3.1. The time of evidence question: assessing the answers

Now, let’s turn to assess each of these answers.

The posterior answer, that the subject should adopt some amount of time after receiving E, leads to implausible verdicts. Consider a subject who receives evidence E at . Given the posterior answer the subject shouldn’t adopt until some later time . And, if she adopts at she’s epistemically deficient for having ‘jumped the gun’. But this seems like an odd verdict. After all, a subject who adopts at is in a strictly better position, epistemically speaking, than the subject who doesn’t – she’s taken all of her evidence into account. And, it’s hard to see why it’s epistemically irrational to take all of one’s evidence into account as soon as one has it.Footnote ⁸

The posterior answer is also in tension with the popular account of evidence endorsed by Howson and Urbach (Reference Howson and Urbach1993), where a subject’s evidence is the strongest proposition in which they have a credence of 1. Suppose a subject receives evidence E at . What should her credences be at ? Since she isn’t required to adopt her new credences until , the most natural option is to maintain that her credences at are still cr. But, since cr generally won’t assign 1 to E, this option is incompatible with Howson and Urbach’s picture of evidence. Of course, there are other options one might try, but none of them are satisfactory. For example, one might maintain that at a subject should assign a credence of 1 to E (as Howson and Urbach’s account requires), but should otherwise assign the same values as cr does. But since this credence function will generally violate the probability axioms, this option’s incompatible with Probabilism. Alternatively, one might maintain that at a subject should adopt . But this is to abandon the posterior answer for the concurrent answer, as this entails that one should receive E and adopt simultaneously. So if we want to hold on to the posterior answer, it seems we must say that the subject’s credences at should be cr. And this is incompatible with Howson and Urbach’s account of evidence.

Now consider the concurrent answer, that subjects should adopt at the same time as they receive E. One worry for the concurrent answer is that it places too strong a demand on subjects like us. In particular, one might worry that this answer is incompatible with the principle that ought implies can. If subjects like us aren’t capable of updating instantaneously, then how can we be obligated to do it?

Of course, this is not a fair criticism of the concurrent answer if the posterior answer also has this problem. And, as given, the posterior answer doesn’t take the cognitive capacities of subjects into account either. It states that a subject’s new credences should be adopted some amount of time after the subject gets her evidence, but it doesn’t say anything about what this later time is, or whether it’s possible for the subject to adopt those credences at that time.

But one might naturally think that one can modify the posterior answer so that it does take the cognitive capabilities of subjects into consideration. And if, by modifying the posterior answer, we can get a plausible formulation of Conditionalization that avoids these kinds of ought-implies-can worries, then we have a reason to favor the posterior answer over the concurrent answer. Let’s see whether this is true: by modifying the posterior answer, can we get a plausible version of Conditionalization that avoids these kinds of ought-implies-can worries?

Here’s a natural way to modify the posterior answer to take the updating time lag of cognitively limited subjects into account:

• Answer 1a (Posterior-a). The subject should adopt her new credences as soon as possible after receiving her evidence.

• Conditionalization (v1.1a): If a subject with credences cr gets evidence E at , she should adopt new credences at such that (if defined), where is the earliest time following at which the subject is capable of adopting .

This formulation of Conditionalization is sensitive to the temporal limitations of subjects with limited cognitive capacities. But it yields inconsistent prescriptions. Suppose a subject with prior credences cr gets evidence E at , and evidence F at . And suppose the subject is cognitively limited in such a way as to not be able to update her credences until . But at , she’s capable of changing her credences in any way she likes. Because the subject gets E at , Conditionalization (v1.1a) requires her to adopt new credences equal to at , since is the first time at which she’s capable of updating on E. Likewise, because the subject gets F at , Conditionalization (v1.1a) requires her to adopt new credences equal to at , since her credences at will still be cr, and is the first time at which she’s capable of updating on F. But these two prescriptions will usually be inconsistent.

The problem here is that Conditionalization (v1.1a) doesn’t take into account the possibility that a subject might receive further evidence besides E between and . So, we might repair our formulation of Conditionalization by adding a clause which rules out this possibility:

• Conditionalization (v1.1b): If a subject with credences cr gets evidence E at , she should adopt new credences at such that (if defined), where is the earliest time following at which the subject is capable of adopting , and the subject doesn’t get any other evidence between and .

