus have come to know that the tree is taller than n + 1 cm. But this contradicts our assumption that n is the largest integer such that G¨odel knows that the tree is taller than n cm. This looks like it might be a standoff, where one person’s modus ponens is another person’s modus tollens. KK acceptors might conclude that this shows that (Margin) is false in this scenario, whereas KK deniers might conclude that 8

this shows the falsity of the objection’s crucial assumption that G¨odel knows that he knows that the tree is taller than n cm. But I think we can make progress here. (Bacon unpublished) contains a very interesting way of trying to resolve the standoff. He points out the following. Say that we concede to the KK acceptor that G¨ odel knows that he knows that the tree is taller than n cm, and we no longer hold that G¨odel knows (Margin). The KK acceptor should still accept, however, the weaker principle that G¨odel does not know the exact limits of his knowledge here. G¨odel is no laboratory scientist, and has not done the right experiments on himself to know the exact limits of his ability to judge tree heights. This means that, while he may know that he knows the tree is taller than n cm high, he does not know that this is the largest integer such that he knows the tree is taller than it. He may in fact know that he knows it’s taller than n cm, but for all he knows, he also knows that it’s taller than n + 1 cm: (No Scientist) For all n, If G¨odel knows that he knows that the tree is taller than n cm tall, then, for all he knows, he knows that it’s taller than n + 1 cm. This principle alone, however, creates trouble for the KK principle. We can replace G¨ odel’s knowledge of (Margin) by the more cautious claim that, in this scenario, he knows (No Scientist). If we assume the KK principle then, if it’s possible for G¨ odel to know (i) that the tree is taller than 1 cm, (ii) that the tree is not taller than some really big height, say 10000 cm, (iii) the factivity of knowledge, (iv) the (No Scientist) principle, and (v) the KK principle (note this crucial assumption!), while competently doing logic and coming to know what he deduces, we get another contradiction.10 Note that Bacon’s argument requires the crucial assumption that it is possible for there to be one of these G¨odel scenarios where G¨odel also knows the KK principle. So the KK acceptor could pursue the tiny escape route of accepting the KK principle while denying that in these scenarios G¨odel can know the KK principle. But remember that the only reason we’re examining the KK principle in this section, is that in the previous section I speculated that some KK acceptors might argue that we are never at a transfinite level of knowledge, because we always know the KK principle. So this is no escape route for 10 Here’s the way I find it easiest to grasp the proof. Let ‘’ stand for ‘G¨ odel knows that’, and let ‘♦’ stand for ‘¬¬’, i.e., ‘For all G¨ odel knows,’. We assume that at the actual world all of the following assumptions (or assumption schemas) are true (or have all true instances): (i) p1 ; (ii) ¬p10000 ; (KG T) (φ → φ); (NS) pi → ♦pi+1 ; (KG NS) (pi → ♦pi+1 ); (4) φ → φ; (KG 4) (φ → φ). Say that a world ‘sees’ another world if the first world bears the accessibility relation to the second world. Say that from one world you can ‘eventually get to’ another world if the first world bears the ancestral of the accessibility relation to second world. By (ii) and (4), there’s no way to eventually get to a world where p10000 . By (i) and (4), at the actual world p1 . So by (NS), ♦p2 . So the actual world can see a world where p2 . But by (KG 4), at this world, p2 . So by (KG NS), at this world, ♦p3 . So this world can see a world where p3 . But by (4) and (KG 4), at the world where p3 , we also have p3 . And by (4) and (KG NS), at that world, ♦p4 . So that world can see a world where p4 . . . So there is a way to eventually get from the actual world to a world where p10000 . Contradiction.

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someone claiming that we are never either at a finite level of knowledge or at a transfinite level of knowledge. If the KK acceptor tries to escape Bacon’s argument by denying that G¨ odel knows the KK principle, then they can no longer use the argument I offered them in the previous section that people like G¨odel are never at a transfinite level of knowledge. So I conclude that we know the following disjunction: either (i) G¨odel’s knowledge here is at some finite level, because the KK principle is false; or (ii) while the KK principle is true, G¨odel’s knowledge here is at some transfinite level, because he does not know that the KK principle is true. Either way, we have our unknowability result: G¨odel’s knowledge is at some (finite or transfinite) level or other, though he can never know which level this is. The G¨ odel argument only concerns sentences of the form ‘the tree is taller than n cm’. Much of our knowledge is of a similar form, though of course it is also true that much of our knowledge is not. It can be argued, however, that margin-for-error principles like (Margin) apply to almost all sentences containing vague terms (Williamson 1994, c. 8). For example, it is plausible that, if I know that 9 is small, then 10 is small. I do not recite the general argument here, for I have nothing worthwhile to add. If the argument is sound, however, this would mean that almost all of our ordinarily-attended-to knowledge is only at a certain level. As section 2 has shown, we could never know what these levels are.

