2. Lange’s argument

Standard Bayesianism assumes that an agent’s degrees of confidence, or credences, in the propositions she entertains can be represented as an assignment of real numbers in the unit interval to those propositions. It further assumes that the following synchronic norm governs this assignment:

More importantly, for our purposes, most Bayesians also take an agent’s credences to satisfy a diachronic constraint. This constraint tells us how we should update our beliefs when we are presented with some new evidence:

Since our evidence always ought to get a credence of one on this updating framework, a commitment that this framework requires us to take on is that we should always be certain of our evidence. But, to many, it seems as though the evidence we get from our sensory experience is often uncertain. Indeed, to some it seems as though the experiences that underwrite our belief revisions almost never leave us certain of anything at all.

To accommodate this intuitive idea, Jeffrey (Reference Jeffrey1965) proposed an alternative to Conditionalization that generalizes this model. Jeffrey suggests that our evidence takes the form of a weighted partition of propositions: an n-tuple of propositions $ \Big\langle $ B $ {}_1 $ ,…,B_n $ \Big\rangle $ that partitions the agent’s credal state, and that are assigned an n-tuple of credences:

It’s easy to see that, in many cases, the order in which two Jeffrey updates happen will determine the credence distribution one ends up with. Consider the case where I see a raven and this leads to me directly changing my credence in the proposition that the raven is black to .9, i.e., p(RB) $ = $ .9. And then after a second glance, I come to directly change my credence in this proposition to .7. After I update on these two pieces of evidence, I am left with a credence of .7 in RB. But had I made these two updates in reverse order, I would have been left with a credence of .9 in RB. Therefore, the order in which these two pieces of evidence are received makes a difference to my post-observational credence distribution. Jeffrey Conditionalization is non-commutative over weighted evidence partitions.

In some cases, this feature of the framework seems exactly right: the probative value of my most recent evidence seems to swamp the value of the evidence I’ve gotten before. If I glance an object from afar, and then glance it again at a much closer distance, the information I get from the second glance should make my first glance irrelevant. But in just as many cases, this seems like the wrong result. Two consecutive glances taken from the very same vantage point shouldn’t necessarily lead us to discount the deliverances of the first.

An assumption made by those who criticize Jeffrey Conditionalization for being non-commutative is that the elements for which the commutative property ought to hold are weighted evidence partitions.Footnote ¹ In his paper, Lange calls into question these criticisms by pointing out that updates that don’t commute in this way will be underwritten by different experiences, in the original sequence and its permutation. If experiences are different in two sequences of updates where the order of the weighted evidence partitions have been reversed, then the conditions required to generate commutativity failure at the level of experience will fail to hold in cases where we get commutativity failure at the level of weighted evidence partitions. Therefore, there’s a more fundamental type of information that isn’t non-commutative under the framework: experiential information.

Why think that experiences will be different in cases where weighted evidence partitions don’t commute? Lange claims that the answer to this question lies with the agent’s background beliefs. It lies especially with one kind of background belief: the agent’s prior opinion about the proposition she gets as evidence. It’s easy to see why such beliefs will be important. If my prior credence in the proposition that the raven I will see a moment from now will be black is pretty low, an experience of a raven that’s not so clear or intense will result in my still having a low credence in the proposition that the raven is black. By contrast, if my prior opinion in the proposition that the raven will be black had been higher, this same mildly informative experience would have resulted in my having a relatively higher credence in this same proposition. The same experience in the presence of different prior opinions will yield different posterior credence distributions.

This same reasoning seems to suggest something else. It suggests that the same posterior credence distribution arrived at from different prior opinions implies that the agent has had different experiences. As Lange (Reference Lange2000, 398) puts it:

We can walk through the reasoning in this passage to see how it is supposed to lead to the conclusion that the Jeffrey framework isn’t defective in virtue of being non-commutative. First, consider that since any update involves the agent changing her credence in the evidence proposition, a non-commutative update will entail that the agent’s priors in the evidence proposition are different in the original sequence and its permutation, provided we begin from the same credence function (p(e) $ \ne {q}^{\prime } $ (e) and p(e) $ \ne $ q(e)). If an agent has lowered her credence in e from 0.99 to 0.8, then the further update on the evidence that leaves her with a credence of 0.75 in the original sequence will begin from 0.8, whereas the update on the same evidence in its permutation will begin from 0.99, and vice versa (Figure 1).