This rule avoids making inconsistent prescriptions in the kinds of ‘multiple evidence cases’ described above by simply falling silent in such cases. But an adequate formulation of Conditionalization should give us some guidance in these cases – it shouldn’t just fall silent. So this rule is too weak.

We can get around this problem by adding a further clause which tells us what credences to adopt in these cases:

• Conditionalization (v1.1c): If a subject with credences cr gets evidence E at , she should adopt new credences at such that (if defined), where is the earliest time following at which the subject is capable of adopting , and assuming she hasn’t gotten any other evidence between and . If she has gotten other evidence, she should update on the conjunction of all of the evidence she’s received up to ; i.e. she should adopt credences .

This formulation avoids the inconsistent prescriptions worry, and offers prescriptions in multiple evidence cases. But it avoids these worries by sliding back toward the concurrent answer to the time of evidence question. For, like the concurrent answer, Conditionalization (v1.1c) effectively takes into account all of the evidence the subject receives up until the time at which she should adopt her new credences. And in doing so, Conditionalization (v1.1c) runs afoul of the same kinds of ought-implies-can worries that prompted these modifications of Conditionalization in the first place. Consider a multiple evidence case like the one described above, but with the following modification: at , the subject isn’t capable of changing her credences in any way one likes. Instead, she can only update on one of the pieces of evidence she’s received, and so can only adopt new credences equal to either or . Since Conditionalization (v1.1c) requires the subject to adopt , such a subject isn’t capable of satisfying the prescriptions the rule makes.

One might continue to finesse the formulation of Conditionalization, but there are some more general reasons why this strategy won’t work. First, the intuitive idea we’re trying to capture is that subjects should update on as much as they can, as soon as they can. But in order to capture this idea, a rule would need to be considerably more complex than the formulations offered above, having to provide verdicts regarding choices between updating on different batches of evidence, choices between updating sooner on less evidence versus updating later on more, and so on. And it’s hard to see how any such rule could remain similar enough to the formulations given above to plausibly be identified with Conditionalization, the norm that people have been talking about in the Bayesian literature.

Second, it’s not clear that subjects should update on their evidence as soon as they can. For example, suppose a subject who gets evidence E also violates some other epistemic norm, such as the Principal Principle – roughly, the claim that your credences should line up with what you think the chances are.Footnote ⁹ And suppose she is only capable of doing one thing at a time – either updating on E, or altering her credences so that they satisfy the Principal Principle. In this case, it’s not clear that she should update as soon as she can – it could well be that she should alter her credences to satisfy the Principal Principle first. So even the general idea that subjects should conditionalize on their evidence ‘as soon as they can’ seems too simplistic to yield the desired prescriptions, given the variety of situations and cognitive limitations facing imperfect subjects.

To sum up, in order to make allowances for the restricted capacities of cognitively limited subjects, we need a very complex updating rule. And no plausible understanding of Conditionalization is this complex.

The moral is that it’s a mistake to think of Conditionalization as a norm which is supposed to provide guidance to cognitively limited subjects. Instead, following Christensen (Reference Christensen2004), we should think of Conditionalization as an ideal toward which to aim, a description of optimal performance in the epistemic realm.Footnote ¹⁰

To borrow Christensen’s analogy, we should think of Bayesian norms as like the norms describing perfect tournament chess play. Better tournament chess players can make better moves in less time, and in the limit, ideal tournament chess players would make perfect moves in no time. But even though this is a good description of perfect tournament chess play, we wouldn’t expect any actual subject to be able to live up to this ideal. These ‘ideal performance’ norms aren’t the kinds of norms to which ought implies can generally applies. And once we properly understand Conditionalization as an ideal performance norm, we can see that ought-implies-can worries regarding it are misplaced.Footnote ¹¹

A different worry that one might raise for the concurrent answer is that it’s incompatible with plausible ‘procedural’ requirements on rational belief. Following Simon (Reference Simon1976), let us distinguish between two kinds of rational requirements:substantive rational requirements, which concern the rationality of the result, and procedural rational requirements, which concern the rationality of the process by which one obtains those results. Thus, when assessing whether a subject is epistemically rational, we might not only want her to have the right beliefs (i.e. to satisfy the relevant substantive requirements), but also to come to have those beliefs in the right way (i.e. to satisfy the relevant procedural requirements).