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Unknowability and knowledge de se

Here is a possible objection to my claim that one cannot know these facts about oneself. The objection relies on cases similar to those in the Frege’s Puzzle and knowledge de se literature. Imagine Bono has some sort of dementia, and he wakes up not remembering his name or who he is. God comes down to Bono and says, “I won’t tell you who you are. I want to talk about something else. There’s this guy called ‘Bono’, and: (i) Bono knows that it’s cold, but (ii) Bono doesn’t know that he knows that it’s cold.” What God says is true, Bono trusts God, and Bono believes what God says. It seems intuitive that in this scenario Bono thus knows what God has said, and that this whole scenario is possible. In particular, it seems that Bono knows conjunct (i). Bono knows that Bono knows that it’s cold. Despite this, Bono does not know that he himself knows it’s cold; for Bono has dementia, so does not know that he is Bono, and so cannot transform his knowledge of (i) into knowledge about himself. Also, Bono knows statement (ii). Bono knows that Bono doesn’t know that he knows that it’s cold. So Bono knows (Bono knows that it’s cold, and Bono1 doesn’t know that he1 (‘he1 himself’) knows that it’s cold).11 In other words, Bono knows that Bono’s knowledge that it’s cold is at level 1. In other words, Bono knows that his knowledge that it’s cold 11 Here I’ve put subscripts below ‘he’ and the last occurrence of ‘Bono’ to show that it’s this occurrence of ‘Bono’ that ‘he’ is supposed to ‘link back to’, or in more technical language ‘be anaphoric on’.

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is at level 1. Furthermore (so the objection goes), we can see that this is no contradiction, for see what happens when we try to run the earlier argument that this kind of scenario is impossible. Since knowledge distributes over conjunction, Bono knows that Bono knows that it’s cold. And because knowledge is factive, Bono doesn’t know that he (‘he himself’) knows that it’s cold. But both of these things can be true at the same time, thanks to Bono’s dementia. So Bono can know that he is at level 1 knowledge in some particular proposition, and these facts about ourselves can be known by us after all. I think this objection is not ultimately convincing, but it is worth discussing because it shows how one must pay attention to details when talking about unknowable self-truths in the vicinity of de se and Frege’s Puzzle cases. These details become relevant in another plausible example of an unknowable selftruth that I will briefly discuss below: Elga’s case of Dr. Evil. But first, here is my diagnosis of the objection. Call it ‘switching’ when, in the ‘that’-clause of a knowledge ascription, you replace a name of the knower with a pronoun ‘linked’ to the subject of the ascription, or vice versa. So, for example, replacing ‘Bono knows that Bono is famous’ with ‘Bono knows that he is famous’ is switching. The problem with the objection is that, in its statements, it switches in an inconsistent manner: at times it switches in certain circumstances, while at other times it disallows switching in those same circumstances. In the literature on propositional attitude ascriptions, it is controversial exactly when a switching move (from a premise to a conclusion that is a switched version of the premise) is logically valid, but we can see what is going wrong with the objection by stepping through the scenario with three different theories: first, one where switching is always truth-preserving with the sentences under discussion applied to this scenario here; second, where switching is never truth-preserving; third, where switching is only sometimes truth-preserving. First, if switching is always truth-preserving here, then it is impossible for Bono to know what the situation claims. Bono cannot know that (Bono knows that it’s cold, and Bono1 doesn’t know that he1 (‘he1 himself’) knows that it’s cold). For if he did know this, since knowledge distributes over conjunction, Bono knows that Bono knows that it is cold. But since switching moves are always allowed, this entails that Bono knows that he knows that it’s cold. By factivity however, Bono knowing the conjunction entails that Bono doesn’t know that he knows that it’s cold. In other words, it entails that it is not the case that Bono knows that he knows that it’s cold. This is clearly a contradiction. So if switching is always allowed, Bono cannot have the knowledge that the objection envisions in this scenario. For the objection to work, switching must be sometimes disallowed, so that it can both be true that (i) Bono knows that Bono knows it’s cold, while (ii) Bono does not know that he knows that it’s cold. Second, if switching is never truth-preserving here, then Bono cannot know that his knowledge (of the proposition that it’s cold) is at level 1. For Bono to know that he is at level 1 is for Bono to know that (he knows that it’s cold 11

