2.1 Insensitive higher-level beliefs

Vogel (Reference Vogel2000: 609–10) argues that sensitivity accounts of knowledge are flawed because they preclude us from having any higher-level knowledge about the truth of our beliefs. He presents the following inference for arguing that higher-level beliefs are insensitive.

1. (1) You know Omar has a new pair of shoes.

2. (2) You know that your belief that Omar has a new pair of shoes is true, or at least not false.

Vogel (Reference Vogel2000: 610) claims that if sensitivity is necessary for knowledge then (2) is false i.e. “you do not know that your belief that Omar has new shoes is true, not false.” He argues that this result is unacceptable and concludes that sensitivity is not necessary for knowledge.

Vogel presents the following argument for his claim that (2) is false if sensitivity is necessary for knowledge: He takes the belief that one does not falsely believe that p to be equivalent to:

• • B(¬(B(p) ∧ ¬p))

Since the believed proposition ¬(B(p) ∧ ¬p) is equivalent with the disjunction ¬B(p) V p, I will abbreviate it as d. In the nearest possible world w, where d is false, S believes that p. Vogel claims that if one believes that p, then one believes that one does not falsely believe that p, so if B(p), then B(d). Hence, in w both B(d) and ¬d hold. Therefore, B(d) is insensitive. This is Vogel's argument.

Notably, B(d) is insensitive, even if B(p) is sensitive. The reason is that for evaluating the sensitivity of the two beliefs, we consider different possible worlds. In the case of B(p) we consider a world w ₁ where just p is false. In the case of B(d) we consider w ₂ where ¬d is true which is farther away from the actual world than w ₁. However, p entails d, since p is one of the disjuncts of the disjunction d. Hence, even if the belief in a stronger proposition p is sensitive, the belief in d is still insensitive.Footnote ³

2.2 Sensitive higher-level beliefs

Beliefs of the form B(d) are insensitive higher-level beliefs about the truth-value of one's own beliefs. Notably, there are some closely related higher-level beliefs that are sensitive. Take higher-level beliefs which have the following propositions as content.

• • I believe that p and p

The underlying formal structure of this proposition is the following conjunction.

• • B(p) ∧ p

Since this proposition is a conjunction, I will abbreviate it as c. The following propositions have a similar underlying structure:

• • I believe that p and p is true

• • My belief that p is true

In the following, I will use c to abbreviate any of the propositions ‘I believe that p and p,’ ‘I believe that p and p is true’ and ‘My belief that p is true.’

Notably, beliefs in this conjunction are sensitive if B(p) is sensitive; i.e. sensitivity carries over from B(p) to B(c).Footnote ⁴ For determining whether B(c) is sensitive, we have to look at the nearest possible world where c is false. If B(p) is sensitive, then, by definition, any world where B(p) ∧ ¬p is true is far off. Thus, any nearby world where c is false is such that ¬B(p) ∧ ¬p or ¬B(p) ∧ p is true. Moreover, we can assume that any world where S believes a conjunction without believing each of the conjuncts is also far off. Therefore, S does not believe c since she does not believe that p. Therefore, B(c) is sensitive, if B(p) is sensitive. Simply said, B(My belief that p is true) is sensitive, if B(p) is sensitive because in the nearest possible world where ‘My belief that p is true’ is false, I do not believe that p and, therefore, do not believe that I truly believe that p.

The reason why B(c) is sensitive but B(d) is insensitive is simply that in evaluating their sensitivity we take different possible worlds into account. In the nearest possible world where d is false, B(p) ∧ ¬p is true, i.e. S still believes that p. In the case of B(c), the nearest possible world is one where either ¬B(p) ∧ ¬p or ¬B(p) ∧ p is true, i.e. a world where S does not believe that p.

If the bi-valence principle is true, then B(p) ∧ p is equivalent to B(p) ∧ ¬¬p, which can be understood as a formalization of the following propositions:

• • I believe that p and p is not false

• • My belief that p is not false

Beliefs in these propositions are sensitive for the same reason as B(c) is sensitive.