Conditionalization imposes a substantive requirement on rational updating: it requires subjects to come to have the right beliefs given their evidence and prior beliefs. Conditionalization doesn’t care about how subjects came to have those beliefs – it doesn’t care whether their beliefs were formed by reflecting on their evidence or whether their beliefs were formed by random quantum mechanical fluctuations. Now, one might think there are further requirements on rational updating beyond those imposed by Conditionalization. For example, one might also take there to be procedural requirements which require that a subject’s evidence be the cause of her changing her beliefs in this way, or require that she come to have her new beliefs by reasoning about her evidence. And, one might worry that the concurrent answer to the time of evidence question is incompatible with such procedural requirements. For if a subject needs to update on her evidence instantaneously, then it doesn’t seem like she can come to have those credences in the right way – via some causal process initiated by the receipt of her evidence, or by reasoning in light of her evidence. For any such process requires a non-zero amount of time.Footnote ¹²

But this worry is misplaced. For these substantive and procedural requirements aren’t actually in conflict. There isn’t anything logically incoherent about a subject who instantaneously comes to have the right beliefs for the right reasons. It’s true that subjects like us can’t do this. But that’s only a reason to think that these can’t be rational requirements if we’re assuming something like ought implies can. And, as we’ve seen, once Conditionalization is understood in the right way – as a description of optimal performance – this kind of reasoning is not compelling. For ideal performance norms aren’t the kinds of norms to which ought implies can generally applies.Footnote ¹³

Thus, all things considered, the concurrent answer is the best way to understand Conditionalization. The posterior answer yields implausible prescriptions, by effectively requiring subjects not to take all of their evidence into account, and is in tension with some popular accounts of evidence. And while the concurrent answer appears to face ought-implies-can worries and to conflict with procedural requirements, further reflection makes it clear that these worries are unreasonable (in the first case) and mistaken (in the second).Footnote ¹⁴

Adopting the concurrent answer also settles the relationship between t(cr), t(E) and . As we noted earlier, we want . If we adopt the concurrent answer, so that , then it follows that as well. Thus, the right relationship between these three times is: .

Building this into our formulation gets us the following rule:

• Conditionalization (v1): If a subject with credences cr at gets evidence E at , she should adopt new credences at such that , if defined.

4.1. The sequential vs. interval updating question: assessing the answers

Both of these answers to the sequential vs. interval updating question have uncomfortable consequences.

If we adopt the sequential answer, and think of Conditionalization as telling subjects how to update whenever they get a piece of evidence, then the rule can’t accommodate cases in which subjects get evidence continuously. This understanding of Conditionalization requires a ‘no evidence between and clause. Thus, the rule requires a pair of distinct times and between which the subject doesn’t get any evidence. But if the subject continuously receives evidence, then there is no such pair of times – between any two distinct times, there will be times in-between at which the subject gets evidence. So if we adopt a sequential understanding of Conditionalization, the rule simply goes silent in cases in which subjects receive continuous evidence.

But we also face some uncomfortable consequences if we adopt the interval answer, and think of Conditionalization as telling subjects how to update over arbitrary intervals, given the cumulative evidence they receive during that interval. In particular, the interval answer is incompatible with the conjunction of two widely held claims:

1. (1) Conditionalization is a special case of Jeffrey Conditionalization.

2. (2) The formalism of Jeffrey Conditionalization is neutral with respect to whether evidence is ‘credence-dependent’.

First, there’s the claim that Conditionalization is a special case of Jeffrey Conditionalization. The thought here is that Jeffrey Conditionalization is just a generalization of Conditionalization, one that returns Conditionalization in the special case in which all of one’s evidence partition takes the form . This understanding of the relationship between Conditionalization and Jeffrey Conditionalization is a standard part of the Bayesian lore.Footnote ¹⁸