Figure 1. Two Glances at a Raven.

Second, notice that in order for an update to be non-commutative, we must have had the same evidence in the original sequence and its permutation. Therefore, it must be the case that the agent’s posterior credence in the evidence proposition after the first update in the first sequence is identical to her posterior credence in the evidence proposition after the second update in the second sequence, and vice versa (q(e) $ ={r}^{\prime } $ (e) and $ {q}^{\prime } $ (e) $ = $r(e)).

By Lange’s reasoning in the previous passage, the experiences that trigger the two updates in question must have been different in the original sequence and its permutation ( $ {\xi}_1\ne {\xi}_4 $ and $ {\xi}_2\ne {\xi}_3 $ ) in order to offset the difference in prior opinions to yield the same posterior credence: $ {\xi}_1 $ must have “not looked much the way a black bird would be expected to look at twilight,” whereas $ {\xi}_4 $ must have “looked about the way that any dusky colored object would be expected to look under those conditions.” A similar story could be told about $ {\xi}_2 $ and $ {\xi}_3 $ .

And this gets us Lange’s conclusion. Since the conditions required to get a violation of experience commutativity (two experiences whose orders have been reversed) fail to hold in cases where we have reversed the order of the weighted evidence partitions, the non-commutativity of weighted evidence partitions does not entail the non-commutativity of experiences. Since the framework isn’t non-commutative over a more fundamental set of elements, we ought to conclude that it is not defective.Footnote ²

Let’s consider a final example. Imagine that a whiff of pie leaves you with a credence of .3 in the proposition that the pie in the oven is rhubarb (R) at t₁ and then, a moment later, another whiff leaves you with a credence of .7 in the same proposition at t₂. Now imagine that you’d gotten the previous evidence in reverse order (Figure 2).

Figure 2. The Smell of Pie.

Here, again, we have a case where the fact that weighted evidence partitions don’t commute under the framework doesn’t seem like a defect of the framework, since the experiences underlying these partitions in the original case and its permutation aren’t likely to be identical. The fact that we have raised our credence slightly at t₁ from p(R) $ = $ .1 to q(R) $ = $ .3 in the first sequence, in response to the evidence gotten in between t₀ and t₁, suggests that the experience we’ve had is something like a decisive whiff of rhubarb pie. By contrast, where we decrease our credence from $ {q}^{\prime } $ (R) $ = $ .7 to $ {r}^{\prime } $ (R) $ = $ .3 in the second sequence, in response to having gotten the same weighted evidence partition in between t₁ and t₂, that must have been because we had a “disconfirming” experience—maybe a whiff of lemon (which is not an ingredient in rhubarb pie). A similar story could be told about the second update in the first sequence, and the first update in the second sequence.

Though weighted evidence partitions don’t commute under Jeffrey Conditionalization, then, that’s altogether appropriate, since they don’t supervene on the same experiences in the original case and its permutation. If we begin both sequences from the same initial, p, and if p(R) $ \ne $ $ {q}^{\prime } $ (R) and p(R) $ \ne $q(R), but q(R) $ ={r}^{\prime } $ (R) and $ {q}^{\prime } $ (R) $ = $ r(R), then the experience that triggered the first update in the first sequence must have been different than the experience that triggered the second update in the second sequence, and vice versa. Therefore the experiences underlying these sequences aren’t the same. Therefore there’s no reason we would want it to be the case that r $ ={r}^{\prime } $ . Although Jeffrey Conditionalization isn’t commutative over weighted evidence partitions, then, examples like this one suggest that it’s not defective in virtue of this feature of it.

