1. The Principal Principle and Initial Vindication

In this section, I present a variant of the argument provided in Pettigrew (Reference Pettigrew2012) for the Principal Principle. Before doing so, however, let me highlight a few assumptions that I will be making.

First, unless otherwise noted, talk of ‘possible worlds’ throughout should be understood as referring to maximally specific ways that a world might be that cannot be ruled out a priori. Second, I assume here and in the following section that credal states and chance functions are defined over algebras of eternal propositions. Later, we will relax this assumption and allow that credal states and chance functions may be defined over algebras of temporal propositions. To avoid unnecessary prolixity, however, in this and the following two sections the term ‘proposition’ will be reserved for eternal propositions.

Finally, I assume that each possible ur-chance function ch(·) is a probability function such that ch(C_ch) = 1.Footnote ⁴ That is, I take it that ur-chance functions are non-self-undermining probability functions.Footnote ⁵

Now on to Pettigrew’s argument. Let be a finite algebra of propositions, and let , that is, the set of functions mapping the members of to real numbers. This set will serve as a representation of possible credal states. Let us say, to a first approximation, that an agent’s credal state is vindicated just in case it is objectively as it ought to be. We will say that a credal state Cr(·) is potentially vindicated just in case it is vindicated relative to some epistemically possible world. We let serve as a representation of the potentially vindicated credal states. We let be the convex hull of .Footnote ⁶ And, finally, we let be a function measuring the distance of a credal state b(·) from a potentially vindicated credal state v(·). We call this the inaccuracy measure.

Jim Joyce has shown that, given certain plausible constraints on the inaccuracy measure, the following theorem holds:Footnote ⁷

This theorem may be used to argue for the following normative constraint on credences:

How might we use the Dominance Theorem to argue for Convexity? The argument, I think, should go as follows. We begin with the following plausible thought:

Next we assume that it is a priori knowable what vindication consists in. Given this assumption, it follows that it is a priori knowable which credal states are potentially vindicated. And so the Dominance Theorem assures us that if the convex hull of the set of potentially vindicated credal states is closed and bounded, then if one has a credal state Cr(·) that is not in this set, then one is in a position to tell a priori that there is some other credal state Cr′(·) that is closer to being vindicated than Cr(·) no matter what the world is like.Footnote ⁹ And so it follows from Vindication Dominance that an agent will always be rationally prohibited from having a credal state that is not in the convex hull of the set of potentially vindicated credal states, and so will be rationally required to have a credal state that is in the convex hull of the set of potentially vindicated credal states, on the assumption that this set is closed and bounded.

Convexity, then, provides us with the following two-stage strategy for justifying some putative norm of rationality N:

Joyce (Reference Joyce1998, Reference Joyce, Huber and Schmidt-Petri2009) offers just such a justification for the following putative norm:

Joyce’s argument depends on the following two claims:

Since is countable, it follows that is closed and bounded. Given Convexity, then, it follows from these two claims that an agent should always have probabilistically coherent credences.

Pettigrew (Reference Pettigrew2012) employs the same general strategy in service of justifying both the Principal Principle and the claim that a rational agent’s initial credences should always be probabilistically coherent. Our presentation of Pettigrew’s argument will work under the assumption that the possible ur-chance distributions over is a closed and bounded set.Footnote ¹¹ This seems, indeed, quite plausible. This set will clearly be bounded, since the set of probability functions over any algebra is bounded and each chance function is a probability function. And it seems quite plausible that this set should also be closed. For if some probability function p over finite can be arbitrarily approximated by probability functions that are a priori possible ur-chance functions over , it is hard to see how we could a priori rule out p itself as a possible ur-chance function over . But given that we cannot rule out such functions, it follows that the set of possible ur-chance distributions over will be closed. And given this, the convex hull of this set will also be bounded and closed. Given this assumption, Pettigrew’s argument depends on the following two claims:

Since we want to repurpose a modified version of this theorem later on, it is worthwhile presenting the proof of this result here.Footnote ¹³

Now it might not seem obvious that 1 and 2 suffice to show that every element of satisfies conditions i and ii. For all that 2 explicitly shows is that convex combinations of finite elements satisfying i and ii will also satisfy i and ii. But has uncountably many members. The following theorem, however, assures us that 1 and 2 are, indeed, sufficient for this result.Footnote ¹⁴

Since by 1 every element of satisfies i and ii, and since by 2 finite convex combinations of points satisfying i and ii also satisfy i and ii, it follows—given Carathéodory’s Theorem and the fact that is finite and so the space for some finite n—that every element of satisfies i and ii.

Proof of 2. Let b and b′ satisfy i and ii. Given this, we have

Given Convexity, then, it follows from Initial Vindication and Pettigrew’s Theorem, together with the assumptions that each chance function is a non-self-undermining probability function and that the set of possible ur-chance functions is closed and bounded, that an agent should always have an initial credal state that is probabilistically coherent and that, in addition, satisfies the constraints imposed by the Principal Principle.

2. Against Initial Vindication

We have seen that if we accept Initial Vindication, then we have good reason to also accept the Principal Principle. In this section, however, I argue that we should not accept Initial Vindication. Thus, Pettigrew’s argument does not provide us with good reason to accept the Principal Principle.

I offer two arguments for this claim. First, I argue that there is another more prima facie plausible account of how credences may be vindicated in light of certain facts about chance. Second, I argue that Initial Vindication has certain odd features that should make us skeptical of the claim that it provides the correct account of credal vindication.

To begin with, consider the following alternative account of credal vindication.

The first point to note is that in certain cases Initial Vindication and Chance Vindication will offer incompatible verdicts about which credal states are vindicated. To see this, consider some agent Q who is at the beginning of her epistemic life. Let w_q be the world in which Q is located and t_q the time at which her epistemic life begins. Initial Vindication says that Q’s credal state Cr_q(·) is vindicated just in case, for every proposition ϕ over which it is defined, , where is the ur-chance function at w_q. Chance Vindication, however, says that Cr_q(·) is vindicated just in case, for every proposition ϕ over which it is defined, . There is, however, no guarantee, in general, that will be identical to . For the chances may change over time. Thus, Initial Vindication and Chance Vindication will, in certain cases, indeed in most cases, issue incompatible verdicts about which credal states count as vindicated.

Initial Vindication and Chance Vindication, then, provide incompatible accounts of how a credal state may be vindicated in light of certain facts about chance. But, prima facie, I think, Chance Vindication is more plausible than Initial Vindication. According to Chance Vindication, what is relevant to whether one’s credal state is vindicated is whether one’s credences match up with the chances where and when one is located. According to Initial Vindication, however, it is some other chance function than the one that correctly describes the chances as they are that is relevant to vindication. But why should some other chance function be relevant to whether one’s credences held in world w and time t are vindicated?

To help bring out the oddity of this, consider the case of belief. In the case of belief, the following seems quite plausible:

Consider, however, the following alternative:

I think that we should find Belief Vindication* quite incredible. Why should the truth-values of propositions at some other world w* matter for whether one’s beliefs in w count as vindicated? At the very least, one would need strong reasons to prefer Belief Vindication* over Belief Vindication. Similarly, I suggest that there is, at the very least, a challenge for the proponent of Initial Vindication to say why it is the ur-chances that are relevant to the vindication of initial credences, even if those credences are held at times when the chances differ from the ur-chances.

One attempt to meet this challenge might appeal to the distinction drawn in Hall (Reference Hall2004) between database and analyst experts.Footnote ¹⁵ A database expert is one that you defer to because he or she has more information than you about a given domain. An analyst expert, however, is one that you defer to because he or she is better at assessing evidence than you are. If one thought that chance functions were valuable solely in virtue of their being analyst experts, then one might provide the following rationale for Initial Vindication. Insofar as an initial credal state ought to match up with some chance function it ought to do so in virtue of the chance function being ideally epistemically responsive. But ur-chance functions are those chance functions that capture the ideally epistemically responsive features of chance functions without packing in any additional information. Thus, insofar as an initial credal state ought to match up with some chance function, it ought to match up with the ur-chance function.