Second, there’s the claim that the formalism of Jeffrey Conditionalization is neutral with respect to whether evidence is ‘credence-dependent’. On one way of thinking about evidence, the evidence partition a subject receives is determined by factors that are independent of her credences. For example, one might take a subject’s evidence partition to be determined solely by her sensory information. On another way of thinking about evidence, the evidence partition a subject receives is determined by factors that are dependent on her credences. For example, one might take the evidence partition a subject receives to be a function of both her sensory information and her beliefs about what this kind of sensory information suggests. Thus, if the subject sees a figure through the window who looks like her friend, the proposition that her friend is outside might be assigned a large value by her evidence partition if she believes her friend is coming over to visit, but a small value if she believes her friend is out of the country. But while various considerations have convinced many people to favor the second picture of evidence over the first, the formalism of Jeffrey Conditionalization itself has generally been taken to be neutral with respect to these two ways of thinking about evidence.Footnote ¹⁹

We can see that these two claims and the interval answer are inconsistent as follows. Suppose that the interval answer is true: we should understand Conditionalization as a rule which tells us how to update over arbitrary intervals, given the cumulative evidence we’ve received during that interval. Given the first claim – that Conditionalization is a special case of Jeffrey Conditionalization – Jeffrey Conditionalization should also be understood this way; as a rule which tells us how to update over arbitrary intervals, given the cumulative evidence we’ve received during that interval. Given the second claim – that the formalism of Jeffrey Conditionalization is neutral with respect to whether evidence is credence-dependent – it follows that this interval understanding of Jeffrey Conditionalization should be consistent with credence-independent evidence.

But it isn’t. If we adopt this interval understanding of Jeffrey Conditionalization, then Jeffrey Conditionalization must yield (on pain of inconsistency) the same prescriptions regardless of what intervals we choose to update on. (For instance, updating on the evidence received during the interval and then updating on the evidence received during the interval had better yield the same result as updating all at once on the evidence received during the interval.) But as I show in Appendix A, if evidence is credence-independent, then Jeffrey Conditionalization’s prescriptions will depend on what intervals we choose to update on. So the interval understanding of Jeffrey Conditionalization is not consistent with credence-independent evidence.

Thus the interval answer and these two claims are jointly incompatible. Given this answer, Conditionalization is an interval rule. Given the first claim, it follows that Jeffrey Conditionalization is also an interval rule. But given the second claim, it follows that Jeffrey Conditionalization can’t be an interval rule – for if evidence is credence-independent, Jeffrey Conditionalization will yield inconsistent verdicts depending on what intervals we choose to update on.

To sum up, there are compelling reasons in favor of both answers. The sequential answer, which adopts a sequential-updating understanding of Conditionalization, allows us to retain the familiar picture of how Conditionalization and Jeffrey Conditionalization are related, without having to accept the surprising claim that the formalism of Jeffrey Conditionalization requires evidence to be credence-dependent. The interval answer, which adopts an interval-updating understanding of Conditionalization, allows us to apply Conditionalization to cases in which subjects get evidence continuously. And neither of these reasons clearly trumps the other.

So the sequential vs. interval updating question leaves us with two viable ways to understand Conditionalization. There’s the understanding suggested by the sequential answer:

• Sequential Conditionalization (v2): If a subject with credences cr at gets evidence E at (and no evidence between and ), then her credences should remain cr between and , and she should adopt new credences at such that , if defined.

And there’s the understanding suggested by the interval answer:

• Interval Conditionalization (v2): If a subject with credences cr at gets cumulative evidence E in the interval, then she should adopt new credences at such that , if defined.

6.1. The narrow vs. wide scope question: assessing the answers

At least at first glance, both answers seem reasonable. The narrow answer is natural when we’re thinking of Conditionalization primarily as an updating rule, a constraint on what one’s posterior credences should be given one’s prior credences. The wide answer is natural when we’re thinking of Conditionalization primarily as a diachronic credence constraint, a constraint on how one’s prior and posterior credences should line up.

Some people have argued that we should understand all fundamental rational requirements as narrow scope norms.Footnote ³⁰ Perhaps the most popular argument against wide scope rational requirements is that such rational requirements make symmetrical prescriptions, and it’s argued that this symmetry is implausible. Thus, in the case of Conditionalization, the wide scope version of the rule requires that subjects either satisfy or satisfy C, and the rule treats these two options symmetrically: the rule gives us no reason to prefer satisfying C over satisfying . But, it’s argued, satisfying C should be preferred to satisfying . If, somehow, we were given a choice between adopting the appropriate new credences given our prior credences and evidence, or adopting whatever new credences we like and having whatever prior credences and evidence would yield these new credences, we should prefer the former option to the latter. And it’s argued that the wide scope formulations of rational requirements cannot accommodate this fact. Insofar as these kinds of arguments are compelling, and insofar as we want Conditionalization to be a fundamental norm, this gives us a reason to favor the narrow answer.Footnote ³¹