2.3 The belief forming method of higher-level beliefs

So far, we have not taken the belief forming method into account for determining the sensitivity of beliefs, as Nozick suggests. In this case, we achieve the same results. Take the following case of forming B(d) by inferring it from the believed proposition p.

1. (1) B(p)

2. (2) B(d)Footnote ⁵

If I form B(d) by inferring it from the believed proposition p, then sensitivity does not carry over from B(p) to B(d), since in the nearest possible world where I falsely believe that p and where I use the inference from p to come to believe whether I falsely believe that p, I still come to believe that I am not falsely believing that p.Footnote ⁶

Let's also have a closer look at processes of coming to believe that c on the basis of believing p. Take the following deductive inference from (1) and (2) to (3):

1. (1) B(p)

2. (2) B(I believe that p)

3. (3) B(My belief that p is true)

If (1) is sensitive, then (3) is also sensitive, if formed via inference from the propositions believed in (1) and (2). Furthermore, sensitivity carries over from B(p) to B(B(p) ∧ ¬¬p). Thus, we acquire the same results even when taking plausible belief forming methods into account.

2.4 The heterogeneity-problem for higher-level knowledge

Vogel (Reference Vogel2000) argues that if sensitivity is necessary for knowledge, then we cannot have any higher-level knowledge about the truth of our own beliefs. His line of argumentation is that B(d) is insensitive and, therefore, we do not know c. Becker (Reference Becker2006, Reference Becker2007) and Salerno (Reference Salerno, Campbell, Rourke and Silverstein2010) are right that Vogel is mistaken in this respect, for we can know c without knowing d, if knowledge is sensitive true belief.Footnote ⁷ Moreover, we can also acquire sensitive higher-level beliefs that our beliefs are not false, if the underlying formal structure is equivalent to c. However, this reply to Vogel does not solve the problem of higher-level knowledge for sensitivity accounts but rather displaces it.

Some higher-level beliefs about the truth of single beliefs are sensitive and some are insensitive. Beliefs with the formal structure B(c), which are beliefs of the conjunction B(p) ∧ p turn out to be sensitive, but beliefs of the form B(d), which are beliefs equivalent to the disjunction ¬B(p) V p are insensitive. Since c entails d, the sensitively believed propositions are logically stronger than the insensitively believed ones. Sosa (Reference Sosa1999) argues that the fact that B(p) is sensitive although the belief in the disjunction B(d) is insensitive provides evidence against the claim that knowledge is sensitive true believing. Since c is stronger than p, the fact that B(c) is sensitive whereas B(d) is insensitive provides even stronger evidence for Sosa's point. We want an account of knowledge that allows one to know the weaker propositions d if we know the stronger propositions c. Thus, the outcome for higher-level beliefs in terms of sensitivity is too heterogeneous to provide a useful explanation.Footnote ⁸

4.1 Insensitive Moorean reasoning

Take the following belief forming process via deductive inference.

1. (1) B(p ₁)

2. (2) B(I do not falsely believe that p ₁)

3. (3) Repeat for p ₂ … p _n

4. (4) B((I do not falsely believe that p ₁) and … and (I do not falsely believe that p _n))

Belief (4) is understood to have the following formal structure:

• • B(¬(B(p ₁) ∧ ¬p ₁) ∧ … ∧ ¬(B(p _n) ∧ ¬ p _n))

In short:

• • B(d ₁ ∧ … ∧ d _n)

Belief (4) is insensitive. The nearest possible world w where (4) is false is one where one of its conjuncts d _i is false. But in w S still comes to believe that d _i via bootstrapping. Moreover, S also comes to believe that (d ₁ ∧ … ∧ d _n) via steps (1)–(3). Sensitivity does not carry over from step (1) to step (2) and neither to step (4).

In the case of Moorean reasoning, p is an experience-based external world proposition such as There is a computer in front of me, in short COMPUTER.Footnote ¹⁶ Take the following example of Moorean reasoning.

