2020

2020, Jul 9

Free Speech and ‘Cancel Culture’

With Harper’s open letter in the news, I’m seeing lots of discussion about the relationship between free speech and ‘cancel culture’. I thought I’d write something up briefly explaining what a free speech principle commits us to (in section 1); rehearsing a traditional argument for the principle (in section 2); and talking about why, if you accept the principle, you should be concerned about ‘cancel culture’, and what exactly you should be worried about (in section 3). (Section 3 presupposes some of what’s discussed in sections 1 and 2, but any section can be read independently of the others.)

This isn’t intended to be an argumentative essay, by the way. My goal isn’t to persuade anybody to change their views about free speech, or about ‘cancel culture’. The goal is just to lay out the principle of free speech, as I understand it, give one reason for endorsing it, and explore how it bears on ‘cancel culture’.

1. The Principle of Free Speech

The principle of free speech says, roughly, that we should not punish an individual for the expression of an opinion. We’ll have to be a bit careful when interpreting ‘punish…for the expression of an opinion’ in this principle. We may, of course, punish an individual for an act which just happens to be achieved through the expression of an opinion. To take a fanciful example, suppose I create a bomb will explode whenever somebody says “We should abolish the FED” in its presence, and then I express this opinion in its presence. A principle of free speech offers me no defense. I can be punished. What I can be punished for, however, isn’t the opinion I expressed, but the act of setting off the bomb. The general lesson is this: we don’t only say things with words. We also, and at the same time, do things with words. According to the free speech principle, the things we do with words are apt targets of collective punishment; the things we say with words are not. (For the cognoscenti: illocutionary and perlocutionary acts are apt targets of punishment, but locutionary acts are not.)

Once we’ve distinguished the things we say with words from the things we do with words, and allowed that the latter are apt targets of punishment, there’s a serious worry that the principle has been bled of any real force. After all, whenever anyone expresses a controversial opinion, one thing they will do is cause someone else’s blood pressure to rise. If causing someone’s blood pressure to rise is enough to justify punishment, then no controversial opinion will ever be protected by the principle.

There are really two worries here. The first worry is that we could custom-tailor our punishments so as to only punish the expressions of some controversial opinions, while leaving others’ unpunished. For instance, if I decide to criticize the Catholic Church, one thing I will thereby do is undermine the authority of the Church. If we decide to punish anybody who undermines the authority of the Church, this will have the same effect as simply punishing anybody who criticizes the Church. Because of this first worry, the free speech principle is also committed to what’s often called a principle of ‘content neutrality’. Content neutrality requires that our reasons for punishment shouldn’t have anything to do with the opinions a person has expressed. Think of it like this: leaving everything else the same, exchange the expressed opinion for another. This shouldn’t make a difference to our punishment. If we collectively punish someone for setting off a bomb by expressing the opinion “We should abolish the FED”, then we should equally well collectively punish someone for setting off a bomb by expressing the opinion “We should not abolish the FED”.

The second worry is that the principle may allow us to shut down all debate and discussion. Suppose that we decide to punish anyone who upsets someone else with their opinions. This is a content-neutral policy. But it is antithetical to the reasons we have for endorsing a free speech principle in the first place (see section 2 below). So, in addition to content neutrality, we should also endorse a principle of content universality: whatever collective punishments we devise, they should permit the expression of every opinion, in some manner or other.

In summary: a free speech principle says that, while we may punish people for what they’ve done, we may not punish people for what they’ve said, or what they think; our reasons for punishment should be insensitive to people’s opinions; and those reasons should allow the expression and defense of any opinion.

It’s worth noting some things which this principle does not exempt from punishment. It does not exempt threats or intimidation. In my opinion, many of the acts which recent opponents of free speech point to—for instance, sending an email reading “I’m going to kill you and your family” to an author you disapprove of, burning a cross on a black family’s lawn, or telling a person speaking Spanish on the street that should “go back to where they came from”—are rightfully understood as threats and intimidation. So a law against intimidating people by sending them death threats, burning a cross on their lawn, or accosting them on the street, does not run afoul of the principle of free speech (in the case of cross burning, the US Supreme Court agreed in Virginia v. Black).

