Bootstrapping, Dogmatism, and the Structure of Epistemic Justification

Consider the following quixotic attempt to determine whether your color vision is reliable. You ask your friend to set up a slide show in which every few seconds a colored slide will appear on the screen. Your friend does not tell you what color the slides are or in what order they will appear. You sit down and the slideshow begins. The first slide comes up and you look at it and, on the basis of your visual experience, you come to believe:

You next notice that you are having a visual experience of the slide being red and come to believe:

You then reason from these two beliefs to form the belief:

Having gotten this far, you conclude:

The next slide comes up. And you go through the same reasoning for this slide. And so on for the whole slideshow. From this stock of beliefs together you conclude:

where n can be as large as we like. Based on Track Record you finally conclude:

Despite your efforts, you have not made progress toward figuring out whether your color vision is reliable.[1] If you did not know or were not justified in believing Reliability in the first place, this process of reasoning does not provide you with justification or knowledge now.

The bootstrapping problem in epistemology arises for views that seem to be committed to the implausible result that this form of reasoning does generate justified belief in or knowledge of Reliability.[2] Though this problem has been raised for a number of different kinds of theories, in this paper I will concentrate on the problem as it arises for so-called dogmatist theories of epistemic justification.

My argument will be that the only way for the dogmatist to avoid the bootstrapping problem is to claim that epistemic justification fails to have a structural property, which I will describe in detail later, known as cut. This allows the dogmatist to admit that each step in this reasoning considered on its own is acceptable, but when stitched together, these pieces of reasoning are unacceptable (§2). The fact that this is the only plausible solution to the bootstrapping problem is, in one way, bad news. This is because, as I will show, it adds another member to a family of recently uncovered results that show dogmatism is incompatible with certain connections between epistemic justification and probabilities (§3). But instead of stopping here and concluding that dogmatism is false, I try to make the best of it on the dogmatist’s behalf. I show that within a certain kind of foundationalist framework, we can make good on this idea that epistemic justification fails to satisfy cut in a way needed to solve the bootstrapping problem (§4–5). But let’s begin by more clearly explaining what dogmatism is and what the bootstrapping problem is.

1. Dogmatism and Bootstrapping

Though there are different versions of dogmatism, there are three core theses that I will take to be definitive of this family of views. The first thesis is that that a visual experience, for example, of a table being red, provides defeasible justification for believing, for example, that table is red. The second thesis is that this justification does not require the agent to have any prior knowledge or justified belief that in the present situation appearances are not deceiving. And the third thesis is that this justification does not require the agent to have any prior knowledge or justified belief that her visual system is reliable. Instead, dogmatism allows that an agent is defeasibly justified simply in virtue of having the visual experience (see Huemer 2001; Pollock & Cruz 1999; and Pryor 2000).

I will also take these theses to be commitments of dogmatism whether we read ‘justification’ as so-called propositional justification or so-called doxastic justification. This is not a universally followed practice. Some theorists identify as dogmatist but only accept these theses understood as claims about doxastic justification; they reject thesis two and three understood as claims about propositional justification (Silins 2008). This paper however will use ‘dogmatism’ and its cognates to refer to the stronger view that accepts the three theses on both interpretations. I focus on this stronger view because it cleaves more closely to certain motivations for dogmatism related to avoiding so-called deeply contingent a priori knowledge that we will discuss in greater detail below (§3.2.4, §C.2) and because solving the bootstrapping problem for these views is especially hard.[3]

So in what follows, I will not belabor the distinction between these two kinds of justification because my claims apply to both. That said, at times, I will put things in terms that are more congenial to one interpretation than the other for simplicity. I leave it to the reader to do the simple translation required to fit my comments with the other interpretation of justification.

A second clarification is also in order: the justification at issue is defeasible. That is, the agent could learn some new information that makes her no longer justified in believing that the table is red. For example, the agent might learn that her perception is unreliable or that in the current situation appearances are deceiving. But when the agent simply lacks knowledge that her perception is reliable and lacks knowledge that in the current situation appearances are deceiving, she is justified in believing the table is red on the basis of her visual experience as of it being red. And this is true whether or not as a matter of fact her visual perception is reliable or as a matter of fact in the current situation, appearances are deceiving.

The bootstrapping objection to dogmatism is that it entails that the reasoning that I began the paper with provides new justification for believing Reliability. In order to present this objection and my solution perspicuously, it will help to adopt a bit of formalism for representing claims about epistemic justification.

Officially, the dogmatist says that having a certain visual experience, for example, of the table being red justifies you in believing, for example, that the table is red. This suggests that a natural way to think of epistemic justification for an agent according to the dogmatist is as a relation that holds between mental states. While this is the natural and most direct way to think of the dogmatist view, it will ultimately prove convenient to formalize this view in a less direct way that allows us to connect our discussion to simple formal properties of relations that are familiar from philosophical logic.

So we will model epistemic justification for an agent, a, at a time, t, using a relation ⊩a_, t and in fact, we will from now on leave reference to the agent and time implicit and just write ⊩. This relation will not be used to relate mental states, but instead will be used, like a logical consequence relation, to relate a set of sentences and a sentence. It will be useful later to use lower case Greek letters to denote sentences (α, β, γ, etc.) and uppercase Greek letters to denote sets of sentences (Α, Β, Γ, etc.).

So for example, we can write:

Or if we wish to be more succinct, we may for convenience omit the braces and write:

This is intended to represent the claim that an agent who is epistemically justified in believing that the table is red and is epistemically justified in believing that the table is round is epistemically justified in believing that the table is round and the table is red. So more generally, we read this as saying an agent who is epistemically justified in believing each of the propositions expressed by the sentences on the left is epistemically justified in believing the proposition expressed by the sentence on the right.

This smoothly covers cases in which beliefs justify other beliefs. But the dogmatist treats cases of perceptual justification differently than cases of beliefs justifying other beliefs. As we saw, the dogmatist says that merely having a perceptual experience can provide epistemic justification for beliefs. In order to capture this in our formalism in a way that will still allow us below to make use of some simple formal properties of relations, we will have to engage in a bit of harmless equivocation. So we will write:

and this will be used to represents the claim that an agent who is in the state reported by the sentence on the left is epistemically justified in believing the proposition expressed by the sentence on the right. For those concerned that this equivocation is problematic, Appendix A introduces the complexity needed to show how our discussion can be successfully conducted in a non-equivocating formalism

With this formalism in hand, let’s return to the bootstrapping reasoning and consider which claims about epistemic justification would have to be true for that reasoning to lead to you having a justified belief in Reliability. To start, notice that since dogmatism does not require agents to be justified in believing Reliability from the start, it entails that there are some agents who are not justified in believing Reliability before the bootstrapping reasoning. Imagine you are such an agent who is engaging in the bootstrapping reasoning. It begins with you believing that the slide is red based on your visual experience of the slide being red. If we let LR be ‘the slide looks red’ and R be ‘the slide is red’ the claim about epistemic justification that would have to be true for this step is the following:

The next step is noticing that the slide looks red to you. For simplicity, we will just assume you are justified in believing that the slide looks red to you from the start and so omit this step.