1. (1) B(COMPUTER)

2. (2) B(I do not falsely believe that COMPUTER)

3. (3) Repeat for DESK, GLASS OF WATER …

4. (4) B((I do not falsely believe that COMPUTER), (I do not falsely believe that DESK), …)

In the nearest possible world where (4) is false, I still come to believe (4) via Moorean reasoning.

Beliefs are also insensitive, if an explanation for my potential deception such as a skeptical hypothesis is incorporated into the conclusion as in the following case.

We can also form beliefs that all my beliefs from a certain source are true or reliably formed via bootstrapping and Moorean reasoning. It turns out that such beliefs are also insensitive. One can reach such a conclusion via the following deduction.

1. (1) B(p ₁)

2. (2) B(I believe that p ₁)

3. (3) B(My belief that p ₁ is true)

4. (4) Repeat for p ₂ … p _n

5. (5) B(B(p ₁) … B(p _n) are all my beliefs from source O)

6. (6) B(All my beliefs from source O are true)

In case of Moorean reasoning, such an inference can take the following form.

1. (1) B(COMPUTER)

2. (2) B(I believe that COMPUTER)

3. (3) B(My belief that COMPUTER is true)

4. (4) Repeat for DESK, GLASS OF WATER …

5. (5) B(B(COMPUTER), B(DESK) … are all my beliefs based on my current experience)

6. (6) B(All my beliefs based on my current experience are true)

Beliefs of (6) formed via inference from (1)–(5) turn out to be insensitive.

We obtain the same results about insensitivity via inductive bootstrapping, i.e. by directly reaching the following conclusions (5’) and (5’’) from (1)–(4).

The inductive inference from (1)–(4) to (5’’) is Vogel's version of bootstrapping. One might claim that sensitivity accounts of knowledge deliver the expected result for the bootstrapping case that Vogel originally presented, since these beliefs are insensitive and, therefore, do not constitute knowledge. However, sensitivity accounts fail insofar as they deliver opposite results for very similar cases, as we will see.

4.2 Sensitive Moorean reasoning

Notably, there are also sensitive beliefs formed via bootstrapping or Moorean reasoning. Take the following inference that is a repetition of the inference leading to sensitive higher-level beliefs B(c).

1. (1) B(p ₁)

2. (2) B(I believe that p ₁)

3. (3) B(My belief that p ₁ is true)

4. (4) Repeat for p ₂ … p _n

5. (5) B((My belief that p ₁ is true) … (My belief that p _n is true))

Belief (5) has the following formal structure:

• • B((B(p ₁) ∧ p ₁) ∧ … ∧ (B(p _n) ∧ p _n))

In short:

• • B(c ₁ ∧ … ∧ c _n)

Belief (5) is sensitive, if each of the beliefs B(p ₁) … B(p _n) is sensitive. (5) is false, if at least one of its conjuncts c _i is false. We have seen that sensitivity carries over from B(p _i) to B(c _i) via inference from B(p _i) and B(B(p _i)). But if I am minimally coherent, then I do not believe a conjunction, if I do not believe each of the conjuncts. So if I do not believe c _i, then I do not believe the conjunction B(c ₁ ∧ … ∧ c _n). Therefore, in the nearest possible world, where (5) is false, I do not believe it anymore and the belief is sensitive.

Take again a concrete example of Moorean reasoning, where beliefs of type (5) are sensitive.

1. (1) B(COMPUTER)

2. (2) B(I believe that COMPUTER)

3. (3) B(My belief that COMPUTER is true)

4. (4) Repeat for DESK …

5. (5) B((My belief that COMPUTER is true), (My belief that DESK is true) …)

Plausibly, a well-functioning perceptual apparatus delivers sensitive beliefs. Thus, our perceptual beliefs are typically sensitive and, therefore, our beliefs about these beliefs are also sensitive.

The types of beliefs that are involved in this inference are about the world, about what we believe about the world, and about the truth of our beliefs about the world. Together, they provide a comprehensive picture about us in the world, notably one, which can be sensitive throughout.

