Ur-Priors, Conditionalization, and Ur-Prior Conditionalization

3.1. Error Correction: Updating Errors

One complaint about Conditionalization is that it doesn’t properly handle cases in which subjects have updated incorrectly. Consider a subject who has credences cr₀ at t₀, who gets evidence E₁ at t₁, and who updates incorrectly to get credences cr₁(·) ≠ cr₀(· |E₁) at t₁. If she then gets some new evidence E₂ at t₂, what should her credences at t₂ be? Should her t₂ credences be those she’d get by conditionalizing cr₁ on E₂, cr₂(·) = cr₁ (· | E₂)? Or should her t₂ credences be those she’d get by conditionalizing cr₀ on the evidence she’s gotten since t₀, cr₂(·) = cr₀(· | E₁ ∧ E₂)? The first answer seems funny: if a subject makes a mistake while updating her credences, one would think that the right thing to do is to correct the mistake, not to ignore it. The second answer seems better: it tells the subject to correct her error, and adopt the credences she would have adopted if she’d updated correctly all along. So one might like one’s updating rule to tell them that after making an updating error, a subject should adopt the credences that she would have come to have if she hadn’t made that error.

But no understanding of Conditionalization yields this result, though the way in which it fails to yield this result depends on which understanding we adopt. First consider the sequential narrow scope understanding of Conditionalization. This rule tells us that upon receiving new evidence, subjects should adopt credences equal to their credences right before they got their new evidence, conditional on that new evidence. So in the case above, this rule tells us that the subject’s t₂ credences should be those she’d get by conditionalizing cr₁ on E₂, cr₂(·) = cr₁(· | E₂). But this isn’t the prescription we want: for it tells the subject she shouldn’t bother correcting any errors she’s made.

Next consider the interval narrow scope understanding of Conditionalization. This rule says that for any earlier credence function cr, the subject’s new credences cr⁺ should be equal to cr conditional on the cumulative evidence she’s received in the interim. So in the case above, this rule tells us that the subject’s t₂ credences should be equal to both her t₀ credences conditional on E₁ ∧ E₂ and her t₁ credences conditional on E₂. But these prescriptions are inconsistent.

Now consider the wide scope understandings of Conditionalization, whether sequential or interval. These rules take the form O(A → C), or equivalently O(¬A ∨ C), and so only require subjects to satisfy a disjunction. Because of this, these rules aren’t committed to the problematic prescriptions of the narrow scope understandings. For example, instead of requiring the subject to adopt the wrong t₂ credences or requiring her to adopt two incompatible sets of t₂ credences, these rules will merely require her to adopt those t₂ credences or to have had different earlier credences and evidence. But these wide scope understandings still won’t yield the desired results: they don’t tell the subject to correct the errors she’s made. For example, given the sequential wide scope understanding, Conditionalization tells the subject to either not have had credences cr₁ and have gotten evidence E₂, or to adopt new credences cr₂(·) = cr₁(· | E₂). So this rule effectively tells her to either not to have made a mistake, or to adopt new credences which effectively ignore the mistake. Neither option is what we’re looking for: what we’re looking for are prescriptions which tell the subject to adopt t₂ credences which correct the mistake she’s made.[10]

One might try to defend Conditionalization (on any of these understandings) by suggesting that these rules should be thought of as norms that describe the beliefs of ideal subjects. After all, it’s well-known that Conditionalization only makes prescriptions to subjects who are at least somewhat idealized. For example, Conditionalization only makes well-defined prescriptions for subjects with probabilistic credences.[11] With this in mind, one might propose to understand Conditionalization as only making prescriptions to subjects who are ideal in another respect: they have always updated correctly. Or, to put it another way, one might understand Conditionalization as only directly telling us how ideal subjects should update their beliefs, and only indirectly (if at all) telling us anything about how non-ideal subjects should update their beliefs. And since ideal subjects never fail to update correctly, the difficulties described above will never arise.

Working out rules which describe the behavior of ideal subjects is an interesting and worthwhile project. And it’s reasonable to understand Conditionalization as such a rule.[12] But even granting this, one might prefer a rule that requires less idealization and makes plausible prescriptions regarding less idealized subjects than Conditionalization does. Indeed, the verdicts described above—that the subject should correct her mistake by adopting cr₂(·) = cr₀(· | E₁ ∧ E₂)—require a rule that applies to less idealized subjects. So insofar as we want a rule which captures these verdicts, we’ll be unhappy with a rule like Conditionalization.

In light of Conditionalization’s inability to yield the desired verdicts in these cases, one might pursue a more radical option: replacing Conditionalization with a rule whose prescriptions aren’t affected by updating errors, such as a version of Ur-Prior Conditionalization. For example, one might adopt a rule whose prescriptions are determined by an array of facts associated with a subject that encode what she should believe given any particular batch of evidence, e.g., some kind of ‘ur-priors’. Such a rule would avoid the above complaint because it would yield the same prescriptions regardless of whether the subject has made any updating errors. And it could yield these prescriptions without weakening the normative scope of the framework.

3.2. Error Correction: Auxiliary Norm Violations

A second complaint about Conditionalization is that it doesn’t properly handle cases in which subjects have violated auxiliary Bayesian norms, such as Lewis’s (1980) Principal Principle. For example, consider a subject whose credences at t₀, cr₀, violate the Principal Principle, and who gets some new evidence E₁ at t₁. Conditionalizing on a credence function that violates the Principal Principle will typically yield another credence function that violates the Principal Principle. So what should her credences at t₁ be? Should her t₁ credences be those she’d get by conditionalizing cr₀ on E₁, cr₁(·) = cr₀(· | E₁), even though these credences will typically violate the Principal Principle? Or should she consider what her t₀ credences should have been in order to satisfy the Principal Principle, and then conditionalize those corrected credences on E₁ to generate her t₁ credences?[13] The first answer seems funny: if a subject’s credences violate the Principal Principle, one would think the right thing to do is to correct this violation, not ignore it. The second answer seems better: it tells the subject to correct her error, and adopt the credences she would have adopted if she’d successfully abided by the Principal Principle all along. So one might like one’s updating rule to tell them that, given an auxiliary norm violation, a subject should adopt the credences that she would have come to have if she hadn’t violated those auxiliary norms.

Conditionalization won’t yield that result, though the way in which it fails to do so depends on which understanding of the rule we adopt. First, consider the narrow scope understandings, whether sequential or interval. Given these understandings, Conditionalization and the Principal Principle will yield conflicting prescriptions. Conditionalization will tell the subject in the case above to adopt the t₁ credences she’d get by conditionalizing her t₀ credences on E₁. But the Principal Principle will tell her not to adopt those credences, since they violate the Principal Principle.

Next, consider the wide scope understandings, whether sequential or interval. Unlike the narrow scope understandings of the rule, the wide scope understandings won’t yield prescriptions which directly conflict with the Principal Principle. For instead of requiring the subject above to adopt the Principal Principle-violating t₁ credences, they’ll merely require her to adopt those t₁ credences or to have had different earlier credences and evidence. But these wide scope understandings don’t yield the desired prescriptions. For these rules effectively tell the subject to either adopt new credences which effectively ignore the mistake, or to not have made a mistake. Neither option is what we’re looking for. What we want is a rule which, holding fixed the mistakes the subject made in the past, prescribes future credences which correct those mistakes.