From R and LR you then come to believe that the slide looks red and the slide is red. If we let R ∧ LR be ‘the slide is red and looks red’, the claim about epistemic justification that corresponds to this step is the following:

From R ∧ LR, you conclude that your color vision worked. If we let Worked be ‘my color vision worked’, the claim about epistemic justification that corresponds to it is the following:

Next after reasoning analogously to n instances of something like Worked that we will represent as Worked₁, Worked₂, …, Worked_n, you are able to conclude that your color vision worked n times which is the proposition expressed by Track Record:

Finally from Track Record you conclude that your color vision is reliable which is the proposition expressed by Reliability:

So steps one through five lead to the result that you are epistemically justified in believing Reliability based on the bootstrapping reasoning.

The objection to dogmatism is that each of steps two through five is acceptable and yet nonetheless an agent is not justified in believing the proposition expressed by Reliability in virtue of the bootstrapping reasoning, so Step 1 must be incorrect. But Step 1 is a commitment of dogmatism.

Let’s look at this. To see that Step 1 is a commitment of dogmatism, we need only note that it is an application of the dogmatist’s general idea that having a visual experience as of p epistemically justifies you in believing p. The remaining steps are quite plausible. Step 2 and Step 4 involves performing “conjunction introduction”. While there are contexts in which this is not acceptable that we will discuss in greater detail later, in the present context it is acceptable (§3.1). Step 3 involves a simple analytic entailment. Finally, Step 5 is an instance of concluding by enumerative induction that your color vision is reliable based on many instances of it working correctly.

In sum, it is unacceptable to think that the bootstrapping reasoning leads us to be newly justified in believing the proposition expressed Reliability. So we must reject the conclusion of this reasoning. But this triggers what Jonathan Vogel (2007) calls “rollback”: We must say which step in the reasoning goes wrong. But since conjunction introduction, the simple analytic entailment, and induction looks overwhelmingly plausible when considered alone, it looks like we roll all the way back to saying the first step in the reasoning is faulty and that’s just a commitment of dogmatism.

2. The Structure of Epistemic Justification

The only way out of this result is to somehow claim that while each step is acceptable on its own, you cannot perform the steps back-to-back.[4] While our formalism may have been initially unnatural, what makes it worthwhile is that we can now state the idea that you cannot do the steps back-to-back simply and precisely as the idea that ⊩ is not a transitive relation, where we say:

If the dogmatist were to deny this claim, then she could accept each step in the reasoning, but deny that they can be performed one after another.[5]

As it turns out, transitivity follows from the following two properties that are commonly called monotonicity and cumulative transitivity or for short, cut:

So if the dogmatist wishes to solve the bootstrapping problem, she must reject at least one of these claims.[6] And if this solution is to be anything but ad hoc the dogmatist must explain why epistemic justification fails to satisfy either cut or monotonicity in a way that solves the problem. The remainder of the paper is dedicated to exploring whether such an explanation can be provided.

In particular, I will explore whether the dogmatist can provide an explanation of why epistemic justification fails to satisfy cut in a way that solves the problem. I start with some bad news for the dogmatist. Recently, Jonathan Weisberg has attempted to use broadly Bayesian considerations to argue that epistemic justification fails to satisfy cut in a way that solves the bootstrapping problem. But I argue that in fact dogmatism is incompatible with this Bayesian picture (§3).

While I am sympathetic to the reaction that this bad news is a reductio of dogmatism, I spend the rest of the paper trying to make the best of it on the dogmatist’s behalf. In particular, I will develop an alternative explanation of why epistemic justification fails to satisfy cut within a certain foundationalist framework and apply it to the bootstrapping problem (§4–5). In doing this, I will be assuming that if the dogmatist can deny that the bootstrapping reasoning goes all the way through to the end of Step 5, this will suffice to solve the problems. I however discuss in detail some recent arguments that suggest that there is a problem much earlier in the reasoning in §5.4 and Appendix C.

But before turning to these tasks, it is worth pausing for a moment to consider whether the problem can be solved by claiming that epistemic justification fails to satisfy monotonicity over the bootstrapping reasoning. This is especially important to do because it is a well-known and relatively uncontroversial commitment of dogmatism that epistemic justification fails to satisfy monotonicity in general.

To see this, we need only note that monotonicity essentially says that if the states corresponding to the sentences in Α justify you in believing the proposition expressed by γ then no matter what new information you may learn from some external source you will still be justified in believing the proposition expressed by γ. But dogmatism says that while something looking red epistemically justifies you in believing it is red, if you were to find out from some external source that the lighting conditions are bad for distinguishing red objects from other objects, you would no longer be justified in believing it is red. So dogmatism already entails that epistemic justification fails to satisfy monotonicity.

Despite this fact, it is nonetheless unsurprising that theorists have not thought that rejecting monotonicity could be used to solve the bootstrapping problem. As I noted, monotonicity fails because learning some new information from an external source might lead an agent to be no longer justified in believing something that she was originally justified in believing. Now consider that the bootstrapping reasoning begins with the agent learning that the slide is red and that it looks red. It is a commitment of dogmatism that the agent learns the slide is red. And it is obvious that an agent can learn that the slide looks red from introspection.

For the failure of monotonicity to solve the problem at this stage in the reasoning, we would have to implausibly claim that learning that the slide looks red makes you no longer justified in believing that it is red. This way of trying to solve the bootstrapping problem is especially implausible for the dogmatist. She would be saying that noticing the perceptual state that justifies you in believing the slide is red defeats the justification for that belief.[7]

Having seen why monotonicity failure cannot help us at the start of the bootstrapping reasoning, we can also see why it will not help in later stages in the argument. It won’t help because these steps do not involve learning any new information from an external source. Instead, all of the new claims are inferred from previous claims. And this in turn shows us why rejecting cut looks to be a more initially promising candidate than rejecting monotonicity. Rejecting cut says that the fact that you inferred each of the sentences in Β from Α rather than having them both from the start makes a difference to whether you can conclude γ. In other words, it says that how you inferred a claim may matter for what it can justify.