3. A partial account of experience

The crucial move in Lange’s argument is his claim that Jeffrey Conditionalization is not defective because it doesn’t fail to commute experiences. A natural question to ask is whether this defensive move commits Lange to some positive thesis about commutativity. It seems as though it must. In particular, it seems as though it must commit Lange to the claim that experiences ought to commute, and that they do commute. Without this assumption, his reasoning that Jeffrey Conditionalization is not defective because it doesn’t fail to commute experiences seems to make no sense. Lange’s argument seems to be committed to the following principle:

But maybe this is too quick. Maybe not everyone committed to Lange’s argument needs to be committed to Lange’s Assumption. Maybe the reason that Jeffrey Conditionalization does not fail to commute experiences is that experiences aren’t the sorts of things that could fail to commute. One might maintain a view about the identity conditions for experience that make it impossible to get the same two experiences in reverse order. Consider a view according to which each experience’s identity is determined by its qualitative character along with the totality of the agent’s background beliefs, which are, in turn, determined by all of that agent’s prior history. On this sort of story, it will be impossible to get the same two experiences in reverse order. If my experiences are individuated by the totality of what I have learnt up until the time in question, then experiences could not be had in reverse order: no two experiences could have been informed by the same history if they occur at different times in this history. On this sort of story, the conditions required to generate a failure of commutativity—two experiences taken in reverse order—could not possibly be met. Therefore, on this sort of story, the nature of experience itself ensures that experiences could not fail to commute under the Jeffrey framework. No additional norm is required.Footnote ³

I think there are a couple of reasons for resisting this sort of story. First, the account just described advocates for a radically holistic view, one whereupon every background belief matters for individuating experiences or, at least, determining their probative value. But this seems implausible. My belief that the lighting is tricky might matter for determining the probative value of my experience of seeing a red shoe on the floor. But my belief that Barack Obama was the forty-fourth president of the United States is surely irrelevant here. One might instead opt for a more modest form of holism that says not all background beliefs are relevant for individuating experiences or for determining their probative value. But this more plausible form of holism isn’t going to get us the result we need. If I have two experiences, and no background beliefs that are relevant to the identity of these experiences, then I may very well have these experiences again, in reverse order. Adopting a more plausible, moderate form of holism that says that only relevant background beliefs individuate experiences will not make it impossible for experiences to fail to commute by making it impossible to get the same two experiences in reverse order. Therefore, adopting a more modest form of holism does not seem to preclude the need for Lange’s Assumption—though, as we will see in section 5, it may inform the norm that this assumption justifies.

One might still object that a more moderate form of holism will make it impossible to get, in reverse order, the particular pairs of experiences featured in the last section: pairs of experiences that both bear on at least some of the same evidence propositions. Therefore, it may be that a more moderate form of holism will render a commutative norm unnecessary for precisely those experiences that are in danger of not commuting. However, I think there’s a simpler reason why anyone, including Lange, should still endorse what I’ve called “Lange’s Assumption.” It seems close to a truism that we would expect identical information to be treated the same by our updating framework. The idea that information, understood in the most fundamental sense, should commute is surely more intuitive than any particular account of experience that would preclude the need for such a norm. I want to suggest, then, that Lange’s Assumption is the weakest assumption that we need in order to be committed to Lange’s argument.

It looks like we have reason to accept Lange’s Assumption. Does the Jeffrey framework vindicate Lange’s Assumption? Let’s focus on the second conjunct: that Jeffrey Conditionalization is commutative over experiences. Some remarks Lange (Reference Lange2000, 400–402) makes near the end of the paper about the account of experience his argument assumes suggest how he hopes to secure this part of the assumption. Recall that Lange’s conclusion is that experiences must have been different in two sequences of updates that feature different priors, but identical posterior credence distributions. This implies that had both prior and posterior credence distributions been the same, the experiences involved could have also been inferred to have been the same. We get something like this idea near the end of Lange’s discussion:

This passage makes it sound as though Lange is putting forth necessary and sufficient conditions for experience identity. In the next section, we will consider whether or not this passage should indeed be read in this way. For now, we can say that, at the very least, this passage says that if two experiences are identical, then they will have the same impact on an agent’s credence distribution.Footnote ⁴ While there are a number of ways this claim might be read, one interpretation of it seems natural. The impact, or degree of change, that some experience induces in an agent’s credence distribution is often identified with a Bayes factor. Formally, two updates have the same Bayes factor, $ \mathrm{\mathcal{BF}} $ , just in case the ratios of the new-to-old odds of the elements of each of the evidence partitions, taken pairwise, are identical. More formally still, for experiences, $ \xi $ and $ {\xi}^{\ast } $ , and credence functions, p( $ \cdot $ ) and q( $ \cdot $ ) defined over every proposition, X, in some $ \sigma $ -algebra, $ \mathcal{A} $ (and where p $ {}_{\xi } $ ( $ \cdot $ ) comes from p( $ \cdot $ ) by updating on the weighted evidence partition induced by $ \xi $ , and q $ {}_{\xi^{\ast }} $ ( $ \cdot $ ) comes from q( $ \cdot $ ) by updating on the weighted evidence partition induced by $ {\xi}^{\ast } $ ), the claim that two updates have the same Bayes factors amounts to the following:

$$ {\mathrm{\mathcal{BF}}}_{p,{p}_{\xi }}={\mathrm{\mathcal{BF}}}_{q,{q}_{\xi}\ast}\frac{p_{\xi}\left({E}_{i_1}\right)/{p}_{\xi}\left({E}_{i_2}\right)}{p\left({E}_{i_1}\right)/p\left({E}_{i_2}\right)}=\frac{q_{\xi}\ast \left({E}_{i_1}\right)/{q}_{\xi}\ast \left({E}_{i_2}\right)}{q\left({E}_{i_1}\right)/q\left({E}_{i_2}\right)}, $$

for all elements, i ₁, i ₂, of the evidence partition, $ \left\{{E}_i\right\} $ .

With this in mind, an account of the impact of an experience can be formulated in the following way (for ease of exposition, let us simply formulate experience identity as $ \xi ={\xi}^{\ast } $ ):

An Account of the Impact of an Experience:

$ \forall p\forall q\left[\forall \xi \forall {\xi}^{\ast}\left(\xi ={\xi}^{\ast}\right)\to \left({\mathrm{\mathcal{BF}}}_{p,{p}_{\xi }}={\mathrm{\mathcal{BF}}}_{q,{q}_{\xi}\ast}\right)\right] $

This way of understanding the passage from above gets us the connection to experience commutativity we are after. This is because a series of results confirm that two updates will commute just in case they yield the same Bayes factors in the original case and its permutation.Footnote ⁵ Since An Account of the Impact of an Experience says that a necessary condition for experience identity is a property that is necessary and sufficient for experiences to commute, it vindicates the second conjunct of Lange’s Assumption. Therefore, An Account of the Impact of an Experience is a necessary condition for the account of experience Lange’s argument relies upon.

4. A complete account of experience

Is An Account of the Impact of an Experience also a sufficient condition for the account of experience Lange’s argument relies upon? Although Lange certainly makes it sound as though he is putting forth necessary and sufficient conditions in the passage that points us toward this account, at the very end of the paper, he says the following: “Of course, a fuller account of the sameness of sensory experiences would be very welcome. But I cannot offer one at present” (Reference Lange2000, 402).

How do we reconcile these two seemingly inconsistent passages? Plausibly, Lange makes the latter statement because he believed that a necessary condition for experience identity is some sort of qualitative constraint—one that we might think of as distinguishing experiences with different phenomenal characters. In this section, I argue that not only is such a constraint plausible, but that some such constraint on experience identity looks like it’s required to secure the conclusion that Jeffrey Conditionalization is not defective. This means that the norm that Lange’s argument defends as not defective is actually a stronger norm than Jeffrey Conditionalization, for it must include a constraint that Jeffrey Conditionalization does not entail.

To begin to see this, recall again the following assumption that Lange’s argument relies upon:

Let’s now focus on the first conjunct: that the elements that ought to commute under Jeffrey Conditionalization are experiences. Lange is careful in his discussion to distinguish the claim that: (1) Jeffrey Conditionalization is formally non-commutative over weighted evidence partitions, from the claim that (2) Jeffrey Conditionalization is defective in virtue of being non-commutative over weighted evidence partitions (Reference Lange2000, 393, 397). The first is a mathematical claim; the second is a normative claim. And, of course, it is the second claim that Lange’s argument targets.

In order for experience commutativity to save Jeffrey Conditionalization from being defective, then, we must assume that, unlike arithmetic operands, the identity conditions for experience make experience commutativity a good thing. The most obvious thing to say here is that the reason why experiences ought to commute is that they are phenomenally indistinguishable. In other words, what saves Jeffrey Conditionalization from being defective is that it treats phenomenally indistinguishable experiences the same, as we would want. But maybe this is too quick. Maybe we don’t actually need to identify experiences with anything phenomenal or qualitative to do justice to the idea that treating experiences consistently makes the Jeffrey framework nondefective. Maybe there’s something about the degree of an impact on a credence distribution that makes this the thing that ought to be treated consistently, and so that ought to commute.