and he does not know that he knows that it’s cold). In the argument, Bono knows that (Bono knows that it’s cold, and that Bono1 does not know that he1 knows that it’s cold). But it is switching to move from this premise to a conclusion replacing the ‘Bono’s in the ‘that’-clause with pronouns referring to the subject of the ascription. So if switching is never allowed, the scenario is only one in which Bono knows that Bono is at level 1 knowledge. It is not one where Bono knows that he (‘he himself’) is at level 1 knowledge (for this would be a contradiction). It is not, in the relevant sense, ‘self-knowledge’. The disallowing of switching allows Bono to have knowledge, but at the cost of it not being self-knowledge, which is why the dementia is an integral part of the scenario here. Third, let’s look at the objection if switching is only sometimes truthpreserving here. Different theories will say different things about when switching preserves truth. Regardless of what a theory says, if it is going to be plausible, it needs to uphold the law of non-contradiction. It needs to say that, for any proposition p, it is not the case that both p is the case and p is not the case. So it needs to somehow block the following objection to the law of non-contradiction: Bad Reasoning: In the above scenario, Bono knows that Bono knows that it’s cold. So (here we do a switching move), Bono knows that he knows that it’s cold. But also in the above scenario, Bono doesn’t know that he knows that it’s cold, for he has dementia (here we disallow switching). So, there is a counterexample to the ‘law’ of non-contradiction: Bono knows that he knows that it’s cold, and Bono doesn’t know that he knows that it’s cold. There must be something bad going on with this reasoning, and different theories will have different things to say about what it is. It seems to me that whatever one says to block Bad Reasoning here, however, should also block the argument of the objection to my claim. For example, perhaps there are some conversational contexts that always allow switching ‘Bono’ and ‘he’ and other contexts that never allow it, and the problem with Bad Reasoning is that there is no single context in which the whole reasoning is valid. Then similarly, it seems to me there should be no single context that validates the reasoning in the initial objection to my claim. In sum, if Bad Reasoning is not a good objection to the law of non-contradiction, the above reasoning about the Bono example is not a good objection to my unknowability claim. These considerations are not only relevant to this particular objection, but are also relevant to other plausible cases of unachievable self-knowledge. For example, take the case of Dr. Evil in (Elga 2004). Dr. Evil is about to blow up the world from his space station. Before pressing the button, however, he receives a letter from the Philosophical Defense Force saying that they have constructed an intrinsic duplicate of Dr. Evil in their headquarters, and have given this intrinsic duplicate a qualitatively identical environment, so that the duplicate thinks it is in a space station. The letter explains that both Dr. Evil and the duplicate are receiving this letter, and if the duplicate does not disarm, the Philosophical Defense Force will do horrible things to the duplicate. Dr. Evil 12

knows the Philosophical Defense Force never lies. In this scenario, Elga argues that Dr. Evil should be indifferent between the belief that he is Dr. Evil and the belief that he is the duplicate, and so he should disarm. It is controversial whether Elga is right (see e.g. Weatherson 2005), but if he is, then it is plausible that we have another instance of necessary selflocating ignorance. Arguably, in this scenario, Dr. Evil cannot know that he (‘he himself’) is Dr. Evil, though he can know that Dr. Evil is Dr. Evil. Now, it is not, in the broadest sense, unknowable that one is Dr. Evil, because Dr. Evil could’ve been in a very different scenario where it would’ve been easy for him to know that he is Dr. Evil. But it seems plausible that it is impossible for there to be someone who knows that both (i) they themselves are one of the two duplicates in a Dr. Evil scenario, and (ii) they are Dr. Evil. Here the above discussion about switching is very relevant, for there does seem to be a good sense in which Dr. Evil does know that Dr. Evil is one of the two duplicates in a Dr. Evil scenario, and that Dr. Evil is Dr. Evil. Nevertheless, this seems like another plausible example where there is some claim about oneself that one can never know. The difference between this case and mine is that (i) there is a more conclusive argument that one cannot know the level of one’s knowledge, and (ii) practically all of us have knowledge that only reaches a certain level, whereas none of us are in this particular Dr. Evil scenario. Consideration (ii), however, may not be that important. Using thoughts from the Problem of the Many literature (Weatherson 2009), one might try to argue that we are all in less dramatic yet structurally similar versions of the Dr. Evil scenario. If so, there are yet more facts about ourselves that we cannot know.

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