2. Mill’s Argument for the Free Speech Principle

J.S. Mill argues for a principle like this in Chapter 2 of his “On Liberty”. Mill makes several argumentative moves in Chapter 2, but I’m going to focus on what I see as the most central argument in favor of the free speech principle. This argument is redolent of Karl Popper’s views in the Philosophy of Science. Popper thought that, in order for a theory to be scientific, it must expose itself to the possibility of refutation and falsification. Insofar as a theory shields itself from refutation, that theory is unscientific. Mill holds a similar view, except that he’s not concerned with what’s ‘scientific’ or not, but rather with what it takes for us to have collective knowledge. Mill thinks that a society cannot achieve knowledge of some claim unless we have exposed ourselves to the possibility of being refuted. At least, Mill thinks that’s the case when it comes to matters of ethics, politics, and religion. His reason for thinking this has to do with the way that he thinks societies paradigmatically come by true beliefs in the realms of ethics, politics, and religion. As he points out, human societies have been wrong—and badly wrong—about these matters time and again throughout history. Widespread error in ethics, politics, and religion isn’t some unusual occurrence. It is the norm. Intelligent groups of people with good intentions have excused rape and murder, justified slavery, discrimination, and oppression of all kinds; intelligent and well-intentioned groups have favored political actions which led to bloodshed, terror, tyranny, mass starvation, and worse. When a society arrives at truth in the realms of ethics, politics, and religion, it could do so in two ways: firstly, through sheer accident. Secondly, through reasoned discourse in which their errors are corrected. The first route is too capricious, too accidental, to qualify as knowledge. Collective knowledge cannot be ‘lucky’. So, if we are to attain knowledge, we must take the second route. And the second route requires the freedom of opinion and discussion.

In sum: to silence those who might disagree with you is to forestall the possibility of having your errors corrected; but this possibility is needed in order for you to know that your opinions are correct. The final premise of the argument is this: society is justified in collectively punishing somebody for the expression of an opinion only if they know that that opinion is false. Mill concludes that society is never justified in punishing somebody for the expression of an opinion. For adopting a policy of punishing those who express heterodox opinions takes away your knowledge that those opinions are false. And if you don’t know that what they are saying is false, then your punishment isn’t justified.

A subtlety with this argument is worth raising. Think about a case in which society has already attained knowledge of some moral, political, or religious truth. For those kinds of cases, Mill needs to say that the continued possibility of refutation is needed for us to continue to know this truth. I think that this is plausible. Think about a scientific analogue: suppose we currently know that general relativity is at least approximately correct. If we outlawed experiments designed to test general relativity, would this collective knowledge persist? I’m inclined to think it wouldn’t persist for very long. So I’m inclined to agree with Mill about this. But it’s worth noting nonetheless that Mill needs something more than the claim that collective knowledge isn’t ‘lucky’. He also needs to claim that dogmatically shielding an opinion from criticism defeats collective knowledge.

In summary: Mill argues for a free speech principle on the grounds that such a principle is needed for us to attain moral, political, or ethical knowledge. The argument isn’t that this kind of knowledge is a valuable end worth pursuing in-and-of itself (though he thinks it is). The argument is instead that this kind of knowledge would be needed for the punishment of an opinion to be just. Since punishment for the expression of an opinion would undermine and defeat the very knowledge needed for the punishment to be just, this kind of punishment is never just.

3. Free Speech and ‘Cancel Culture’

I’ve been talking throughout about ‘punishment’. By this, I mean a bit more than punishment by the state, or legal punishment. Legal punishment is one kind of punishment, but it is not the only kind. For instance, boycotts, tarring-and-feathering, and smear campaigns can also count as punishment.