The worry I have for the account is that while it is plausible that chance is valuable as an analyst expert, it seems to me that it is also valuable, in many cases, as a database expert. That is, one reason to defer to chance is that chance often incorporates information about the world. But, if at least part of what is valuable about chance is that it contains information about the world, then what we should, at least from a certain idealized point of view, care about are the chances of where and when one is located, since these not only will be appropriately responsive to evidence but will also include more information than the ur-chances.

There is, however, another reason, apart from its conflict with Chance Vindication, to be skeptical of Initial Vindication. To see this, first note that there is a striking disanalogy between, on the one hand, the picture of vindication for credal states that emerges from Alethic Vindication and Chance Vindication and, on the other hand, the picture of vindication for credal states that emerges from Initial Vindication. Given Alethic Vindication and Chance Vindication, whether an agent’s credal state is vindicated is an objective matter in the following sense. Any two agents with the same credal state who are located in the same world and same time will be such that either both of their credal states are vindicated or neither is.Footnote ¹⁶ Given Initial Vindication, however, whether an agent’s credal state is vindicated depends not only on objective facts about where in logical and temporal space the agent is located but also on facts about what the agent’s evidence is.

Here’s why. According to Initial Vindication, an agent’s initial credal state, that is, her credal state before receiving any evidence, is vindicated just in case it matches the ur-chances. But the proponent of Initial Vindication cannot reasonably hold that, in general, an agent’s credal state is vindicated just in case it matches the ur-chances. That is, the proponent of Initial Vindication should not endorse the following general principle:

For, given certain plausible assumptions, Ur-Chance Vindication can be shown to be incompatible with the following principle:

For, given Ur-Chance Vindication, it turns out that there will be cases in which an agent starts out as rational and updates in a manner that makes her certain of her new evidence and preserves her prior certainties, and yet the agent comes to have a credal state that is irrational. But the possibility of such a case is ruled out by Certainty Preservation. Since Certainty Preservation is extremely plausible, this incompatibility between Ur-Chance Vindication and Certainty Preservation should, I think, lead one to reject the former principle.

Definition Let Cr(·) be a credal state such that there is a finite set of propositions such that . Let be the set of indices for such propositions. We say that the expected inaccuracy of a credal state Cr′(·), given Cr(·), is

The proponent of Initial Vindication, then, cannot say that, in general, vindication consists in matching the ur-chances. Thus, if one endorses Initial Vindication, one must hold that whether a credal state is vindicated depends both on facts about what the world is like and on additional facts about the agent’s epistemic situation, that is, whether she has received any evidence.

Why might this be a problem? Well, I take it that we have a grip on an objective sense of what an agent ought to believe or what credences an agent ought to have. That is, we have a grip on a sense of ought that abstracts away from facts about an agent’s evidence and simply considers what would be the ideal belief or credal state given what the world is like where and when the agent is located. Given this sense of ought, agents located at the same world, or perhaps the same world and time, ought to have the same doxastic states. We also, I take it, have a grip on a subjective sense of what an agent ought to believe or what credences an agent ought to have. That is, we have a grip on a sense of ought that abstracts away from what the world is in fact like where and when the agent is located and simply considers what doxastic or credal states are best given the agent’s evidence. Given this sense of ought, agents cannot differ with respect to what credences or beliefs they ought to have without also differing with respect to their evidential situation. But neither of these corresponds to the sense in which, according to Initial Vindication, an agent who has received no evidence ought to set her credences in line with the ur-chances.