If the principle of bi-valence is true, then the proposition my belief that p is true is equivalent to my belief that p is not false, with the formal structure B(p) ∧ ¬¬p. Accordingly, beliefs of the following structure are also sensitive when acquired via steps (1)–(4):

(5’’) is remarkably close to a direct denial of the skeptical hypotheses of the form (I am not a BIV deceived in falsely believing that COMPUTER …), which is, as noted, insensitive.

4.3 Summary: The heterogeneity problem

Table 1 summarizes sensitive and insensitive higher-level beliefs and sensitive and insensitive beliefs acquired via bootstrapping and Moorean reasoning.

Table 1

Summary

These insensitive and sensitive beliefs are closely related. Moreover, some of the sensitively believed propositions are stronger than their insensitively believed counterparts. This outcome is a mess. Thus, we see how the heterogeneity problem for higher-level beliefs also affects bootstrapping and Moorean reasoning.

It is worth noting that I do not assume that all cases of higher-level beliefs and beliefs based on bootstrapping need to be treated the same in terms of justification or alleged justification. Rather, I am open to both accounts: first, that we need to treat them equally, and, second, that we do not need to. The point is that in the second case sensitivity does not offer the right criterion for distinguishing the good cases from the bad ones. Intuitively, instances of simple higher-level beliefs like B(d) or B(c) seem less problematic than instances of bootstrapping from B(p) to B(I am not a BIV deceived in falsely believing that p) or to B(p ∧ I am not a BIV deceived in falsely believing that p).Footnote ¹⁷ Thus, if we allow treating different instances of bootstrapping differently, then we want an account that is in accordance with these intuitions, but sensitivity accounts are not. They allow some complex instances of bootstrapping to constitute knowledge, whereas they prohibit some simple instances from constituting knowledge.

5.1 Nozick's sensitivity account

Nozick argues that sensitivity is a necessary condition for knowledge and that knowledge is not closed under known entailment. The simple Nozickian view is that any anti-skeptical hypothesis is insensitively believed and therefore not known. This view is mistaken. If sensitivity is necessary and – in addition to Nozick's conditions (1), (2) and (4) – sufficient for knowledge, then there are some anti-skeptical hypotheses which we know and some which we do not know. Nozick (Reference Nozick1981) was well aware of the fact that his account implies that we know the conjunction ¬sh ∧ p, though we do not know ¬sh. Thus, Nozick was already explicitly holding a differentiated view with respect to this conjunction, a fact that is often neglected in literature.

According to this differentiated view, skepticism is a more isolated phenomenon than according to the simple view, since it is only some specific anti-skeptical hypotheses that we cannot know. This can be seen as an advantage of the differentiated view. However, the simple view has it that knowledge closure does not hold between external world propositions and any kind of anti-skeptical hypotheses, which is already highly controversial. According to the differentiated view, knowledge closure fails between strong and weak anti-skeptical hypotheses. This is an even more problematic instance of closure failure. Astonishingly, Nozick is willing to bite this bullet.

We can find a more detailed explanation for Nozick's conception of closure-failure, if we consider his often ignored account of inferential knowledge. Nozick proposes:

S knows (via inference from p) that q iff

1. (1) S knows that p

2. (2) q is true, and S infers q from p (thereby, being led to believe that q)

3. (3) if q were false, S wouldn't believe that p (or S wouldn't infer q from p)

4. (4) if q were true, S would believe that p (and would infer q from p if she were to infer either q or ¬q from it).Footnote ²⁰

Nozick argues that I cannot acquire knowledge that ¬sh from my external world belief that p, because condition (3) is violated. If I were a BIV deceived in falsely believing that p, I would still believe that p and would infer from p that ¬sh. Hence, Nozick not only claims that one can know that p without knowing that ¬sh, he also provides an explanation for why we do not have this inferential knowledge via Moorean reasoning.