If you’re persuaded by the Millian argument, you shouldn’t want to distinguish between these different forms of punishment. For, if that argument works, it works both for legal punishment for other kinds of punishments. American revolutionaries tarring-and-feathering someone for being a British sympathizer is only just if the revolutionaries know that the British sympathizer is incorrect; but if British sympathizers are tarred and feathered, then they will not share their opinions, and the American revolutionaries will have undermined their collective knowledge that these British sympathizers are wrong (supposing, for the sake of argument, that they were wrong).

Of course, not just anything which harms counts as a punishment. When a large group of people strongly disagree with and criticize me, this raises my blood pressure and leads to psychological distress, which is plausibly a harm. But it is not a punishment. Some people seem to understand ‘cancelling’ as a widespread expression of disagreement. If this is what ‘cancelling’ means, then cancelling as such does not conflict with a free speech principle. (In fact, commitment to free speech means promoting these kinds of ‘cancelling’.) What makes something a punishment? It’s not entirely clear, but I believe that intent must play a role—if a harm is a punishment, then your reason for harming somebody must be that you take them to have done something wrong. If your intent is simply to persuade or express disagreement, then even widespread and vehement criticism does not constitute punishment.

Nor is calling for somebody to be fired from their job necessarily calling for them to be punished. Consider a police officer who is caught on tape expressing racist views. In my opinion, a widespread call for the police officer to lose his job is not necessarily in conflict with a principle of free speech. If those calling for the officer to lose their job are doing so because, in their view, these opinions make him unfit for the job, then I don’t think that they are seeking punishment, and the calls are not in conflict with a free speech principle.

Nonetheless, some activities which have been called ‘cancelling’ do seem, in my opinion at least, to constitute punishment. In my opinion, there are some instances of groups engaging in a form of bullying with the intent of bringing harm—psychological distress, loss of income, loss of social status, or what-have-you—in retribution for expressed views. And this, I think, is inconsistent with a free speech principle. Some of us feel that these kinds of punishments have been happening with an increasing frequency—though I don’t have any hard data to verify that sense.

If that’s right, and if you endorse the free speech principle, then you have reason to be worried about these more extreme instances of ‘cancelling’. The principle of free speech can effectively serve its purpose of exposing us to the possibility of refutation only to the extent that a preponderance of society abides by the principle. As more and more people advocate punishment for increasingly banal and milquetoast opinions on certain topics, you will see fewer and fewer people willing to address ideological errors on those topics.

Notice that, to the extent that acts of ‘cancelling’ are carried out with the intent of intimidating and threatening those who disagree, they are not themselves covered under a principle of free speech—anymore than intimidating and threatening through cross-burning or accosting somebody on the street with racist taunts is covered. Of course, a principle of free speech must permit, and even encourage, criticizing the principle of free speech itself. So it must allow people to defend acts of ‘cancelling’. But it does not follow that it must permit the acts of ‘cancelling’ themselves. So I must disagree with those who characterize these acts of ‘cancelling’ as ‘just more speech’, or who characterize its criticism as suppressing speech. Opposition to ‘cancelling’ isn’t an objection to the things that people are saying with their words; it is an objection to the things that people are doing with their words.

2020, Mar 25

Deference and updating

Bas van Fraassen’s principle of Reflection tells you to defer to your future credences. A natural generealization of Christensen’s principle of Rational Reflection tells you to defer to whichever future credence will be rational. Elga’s principle New Rational Reflection is like Christensen’s principle, except that it allows that the rational credences may not be certain that they are rational.

Each of these deference principles is equivalent to a claim about updating—a claim about how your credences should be disposed to change when you learn that some proposition, e, is true. Reflection is equivalent to the claim that you should be disposed to update by conditioning on the proposition that your credences have been updated on e. Rational Reflection is equivalent to the claim that you should be disposed to update by conditioning on the proposition that e is your total evidence. And New Rational Reflection is equivalent to the claim that you should be disposed to update with a Jeffrey shift on the partition of propositions about what your total evidence may be.