So it is possible that by rejecting cut we may say that the bootstrapping way of inferring is what makes it so we cannot perform the steps back-to-back. Our question, then, is what could explain why cut fails in this way.

3. The Bad News

Jonathan Weisberg (2010) has a simple answer to this question. He points out that so-called Bayesian epistemologists have a number of different probabilistic models of epistemic justification. And an interesting fact is that all of these different models of epistemic justification entail that epistemic justification does not satisfy cut.[8]^, [9] In this section, I will present the basic idea behind why these Bayesian approaches entail that epistemic justification fails to satisfy cut and how this is supposed to solve the bootstrapping problem (§3.1). I will then argue that Bayesian approach is incompatible with dogmatism (§3.2). This argument is similar to and is another member of a family of results demonstrating tension between dogmatism and probabilistic models of justification (§3.2.5).

4. cut in Foundationalist Epistemology

One reaction to this result is that it is a reductio of dogmatism: Bayesianism is a well-understood, powerful, formal theory of rational belief and dogmatism either suffers from the bootstrapping problem or is incompatible with it. So much the worse for dogmatism.

While I’m sympathetic to this reaction, I wish to make the best of it on the dogmatist’s behalf and explore an alternative perspective. Indeed, I believe that this alternative perspective is one that is, in any case, a more natural fit with dogmatism. This is because prominent dogmatists such as John Pollock and James Pryor are also critics of Bayesianism.[23]

Now since we are rejecting Bayesianism and since Bayesianism is the only model that we have looked at so far that allows epistemic justification to fail to satisfy cut, the first thing that we must do is develop an alternative picture of how it could be so much as possible that epistemic justification fails to satisfy cut.[24] While I cannot develop a theory as powerful and mathematically precise as Bayesianism here, what I will do is show how a broadly foundationalist epistemology has the resources to explain why epistemic justification fails to satisfy cut.

I begin by developing a minimal account of what foundationalism is (§4.1) and then frame the question of whether epistemic justification satisfies cut within this perspective (§4.2). After this, I will show how it could be that epistemic justification fails to satisfy cut (§4.3). This discussion of how it is possible for epistemic justification to fail to satisfy cut closely follows ideas that I have presented in earlier work (Nair 2019: §5). The task of Section 5 will be to explain how this possibility can be applied to the bootstrapping problem.

4.2. Interpreting Structural Properties of Justification

I now want to explain how to think about the different claims about the structure of justification that we have been discussing. In order to do this, it helps to start by looking at the structure of what we might think of as standard cases of learning.

Consider, then, the following:

Begin by focusing on what is on the left. As before, Α is a set of sentences and we are meant to understand this diagram as saying the set of mental states corresponding to the sentences in Α are the foundational states that the agent has. The ⇑ represents epistemic justification. So it says that the mental states corresponding to Α justify the non-foundational mental states corresponding to α and β respectively. As we move to the right, we have Add γ and this tells us that the agent comes to be in the state corresponding to γ and this state is a foundational mental state for that agent. We then see that this new collection of foundational states justify not just the non-foundational states corresponding to α and β but also a new non-foundational state corresponding to δ.

This is the structure of standard cases of learning in the sense that typically when we learn some new information we acquire additional new justified non-foundational states. With this structure in mind, let’s consider the two properties that I mentioned earlier, monotonicity and cut.

As I said earlier, it is relatively uncontroversial that epistemic justification does not satisfy monotonicity. In foundationalist terms, this means that we can have a situation like this:

That is, adding a new foundational state can make it so you lose justification for being in the non-foundational state corresponding to β.

The easiest way to understand what cut is about is to compare it to standard cases of learning. The natural way to interpret the diagram of standard cases of learning is so that γ is distinct from the sentences in Α as well as α and β. But cut concerns cases where γ is the same as one of these, say the mental state corresponding to β:

If epistemic justification satisfies cut, then the situation depicted by this diagram cannot arise. If it fails to satisfy cut then it can arise.

So far then, we have an interpretation of each of the structural claims about epistemic justification that we are interested in. With this in hand, I want to turn to showing how it might be that epistemic justification fails to satisfy cut.

4.3. How Justification Might Not Satisfy cut

The first thing to notice is that the foundationalist framework is compatible with epistemic justification satisfying cut as well as epistemic justification failing to satisfy cut. Whether epistemic justification satisfies cut turns on the following question: can adding a state corresponding to β to the foundations lead you to have at least one new justified non-foundational state if you were already justified in having a state corresponding to β among your non-foundational states? As far as I know, no major foundationalist has posed or answered this question. Nonetheless, it is a substantive question that is not answered by the bare bones foundationalist framework.

And in fact, it is somewhat unsurprising that foundationalism is compatible with epistemic justification failing to satisfy cut. After all, according to foundationalism, the foundational states are epistemically special in that they get their epistemic status in a different way than non-foundational states. Epistemic justification failing to satisfy cut suggests that they are special not just in how they get their epistemic status but also in what they are capable of justifying. In light of this, it is natural to wonder what we would have to add to the bare bones foundationalist theory in order for it to entail that epistemic justification fails to satisfy cut. I will answer this question in an abstract in this subsection. And then in Section 5 I apply this answer to the bootstrapping problem.

To begin, let’s consider an informal way of thinking about defeasible justification. One way to think about the idea that having a visual experience as of something being red defeasibly justifies you in believing that it is red is as saying that the visual experience justifies you in believing that it is red unless you believe that the lighting conditions are bad, believe that your visual system is malfunctioning, etc. In general, foundationalism can be informally thought of as making claims like: the proposition expressed by α justifies you in believing the proposition expressed by β unless such and such conditions holds. And according to foundationalism what goes in these ‘unless’-clauses are some claims about what other epistemic states you are in.

With this informal picture in mind, we can notice that the foundationalist psychology does not just claim that agents have epistemic states. It also claims that the epistemic states of an agent are organized in a certain way. And this means that the foundationalist could mention these facts about the organization of our beliefs in the ‘unless’-clauses as well.

To explain why this might be a promising idea, I want set dogmatism aside for a moment and work through an example in terms that will be most congenial to more traditional forms of foundationalism. In particular, one form of foundationalism says that the foundational states are the beliefs that things seem to you a certain way that are the direct result of perception (rather than the foundational states being the perceptual states themselves). According to this theory, we have foundational beliefs like the belief that o seems to have a rough surface. And this belief counts as defeasibly justified because it is the direct result of a perceptual experience. This belief in turn defeasibly justifies the non-foundational belief that o has a rough surface. Now suppose also that you are justified in believing that your sense of touch is malfunctioning. Are you justified in believing that o has a rough surface or not? One natural (albeit not inevitable) answer is that it depends on which perceptual experience your belief that o seems to have a rough surface was the direct result of. If that belief was the direct result of visual perception, then it is plausible to think that you are still justified in believing that o has a rough surface. But if it was the result of touching, you are not justified in believing this.