We cannot understand this as a subjective ought. For, according to Initial Vindication, what credences an agent with no evidence ought to have will vary from world to world. But we also cannot understand this as an objective ought.Footnote ²⁰ For, as we have seen, the demands imposed by Initial Vindication cannot be generalized in a way that ignores facts about an agent’s evidence.

The proponent of Initial Vindication, then, would seem to require that there be some nonobjective and nonsubjective sense in which certain agents ought to set their credences in line with the ur-chances. Such a sense of ought must be partly subjective, depending in part on the agent’s evidential situation. But it cannot be understood, for example, in terms of it being the case that certain credences seem best in light of the agent’s evidence. And such a sense of ought must be partly objective, depending in part on where in logical space the agent is. But it cannot be understood, for example, in terms of the ur-chances being the feature of reality that it is objectively best for a credal state to match.

It is, however, quite opaque to me whether there really is a coherent hybrid normative notion meeting this description. At the very least, the proponent of Initial Vindication owes us a story about why we should suppose that there are normative facts that are not, as we standardly think of such facts, objective or subjective. Without such a story, I think there is good reason to be skeptical of Initial Vindication.

Note, however, that the same worry clearly does not apply to Chance Vindication. For we can understand this principle as employing an objective sense of ought. Chance Vindication tells us that objectively an agent ought to have credences that match the chances as they are at the time those credences are held. Such a principle imposes the same demands on all agents located in the same world at the same time, regardless of what evidence they possess.

The fact that we can understand Chance Vindication, but not Initial Vindication, by appeal to an antecedently intelligible notion of how an agent’s credences ought to be from an objective point of view gives us, I think, good reason to prefer the former principle to the latter. We have, then, good motivation to explore what normative constraints on credal states follow on the assumption that Chance Vindication, instead of Initial Vindication, is correct. It is to this question that we now turn.

3. Chance Vindication and the Temporal Principle

In this section, I show that both Probabilism and an interesting chance-credence norm follow from Chance Vindication. To show this, we follow Pettigrew’s strategy and establish that the demands imposed by these norms are satisfied by every credal state in , that is, the convex hull of the set of potentially vindicated credal states given Chance Vindication. As we will see, however, certain modifications to our current framework will be required in order to establish, in this manner, that there is an interesting chance-credence norm that follows from Chance Vindication.

It will be illuminating, however, to begin by considering some plausible chance-credence norms that do not, in fact, follow from Chance Vindication. Seeing why these fail to be entailed by Chance Vindication will lead us naturally to the amendments that are required to our current framework in order to derive an appropriate chance-credence norm.

Let us begin with the Principal Principle. Pettigrew (Reference Pettigrew2012) demonstrated that this norm follows from Initial Vindication by showing that the constraints that it imposes are satisfied by every credal state in . We will now show that there are credal states in , and so in , that do not satisfy the constraints imposed by the Principal Principle on credal states. The Principal Principle, then, cannot be shown to follow from Chance Vindication in the same manner in which it was shown to follow from Initial Vindication.

We can generalize this proof in the following way. Let ϕ_ch be an eternal proposition concerning a chance function ch(·). Perhaps ϕ_ch says that the ur-chances are given by ch(·), or perhaps it says that the chances at t are given by ch(·), or perhaps it says that God’s favorite chance function is ch(·). Given minimal assumptions, we can show that there is some such that Cr(· | ϕ_ch) ≠ ch(·).

Now many eternal chance propositions that might be thought, at least prima facie, to rationally constrain our credences in the manner in which the Principal Principle claims ur-chance propositions rationally constrain our initial credences will satisfy conditions i and ii. Not only can the Principal Principle, then, not be shown to follow from Chance Vindication in the manner in which it was shown to follow from Initial Vindication but neither can other formally analogous principles that appeal to alternative chance propositions.