(As before, one might try to defend Conditionalization by urging us to think of the rule as a description of how ideal subjects update their beliefs. Since ideal subjects will never violate auxiliary norms like the Principal Principle, the worries discussed above won’t arise. But again, while this is a reasonable way to think of Conditionalization, one might prefer a rule that requires less idealization, and makes plausible prescriptions regarding less idealized subjects. And insofar as we want a rule which captures the verdicts described above, we’ll be unhappy with a rule like Conditionalization.)

Given Conditionalization’s inability to yield the desired results in these cases, one might want to replace Conditionalization with a rule whose prescriptions aren’t affected by the mistakes a subject made in the past, such as a version of Ur-Prior Conditionalization. If we adopt a rule whose prescriptions are determined by some ‘ur-priors’ which tell us what credences a subject ought to adopt (all things considered) given an arbitrary batch of evidence, then we can avoid these problems. Since the prescriptions of auxiliary norms like the Principal Principle would already be built into these facts, the prescriptions of this rule and the auxiliary norms will never conflict.

3.3. Changing Desiderata of Theory Choice

A third complaint about Conditionalization is that it doesn’t provide a satisfactory way of handling changing desiderata of theory choice.[14] Consider two theories regarding a certain phenomenon, T1 and T2. Suppose none of the available evidence distinguishes between these two theories.[15] And suppose that while T1 is better than T2 with respect to some theoretical virtues (e.g., simplicity), it is worse with respect to others (e.g., unification). It seems like reasonable subjects could disagree as to which of these two theories is more plausible.[16] Which theory a subject finds more plausible hangs on the relative importance she attaches to different desiderata of theory choice. And one reasonable subject could take simplicity or unification to be more important than another.

Sometimes subjects change their attitudes about the relative importance of the desiderata of theory choice. They come to think that certain desiderata of theory choice are more or less important than they were initially inclined to think. And one might think that these changes needn’t be irrational. If so, it seems reasonable for a subject who undergoes such a change to readjust their credences in theories like T1 and T2. But Conditionalization won’t allow for such changes. These are changes in one’s credences without a corresponding change in one’s evidence, and this is something that Conditionalization forbids.

Now, for the most part, that’s surely the right verdict—changes in one’s credences without changes in one’s evidence are usually irrational. But one might think there are exceptions, like the one just described above.[17] And insofar as one thinks such changes can be rational, one has a reason to be dissatisfied with Conditionalization.

In light of this worry, one might be motivated to replace Conditionalization with a version of Ur-Prior Conditionalization. In particular, one might adopt a rule according to which the credences one ought to have is determined by something like (i) one’s evidence, and (ii) what hypotheses one takes to be inherently plausible, evidence aside. Such a rule would allow for the kinds of Conditionalization-violating belief changes described above, where belief changes result from changes in what the subject finds inherently plausible. But it would still make it the case that most evidence-less belief changes are impermissible, since most evidence-less belief changes won’t reflect changes in what the subject finds inherently plausible.

3.4. Evidence Loss

A fourth complaint about Conditionalization is that it doesn’t adequately handle cases of evidence loss, like memory loss.[18] Suppose you know a fair coin toss is going to take place, and you have a credence of 0.5 that it will land heads. Then you learn that the coin toss landed heads, and you increase your credence in heads to 1. Finally, you suffer from memory loss, and you forget the outcome of the coin toss. What should your credence now be that the coin landed heads?

It seems plausible that your credence in heads should no longer be 1. After all, you don’t remember how the coin landed, and so it seems unreasonable to be certain the coin landed heads. Instead, it seems your credence in heads should return to something like 0.5.

But Conditionalization won’t allow this kind of belief change. Conditionalization makes certainties permanent: if you come to have a credence of 0 or 1 in a proposition, Conditionalization will require you to always have a credence of 0 or 1 in that proposition. Thus, once you’ve adopted a credence of 1 in heads, Conditionalization will require you to continue having a credence of 1 in heads, even if you forget the evidence that justified this certainty in the first place. Insofar as one takes this to be implausible, one has a reason to be dissatisfied with Conditionalization.[19], [20]

One response to this worry is to maintain that Conditionalization is a description of how ideal subjects update their beliefs, and that ideal subjects never lose evidence. If so, then these evidence loss worries are not a problem for Conditionalization. But while it might be reasonable to understand Conditionalization as a rule describing the belief updates of this kind of ideal subject, one might still prefer a rule that requires less idealization, and makes pleasing prescriptions regarding less idealized subjects. And insofar as we want a rule that yields the kinds of plausible verdicts described above, we have a reason to be unhappy with Conditionalization.

Another response is to argue that we should accept Conditionalization’s prescriptions in cases of evidence loss. Meacham (2010) and Schwarz (2012) have both argued that the verdict that Conditionalization’s prescriptions in cases of evidence loss are implausible stems from implicit internalist or evidentialist intuitions. And, they’ve argued, these views are incompatible with diachronic norms, like Conditionalization. So insofar as one takes there to be diachronic norms, one should reject these intuitions, and thus reject these kinds of complaints about Conditionalization.

Those dissatisfied with these responses might concede that this is a worry for Conditionalization. And thus they might take this to be a reason to replace Conditionalization with a rule that can take evidence loss into account, such as a version of Ur-Prior Conditionalization. Forgotten evidence shows up in Conditionalization’s prescriptions for two reasons: (i) Conditionalization updates using the subject’s cumulative evidence, which is not affected by evidence losses, and (ii) Conditionalization updates on a subject’s prior credences, credences which take her prior evidence into account, and thus implicitly imports evidence that may have been lost into its prescriptions. And if one adopts a rule which diverges from Conditionalization in each of these two respects, then one can avoid these worries.[21] First, the rule would update using a subject’s current evidence, instead of her cumulative evidence, which allows it to update using a form of evidence that’s sensitive to evidence loss. Second, the rule would update on something that encodes how a subject ought to respond to evidence without taking any of her prior evidence into account, which ensures that there is no prior evidence being implicitly imported.

3.5. Self-Locating Beliefs

A fifth complaint about Conditionalization, related to the complaint about evidence loss, is that Conditionalization is unable to provide a satisfactory account of how to update self-locating beliefs, such as beliefs about one’s temporal location or identity.[22] Consider a subject who is looking at a clock that she knows to be accurate. The clock currently reads 9 AM, and so the subject adopts a credence of 1 that it’s 9 AM. A minute later the clock reads 9:01 AM. At this point it seems the subject should adopt a credence of 1 that it’s 9:01 AM, and decrease her credence that it’s 9 AM to 0. But, as we’ve seen, Conditionalization can’t allow this kind of belief change. Conditionalization makes certainties permanent, so once the subject has adopted a credence of 1 that it’s 9 AM, it will require her to continue to be certain that it’s 9 AM forever.