1. Learning Dispositions

To set the stage, I’m going to suppose that you’ve got some (prior) credence function $C$, and that there’s some (finite) set of propositions, $\mathscr{E} = \{e, f, g, \dots \}$, such that exactly one of the members of the set $\mathscr{E}$ will be your total evidence. That is: for each $e \in \mathscr{E}$, $e$ might be your total evidence. And your total evidence must be some member of $\mathscr{E}$. Write ‘$\mathbf{T}e$’ for ‘your total evidence is $e$’. Then, note that, since you must learn exactly one of the propositions in the set $\mathscr{E} = \{e, f, g, \dots \}$ the set $\mathbf{T}\mathscr{E} = \{ \mathbf{T}e, \mathbf{T}f, \mathbf{T} g, \dots \}$ will form a partition.

Your learning dispositions are dispositions to respond to each of these possible conditions: the condition of having t/otal evidence $e$, $\mathbf{T}e$, the condition of having total evidence $f$, $\mathbf{T}f$, and so on. I’ll write ‘$D_e$’ for the credence function that you’re disposed to adopt in the condition $\mathbf{T}e$, for each $e \in \mathscr{E}$. If you take your learning dispositions to be perfectly attuned to your potential evidence, then you’ll foresee no possibility of failing to adopt $D_e$ in the condition $\mathbf{T}e$. In that case, we needn’t distinguish the condition of you having $e$ as your total evidence and the condition of you updating on the proposition $e$. But what if you do forsee the possibility of not recognizing that $e$ is your total evidence in the condition $\mathbf{T}e$, or not responding to that total evidence by updating appropriately? What if you don’t take your learning dispositions to be perfectly attuned to your evidence? In that case, we should distinguish the condition of having $e$ as your total evidence and the condition of you updating on the total evidence $e$. I’ll use ‘$\mathbf{U}e$’ (read: you update on $e$) to stand for the condition of you taking $e$ to be your total evidence, and responding accordingly. If $D_e$ is the credence function you’re disposed to adopt when your total evidence is $e$, then $\mathbf{U}e$ says that you have taken your evidence to be $e$ and adopted the credence function $D_e$ in response. If you think that your learning dispositions are not perfectly attuned to your potential evidence, then you’ll think that, for some $e$, $\mathbf{U}e$ could be true, even when $\mathbf{T}e$ is not. Equivalently: you’ll think that, for some $e \neq f$, you might update on $f$ even when your total evidence is $e$.

2. Deference and Updating

2.1 Deference

When you defer to some other, expert, credence function, you use its probabilities to determine your own. If you’re certain of what credence function the expert has, then deference is simple: simply adopt the known expert credence function as your own. Sometimes, however, you don’t know precisely what the expert function’s credences are. In that case, you should use your credences about the expert function to determine your own credences.

Principles of expert deference tell you exactly how your credences about the expert function should determine your own credences. The simplest such expert deference principle says that, conditional on a probability function, $E$, being the expert, your credences should agree with $E$'s. That is:

Immodest Expert Deference You defer to an (immodest) expert iff, for every proposition $p$ and every probability function $E$, $$ C(p \mid E \text{ is the expert }) = E(p), \text{ if defined} $$

(Note: If $E$ is certain to not be the expert, $C(E \text{ is the expert })= 0$, then the conditional probability on the left-hand-side may not be defined; in that case, the principle will impose no constraint. That’s why I’ve written ‘if defined’ above; I’ll leave this proviso implicit in what follows.)