Notably, the results concerning insensitive and sensitive beliefs remain the same, if we consider Nozick's concept of inferential knowledge. Take the example of B(d) and B(c). Beliefs of type B(d) acquired via inference from B(p) cannot constitute inferential knowledge, because if d were false, S would still believe that p. However, beliefs of type B(c) acquired via bootstrapping are sensitive if B(p) is sensitive because if c were false, S would not believe that p.

A particularly counter-intuitive result of Nozick's account of inferential knowledge is that knowledge does not transmit via conjunction elimination. Thus, we can know the conjunction p ∧ ¬sh without knowing the conjunct ¬sh, even if we form the belief in the conjunct via inference from the conjunction. However, the view that knowledge does not transmit via conjunction elimination seems very counter-intuitive.Footnote ²¹ Thus, given Nozick's account of inferential knowledge, the heterogeneity problem remains the same. Nozick is not ignorant about the consequences of his account. Rather, he offers an additional explanation for why this is so. However, his explanations are highly implausible.Footnote ²²

5.2 DeRose's indirect sensitivity account

DeRose's (Reference DeRose1995, Reference DeRose2010) indirect sensitivity account has it that the insensitivity of certain beliefs explains why they fail to be knowledge. DeRose (Reference DeRose2010: 163) argues that we have “some at least fairly general – though perhaps not exceptionless – tendency to judge that insensitive beliefs are not knowledge.” Accordingly, the insensitivity of our beliefs in anti-skeptical hypotheses can explain our intuitions that these beliefs fail to constitute knowledge, even if sensitivity is not a necessary condition for knowledge in general. Moreover, DeRose (Reference DeRose1995) rejects conjunctions of the form “S knows that she has hands but S does not know that she is not a handless BIV.” His sensitivity based contextualist solution to the skeptical problem takes the following form: In ordinary contexts, persons have external world knowledge and knowledge that the skeptical hypotheses are false. However, when it is asserted that S knows that p (or that she does not know that p), the standards for knowledge tend to be raised up to a level that S's belief has to be sensitive for constituting knowledge. By asserting that S knows that she is not a BIV we establish such high standards and the assertion turns out to be false. In these more demanding contexts, a person has neither external world knowledge nor knowledge that she is not a BIV.

Let's take belief-forming method into account, as sensitivity accounts suggest. We have seen that some anti-skeptical beliefs formed via Moorean reasoning are insensitive, as DeRose assumes, but that some are sensitive. However, Nozick's claim that we know the sensitively believed anti-skeptical hypotheses, but do not know the insensitively believed anti-skeptical hypotheses, is not in line with our intuitions about skepticism. In contrast, we tend either to accept that we know insensitively believed anti-skeptical hypotheses and sensitively believed ones or neither. Moreover, we maintain these intuitions even if we become aware of the fact that some beliefs of anti-skeptical hypotheses are sensitive and that some are insensitive. Hence, our intuitions about whether we know anti-skeptical hypotheses cannot be based on the sensitivity or insensitivity of our beliefs of anti-skeptical hypotheses.

This is a serious problem for DeRose, who thinks that his contextualist account also contains an explanation for why we raise the standards for knowledge. However, his sensitivity-based explanation is simply inadequate. DeRose can still defend a contextualist solution to the skeptical problem, but without being able to explain in terms of sensitivity why we shift the standards. Hence, he is mistaken that his indirect sensitivity account is ‘on the right track.’Footnote ²³

5.3 Black's Moorean invariantism

Black (Reference Black2002, Reference Black, Hendricks and Pritchard2008a) defends a Moorean invariantist response to the skeptical argument according to which the standards for knowledge are always comparatively low and, therefore, we always have external world knowledge and knowledge that we are not BIVs. Black criticizes Nozick's (Reference Nozick1981: 184) specification of belief forming methods that any method that is experientially the same, the same “from the inside,” counts as the same method. According to Nozick's account, a normal person and a BIV share the same method. Black objects that our method of coming to believe that we are not BIVs is partly perception, but that perception is in part characterized “from the outside,” i.e. as a method based on using our sense apparatus. Thus, normal persons and BIVs do not share the same method, since BIVs, ex hypothesi, do not use any sense apparatus.Footnote ²⁴

Black argues that given this reformulation of belief-forming methods, our beliefs that we are not BIVs turn out to be sensitive. S's belief via method M that p is sensitive iff in the nearest possible world where p is false and S uses M, S does not come to believe that p via M. Since BIVs do not share our method of perception, there simply is no nearest possible world where I am a BIV and where I use the same method that I use in the actual world. Hence, my belief that I am not a BIV is sensitive. Since Black defends a sensitivity account of knowledge, he concludes that we can know that we are not BIVs.