One might take this worry to be a reason to replace Conditionalization with a different updating rule. There are a number of different kinds of rules one might employ, but one option is familiar from the preceding sections: one could adopt a version of Ur-Prior Conditionalization. In particular, one could adopt a rule whose prescriptions are determined by some facts (‘ur-priors’) that encode how a subject should respond to evidence, which are independent of her prior credences and evidence. Such a rule would be able to yield the kinds of temporally-shifting prescriptions we want in the clock-watching case described above.[23]

3.6. Learning about New Theories

A sixth complaint about Conditionalization is that it doesn’t provide a satisfactory account of what subjects should believe when they learn about new theories they hadn’t considered before.[24] Let T, U and V be three mutually exclusive theories that could explain some phenomenon. And suppose a subject who satisfies Probabilism has a credence of 0.4 that theory T provides the correct explanation of that phenomenon, and a credence of 0.6 that theory U provides the correct explanation. Finally, suppose the subject then discovers that theory V, a theory she was not previously aware of, would also explain the phenomenon. If V is a plausible and attractive theory, it would seem reasonable for her to now assign some non-zero credence to V. But Conditionalization won’t allow this. Since the subject’s prior credence in T ∨ U was 1, Conditionalization will require her to continue to be certain that one of these two theories is correct. And since V is incompatible with T and U, it follows that her credence in V must be 0.

One response to this complaint is to again appeal to the idea that Conditionalization is a description of how ideal subjects update their beliefs. If ideal subjects are aware of every possible theory, then the worry that they could discover some new theory won’t arise. Of course, it’s not clear how the subject described in the case above fails to be ideal. Being unaware of certain theories doesn’t make one violate Probabilism or Conditionalization. But one could, perhaps, appeal to some further auxiliary norm—perhaps a norm that requires subjects to be ‘modally omniscient’—in order to maintain that subjects who are unaware of certain theories fail to be ideal.

In any case, if we adopt this response, then cases in which subjects learn about new theories become cases of error correction. And as we’ve seen, Conditionalization (arguably) fails to provide a satisfactory account of error correction. If that’s right, then this response does little to vindicate Conditionalization.

A second response is to embrace the thought that cases in which subjects learn about new theories should be treated as cases of error correction, and appeal to a version of Ur-Prior Conditionalization to deal with cases of error correction, in the manner described earlier. One might go about this in one of two ways. First, one might take there to be auxiliary norms which require rational subjects to be aware of every possibility, and take cases in which subjects learn about new theories to be cases in which they violated this norm. That would make these cases instances of the kind of auxiliary norm error correction discussed in Section 3.2. Second, one might take there to be some array of facts (‘ur-priors’) that encode how a subject should respond to evidence, and take cases in which subjects learn about new theories to be cases where subjects haven’t been generating their credences from these facts in the right way. That would make these cases instances of the kind of updating error correction discussed in Section 3.1 (though in this case the updating rule being violated is a version of Ur-Prior Conditionalization, not Conditionalization).

That said, this response does face one potential demerit. Some will want an account that not only tells a subject what credences to adopt when she learns about new theories, but allows subjects who do so to be rationally blameless. That is, some will want an account on which the subject described above never does anything wrong, because both her initial certainty in T ∨ U and her subsequent uncertainty in T ∨ U are rational. And this second response—which reduces cases of learning about new theories to cases of error correction—won’t yield this result.

A third response is to treat learning about new theories as an expansion of a subject’s conceptual space, and supplement Conditionalization, which tells you how to update when you get evidence, with a second rule, which tells you how to update when you change your conceptual space. On this approach, we wouldn’t take the subject described above to have a prior credence of 0 in V. Instead, we’d take the subject to not have a prior credence in V; before she learned about V, V wasn’t a part of the possibility space over which her credences were defined.[25]

A fourth response is to again handle these cases by appealing to Ur-Prior Conditionalization, but not by treating these as cases of error correction. Instead, one might treat these as cases where learning about new theories changes one’s ur-priors. This way of treating these cases would allow subjects to change their beliefs upon learning about new theories in a way that doesn’t imply any rational deficiency on their part.

3.7. The Problem of Old Evidence and Confirmation

A seventh complaint about Conditionalization is that it’s subject to the problem of old evidence.[26] The literature on this issue has recognized at least two distinct ‘problems of old evidence’—a synchronic problem and a diachronic problem.[27] And only the synchronic problem of old evidence raises a new complaint about Conditionalization. But it’s worth briefly discussing both problems.

The diachronic version of the worry is that Conditionalization won’t yield the desired prescriptions in cases where you learn that some old evidence—that is, evidence you’ve already received—is entailed by a theory. So, for example, suppose you have probabilistic credences, and you learn that some theory H (such as the General Theory of Relativity) entails some already observed behavior E (e.g., Mercury’s perihelion).[28] Since E has already been observed, your credence in E is 1, i.e., cr(E) = 1. Thus discovering that H entails E can’t raise your credence in H, according to Conditionalization, since if cr(E) = 1 then cr^+E(H) = cr(H | E) = cr(H). But it seems like discovering that H entails E should raise your credence in H. Certainly, Einstein took the discovery that the General Theory of Relativity explains Mercury’s perihelion—behavior that was well-known to the scientific community—to be a powerful mark in its favor.[29]

There are a couple of ways to flesh out the diachronic version of the worry, but none of them yield new complaints about Conditionalization. If we understand the case as one where you didn’t realize that H entails E because theory H hadn’t occurred to you yet, we get an instance of the “learning about new theories” worry discussed in Section 3.6. If we understand the case as one where you failed to notice the relationship between H and E when you originally received E, and so you failed to update in the appropriate way, we get an instance of the updating error correction worry discussed in Section 3.1. A third way to flesh out the diachronic worry is as a complaint about the logical omniscience requirement, something entailed by typical formulations of Probabilism. But this is a complaint about Probabilism, not Conditionalization. Since none of these ways of understanding the diachronic worry raise new concerns for Conditionalization, I’ll put the diachronic worry aside.

The synchronic version of the worry is that Conditionalization, when paired with a natural claim about the relation between updating and confirmation or evidential support, will yield unsatisfactory verdicts. Suppose we want to assess whether evidence E supports theory H. A natural thought is to look at whether the credence in H a given subject would adopt after receiving E is greater than her current credence in H. That is, it’s natural to maintain that, for a subject with credences cr, E confirms H (or E supports H) iff cr^+E(H) > cr(H). But if we adopt Conditionalization, then this account of confirmation seems to yield the wrong results in cases in which E is old evidence. If a subject already has E as evidence, then cr(E) = 1, and thus cr^+E(H) = cr(H | E) = cr(H). So it would seem to follow that no evidence a subject already has can support a hypothesis, which seems incorrect. This is the synchronic problem of old evidence: given Conditionalization and a natural claim about the relation between updating and evidential support, how do we make sense of the fact that evidence can still support a hypothesis after it’s been updated on?[30]

One response to this worry is to argue that we should characterize confirmation using the counterfactual credences a subject would adopt if she did not have E as evidence (call this cr_{cf (¬E)}) instead of her actual credences, cr.[31] So, on this account, E supports H for a subject iff \(cr_{cf(\neg E)}^{+E}(H) > cr_{cf(\neg E)}(H)\). This response faces a number of obstacles, however.[32] For one, cr_{cf (¬E)} seems underdetermined, since there are a number of different ways to evaluate this counterfactual. For another, there are natural ways of evaluating such counterfactuals that seem to yield the wrong results. To borrow an example from Maher (1996), consider an author who knows her work is popular, and who believes that the popularity of her work supports the claim that her work is important, but who would not believe that popularity is evidence for importance if she did not know her work was popular. The counterfactual account entails that, given the author’s beliefs, popularity is not evidence for importance. But this is the wrong result.