And one plausible (albeit not inevitable) way of getting this result is to allow ‘unless’-clauses that make reference to what your belief was the direct result of. In particular, we might say that the belief that o seems to have a rough surface and the belief that your sense of touch is malfunctioning justify you in believing that o has a rough surface unless your belief that o seems to have a rough surface is based on touching.

This case illustrates that how your belief came about might affect what it can justify. There are, of course, alternative ways of treating this case. But my only goal here is to sketch one plausible way of treating this case that involves an ‘unless’-clause that mentions not just other epistemic states but how those states came about as well. What’s more, I believe that there is some plausibility to the idea that there are true claims about defeasible justification that have ‘unless’-clauses of this sort.

This is because these kinds of claims occupy a strategic position between two alternative kinds of clauses that we could use to handle these cases. In these cases, I have been suggesting (and here I am simplifying a bit) that an epistemic state corresponding to α justifies an epistemic state corresponding to β unless the epistemic state corresponding to α came about in a certain way. The two main alternatives to this suggestion would be to simply deny that the epistemic state corresponding to α justifies the epistemic state corresponding to β or to claim that the epistemic state corresponding α justifies the epistemic state corresponding β no matter how it came about. The proposal I am sketching strikes a strategic balance between these by not being as conservative as the first option or as liberal as the second.

So the example that I sketched together with the fact that this claim occupies a kind of strategic position argue in favor of thinking that claims about defeasible justification might have ‘unless’-clauses that mention how an epistemic state comes about. Now the example that I gave does not involve a failure of cut but it does illustrate the significance of how a belief came about by considering the significance of two different ways the belief that o’s surface appears to be rough can come to be a foundational state.[25] So we might also consider different ways in which a state can come about. For example, we can consider the difference between a state coming about in the way a foundational state comes about and a state coming about in the way a non-foundational state comes about.

Here is how that might go. Suppose (1) the epistemic states corresponding to Α defeasibly justify the epistemic state correspond to β and (2) the epistemic states corresponding to Α together with the epistemic state corresponding to β defeasibly justify the epistemic state corresponding to γ. And suppose that if we unpack (2)’s ‘unless’-clause it would say the epistemic state corresponding to Α together with the epistemic state corresponding to β justify the epistemic state corresponding to γ unless the epistemic state corresponding to β is solely based on the epistemic states corresponding to Α.

Let’s look at what these claims predict. If we add the supposition that the epistemic state corresponding to δ defeasibly justifies the epistemic state corresponding to β, they predict that we can have a situation like this:

Though this case is not what I called a standard case of learning, it shares its structure. In standard cases of learning the new information plays a more or less direct role in justifying the new epistemic state. But in this case, it is the states corresponding to Α and the state corresponding to β that justify the new state. What the state corresponding to δ does is make it so the state corresponding to β is no longer solely based on the states corresponding to Α and this makes it so the ‘unless’-clause of (2) is no longer true.

So in that example the proposition expressed by β’s justificatory role changes by becoming justified by the state corresponding to δ. But the proposition expressed by β’s justificatory role could also change if the state corresponding to β came to be part of the foundations. That is, we could have a situation like this:

And now notice that this situation is one that demonstrates how epistemic justification might fail to satisfy cut. Thus, once we admit that claims about epistemic justification might have ‘unless’-clauses that mention how beliefs come about, it is easy to come up with abstract claims about epistemic justification that would lead to epistemic justification failing to satisfy cut.

And since I have argued that such claims are plausible, this shows that within a purely qualitative foundationalist epistemology we have the resources to explain why epistemic justification does not generally satisfy cut (cf. Weisberg’s 2010: 536 discussion of evidentialism).

5. The Solution

Of course, the fact that epistemic justification fails to satisfy cut in general does not by itself show that Step 5 cannot be performed after the other steps. But it is not hard to see how it can be applied in a way the secures this result.

Consider the claim that track record defeasibly justifies an agent in believing reliability.[26] There are many well-known defeaters for inductive justification. For example, there are defeaters concerning the existence of competing evidence and concerning certain defective ways of gathering data. So the ‘unless’-clause associated with the relevant claim about justification will mention these defeaters. I wish to add another defeater to this ‘unless’-clause: I posit that the ‘unless’-clause says ‘unless the belief that track record is solely based on the perceptual state of the slide being red and introspective awareness of this state’.[27]^, [28] This solves the bootstrapping problem.

6. Conclusion

Overall, then, the results of this paper are mixed for the dogmatist. We have seen that the only way for the dogmatist to solve the bootstrapping problem is for her to claim that epistemic justification does not satisfy cut (§1–2, §A). And this is some bad news for the dogmatist because probabilistic considerations suggest that this solution is not compatible with dogmatism (§3, §B). But we then tried to make the best of it on the dogmatist’s behalf by developing an alternative non-probabilistic framework in which to implement our solution. The framework we used was a broadly foundationalist epistemology (§4). We saw that in this framework, we can explain why epistemic justification fails to satisfy cut in a way that solves the bootstrapping problem (§5).

Though this doesn’t solve all of the problems for dogmatism (§5.2, §5.4, §C), it, I submit, is the best the dogmatist can do by way of solving the bootstrapping problem. Whether the dogmatist best is good enough depends on whether sufficiently powerful alternatives to probabilistic frameworks for understanding justification can be developed, whether deeply contingent a priori justification is really problematic, and whether alternative theories can capture the attractive feature of dogmatism. For these reasons, it is a question best left for another day.

A. A More Realistic Formalization

In the main text, I simplified things by giving an equivocating formalization of bootstrapping reasoning. Here I give a non-equivocating formalization that gives a richer albeit more complicated picture of the bootstrapping problem and our solution.[42]

To begin, we will need two relations ⊫ and ⊩ to give a non-equivocating formalization. These will both be relations between mental states. To see how the first relation works, let ⌈V(p)⌉ refer to the mental state of visually perceiving p (where we, perhaps, abuse language by treating ‘perceiving’ as non-factive) and let ⌈Bel(p)⌉ refer to the mental state of believing p. Then using ⊫ we formalize the dogmatist idea that perceiving justifies you in believing, as accepting each instance of the following scheme for suitably restricted values for p:

⊫, then, represents cases where merely being in the state on the left justifies you in being in the state on the right.