One advantage of Black's account is that beliefs in the conjunction ¬sh ∧ p as well as beliefs in the conjunct ¬sh turn out to be sensitive. Thus, Black offers a uniform anti-skeptical solution that allows knowing both. In this respect, he is better off than Nozick and DeRose, who both have to treat beliefs in ¬sh and beliefs in ¬sh ∧ p differently.

Black (Reference Black2002) concedes that his solution to the skeptical problem only applies to skeptical hypotheses that rely on subjects like BIVs whose belief forming methods are essentially different from our own. However, for many of the discussed cases there is a nearest possible world where the proposition in question is false but where the same belief forming method is still available. Take the inference from B(p) to B(d) and suppose that p is the content of a perception based belief like ‘Mark is on the other side of the street.’ In this case, there are possible worlds w, where d is false i.e. where I falsely believe that Mark is on the other side of the street, and where I use the same method of perception and inference as in the actual world. In w, I take, for example, someone else to be Mark. Hence, my belief B(d) is still insensitive. The same is true for the following beliefs formed via bootstrapping:

In both cases, we could still use the same method ‘from the outside’ of consulting the gas gauge plus bootstrapping, if these conclusions were false. Therefore, these beliefs still turn out to be insensitive.

If we apply Black's sensitivity account to all instances of bootstrapping, then we know that we are not BIVs via Moorean reasoning without knowing d and without acquiring knowledge via alternative instances of bootstrapping. This seems counter-intuitive. Thus, Black's sensitivity account might offer a partial solution to the skeptical problem, as he argues, but the heterogeneity problem is only displaced.Footnote ²⁵

5.4 Roush's recursive sensitivity account

Roush (Reference Roush2005) offers a tracking account of knowledge that is inspired by Nozick's account, but that avoids the undesired consequence of closure-failure. Roush (Reference Roush2005: 43) argues, as a first approximation, that S knows that p if and only if S's belief fulfills Nozick's condition (1)–(4) or “p is true, S believes p, and there is a q not equivalent to p such that q implies p, S knows that q implies p, and S knows that q.” According to Roush's account, S knows that she is not a BIV deceived in falsely believing that p, if S knows that p and knows that p implies that she is not a BIV deceived in falsely believing that p.

How does Roush explain our intuition that we do not know that we are not BIVs? Roush (Reference Roush2005: 54f) argues that the fact that we cannot find conclusive reasons for believing that we know, even if our beliefs constitute knowledge, explains our experience of helplessness when confronted with the skeptical hypothesis and our frustration about our inability to refute the skeptic. Since sensitivity is not part of an explanation for why Moorean reasoning seems flawed according to Roush, her view is partly in accordance with the main theses of this paper.

However, Roush's account faces two problems concerning bootstrapping. Roush (Reference Roush2005: 120) claims that bootstrapping obviously does not constitute knowledge. She claims that her recursive sensitivity account can explain why: First, the beliefs in the conclusion are insensitive and, second, the last step of inference of bootstrapping is not deductive.

However, we have seen that we can construe instances of bootstrapping that violate each of these conditions. First, there are beliefs based on bootstrapping that are sensitive. Second, we can also construe deductive instances of bootstrapping about the reliability of a source O. Hence, Roush cannot argue that we fail to know in these two cases because the resulting belief is insensitive and not all inferences involved are deductive. Without providing further arguments, she is committed to accepting that we do know in these cases of bootstrapping, although we do not know in Vogel's original example. Again, the heterogeneity problem is only displaced.