Another response is to argue that we should assess claims about what E supports using the subject’s credences before they got E as evidence.[33] Doing so would allow old evidence to support hypotheses, since a subject’s credences cr^∗ before they got E as evidence could be such that cr^∗+E(H) > cr^∗(H). But this proposal also faces worries.[34] For one, this proposal entails that the order in which a subject receives evidence can bear on what hypotheses that evidence supports. Consider the hypothesis H = E ∨ F, and consider a subject who initially has some middling credence in H. If that subject receives E and then F as evidence, this proposal will entail that E but not F supports H. Whereas if the subject receives F and then E as evidence, this proposal will entail that F but not E supports H. But these seem like the wrong results: both E and F should support H, regardless of the order in which they were received. Another worry for this proposal is that questions of support often seem to presuppose some subset of the subject’s evidence, with the evidence being presupposed varying from context to context. And since this proposal makes no appeal to context, it can’t explain this phenomenon.[35] A final worry for this proposal is that it can’t handle subjects who come into existence with non-trivial (i.e., non-tautological) evidence. Since there is no credence function the subject had before she received such evidence, this proposal can’t assess what such evidence supports. And there are a number of accounts of evidence which allow subjects to come into existence with non-trivial evidence. For example, the popular account endorsed by Howson and Urbach (1993), in which a subject’s evidence is the logically strongest proposition in which she has a credence of 1, allows subjects to come into existence with non-trivial evidence. Likewise, the account sketched by Lewis (2010), in which a subject’s evidence corresponds to the set of possibilities in which her perceptual experience and memory are the same as they actually are, allows subjects to come into existence with non-trivial evidence.

A third response follows the second in appealing to historical credences, but does so in a more nuanced manner. Following Eells and Fitelson (2000), one might argue that we should assess claims about whether E supports H using a credence function that we construct in the following way.[36] First, we select some credence function cr₀ the subject had before she received any evidence relevant to the hypothesis H.[37] Second, take the context in which the question of confirmation is being assessed to pick out some (possibly empty) subset B of the evidence relevant to H that the subject has received—this is the background evidence that the question is presupposing—and update cr₀ on B. Then E supports H (for this subject, in this context) iff \(cr_0^{+B\wedge E}(H) > cr_0^{+B}(H)\).[38]

This response almost gets us what we want; it avoids the order-dependence and contextual worries facing the second response, and it yields plausible verdicts in almost every case. But unfortunately it’s still subject to a version of the third worry facing the second response. Namely, if a subject came into existence with some evidence relevant to H, then there will be no pre-relevant-evidence credence function cr₀ for us to use to assess claims regarding what evidence supports H. So, like the second response, it has difficulties handling old evidence if a subject has had such evidence from the moment she came into existence.

These difficulties might motivate a different kind of reply to these worries. One might adopt something like the third response, but make one key change: replace the appeal to cr₀—some early credence function the subject has before she’s received any evidence relevant to the hypothesis—with an appeal to some ‘ur-prior’ function up that encodes how a subject ought to respond to evidence.[39] Thus, on this account, E supports H (for a subject, in a context) iff up^+B∧E(H) > up^+B(H). Now, something like up^+B isn’t well-defined given Conditionalization. Conditionalization tells us how to go from a credence function the subject has at one time, and the evidence she’s received since then, to the credence function she should adopt now. But it doesn’t know what to do if you give it some up function that doesn’t correspond to any credence function the subject has ever actually had. So to finish off this proposal, we need to replace Conditionalization with a rule that can make sense of updating on up, i.e., a version of Ur-Prior Conditionalization. Once we do this, we have a proposal that yields the desired verdicts regarding confirmation.[40]

3.8. Epistemic Continuity

A final complaint about Conditionalization is that it relies on some notion of personal identity or “epistemic continuity”.[41] Conditionalization tells us what a subject’s current credences should be given what her prior credences were. This requires a way of determining which time slices of doxastic subjects are time slices of the same subject. One might appeal to the ordinary notion of personal identity here, or (following Meacham 2010) one might appeal to an epistemic continuity relation which tracks the notion of ‘same subject’ that’s relevant to diachronic credence constraints like Conditionalization. But either way, providing a precise characterization of Conditionalization requires spelling out what the relevant relation is like.

Meacham (2010) takes considerations regarding epistemic continuity to motivate modifying Conditionalization. For example, as standardly presented, Conditionalization can’t handle cases of fusion—cases in which one person stage is the successor of two person stages with different credences. Thus he recommends replacing Conditionalization with a different kind of diachronic credence constraint.

Hedden (2015), however, has recently appealed to considerations regarding epistemic continuity to argue for a more radical conclusion: that we should reject any kind of diachronic credence constraint.[42] Hedden argues that we should be skeptical of the possibility of providing a satisfactory account of epistemic continuity. And since any diachronic credence constraint like Conditionalization has to appeal to something like epistemic continuity, it follows that we should be skeptical of any rule of this kind.

One response to Hedden’s worry is to meet the challenge head on, and to develop an account of epistemic continuity to pair with diachronic credence constraints. But those persuaded by Hedden’s arguments are unlikely to be satisfied with this response. A more radical response is to follow Hedden and give up on diachronic credence constraints. The most natural way to do this is to replace Conditionalization with a synchronic rule. For example, one might adopt a version of Ur-Prior Conditionalization whose prescriptions are determined wholly by features of the subject’s current state, like her current evidence and what hypotheses she currently finds inherently plausible. Such a rule won’t require a notion of epistemic continuity. And thus it will avoid the sorts of worries Hedden raises.

4.1. Initial Credence Versions

The first two versions of Ur-Prior Conditionalization I’ll consider take a subject’s ur-priors to be her initial credences.[44]

Let’s start with a normatively bare initial credence account, which imposes no further constraints on what one’s initial credences must be like. To formulate the account precisely, we need to introduce some vocabulary.

First, I’ll employ some restricted variables. I’ll take the variable s to range over doxastic subjects, the variable t to range over times, the variable f to range over functions from (at least some) propositions to real numbers, and the variable A to range over propositions.