⊩, on the other hand, is used to discuss cases in which being justified in being in a certain mental state justifies you in being in another mental state. So for example

is to be understood as saying that being justified in believing that snow is white justifies you in believing that grass is green.

With these two relations in hand, we can now give a non-equivocating formalization of the reasoning. To start, we assume that you have a perceptual experience of the slide being red—V(the slide is red)—and an introspective experience as of having a visual perceptual experience of the slide being red—I(the slide looks red). From here, the rest of the reasoning proceeds in the following six steps.

• Step 1 V(the slide is red), I(the slide looks red) ⊫ Bel(the slide is red)
• Step 2 V(the slide is red), I(the slide looks red) ⊫ Bel(the slide looks red)
• Step 3 Bel(the slide is red), Bel(the slide looks red) ⊩ Bel(the slide looks red and is red)
• Step 4 Bel(the slide looks red and is red) ⊩ Bel(my color vision just worked)
• Repeat
• Step 5 Bel(color vision just worked 1), ..., Bel(color vision just worked n) ⊩ Bel(my color vision worked n times)
• Step 6 Bel(my color vision worked n times) ⊩Bel(my color vision is reliable)

Next once we have these six steps, the issue of whether these steps can be performed one after the other turns on whether the two relations satisfy two structural principles. To state these principles in a way that is analogous to the way that we stated them above, let us now use uppercase Greek letters to refer to sets of mental states and use lowercase Greek letters to refer to mental states. With this change of notation, the structural properties are these:

• bridge cut: if Α ⊫ β for each β∈Β and Β ⊩ γ, then Α ⊫ γ
• plain cut: cut for ⊩ as in the main text but adopting the present reading of the Greek letters

In this richer notation, the view that I develop rejects bridge cut (confirming the informal idea in the text that it is the special role of the foundational states that makes the difference). Thus, though slightly more complex, this formalization involves no equivocation and the remaining discussion in the paper fits smoothly with this more complex formulation with some easy translation that I leave to the interested reader.

B. Generalization for Jeffrey Conditioning

How do our results fare if we don’t rely on the assumption that we learn that the slide is red by conditioning on the slide looking red and we don’t rely on the assumption that we know for certain the slide looks red? Suppose in particular, instead, we think the natural modelling choice for learning is Jeffrey conditionalization. Indeed, I believe this is the best probabilistic development of the dogmatist view. According to the view, one directly learns that the slide is red, but does not become fully certain that the slide is red because the belief is only defeasibly justified.

Matters become more complicated in this setting. To begin, Jeffrey conditionalization is defined over a partition of an agent’s epistemic space. Normally, when we think of learning p by Jeffrey conditioning we think of p as an atom of the partition. If we were to apply this approach to our current situation, our model would be that we start out by conditioning on the slide being red, so the partition would include the slide is red as an atom. And then we condition on the slide looking red, so the partition would include the slide looks red as an atom.

But it is obviously possible that the slide looks red and is red, so these propositions cannot be atoms of the same partition. This way of modeling the case, then, assumes that we change partitions as we learn. Now it is known that once we allow switching of partitions over learning episodes, there are no constraints at all provided by your prior probabilities on the final result after conditionalization. This is more or less to say that if we think of the situation this way, the probability of reliability could be anything from 0 to 1 no matter what your priors are.

This, in principle, is one way out for the dogmatist. It would be one which more or less says that we have sequences of Jeffrey updates involving a change of our partition and that probabilities play no interesting role in our epistemology of these cases. Perhaps this is what the dogmatist should say. But if she wishes to say this, the interesting question is what interpretation she can provided of the epistemic role of a partition and of changes in partitions. This is a question that has, to date, been relatively underexplored in traditional epistemology.[43]

In light of this issue involved with shifting partition, the next question to consider is whether we can work with a fixed partition for both Jeffrey updates and what if anything that would tell us. Now in doing this, we should remember that the cases we are discussing are ones where we learn things like the slide looks red and the slide is red. And these, as we said, cannot be atoms of the same partition.

The way to interpret, say, learning the slide is red directly is to think of us as Jeffrey updating our partition (whose atoms does not include the slide is red or the slide looks red) so that the sum of the probabilities of the atom at which the slide is red is greater than or equal to the threshold for justification. There are of course many different kinds of partitions we could use. But the most natural choice is the one that is just as fine grained as we need: {r ∧ lr, r ∧ ¬lr, ¬r ∧ lr, ¬r ∧ ¬lr}.[44] Then we imagine a first Jeffrey update on r here understood to tell us that the sum of the probabilities in r ∧ lr and r ∧ ¬lr after the update are at least as high as t and then a Jeffrey update on lr understood analogously.

So recall now our setup. Our setup is that your probability that your color vision is reliable shouldn’t go up after these two Jeffrey updates. So using Pr₀ for your probabilities before Jeffrey updating at all and using Pr₂ for the results after two Jeffrey updates, we have it as a constraint that:

Next, since Pr₂ is the result of Jeffrey conditioning on the above described partition, we know that:

From here, we can notice that:

This follows from the following two facts

And we can also notice that:

This is because our discussion in Section 3.1 showed that perception and introspection are not just ways of forming justified beliefs in r and in lr respectively but also ways of forming beliefs in r and in lr that are sufficiently good to allow us to safely conjoin these beliefs. This then allow us to get a similar constraint on the probability of reliability:

This is our problematic result again except with more complicated terms summing with t².

This shows that given the natural choice of a fixed partition, our result can be established for Jeffrey conditioning. The best dogmatist response to this would be to identify some other fixed partition and show that given this partition our result cannot be established. And I do not at this time have a general argument that shows this is impossible.

But much like the dogmatist response that involved switching partitions, the best thing to say at this time is that if the dogmatist wishes to adopt this approach it is incumbent on them to say more. In particular to develop this response, the dogmatist must identify the partition that avoids the results of this appendix and give some grounds for thinking that this partitioning of the space is the correct one.

Thus, I conclude that while matters are more complicated if we consider Jeffrey conditioning, dogmatist views about justification continue to have an uneasy relationship with probabilistic models of justification.

C. The Problem Comes Earlier

We saw that the solution I have developed can be adapted to target Step 3 rather than Step 5. Though the paper is officially neutral about this issue, I prefer to adopt the solution that targets Step 5. This appendix considers further arguments for targeting Step 3 rather than Step 5 that I believe are inconclusive (though I leave it to the reader to assess the costs and benefits for themselves).[46]

Acknowledgements

For comments on an early draft thanks to the USC dissertation seminar, several anonymous referees, Andrew Bacon, Kenny Easwaran, John Hawthorne, Ben Lennertz, Jacob Ross, Mark Schroeder, Scott Soames, Gabriel Uzquiano, Ryan Walsh, Ralph Wedgwood, and Jonathan Weisberg. For comments on later drafts, thanks Brad Armendt, Bryan Lietz, Ángel Pinillos, and especially the area editor and anonymous referees at Ergo.