That said, there also seem to be considerations that tell against the narrow answer.Footnote ³² The narrow understanding of the rule takes the form . Thus, it tells us that if a subject actually satisfies A, then they will satisfy C at all of the best worlds. But this claim seems strange, since it requires the subject to satisfy C at all of the best worlds, regardless of whether the subject satisfies A at those worlds. And it’s hard to see why the subject should have to satisfy C at best worlds in which they don’t satisfy A.

In light of this worry, one might modify this answer by adding A to the consequent as well:

• Answer 1a (Narrow-a). The rule has the form .

This formulation of the rule gets around the above worry, since it tells us that a subject who actually satisfies A should satisfy both A and C at all of the best worlds.

But this isn’t what we want the rule to say. If is true at all of the best worlds, then A is true at all of the best worlds. So the above norm entails that . But we don’t want A’s actually obtaining to entail that A ought to obtain. In the case of Conditionalization, this would mean that the subject’s actually having credences cr and getting evidence E entails that she should have credences cr and get evidence E. And this is implausible.

In light of this, one might consider a second way of salvaging the narrow answer, inspired by the ethics literature, that leaves the form of the rule the same: .Footnote ³³ The idea is to avoid the difficulties sketched above by adding certain background assumptions. We will go through this in more detail in a moment, but here’s a preliminary sketch of how the story will go.

One can avoid the first worry – that the rule implausibly requires C to be true at all best worlds, even those where A is false – by adding assumptions which entail that if A is currently true, then A is also true at all of the best worlds. Now, this way of avoiding the first worry leads to the second worry: if the current truth of A entails that A is true at all of the best worlds, then the current truth of A entails that A is obligatory. One can blunt this second worry by distinguishing between ‘trivial’ and ‘non-trivial’ oughts, and showing that A only entails that A is obligatory in a trivial sense.

Let’s go through this in more detail. Standard deontic logic employs a notion of accessibility, call it O-accessibility, that determines what things are permissible and obligatory. Let’s add to this another notion of accessibility, call it C-accessibility, that determines what things we have the capacity to influence.Footnote ³⁴ So while O-accessibility intuitively picks out the normatively ‘best’ worlds, C-accessibility intuitively picks out the worlds that one is capable of getting to.

With these two notions of accessibility in hand, we can impose a constraint that implements the idea that ‘ought implies can’:

• Ought Implies Can: For any world w, the worlds that are - from w are a subset of the worlds that are - from w.

It follows from this that a proposition can only be obligatory for a subject if she’s capable of bringing about a state of affairs in which that proposition is true.

Given Ought Implies Can, there will be many propositions that are strictly speaking obligatory, but only in an uninteresting sense. For example, take the proposition that 1+1=2. This proposition will be true at every world, and a fortiriori at all of the best worlds. Thus the proposition that 1+1=2 will be obligatory. But it is only obligatory in a trivial sense. Since it’s inevitable, nothing a subject can do has any bearing on whether 1+1=2 turns out to be true or not. I’ll call oughts of this kind trivial oughts. A proposition A is trivially obligatory iff it’s true at both every O-accessible world (making it obligatory) and every C-accessible world (making it inevitable).

Contrast this with the interesting obligations that subjects have, obligations towards propositions which are not inevitable. These are the obligations we normally talk about, such as the obligation to save a drowning baby, or to adopt the appropriate credences. I’ll call these non-trivial oughts. A proposition A is non-trivially obligatory iff it’s true at every O-accessible world (making it obligatory) but not true at every C-accessible world (making the subject capable of falsifying it).

Introducing the notion of C-accessibility also allows us to impose a constraint that implements the idea that the past is immutable:Footnote ³⁵

• The past is immutable: For any world w, the only worlds that are - to w are worlds that agree with w about the state of the world before t. It follows from this that we can’t change what’s in the past. (Note that if we didn’t take the accessibility relation to be relativized to times, we couldn’t formulate a constraint like this.)