Second, we need an obligation operator, O. Following various authors in the literature, I’ll take this operator to be subject and time indexed, allowing obligations to vary with subject and time.[45] So O_s,t(A) holds iff for s, at t, it’s obligatory that A. In some cases, when one or both of the indices don’t appear in the scope of the operator and don’t play a substantive role in characterizing the content of the norm, I’ll simplify by omitting the index and the universal quantifier ranging over it. Thus, for example, I might write \(\forall s \, O_s(\exists t \, B(s,t))\) as shorthand for \(\forall s \, \forall t^\prime \, O_{s,t^\prime}(\exists t \,B(s,t))\).

Third, we need a predicate which characterizes a subject’s evidence. Let the evidence predicate E(s, t, A) represent a 3-place relation between a subject s, a time t, and a proposition A. E(s, t, A) holds iff A is the content of subject s’s evidence at t.

Fourth, we need a predicate which characterizes a subject’s initial credences. Let the initial credence predicate IC(s, f ) to represent a 2-place relation between a subject s, and a function f . IC(s, f ) holds iff f is s’s initial credence function.

Fifth, we need a predicate that tracks whether conditionalizing a subject’s credence on their evidence yields a well-defined function. Let the well-defined predicate W( f, A) represent a 2-place relation between a function f and a proposition A. W( f, A) holds iff f(· | A) is well-defined.[46]

Sixth, we need a predicate which characterizes a subject’s credences. Let the credence predicate C(s, t, f) represent a 3-place relation between a subject s, a time t and a function f . C(s, t, f) holds iff f is s’s credence function at t.

With this in hand, we can formulate the bare initial credence version of Ur-Prior Conditionalization as follows:[47], [48]

Roughly, this rule says that it ought to be the case that if a subject’s evidence is A, her initial credences were f, and f(· | A) is well-defined, then her credences are f(· | A).[49]

Now let’s look at a normatively loaded initial credence account, which does impose further constraints on what one’s initial credences can be like. Let ∀_R and ∃_R be restricted quantifiers that range over only the rational fs; that is, only those fs which, when plugged into the rule, yield credences that satisfy whatever further normative constraints one takes to constrain belief (e.g., the Principal Principle, Reflection, etc.). At a first pass, we might try to formulate the loaded initial credence version of Ur-Prior Conditionalization by taking the bare initial credence version and replacing ∀f with ∀Rf, as follows:

But this rule won’t deliver the desired prescriptions to subjects whose initial credences are irrational. Subjects with irrational initial credences will automatically satisfy this norm—for such subjects, IC(s, f) won’t hold for any rational f, making the antecedent of the conditional false, and thus the conditional true. And we’d like the rule to tell irrational subjects to adopt credences which correct their errors, not to tell them they don’t need to do anything at all (because they automatically satisfy the norm).

At a second pass, one might instead require subjects to just update in accordance with some rational function; that is, one might adopt a rule like this:

But on this version of the rule the f′ the subject updates on needn’t bear any resemblance to the f representing her initial credences; it’s as viable for her to use a rational function which is radically different from her initial credences as one which is a slight correction of her initial credences.

In order to get a normatively loaded account in which a subject’s initial credences play a role, we need the rule to prescribe updates based on the subject’s initial credences. That is, we need the rule to require subjects to update using something like the closest rational approximation, or closest rational fit, to their initial credences (or one of the closest rational fits, if there are ties). As it turns out, formulating such a rule in a satisfactory manner is difficult. This is because of what I’ll call the Quantifier Problem: there are three compelling desiderata regarding where to place the quantifiers when formulating the rule that are jointly unsatisfiable. Thus every way of formulating such a rule will incur some costs. In the following section, I’ll look more carefully at how the Quantifier Problem arises, and what the options for addressing it are.

4.2. Functional Versions

Next, let’s consider a pair of accounts which adopt a “functional role” understanding of ur-priors.[54] In some ways this is the trickiest understanding of ur-priors, so let’s start by getting a feel for the difference between functional and non-functional understandings.

One could understand Ur-Prior Conditionalization’s appeal to ur-priors in two kinds of ways. First, one could take ‘ur-priors’ to pick out some particular feature of the subject—say, her initial credences. So, on this understanding of the Ur-Prior Conditionalization, the norm requires a subject’s credences to equal her initial credences conditional on her evidence. Alternatively, one could understand the norm’s appeal to ‘ur-priors’ functionally. On this understanding, the norm just requires that there exist some function—call it the ur-prior function—that plays a certain role with respect to a subject’s beliefs. In particular, the norm requires that there exist some function, up, such that her credences at every time are equal to up conditional on her evidence at that time.

On this functional understanding of the norm, there isn’t some particular ‘ur-prior’ function that a subject is required to line up her beliefs with. Rather, it’s just required that there exist some function that her beliefs line up with at every time. And the term ‘ur-priors’ picks out any function that plays this role. (So a subject may have no ur-prior function, if no function lines up with her beliefs over time in the right way. And a subject may have many ur-prior functions, if many functions line up with her beliefs over time in the right way.[55] )

Let’s look at how to formulate this kind of functional understanding of Ur-Prior Conditionalization more precisely. Let’s start with a normatively bare functional account, which imposes no further normative constraints on what up must be like. We can formulate the bare functional version of Ur-Prior Conditionalization as follows:

Roughly, this rule says that for each subject, there ought to be a function f such that whenever her evidence is A, she has (well-defined) credences f(· | A).[56]

This version of Ur-Prior Conditionalization is normatively bare, in that it doesn’t impose any further normative constraints on f other than that it be such that it can line up with the subject’s credences and evidence at various times. Now let’s turn to look at a normatively loaded version of the functional account, one which imposes further constraints on what the f the subject is told to update on should be like. Imposing such constraints is much easier here than it was for the initial credence accounts. Here all we have to do is require f to be rational, i.e., replace ∀f with ∀_Rf. Thus we can formulate a loaded functional version of Ur-Prior Conditionalization as follows:

Roughly, this rule says that for each subject, there ought to be a rational function f such that whenever her evidence is A, she has (well-defined) credences f(· | A).

With respect to the commitments they incur, these functional versions of Ur-Prior Conditionalization are the most attractive of the six versions we’ll consider. For these functional accounts have no more commitments than the standard Bayesian framework. They require subjects to have credences, and to have evidence—albeit evidence at a time instead of cumulative evidence—but these kinds of commitments are familiar in the standard Bayesian framework. The loaded version appeals to some further normative constraints on credences, but these constraints are again familiar to the Bayesian. So in this respect, these functional accounts offer an appealing way to understand Ur-Prior Conditionalization.

4.3. Evidential Standards Versions

Finally, let’s consider a pair of accounts which take a subject’s ur-priors to be what I’ll call evidential standards.[57] People have used the term ‘evidential standards’ to pick out different kinds of things.[58] So the ‘evidential standards’ understanding of ur-priors isn’t a specific proposal, but rather a cluster of related proposals, with the following common feature: on all of these views, a subject’s evidential standards supervene on her mental state at a time. Thus, in contrast to the functional and initial credence understandings of ur-priors, evidential standards are the kinds of things which can (potentially) change over time.