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Notes

1. Here we make a number of simplifications and idealizations for ease of presentation. First, we put aside the idea that you get some evidence that you are reliable simply by not having extremely bizarre perceptual experiences. Second, it would do no harm to assume the first two steps occur simultaneously. Third, plausibly the relevant track record propositions should record the ratio of success to failure as well. In our context, of course, you always succeed and never fail. Third, for the inference to Reliability from Track Record we assume n is large enough to make the inference plausible but not so large that it raises concerns about conjunction introduction (see §3.1 for discussion). Fourth, Reliability perhaps should be qualified so it mentions a context. It does nothing to reduce the force of the problem if we simply make the context the specific one the agent is currently in. Thanks to Brad Armendt and Ángel Pinillos for pushing me to clarify these issues.

2. See Cohen (2002) for a seminal discussion of bootstrapping for dogmatism as well as a closure problem. Following Vogel (2007) and Weisberg (2010), I focus only on the bootstrapping problem in this paper. I do not believe the arguments of this paper on their own are sufficient to solve the closure problem (see §C for discussion). Weisberg (2012) is a survey of work on the bootstrapping problem. Other discussions of the bootstrapping or closure problems include Altschul (2012), Becker (2012), Black (2008), Breisen (2013), Brueckner (2013), Brueckner and Buford (2009), Davies (2004), Douven and Kelp (2013), Cohen (2005; 2010), Hawthorne (2004: 73–77), Kalstrup (2012), Kornblith (2009), Markie (2005), Neta (2005), Scheall (2011), Titelbaum (2010), Weisberg (2010), Vahid (2007), van Cleve (1979; 2003), Vogel (2000; 2008), and Zalabardo (2005).

3. Those who only accept the weaker dogmatist claims may endorse the neo-Cartesian solutions offered by Cohen (2010) and Wedgwood (2013). See §C.3 for discussion.

4. Cf. Vogel (2007: footnote 43) who suggests but does not develop such a solution and Vogel (2000: footnote 30) who is unimpressed by a related strategy. But Vogel’s (2000) pessimism is directed at an application of it that says Step 3 cannot be performed after Step 2 and as we will see, my application of this strategy is different.

5. §5.2 gives an example that shows why epistemic justification is not transitive in general. But the interesting question is whether it is transitive over the bootstrapping reasoning.

6. To see that cut and monotonicity entail transitivity, assume the antecedent of transitivity. That is, assume (i) Α ⊩ β for all β∈Β (ii) Β ⊩ γ. By applying monotonicity to (ii), we have (iii) Α⋃Β ⊩ γ. Finally by applying cut to (i) and (iii), we have the consequent of transitivity, Α ⊩ γ.

7. Cf. Titelbaum’s (2010: 121) discussion of how appeal to defeaters can’t help the reliabilist and Barnett’s (in press: §4.3) discussion of this defeater for dogmatists.

8. More precisely, Weisberg posits a defeater he calls No Feedback (2010: 533–534) and for this defeater obtain it must be that epistemic justification fails to satisfy cut on probabilistic grounds.

9. Probabilistic models are also often used to model a related issue of transmission failure (see Moretti & Piazza 2013; Okasha 2004). For non-probabilistic philosophical discussion of transmission failure, see Wright (2004) for non-dogmatist perspective and Pryor (2012) for a dogmatist perspective.

10. Cf. Cohen who writes “we can stipulate that the number of conjuncts is small enough that I remain justified in believing the entire conjunction, but long enough to justify inductive inference. This is possible on the assumption that enumerative induction is possible” (2010: 143).

11. An anonymous referee points out still another helpful way to think about why this simplification is harmless. Suppose instead of thinking about an ordinary agent we are considering a case where “someone with excellent (maybe super-human) color vision looks at a paint chip and can tell that it’s some very particular shade of blue (out of arbitrarily many possible shades). She can know that it’s arbitrarily unlikely that she’d be right about which very specific shade it is by chance, so if she nevertheless knows which shade it is, this seems like as good evidence for reliability as a series of coarse-grained identifications would be.”

12. But see §5.4 and §C for discussion.

13. §3.2.5 considers how our results fair under weaker assumptions.

14. In general, Pr(p) = Pr(p ∧ q) + Pr(p ∧ ¬q), so Pr₁(reliability) = Pr₁(reliability ∧ r) + Pr₁(reliability ∧ ¬r). And so by the definition of Pr₁, Pr₀(reliability|lr) = Pr₀(reliability ∧ r|lr) + Pr₀(reliability ∧ ¬r|lr).

15. This assumes that Pr(p|q) = Pr(p ∧ q)/Pr(q). So Pr₀(r|lr)Pr₀(reliability|r ∧ lr) = [Pr₀(r ∧ lr)/Pr₀(lr)][Pr₀(reliability ∧ r ∧ lr)/Pr₀(r ∧ lr)]. Next [Pr₀(r ∧ lr)/Pr₀(lr)][Pr₀(reliability ∧ r ∧ lr)/Pr₀(r ∧ lr)] = [Pr₀(r ∧ lr)Pr₀(reliability ∧ r ∧ lr)]/[Pr₀(lr)Pr₀(r ∧ lr)]. Then we have that [Pr₀(r ∧ lr)Pr₀(reliability ∧ r ∧ lr)]/[Pr₀(lr)Pr₀(r ∧ lr)] = Pr₀(reliability ∧ r ∧ lr)/Pr₀(lr). Finally by definition, Pr₀(reliability ∧ r ∧ lr)/Pr₀(lr) = Pr₀(reliability ∧ r|lr). So Pr₀(reliability ∧ r|lr) = Pr₀(r|lr)Pr₀(reliability|r ∧ lr).

16. This assumes that the dogmatist notion of coming to have justified beliefs based on perception can be modeled as conditioning on the claim that you had the perception. This assumption has been rejected by some who discuss dogmatism (see Morreti 2015 and Barnett in press for discussion). I am sympathetic to this idea, so below in §3.2.5 and §B I show how the argument of this section can be adapted so that it does not rely on this assumption.