Given this, a natural response is to look for some other way of deriving normative principles from Chance Vindication. And there is a tempting thought here. For, according to Chance Vindication, vindication must be relativized to a time. It is natural, then, to think that we might be able to justify certain temporally relativized norms by relativizing the set of potentially vindicated credal states to a time and showing that, for each time t, each member of the set of potentially vindicated credal states at t satisfies the constraints imposed by the appropriate temporally relativized normative principle. Indeed, there is an analogue of Pettigrew’s Theorem that would seem to support this thought.

Let For some possible chance distribution at t, , , for every . This is, according to Chance Vindication, the set of potentially vindicated credal states, defined over , given a time t. If ch(·) is a possible chance function given a time t, we will let be the proposition that says that the chances at t are given by ch(·).

Consider, then, the following putative norm:

Now it is tempting, I think, to suppose that this fact can be marshaled to show that, given Chance Vindication, an agent at time t ought to have a credal state that is probabilistically coherent and, in addition, satisfies the T-Principle. To conclude this, however, would be a mistake. For, in order for the preceding to show that Probabilism and the T-Principle follow from Chance Vindication, the following auxiliary principle would have to hold:

But, unlike Convexity, T-Convexity is not at all plausible, at least given Chance Vindication.

To see this, recall the argument for Convexity. That argument rested on the claim that, given Joyce’s Dominance Theorem, if the convex hull of the set of potentially vindicated credal states is closed and bounded, then if an agent has a credal state, Cr(·), that is not in this set it is a priori knowable for the agent that there is some other credal state, Cr′(·), that is closer to being vindicated no matter what the world is like. Given Chance Vindication, however, we do not have parallel reason to endorse T-Convexity. For, given Chance Vindication, it does not follow from the Dominance Theorem that if the convex hull of the set of potentially vindicated credal states at t is closed and bounded, then if an agent located at t does not have a credal state in this set it is a priori knowable for the agent that there is some other credal state, Cr′(·), that is closer to being vindicated no matter what the world is like.

Now it is true that if is closed and bounded, then Joyce’s Dominance Theorem establishes that for any , there is some such that, for every , . But this fact does not ensure that, for an agent located at t whose credal state, Cr(·), is not in , it is a priori knowable that there is some other credal state, Cr′(·), that is closer to being vindicated no matter what the world is like. For, if t ≠ t′, then the set of possible chance functions at t will not be the same as the set of possible chance functions at t′.Footnote ²² And so a credal state may be such that there is some other credal state that is closer to each element of but not closer to each element of . Given Chance Vindication, then, if an agent is located at time t and her credal state is not in but she does not know that she is located at time t, she need not be in a position to tell that there is some other credal state that is guaranteed to be closer to vindication. And so, since one cannot know a priori at which time one is located, it follows that if Chance Vindication is correct, then despite Joyce’s Dominance Theorem an agent whose credal state at t is outside the convex hull of the set of potentially vindicated credal states at t need not be in a position to tell a priori that there is some other credal state that is guaranteed to be closer to vindication.

The lesson to be drawn from this is that if we are going to try to derive some normative principle using Joyce’s Dominance Theorem, then we must show that the constraints imposed by the principle in question hold for every member of the convex hull of some set S such that it is a priori that the vindicated credal state is among S. Given Chance Vindication, however, is the smallest set S such that it is a priori that the vindicated credal state is among S. The problem, though, is that there are, as we have seen, very strict limitations on the sorts of chance propositions that may serve to constrain each member of in the manner in which ur-chance propositions constrain each member of .

There is, though, a way around this problem. We have been assuming, in the preceding discussion, that the objects of credence and chance are eternal propositions, that is, propositions that determine truth-values relative to a world of evaluation. If, however, we allow that the objects of credence and chance are temporal propositions, that is, propositions that determine truth-values relative to a world and time of evaluation, then we can show that there is in fact an interesting class of chance propositions that serves to constrain each member of in the manner in which ur-chance propositions constrain each member of .