One can develop this kind of proposal in a number of ways. One rough thought is that evidential standards represent what a subject finds inherently plausible, independently of any evidence she’s received. Another rough thought is that evidential standards should be understood as analogous to moral standards— something which the subject is, in some sense, committed to or an advocate of.[59] And each of these rough thoughts is compatible with a number of different concrete proposals. For example, one might take evidential standards to be determined by a subject’s dispositions to adopt new beliefs, a subject’s beliefs about what beliefs she should adopt, a subject’s intentions about what beliefs to adopt, some mixture of these factors, or even some primitive propositional attitude. We’ll return to look at some of these proposals more carefully below.

In order to formulate such rules, we need to introduce a predicate which characterizes a subject’s evidential standards. I’ll take ES(s, t, f) to represent a 3-place relation between a subject s, a time t, and a function f from propositions to real numbers. (Unlike IC, ES is relativized to times, since evidential standards, like any other feature of a subject’s current psychological state, might change over time.) ES(s, t, f) holds iff f correctly represents the relevant features of s’s mental state at t.

With this in hand, we can formulate the desired evidential standards versions of Ur-Prior Conditionalization by simply taking the corresponding initial credence formulation of the rule and substituting ES(s, t, f) for IC(s, f). This will yield a normatively bare evidential standards account similar to the bare initial credence account from Section 4.1:

And it will yield three versions of the loaded evidential standards account that mirror the three versions of the loaded initial credences account. As before, I’ll restrict my attention to the first and the third versions of this account in what follows:

The (v1) formulation says, roughly, that at every time there ought to be some best rational fit f′ to a subject’s evidential standards such that, if her evidence at that time is A and f′(· | A) is well-defined, then her credences at that time are f′(· | A). The (v3) formulation says, roughly, that there exists some best rational fit f′ to a subject’s evidential standards such that, at all times, it ought to be the case that if her evidence at that time is A and f′(· | A) is well-defined, then her credences at that time are f′(· | A).

As we saw above, there are a number of different ways to cash out evidential standards. Let’s briefly look at some of these possibilities, before turning to look at some worries facing evidential standards accounts.

One possibility is that evidential standards encode a subject’s doxastic dispositions. In particular, a subject’s evidential standards might represent the credences the subject is disposed to adopt given various batches of evidence (those credences being equal to her evidential standards conditional on that evidence). This understanding of evidential standards trivializes Ur-Prior Conditionalization if we adopt a normatively bare account; for then the rule merely says that subjects should adopt the credences they’re disposed to adopt. But the rule remains non-trivial if we adopt a normatively loaded account, on which the rule says that subjects should adopt the credences suggested by the closest rational fit to their doxastic dispositions.[60]

Another possibility is that a subject’s evidential standards encode a subject’s normative beliefs.[61] In particular, a subject’s evidential standards might represent the credences she believes she should adopt given various batches of evidence (where her evidential standards conditional on the evidence yield those credences). Or, if a subject is unsure about what her credences should be given some evidence E, we can take the expectation of the credences the subject believes she should adopt given E, and require her evidential standards to yield that expectation when conditionalized on E.[62]

A third possibility is that a subject’s evidential standards encode her doxastic intentions. In particular, a subject’s evidential standards might represent the credences the subject intends to adopt given various batches of evidence (those credences being equal to her evidential standards conditional on that evidence). There are also some more open-ended possibilities. For example, one might take a subject’s evidential standards to encode some mixture of the first three possibilities. Or one might take a subject’s evidential standards to represent some primitive propositional attitude that’s not characterizable in terms of a subject’s other attitudes.

Now let’s turn to look at some demerits of these evidential standards versions of Ur-Prior Conditionalization. These evidential standards accounts employ a more substantive notion of ur-priors than the functional or initial credence versions. As we’ll see in Section 5, this allows them to handle the broadest range of complaints about Conditionalization. But it also leaves them open to a broader range of worries. I’ll flag five worries for these accounts here: one which applies equally to both bare and loaded accounts, two which only apply to the bare account, and two which only apply to the loaded account.

The first worry, which applies to both the bare and loaded accounts, arises from the fact that evidential standards can change over time. Ur-Prior Conditionalization is supposed to tell us how a subject should change her beliefs over time. And if evidential standards can change, then part of that story is telling us how a subject’s evidential standards should change over time. But the story provided here doesn’t say anything about the norms that govern evidential standards changes.[63]

The most straightforward response to this worry is to reject the claim that there are further norms governing changes in evidential standards. For one may find it plausible that one’s credences should line up with one’s evidential standards, and (on a loaded approach) that there are synchronic constraints on these evidential standards, but yet not be pulled by the thought that there are further diachronic constraints on these evidential standards.[64] A more concessive response is to grant that there may be further norms governing evidential standards changes, but to note that this doesn’t take away from the substance of the evidential standards versions of Ur-Prior Conditionalization. We can get a reasonable grasp on what evidential standards are, and these updating rules tie a subject’s credences to her evidential standards. The fact that there may be further parts to the story—stories regarding how her evidential standards change—doesn’t show that there’s anything wrong with this part of the story.

The second worry, specific to the bare account, is that it won’t yield substantive prescriptions for doxastic subjects who don’t have evidential standards.[65] And given some understandings of evidential standards (e.g., as representing a subject’s normative beliefs), this seems like a real possibility (e.g., there can be subjects who don’t have normative beliefs of this kind). One reply would be to argue that this result is unproblematic, because subjects without evidential standards plausibly shouldn’t be bound by updating rules like Ur-Prior Conditionalization. The viability of this reply depends on the notion of evidential standards one adopts. E.g., suppose one took evidential standards to be some primitive propositional attitude with no clear relation to belief. It would seem prima facie strange to maintain that the beliefs of some subject shouldn’t be constrained by updating rules because, though she has beliefs, she doesn’t have this belief-independent primitive propositional attitude. (Just as it would be strange to say that a subject’s beliefs shouldn’t have to satisfy Conditionalization because she doesn’t have some prima facie unrelated propositional attitude, like fears. Presumably, whether a subject has fears or not has no bearing on whether we should apply rational constraints to her beliefs!) A second reply would be to argue that doxastic subjects must have evidential standards. Again, the plausibility of this reply depends on the notion of evidential standards one adopts. E.g., it may be tenable to maintain that doxastic subjects must have doxastic dispositions, but it’s far less plausible to maintain that doxastic subjects must have doxastic intentions. A third reply would be to concede that subjects without evidential standards aren’t bound by Ur-Prior Conditionalization, but are bound by some ‘back-up’ rule instead—perhaps some functional version of Ur-Prior Conditionalization. But it seems ad hoc to bring in a different updating rule for such subjects.

A third (and related) worry, specific to the bare account, is that it won’t make substantive prescriptions to subjects whose evidential standards aren’t probabilistic.[66] For example, a subject’s normative beliefs might just be a grab bag of rules encoding what sorts of things the subject thinks she should believe, something much less detailed than a probability function. The three potential replies to the first worry are also available here—one might maintain that this result is unproblematic, maintain that evidential standards must be probabilistic, or adopt a back-up rule for subjects with non-probabilistic evidential standards. And, as above, the plausibility of these replies depends on the notion of evidential standards we adopt.