I’ve called this principle Immodest Expert Deference because it implies that the expert is certain to be certain that it is the expert. In the jargon, it entails that the expert is immodest. To see that this follows from the principle, just let $p$ be the proposition that $E$ is the expert. Then, the principle tells us that, for every $E$, $$ C(E \text{ is the expert } \mid E \text{ is the expert }) = E( E \text{ is the expert }) $$ If $C$ is a probability function, then the left-hand-side must equal 1, so, if you are able defer to the expert in the way this principle advises, then it must be that, for every $E$ which might be the expert, $E$ is certain that it is the expert. So you are certain that $E$ is certain that it is the expert.

Suppose you wish to defer to a modest expert: one who isn’t certain that it is the expert. The simplest such principle governing deference to such an expert says that, conditional on a probability function, $E$, being the expert, your credences should agree with $E$'s, once $E$ is conditioned on the proposition that it is the expert.

Expert Deference You defer to an expert iff, for every proposition $p$ and every probability function $E$, $$ C(p \mid E \text{ is the expert }) = E(p \mid E \text{ is the expert }) $$

This principle subsumes Immodest Expert Deference as a special case: whenever $E$ is immodest, $E(p \mid E \text{ is the expert })$ will be equal to $E(p)$. However, it applies even when $E$ may not be certain that it is the expert. If $E$ isn’t certain that it is the expert, then, when you condition your credence function on the proposition that $E$ is the expert, you’re taking something for granted that $E$ itself has not taken for granted. The solution is to have $E$ also take for granted that it is the expert (by conditioning it on the proposition that it is the expert), and only then align your credences with it. That is: the principle tells you to align your conditional credences with the expert’s conditional credences, where the condition is that it is the expert.

(By the way, there are several other options for how defer to experts. At least 12 different formulations have been floated in the literature, and there are often subtle, unexpected differences between them (see this blog post, for instance). But, in the interests of simplicity, I’m just going to focus on these two here.)

2.2 Reflection

Bas van Fraassen’s principle of reflection says that you should treat your future, posterior credence function as an (immodest) expert. That is: for each proposition $p$, given that you update your credences to the posterior function $D$, your credence that $p$ should be $D(p)$.

Reflection Conditional on $D$ being your updated credence function, your credence that $p$ should be $D(p)$ $$ C(p \mid D \text{ is your updated credence }) = D(p) $$

Suppose that you know that your updated credence function will be one of $D_e, D_f, D_g, \dots$. That is, suppose that you know your updated credence function will be one of the members of the set $\{ D_e \mid e \in \mathscr{E} \}$. Then, the claim that $D_e$ is your updated credence is just the claim that you have updated on the proposition $e$, $\mathbf{U}e$. In that case, we can re-write Reflection like this: for every proposition $p$, and every $e \in \mathscr{E}$, $$ C(p \mid \mathbf{U} e) = D_e(p) $$

Reflection has a straightforward corollary for updating: it entails that you should be disposed to update on $e$ by conditioning on the proposition $\mathbf{U}e$. To see this, instead of thinking of Reflection as a constraint on your prior credence function, $C$, think of it as a constraint on your learning dispositions, $D$. Then, it tells you that, for each $e \in \mathscr{E}$, you should be disposed, upon learning $e$, to adopt a new credence which is equal to your old credence, conditional on $\mathbf{U}e$. $$ D_e(p) = C(p \mid \mathbf{U} e) $$ (To emphasize this shift in perspective—from thinking of Reflection as a constraint on $C$ to thinking of it as a constraint on $D$, I’ve just switched the left- and right-hand sides*.) Obviously, this entailment goes both ways; so **Reflection** is equivalent to the claim that, upon learning $e$, you should be disposed to condition on $\mathbf{U}e$. (In my paper Updating for Externalists, I called this claim ‘update conditionalization’, and I showed that, given some assumptions about accuracy, update conditionalization maximizes evidentially expected accuracy. That is: if you’re an evidential decision theorist, and you want your credences to be as accurate as possible, then you should have the learning dispositions which update conditionalization recommends.)