Now there is some good motivation for thinking that the objects of credence may vary in truth-value across times. In particular, considerations of cognitive significance may be used to motivate this claim. Thus, for any value t, believing say, that it is raining in San Francisco would seem to be different from believing that it is raining in San Francisco at t. For, even if one is located at time t, if one is unaware of what time it is, the latter proposition would seem to be informative, even given belief in the former. And one very simple way to account for this is to allow that the proposition that it is raining in San Francisco varies in truth-value across times and so is distinct from the proposition that it is raining in San Francisco at t, for every value of t.Footnote ²³

It may be less clear, though, that chance functions will be well defined for essentially temporal propositions. There is, however, a perfectly natural way of extending a chance function defined over eternal propositions to a chance function defined over temporal propositions. For simplicity let us think about propositions as being sets of world-time pairs. An eternal proposition can be represented by a set of world-time pairs ϕ such that, for all w, t, t′, 〈w, t〉 ∈ ϕ if and only if 〈w, t′〉 ∈ ϕ. A temporal proposition, however, will be represented a set of world-time pairs such that, for some w, t, t′, 〈w, t〉 ∈ ϕ and 〈w, t′〉 ∉ ϕ.

Let ch_t(·) be a possible chance function for some time t that is defined for eternal propositions. We now show how ch_t(·) may be extended so that it is defined also for temporal propositions. We assume this:

This principle tells us that, at t, the chance of any temporal proposition is just the chance of its t-rigidification. For example, consider the temporal proposition that it will rain. We can think of this as the set of world-time pairs where at the world parameter it rains at some time later than the time parameter. The t-rigidification of this proposition is the eternal proposition consisting of all worlds where at t it rains later than that time. Chance Rigidity, then, claims that the chance at t of the temporal proposition that it will rain is just the chance at t that it rains at some time later than t. This seems to me to be exactly the right thing to say.

Now consider the following principle:

Now just as it seems quite plausible that , and so , is closed and bounded, so too is it plausible that , and so , is closed and bounded. Given this assumption, then, if we accept Chance Vindication, then it follows from Convexity that an agent should have a credal state that is probabilistically coherent and, in addition, satisfies the constraints imposed by the Temporal Principle.

Earlier we showed that the Principal Principle fails to follow from Chance Vindication by showing that there are distinct possible chance functions that assign the value 1 to the same proposition C_ch. Now, as we have seen, the same is not true for propositions of the form T_ch. Indeed, we have seen that every possible chance function ch(·) is such that if ch(·) = ch′(·), then , and otherwise . A natural question, then, is where the parallel attempt to prove that there are distinct possible chance functions that assign the value 1 to some proposition T_ch breaks down.

To show that there are distinct possible chance functions that assign the value 1 to the same proposition C_ch, we began by considering the ur-chance function for a world w, ch_w(·), and the chance function for that world at time t, , such that . We then noted that is identical to , and so it follows that, while , since and since conditional probabilities preserve certainties, we also have . The corresponding argument for propositions of the form T_ch breaks down at the point where we assume that is identical to . Now it is true that the values that assigns to eternal propositions is the same as the values assigned by . However, given Chance Rigidity, the same will not be true for all temporal propositions. To see this, note that, given Chance Rigidity, there will be temporal propositions ϕ such that ch_w(ϕ) = 1 but . This, however, is precluded on the assumption that is identical to . As an example, consider the temporal proposition . Given Chance Rigidity, we have that , but, assuming that , we will also have that .

Let me close by considering how the Temporal Principle relates to the Principal Principle. We can show, given certain auxiliary constraints, that if an agent’s initial credences satisfy the Principal Principle, then they will also satisfy the Temporal Principle. The reverse, however, is not true. There is a precise sense, then, in which the Principal Principle imposes a stronger normative constraint than the Temporal Principle.

The Temporal Principle, then, provides a weaker normative constraint on our credences than the Principal Principle. It is this weaker principle, though, that follows from Chance Vindication, that is, from the assumption that vindication for credences consists in matching the chances.