A fourth worry, specific to the loaded account, is that, as it stands, the account described here is incomplete. For a complete description of this account would need to include a metric for assessing how close a fit one function is to another which we could use to assess what the best rational fits to a subject’s evidential standards are.

A fifth worry, specific to the loaded account, is the Quantifier Problem. As we saw in Section 4.1, it seems that all of the natural ways of formulating this rule have some undesirable features. They either won’t yield the kind of diachronic coherence we’d like (in cases of ties), won’t allow for updating error corrections, or will require us to privilege one best rational fit over the others (in cases of ties).

5.1. Error Correction: Updating Errors

One complaint about Conditionalization was that it can’t properly handle cases in which subjects have updated incorrectly. We’d like an updating rule’s prescriptions to lead subjects to correct previous updating errors. And Conditionalization doesn’t do this.

The functional versions of Ur-Prior Conditionalization are subject to the same complaint. These versions of the rule require a subject’s credences at all times to line up with some f . And if she adopts credences at a pair of times which don’t line up with some f, then these rule’s prescriptions become impossible to satisfy—there’s no credence function she can now adopt which will allow her to satisfy these rules. Thus these functional versions of the rule won’t make the desired prescriptions to subjects who have made updating errors.

The initial credence and evidential standards versions of Ur-Prior Conditionalization, on the other hand, are not subject to this complaint. These norms tell a subject that her current credences should be equal to her ur-prior (or some best rational fit) conditional on her current evidence. Thus they direct her to correct her errors—to adopt the same credences as she would have had if she hadn’t made any updating errors.

5.2. Error Correction: Auxiliary Norm Violations

Another complaint about Conditionalization is that it can’t adequately handle cases in which subjects have violated auxiliary Bayesian norms, like the Principal Principle. We’d like an updating rule’s prescriptions to lead subjects to correct violations of auxiliary norms, or at least to not conflict with such corrections. But Conditionalization will typically direct a subject whose credences violate an auxiliary norm to continue to adopt credences which violate that norm.

The normatively bare versions of Ur-Prior Conditionalization can’t avoid this complaint either. In order to direct subjects to satisfy auxiliary norms, the rule’s prescriptions need to take these auxiliary norms into account. And only the normatively loaded versions of Ur-Prior Conditionalization make prescriptions which take auxiliary norms into account.

But adopting a normatively loaded version of Ur-Prior Conditionalization isn’t enough. Consider the loaded functional account. This rule requires a subject to have credences which line up with some rational f at all times. As a result, if at any time she adopts credences which don’t line up with f —she adopts credences that violate the Principal Principle, for instance—then the rule’s prescriptions become impossible for her to satisfy. So the loaded functional version of the rule won’t yield the desired prescriptions for subjects who have violated auxiliary norms.

By contrast, the loaded initial credence and evidential standards versions of Ur-Prior Conditionalization can avoid this complaint. These rules tell a subject to adopt credences equal to some best rational fit to her ur-priors conditional on her current evidence. Thus these rules effectively tell her to adopt the credences she would have had if she hadn’t violated any auxiliary norms.

5.3. Changing Desiderata of Theory Choice

A third complaint about Conditionalization is that it can’t accommodate cases of rational belief change due to changes in a subject’s assessment of the relative importance of different desiderata of theory choice.

The functional and initial credence versions of Ur-Prior Conditionalization don’t help with this complaint. In order to avoid this complaint, the rule needs to allow the ur-priors that generate the prescriptions to change over time. And on both the functional and initial credence accounts these ur-priors are static. (The functional accounts require a subject’s credences to line up with the same f at all times, and so don’t allow for changing fs. And the initial credence accounts require a subject to line up her credences with her initial credences, which don’t change.)

The evidential standards versions of Ur-Prior Conditionalization, however, can address this complaint. This is because a subject’s evidential standards can change over time. A subject may find certain things inherently plausible which, over time, come to seem less inherently plausible. Thus the evidential standards version of the rule can accommodate this kind of belief change.

5.4. Evidence Loss

A fourth complaint about Conditionalization is that it can’t adequately handle cases of evidence loss. If a subject learns that a fair coin toss landed heads, and then loses this evidence, it seems her credence in heads should revert to 0.5. But Conditionalization won’t allow subjects to go back to their earlier credences in this way.

Every version of Ur-Prior Conditionalization can handle this complaint in most cases. All of these rules tell subjects to set their credences equal to their ur-priors conditional on their current evidence. It follows that if a subject loses evidence, and so comes to have the same evidence as she had at some earlier time, she will be prescribed the same credences as she was at that earlier time.

That said, if one adopts an account of evidence that allows subjects to have (non-trivial) evidence as soon as they come into existence,[67] then there will be some cases of evidence loss which initial credence versions of Ur-Prior Conditionalization can’t handle. Consider a case in which a subject comes into existence with evidence E (e.g., a robot who is turned on for the first time and has E programmed into her memory banks). If she then loses that evidence (has her memory banks deleted), the initial credence versions of the rule will prescribe her the same credences as before. But in many cases this seems like the wrong prescription—it should at least be possible for her credences to reflect this evidence loss in some way.

5.5. Self-Locating Beliefs

A fifth complaint about Conditionalization is that it can’t handle changes in self-locating beliefs, such as beliefs about what time it is.

The initial credence versions of Ur-Prior Conditionalization won’t help with this complaint. Start with the normatively bare initial credence approach. A subject’s initial credences can assign a value of 1 to some self-locating belief, such as that it’s 9 AM. If such a subject gets new evidence that doesn’t rule out the possibility that it’s 9 AM, then the bare initial credence approach will require her to continue to have a credence of 1 that it’s 9 AM. And if a subject gets new evidence that does rule out the possibility that it’s 9 AM, then the bare initial credence approach will fail to impose any constraints on her beliefs at all.[68] Either way, the bare initial credence approach won’t provide a satisfactory account of how the subject should update her self-locating beliefs.

The normatively loaded initial credence approach faces similar problems. It seems an ideally rational subject can have initial credences which assign a value of 1 to some self-locating belief, such as that it’s 9 AM. (E.g., consider a rational robot who’s just been switched on for the first time, and who has been programmed to start with the correct belief about what time it is.) And the loaded initial credence version of Ur-Prior Conditionalization will yield the same implausible prescriptions as the bare initial credence approach given these kinds of initial credences.

Likewise, it’s not clear the bare evidential standards version of Ur-Prior Conditionalization helps with this complaint. After all, a (rather strange) subject might have evidential standards which assign a value of 1 to some self-locating belief, such as that it’s 9 AM. And given such evidential standards, the bare evidential standards approach will run into the same problems as the initial credence approaches.