There are several reasons to reject this suggestion. First, it does not seem to be what Lange had in mind; never once in the examples he provides throughout the paper does he talk about the agent’s doxastic behavior. Instead, he talks about “sensory experiences”—or sometimes just “experiences”—being different in two sequences of updates where the weighted evidence partitions have been reversed (Reference Lange2000, 394–95, 397). Sometimes he talks about particular sensory experiences, like “the way that any dusky colored object would be expected to look under those conditions” (Reference Lange2000, 398). And, of course, there is the passage we were left with in the last section, wherein Lange acknowledges he has failed to offer a complete account of experience in providing an account of how identical experiences impact an agent’s credence distribution.

Second, regardless of Lange’s intentions again, this is clearly not what he should have had in mind. To whatever extent we think that there is something intuitively good about belief revisions commuting, it’s difficult to imagine that this is not because we are assuming that the experiences that ground these revisions are phenomenally identical, i.e., that the feeling that induces these changes is the same. As Bradley (Reference Bradley2005, 6) puts it, “it does not follow from the fact that your Bayes factors represent what you have gleaned from observation, that they have the kind of objectivity which obligates others to modify their beliefs using them as constraints on their posteriors.”

To see Bradley’s point more clearly, consider the lack of rhetorical force Lange’s argument would have if it did not include a description of the experiences that prompted the belief revisions he describes. To claim that Jeffrey Conditionalization is not defective because reversing the order of the weighted evidence partitions involved in two updates does not entail that the magnitudes of these revisions are identical comes pretty close to claiming that Jeffrey Conditionalization is not defective in virtue of being non-commutative because its updates don’t commute. It comes pretty close to being just a description of the fact that Jeffrey Conditionalization is non-commutative, rather than a defense of it.

These considerations suggest that Lange’s Assumption—that Jeffrey Conditionalization is not defective in virtue of commuting experiences—requires that we think of experiences as individuated by something more than merely Jeffrey Conditionalization’s commutative property. Plausibly, this “something more” is an experience’s phenomenal character.Footnote ⁶ But then this means that experience commutativity won’t be trivially preserved by the Jeffrey framework. If we want to ensure that experiences, individuated in a way that would make Jeffrey Conditionalization not defective in virtue of commuting such experiences (as required by the first conjunct of Lange’s Assumption) actually do commute (as required by the second conjunct of Lange’s Assumption), we need an additional norm to secure that experiences with the same phenomenal characters actually commute under the framework. The norm that isn’t defective in virtue of being non-commutative isn’t Jeffrey Conditionalization, then, but an updating rule that comprises two constraints. The first constraint is Jeffrey Conditionalization. The second constraint secures Lange’s Assumption by ensuring that phenomenally indistinguishable experiences yield updates with the same Bayes factors:

Experience-Commuting Jeffrey Conditionalization (ECJC):

1. 1. When an experience, $ \xi $ , directly changes your credences over a partition $ \Big\{ $ E_i $ \Big\} $ from p(E_i) to $ {p}^{\prime } $ (E_i), your new degree of belief in A, for any A, should be $ {p}^{\prime } $ (A) $ ={\sum}_i $ p(A $ \mid $ E_i) $ {p}^{\prime } $ (E_i).

2. 2. For any two experiences, $ \xi $ and $ {\xi}^{\ast } $ , that are identical with respect to phenomenal character, and for any credence distributions, p( $ \cdot $ ), q( $ \cdot $ ), the following should hold: $ \mathrm{\mathcal{BF}} $ (p, $ {p}_{\xi } $ ) $ =\mathrm{\mathcal{BF}} $ (q, $ {q}_{\xi^{\ast }} $ ).Footnote ⁷

Lange’s argument entails that Jeffrey Conditionalization alone does not commute the elements that ought to commute. We need, in addition, ECJC’s second constraint.