17. The comments at the end of §3.1 that explain why conjunction introduction is safe in the bootstrapping context are important for seeing why these assumptions are innocuous given the simplifications that I have made. Pr₀(reliability|r ∧ lr) ≥ t is innocuous because we are treating r ∧ lr as simplifying proxy for Track Record. Pr₀(r|lr) ≥ t then must be understood as saying, in effect, that Track Record is probable on each perceptual experience. This too is innocuous because we have argued that not only must the belief that slide one is red, the belief that slide two is green, etc., and the belief that slide one looks red, the belief that slide two looks green be above t, they must be sufficiently above t to allow for conjoining them all together.

18. Here I put aside the coherentist solution that the dogmatist must also reject that says our justification comes “at the same time”.

19. While Kripke may have introduced convincing cases of contingent a priori knowledge, this example does not fit that model (e.g., it does not involve any reference fixing descriptions). I, following Hawthorne (2002) (who takes the term from Gareth Evans), flag this difference by calling the proposition deeply contingent.

20. Cohen (2005: footnote 1) conjectured that the bootstrapping problem would still arise for views that allowed that one had strong a priori evidence that falls short of knowledge or fully-justified belief. Our discussion of Weisberg suggests that this conjecture is incorrect but nonetheless, that this should be of no comfort to the dogmatist.

21. Other results focus on dogmatism’s tension with other probabilistic constraints. For example, some results focus on showing that the dogmatist idea that a perceptual experience allows us to newly rule out the possibility of a skeptical scenario obtaining is incompatible with certain probabilistic constraints (§C provides a description of these results and discusses their bearing on the issues of this paper).

22. Those who give probabilistic arguments against dogmatism (e.g., White 2006: 534–535) standardly respond to the objection from Jeffrey conditionalization by arguing that there is no reason why it should work relevantly differently from standard conditionalization: it would be bizarre, they say, for there to be a relevant difference between learning directly to degree n that the slide is red because you have a perceptual experience of it being red (the Jeffrey conditioning model) and learning indirectly the slide is red to degree n based on learning directly for certain that you are having the visual experience (the standard conditionalization model). Whatever the merits of this response (I myself am of two minds about it and think it is at least not dialectically effective), it is an interesting fact that the argument of this section does not need this response to succeed in order for it to be sound (in particular, I do not need to assume that the Pr(r|lr) is high). Instead, the argument can be made without making any claim about the connection between Jeffrey conditionalization and standard conditionalization; it can be made directly in terms of Jeffrey conditionalization (see §B). Thanks to John Hawthorne and several anonymous referees for pushing me to consider this issue.

23. Pollock and Cruz (1999) and Pryor (2013) are dogmatist critics of Bayesian approaches. Pryor’s (2013) is closely related to Christensen (1992) and Weisberg (2009) and (2015). Together they develop, in my view, the most serious dogmatist challenge to using a probabilistic framework to model justification.

24. In fact, as I observe in Nair (2019), all formal theories other than probabilistic theories and certain systems of inheritance reasoning entail that epistemic justification satisfies cut. Nair (2019) also considers from an abstract perspective whether this convergence among formal theories reflects a deeper truth about justification and argues it does not.

25. It is also important that this first case applies to a particular form of non-dogmatist foundationalism according to which the causal etiology of our beliefs about appearances may matter. The case to follow (the one that illustrates a failure of cut) is consistent with a dogmatist setting in which our foundational states may be perceptual states themselves and the causal etiology of these states is irrelevant. It has been pointed out to me by an anonymous referee that some dogmatists have argued in response to concerns about cognitive penetration that the causal etiology of our states is irrelevant (see Siegal 2012 and 2013 for the concern and, e.g., Huemer 2013 and Tucker 2010: §6.1 for the response). The case that illustrates the failure of cut does not assume the causal etiology of our states (in general or our perceptual states in particular) is relevant; it only assumes the current basing structure of our states is relevant (a claim which dogmatist are in any case committed to).

26. Whereas ‘Track Record’ refers to the sentence given at the beginning of the paper, ‘track record’ expresses the proposition expressed by that sentence.

27. Though there are a number of counterexamples to this formulation of the ‘unless’-clause, I will work with it in what follows because it is the simplest way to capture the spirit of my proposal. A formulation designed to avoid these counterexamples would read something like ‘unless the belief that the slide is red that is involved in supporting track record is partially essentially based on the perceptual experience of slide being red’. Thanks to John Hawthorne for asking me to consider these examples.

28. I recently learned of the NSC defeater in Butzer (2017: 2090) that is similar to the defeater here. While I do not essentially disagree with Butzer, his work does not situate the defeater in the general context about the structure of justification (with both the problems based on probabilistic results and the prospects based on the informal model developed in the previous section) that is center stage in this paper.

29. This shows how the present account would treat the case of Eliza in Weisberg (2010: 530; cf. White 2006: 530) and thereby avoid a problem that plagues Vogel’s (2007) views.

30. Cf. Weisberg (2010: 538–539). It may also be worth noting that this response assumes (to my mind, harmlessly) that the dogmatist will individuate the reasons that we have from testimony to believe that the slide is red from the reasons we have from the experience of the slide being red to believe the slide is red. See Black (2008; especially footnote 9) and Vogel (2008: footnote 17 and 527–528) for closely related issues concerning individuation.

31. Cf. Weisberg’s (2010: 537–538) response to a similar worry and Vogel’s (2000: 615–619) objection to the claim that bootstrapping is unreliable.

32. This underestimates the resources available to independently motivate this defeater. First, this defeater may be a specific instance of a more well-known defeater (e.g., a biased sample defeater). Second, the defeater that I have posited captures the intuitions that support a number of principles that have been offered in the literature such as The No Self-Support Principle (Bergmann 2000: 168; Cohen 2002: 319; Fumerton 1995: 180), the Independence Principle (Black 2008: 606–609; Cohen 2005: 428; Markie 2005: 414), the No Rule Circularity Principle (Vogel 2007: 531), and certain epistemic difference-making principles (Comesaña & Sartorio 2014) . Third, Butzer (2017: §4ff.) argues for a similar defeater and defends it against objections. I do not myself wish to endorse (or deny) any of these claims as motivations for the defeater. Instead, in work in progress, I try to motivate this defeater by appeal to a certain general feature of the sort discussed in Footnote 35.

33. This is one development of the Reidian non-reductionist approach to testimony that is perhaps best represented by Burge (1993).