The loaded evidential standards approach, on the other hand, is better positioned to handle this complaint. For one could plausibly deny that rational evidential standards will (say) assign a value of 1 to the belief that it’s 9 AM. After all, one would not expect a rational robot’s evidential standards—say, her beliefs about what beliefs she ought to have, given various batches of evidence—to require her, given any batch of evidence, to assign a credence of 1 to it being a particular time. For example, one would not expect a rational robot to believe that her credence that it’s 9 AM should remain 1 given the evidence that an accurate clock reads 10 AM. So the loaded evidential standards approach can maintain that rational evidential standards are such that they always yield plausible prescriptions regarding self-locating beliefs.

Finally, both the bare and loaded functional versions of Ur-Prior Conditionalization can handle this complaint. For on the functional accounts, a subject’s credences can still satisfy the rule even if they’re certain they’re watching an accurate clock, and so become certain that it’s t₁ at t₁, and t₂ at t₂. For all the functional accounts require is that there exist some function which, when conditionalized on her evidence at each of those times, yields the appropriate credences. And there will be plenty of functions which satisfy these constraints.[69]

5.6. Learning About New Theories

A sixth complaint about Conditionalization is that it doesn’t provide satisfactory prescriptions in cases in which subjects learn about new theories. In particular, there seem to be cases where a subject who learns about some new theory V should come to assign V a non-zero credence, even though she hasn’t acquired any new evidence. And Conditionalization won’t make such prescriptions.

The functional versions of Ur-Prior Conditionalization can’t handle this complaint. If a subject didn’t assign a non-zero credence to V before, these rules will forbid her from assigning a non-zero credence to V now (given the same evidence). The initial credence versions of Ur-Prior Conditionalization can’t handle this complaint either. For if a subject’s initial credences fail to assign a non-zero credence to V, then these rules will forbid her from assigning a non-zero credence to V at any point in the future.[70]

On the other hand, the evidential standards versions of Ur-Prior Conditionalization can handle this complaint in a couple of ways. One way is to treat this as a case of updating error correction—as a case where, given the subject’s evidential standards, her initial failure to assign V a non-zero credence was irrational. Given an auxiliary norm that (say) requires subjects to be ‘modally omniscient’, one could treat this as a case where the subject needs to correct for past violations of auxiliary norms. And loaded evidential standards accounts will direct such a subject to adopt credences which correct these errors; i.e., to adopt a non-zero credence in V. Alternatively, one could treat this as a case where the subject satisfies all of the relevant auxiliary norms, but has just failed to update correctly, and both bare and loaded evidential standards accounts will direct such a subject to adopt credences which correct these errors. Of course, both of these options entail that only subjects who are have been irrational in some way can learn about new theories.

Another way evidential standards versions of Ur-Prior Conditionalization can handle this complaint is to treat this as a case of changing evidential standards. This allows us to maintain that subjects who learn about new theories needn’t be rationally deficient. Assuming V is compatible with the subject’s prior evidence, a subject’s prior evidential standards could fail to yield a non-zero credence in V for one of two reasons. First, the subject’s evidential standards might have assigned a zero value to V. Second, the subject’s evidential standards might have been defined over an algebra of propositions that doesn’t include V as a member, and so might have failed to assign any value to V. Either way, if we take learning about a new theory like V to correspond to a change in a subject’s evidential standards—perhaps a boost in the value assigned to V, or the addition of V to the algebra of propositions—then we can maintain that a subject who learns about some new theory V and comes to assign it a non-zero credence needn’t be in any way irrational.[71], [72]

5.7. The Problem of Old Evidence and Confirmation

A seventh complaint about Conditionalization is that, when paired with a natural claim about the relation between rational updating and confirmation, it has a hard time making sense of how evidence can still support a hypothesis after it’s been updated on. Some sophisticated accounts of confirmation, such as the one proposed by Eells and Fitelson (2000), can accommodate much of what we’d like to say here, by appealing to credence functions the subject had before she received the relevant evidence E. But these accounts still run into difficulties in cases in which subjects have E from the very start, and so have no ‘pre-E’ credence function to use when assessing claims regarding the evidential relevance of E.

The initial credence versions of Ur-Prior Conditionalization are subject to similar worries. In cases in which subjects have E from the start, E will be encoded in their initial credences. Thus these accounts don’t provide a ‘pre-E’ function to employ to assess claims about confirmation.

For the functional versions of Ur-Prior Conditionalization, scenarios in which subjects start with E aren’t necessarily problematic. Recall that the functional accounts require there to exist some function up such that one’s credences are equal to up conditional on one’s evidence at every time. So suppose there’s a subject who starts with E as evidence, but who later gets amnesia at some time t, loses all of her evidence, and comes to have a credence cr_t(E) = 0.5. In order for cr_t(E) = up(E | ⊤) = 0.5, as the functional accounts require, it must be the case that up(E) = 0.5. And if up(E) = 0.5, then there can be hypotheses that E can confirm (i.e., hypotheses A such that up(A | E) > up(A)).

While the functional accounts don’t have problems providing ‘pre-E’ functions, in some cases they’ll provide too many ‘pre-E’ functions. Consider, for example, a case in which a subject has E and a credence of cr(E) = 1 at all times. In this case, if any function satisfies the constraint the functional accounts impose, then an infinite number of functions will satisfy this constraint—for every x ∈ (0, 1], there will be some function up(E) = x that yields the subject’s credences when conditionalized on her evidence at every time. Thus in these cases there won’t be a fact of the matter as to whether E confirms other hypotheses or not. For there won’t be a unique up function to appeal to—just a set of them that satisfy the relevant constraints—and the up functions in this set won’t agree about whether E can confirm anything.

The evidential standards versions of Ur-Prior Conditionalization are not subject to either of these worries, however. For these accounts do provide a ‘pre-E’ function to employ; namely, the subject’s evidential standards es. So even if a subject starts with E, and has E at all future times, it can still be the case that es(E) ≠ 1, and thus that E can confirm other hypotheses.[73]

5.8. Epistemic Continuity

A final complaint about Conditionalization is that it relies on something like a personal identity or epistemic continuity relation, and one might be skeptical of the prospects of providing a satisfactory account of this relation.

The functional and initial credence versions of Ur-Prior Conditionalization won’t help with this worry, since they both rely on something like personal identity or epistemic continuity. The functional approaches consider whether there exists some function up such that a subject’s credences at different times can be generated by conditionalizing up on her total evidence at that time. And the initial credence approaches consider whether the subject’s credences are equal to her initial credences conditional on her total evidence. In both cases, these approaches require a way of determining whether time slices of subjects at different times belong to the same subject.

The evidential standards versions of Ur-Prior Conditionalization, on the other hand, do help with this worry. The evidential standards approaches consider whether a subject’s credences are equal to her current evidential standards conditional on her total evidence. And this doesn’t require a way of identifying subjects over time. Thus the evidential standards approaches avoid any complaints regarding how to spell out the notion of epistemic continuity.

5.9. Summing Up

Thus each version of Ur-Prior Conditionalization has a different profile with respect to how it handles these complaints about Conditionalization. Putting these results together, we can map the ability of these accounts to handle these complaints in a single chart (see Figure 1).

Figure 1. Ur-Prior Conditionalization: Assessing The Complaints