5. Commutativity, normativity, and holism

6. Carnap and Field

I’ve argued for two claims. First, I’ve argued that the norm that Lange’s argument targets is not Jeffrey Conditionalization, but the norm that I’ve called ECJC. Second, I’ve argued that, like Jeffrey Conditionalization, ECJC is defective. ECJC is defective in virtue of the nature of its inputs and, in particular, in virtue of the problems that these inputs raise. Before closing, I want to briefly consider the history of previous attempts to formulate a norm like ECJC. This history lends some support to my results. My results, in turn, help to unify some of the ideas in these earlier discussions.

The first worries about a norm that tells us how to get from some particular phenomenal experience to a posterior credence distribution come from Carnap’s 1957 correspondence with Jeffrey (reprinted in Jeffrey [Reference Jeffrey, Maxwell and Anderson1975]), where Carnap tells Jeffrey that he himself had attempted to formulate such a norm. His correspondence expresses two worries about its prospects. Carnap’s first worry is one that our own discussion has circumvented. It arises as a result of assuming that experience ought to be represented, not as a Bayes factor, but as a probability attached to some experientially affected sentence which indicates “the subjective certainty of the sentence on the basis of the observational experience” (Reference Jeffrey, Maxwell and Anderson1975, 42). Carnap noticed that since probabilities, unlike Bayes factors, aren’t defined in terms of the agent’s prior and posterior probabilities, this made it impossible to represent experience in the right way: as the impact on an agent’s credence distribution. Even before Lange, then, there were worries about how to represent the inputs, though these worries took for granted that these inputs were experiences. Call this the representation problem.

Carnap’s second concern is closely related to the one that we were left with at the end of the last section. Even if we were to represent experience as a Bayes factor so that we could describe the Jeffrey framework as representing the agent’s evidence as a function of her experiences and prior expectations, we still would not have a rule for how to map a particular experience to a particular Bayes factor in a way that takes into account how features like “the clarity of the observation (or the feeling of certainty connected with it or something similar)” (Reference Jeffrey, Maxwell and Anderson1975, 42) should determine the latter. Carnap complained to Jeffrey that his framework did not include the latter sort of rule:

As we saw in the last section, the latter sort of rule is not even compatible with a certain understanding of the Jeffrey framework. This is the normative problem.

Carnap’s frustration that he could not adequately address the representation problem and the normative problem led him to abandon the project of crafting a rule like ECJC.Footnote ¹⁴ The first discussion that actually addresses both the representation problem and the normative problem is Field (Reference Field1978). Field distinguishes his approach from Carnap’s attempt at such an account based upon the responses he offers to the representation problem and the normative problem (1978, 363–64). Field’s account resolves the representation problem by assuming, like Lange, that experience should be represented by a Bayes factor. Field’s response to the normative problem is equally decisive. He claims that it is not a legitimate problem at all:

Having identified experience with a Bayes factor, rather than a probability, the normative problem for Field is close to the problem we were left with at the end of the last section. It is the problem that there are cases where a person ought, if he is rational, to come to adopt a certain Bayes factor on the basis of an experience, but that nothing in Jeffrey’s constraints requires him to do this. As the previous passage suggests, Field rejects the normative problem. He argues that the task of providing an account of how experience figures into the updating process should be conceived of as a purely descriptive one, as “the problem of giving a complete psychological theory for a Bayesian agent” (1978, 364). Unlike Carnap, Field believed that a norm for evidence does not belong as part of the framework.

Both Field and Carnap took the representation problem and the normative problem to be independent of each other. Our discussion establishes a relation between them by arguing that it is precisely how we are forced to represent the inputs to the Jeffrey framework—as experiences formalized as Bayes factors—that gives rise to the version of the normative problem we have discussed. Whether or not we side with Field or with Carnap on the question of whether a substantive norm for evidence belongs as part of the Jeffrey framework, I’ve argued that the Jeffrey framework, like the classical framework, should at least be compatible with such a norm. And the fact that we must represent our inputs as experiences, or Bayes factors, entails that it is not. No such norm is going to be able to directly govern these inputs since no such norm is going to be able to directly govern these inputs qua magnitudes. I think, then, we cannot escape the conclusion that the Jeffrey framework is defective. Either it is defective in virtue of not commuting its inputs, or else it is defective in virtue of commuting the wrong kinds of inputs.