34. Lackey (2006) provides a rich introduction to the variety of views about testimony.

35. Begin by noting that one reason why justification is defeasible is that it is ampliative in the sense that a proposition that you justifiably believe, say p, can justify belief in another proposition, say q, without entailing it. This opens the door for an agent to learn new information d that is consistent with p but is inconsistent with q, makes q unlikely, or makes q otherwise unreasonable. Thus, the ampliative relations among propositions are one source of defeaters in the sense that allowing for defeaters is the only way such ampliative leaps could be reasonable. But notice that in forming beliefs, we not only take risks related to the propositional relations among our beliefs but also with respect to our belief forming processes themselves. For example, even when we form beliefs based on deductive inference from other things that we believe, the mental process that subserves this transition is not completely reliable. This makes it natural to allow defeaters that are sensitive to this way in which our rational belief formation incurs risk. And once we admit that justification should be sensitive to the propositional relations among beliefs as well as the nature of the belief forming processes themselves, this opens the door for the details of the structures of those processes to influence which things are defeaters. (One way to think of this is as saying how you have your evidence is epistemically important; cf. Neta 2013.) It is here that within a foundationalist framework which wishes to ward off a priori knowledge of deeply contingent propositions that I believe an explanation for the defeater that I posit can be found. These remarks require further development (of the sort that I give in work in progress), but it should be enough to provide some license for optimism that such an explanation can be provided.

36. As a referee pointed out to me, a different challenge to Markie’s principle comes from the literature on so-called transmission failure, see Footnote 9.

37. Thanks to an anonymous referee and the editor for encouraging me to address this issue and for helpful suggestions about how to address it.

38. For example, White (2006: 538; cf. Cohen 2010: 144) suggests that the best explanation of our color vision working is our color vision being reliable so getting this far is a problem. My approach, of course, would simply posit the same defeater for inference to the best explanation that we posited for enumerative induction.

39. Of course, as we have seen, the model in Section 4 allows for defeaters of a distinctively structural sort as well. And this may give us room for positing defeaters for even deductive inference (see Footnote 35). But the point in the text is that there is no independent prior reason to think Step 3 is a defeasible inference whereas there is for Step 5.

40. It is a delicate matter how best to state the principle of single premise closure. We pass over this important issue.

41. Rejecting this principle will also make it difficult for the dogmatist to offer their typical Moorean response to skepticism. To some (such as myself), this is a serious cost as well.

42. Thanks to Andrew Bacon for pushing me to discuss the problem that this section addresses.

43. But see Greco (2017) for a framework that could be applied to the present context to help to make this option more palatable.

44. This follows Jeffrey’s advice (1983: §11.6) about how to deal with cases where one simultaneously learns several propositions. Since no harm is done by assuming one simultaneously learns the slide is red and the slide looks red in our puzzle, this way of modeling the case is the modeling choice recommended by Jeffrey. Thanks to Brad Armendt for pointing this out to me.

45. This represents inductive reasoning being good reasoning in this context. Notice that we make no assumption about Pr(r|lr) an indeed this probability may even be low. This accords with a dogmatist perspective on which one directly learns the slide is red but one does not accept the inference from the belief the slide looks red to the belief the slide is red. As far as I know, the result here is the only probabilistic objection to dogmatism that can be directed at this kind of dogmatist.

46. See Cohen (2010: 145, 149), Kallestrup (2012: §4), Titelbaum (2010: 120–121, 128–129), White (2006: §7). Though it is actually not clear exactly which steps each of these philosophers find problematic, I focus on Step 3 to fix ideas. See also Butzer (2017: §6) for a response to this objection that usefully clarifies some potential misunderstandings and responds to some reason to reject Step 3 but, in my view, does not adequately address the two arguments below.

47. See Cohen (2010: 145, 149) and Titelbaum (2010: 120–121).

48. Though my official response to this objection is the somewhat concessive one given below, the dogmatist may be able to respond to this objection less concessively. I can only sketch the response here.

    Begin by considering the case where the oracle tells you that if your color vision works once, then it is reliable. This amounts to telling you that your color vision will either never work or it is reliable. If you were fully justified in being confident in this disjunction and spreading your confidence equally over each disjunct, the dogmatist should say this defeats your perceptual justification for believing that the slide is red. This is just an undercutting defeater like learning that in the present circumstances it is as likely as not that the slide is red if it looks red

    This covers the case where n is one. But not all values of n will look like cases of undercutting as this one does. However, we may be able to generalize the strategy to cover these cases. The idea would be that all of them adjust how much justification you get for believing that the slide is red based on it looking red. The amount of justification you get for the slide is red constrains whether you are justified in believing this and whether this belief can be conjoined with other ones. If we hypothesize that the level of justification will never be high enough to admit of conjunction introduction of n conjuncts for n high enough to allow enumerative induction unless the oracle’s claim is sufficient on its own to justify us in believing our color vision is reliable, we avoid the objection. (This response twists Cohen’s, 2010: 146–148, response to the super reliability related objection to be a response to his own objection to reaching Step 3.) Though the details need to be spelled out, the basic idea is that the oracle’s testimony in effect gives you information about your reliability and this constrains the bootstrapping reasoning in a way that allows us to avoid this problem.

49. See Cohen (2005: 424–425), Hawthorne (2004: 73–77), White (2006: §1–6). But these results cannot be established in the Jeffrey conditioning setting (cf. §B) at least in the sense elaborated in Footnotes 22 and 45.

50. It was originally presented as a different problem in Cohen (2003). More recent discussion such as Vogel (2007) and Weisberg (2010) also treat it as a different problem.

51. I only discuss the possibility of generalizing the backstory about why the claim that if the slide is red, then it looks red is a priori justified to explain why the claim that your color vision is a priori justified. I leave to one side direct attempts to explain why the claim that your color vision is reliable is a priori justified such as the one in Wright (2004).

52. See Cohen (2010: 150–156; cf. Hawthorne 2002: §1.2). Wedgwood (2013: §3, especially n. 14) offers a similar backstory but does not claim that we are a priori justified in believing that our color vision is reliable (see Footnote 53 for further discussion).

53. Though Wedgwood’s (2013) ideas are similar to Cohen’s they differ in important ways. In particular, Wedgwood only claims that we are a priori justified in believing that if the slide looks red and my experiences contain no defeaters, then it is red and similarly for all of the conditionals corresponding to rules. This difference leads us to be unable to perform the suppositional reasoning that I sketched on Cohen’s behalf. This is because we cannot coherently add the supposition that no defeaters to enumerative induction are present after inferring the slide is red from the slide looks red because this itself is a defeater of that inductive inference according to the present account. To better understand this point as well as the point that I make about Cohen’s solution, it may be useful to compare how Cohen and Wedgwood would treat the failure of transitivity discussed in Section 5.3.

54. The dogmatist may also claim that supposing a proposition that reports one is having a visual experience only supports believing under the supposition what one is justified in believing when one believes one is having a visual experience. But dogmatism has no commitments about this; it only has commitments about what one is justified in believing when one has the visual experience.