Finally, let’s suppose that the posterior answer to the time of evidence question is true. I argued in Section 3.1 that we should adopt the concurrent answer, and maintain that subjects should get E and adopt simultaneously. But now let’s suppose that we adopt the posterior answer instead, and hold that subjects get E some amount of time before they adopt .Footnote ³⁶

With all this in hand, let’s turn back to the Narrow vs. Wide Scope question. Suppose we adopt the narrow answer to this question, and maintain that the rule has the form . We now have the resources to get around the first worry raised for the narrow answer: that the rule requires C to be true at all of the best worlds, regardless of whether A is true at those worlds.

Given the posterior answer to the time of evidence question, A concerns solely facts that occur at – the subject’s having credences cr at and the subject getting evidence E at . So the truth value of A will be fixed by the state of the world before . Given the past is immutable, it follows that A will be true at all of the - worlds, and thus (given Ought Implies Can) at all of the - worlds. Thus, given this set-up, the first worry no longer arises: if A is actually true, then A will be true at all of the worlds too, and the rule won’t have the awkward consequence of potentially requiring C to be true at worlds where A is not.

Now, this way of avoiding the first worry leads to the second worry raised above for answer 1a: that if A is true at all of the best worlds, then A is obligatory, which seems implausible. But we now have the resources to explain away this worry too. For while it’s true that on this account A is obligatory, it’s only trivially obligatory, since it’s true at all of the C-accessible worlds. So it won’t be the case that A is obligatory in any sense that should bother us.

Thus if we adopt this set-up, we can lay out a way of understanding the narrow answer that avoids the worries raised above. Furthermore, this set-up fits nicely with the motivations for the posterior answer to the time of evidence question. Since the posterior answer is largely motivated by ought-implies-can worries, it’s a natural fit with this way of fleshing out the narrow answer, which also appeals to ought-implies-can-like considerations.

This is, I think, the closest one can get to a viable narrow scope understanding of Conditionalization. But ultimately, even this account does not give us a compelling reason to adopt the narrow answer. First, given this account, we have no reason to favor a narrow scope over a wide scope understanding of Conditionalization. As I show in Appendix C, given the assumptions we’ve been making, the narrow and wide scope formulations of Conditionalization are equivalent.Footnote ³⁷ And this equivalence undercuts the arguments that have been offered in favor of narrow over wide scope understandings of rational requirements. For example, the argument that wide scope understandings yield implausibly symmetric prescriptions while narrow scope understandings do not won’t work since both understandings will yield the same prescriptions.Footnote ³⁸

Second, this account inherits the demerits of the posterior answer to the time of evidence question. So even this version of the narrow answer carries some unpleasant baggage.

Third, all of the versions of the narrow answer we’ve considered, including this one, yield implausible verdicts in cases in which the subject’s credences are irrational.Footnote ³⁹ Consider a variant of an example we discussed in section 3.1. Suppose a subject both violates the Principal Principle and receives evidence E. The subject can conditionalize on E, or adopt credences that satisfy the Principal Principle, but it’s impossible for her to do both (since conditionalizing on E yields credences that violate the Principal Principle). The narrow answer yields the undesirable result that this subject must always update on E. ( will be true iff either A is false or at all best worlds C is true; and since A is true, it follows that at all best worlds C must be true.) By contrast, the wide answer yields the desired result that the subject needn’t update on E. ( will be true iff at all best worlds either A is false or C is true; and since A commits the subject to irrational credences, A will be false at all best worlds. So it doesn’t follow that C must be true at all (or even any) best worlds.)Footnote ⁴⁰

So, when all is said and done, we’re left with little motivation to adopt the narrow answer. In the best case scenario for the narrow answer, the narrow and wide answers are equivalent. And even in this best case scenario, the narrow answer has unpleasant consequences, as it inherits the demerits of the posterior answer and yields implausible verdicts in cases in which a subject’s starting credences are irrational.

So the wide scope understanding of Conditionalization is the better of the two options. Making the wide scope understanding explicit gives us the following formulations:

• Sequential Conditionalization (v3): It be the case that if a subject s with credences cr at gets evidence E at (and no evidence between and ), then her credences will remain cr between and are cr, and she will adopt credences at such that , if defined.

• Interval Conditionalization (v3): It be the case that if a subject s with credences cr at gets cumulative evidence E in the interval, then she will adopt credences at such that , if defined.