2.3 Rational Reflection

Christensen’s Rational Reflection principle says that you should treat your current rational credence as an (immodest) expert. That is, conditional on $R$ being the rational credences for you to hold now, your credence in every proposition should be the same as $R$'s credence in that proposition.

Rational Reflection (present) Conditional on $R$ being the rational credence for you to hold, your credence in $p$ should be $R(p)$, for every proposition $p$. $$ C(p \mid R \text{ is rational } ) = R(p) $$

This formulation of Rational Reflection only says that you should defer to your currently rational credences. However, it is meant to apply at any time. So, in particular, we can think about your posterior credences, after you’ve learnt which $e \in \mathscr{E}$ is true. In that case, since you are certain that your total evidence was one of the propositions in $\mathscr{E}$, and you’re certain that $D_e$ is the rational credence function iff $e$ is your total evidence, Rational Reflection (present) says that, for each $e, f \in \mathscr{E}$, \begin{aligned} D_f(p \mid D_e \text{ is rational }) &= D_e(p) \\\
D_f(p \mid \mathbf{T}e) &= D_e(p) \end{aligned}

If you think that you should defer to your currently rational credences, you should also think that you should defer to your future rational credences. So you should also accept the following principle, govnering your prior credences, before you learn which proposition in $\mathscr{E}$ is true: conditional on $D_e$ being the rational credence function for you to hold after learning which $e \in \mathscr{E}$ is true, your credence in $p$ should be $D_e(p)$, for every proposition $p$. Since $D_e$ will be the rational credence function for you to hold iff $e$ is your total evidence, this means that, conditional on $\mathbf{T}e$, your prior credence that $p$ should be $D_e(p)$, for every proposition $p$.

Rational Reflection (future) Conditional on $e$ being your total evidence, your credence that $p$ should be $D_e(p)$. \begin{aligned} C(p \mid D_e \text{ will be rational }) &= D_e(p) \\\
C(p \mid \mathbf{T}e) &= D_e(p) \end{aligned}

Once again, this deference principle implies a corollary about updating. Instead of seeing Rational Reflection (future) as a constraint on your prior credences $C$, think of it as a constraint on your learning dispositions, $D$. Then, it says that you, upon learning $e$, you should be disposed to condition on the proposition $\mathbf{T}e$, $$ D_e(p) = C(p \mid \mathbf{T}e) $$ This update rule has been defended by Matthias Hild and Miriam Schoenfield. In Updating for Externalists, I called it ‘Schoenfield conditionalization’. Again, the reverse entailment also goes through, so the two claims are equivalent. Rational Reflection is equivalent to Schoenfield conditionalization. In Updating for Externalists, I showed that (given some assumptions about accuracy) Schoenfield conditionalization maximizes causal expected accuracy whenever you are certain that you’ll update on a proposition iff that proposition is your total evidence. That is: if you’re certain that $\mathbf{U}e \leftrightarrow \mathbf{T}e$, for each $e \in \mathscr{E}$, you’re a causal decision theorist, and you want your credences to be as accurate as possible, then you should have the learning dispositions which Schoenfield recommends.

2.4 New Rational Reflection

Because Rational Reflection tells you to treat your rational credence as an immodest expert, it entails that your rational credence is certain to be immodest. Suppose that you deny this. Suppose you’ve persuaded by authors like Williamson that you think you can end up rationally uncertain about what your total evidence is. In that case, since which credence function is rational is a function of what your total evidence is, it follows that you can end up rationally uncertain about whether your credences are in fact rational. That is: it can be rational to be less than certain that your credences are rational, even when they in fact are.

In that case, you cannot treat your rational credences as an immodest expert. So Elga recommends that you treat them as a potentially modest expert. That is: he advises that, conditional on $R$ being the rational function for you to adopt, you match your credences to $R$'s, once $R$ is conditioned on the proposition that it is the rational credence function.

New Rational Reflection (present) Conditional on $R$ being the rational credence for you to hold, your credence in $p$ should be $R$'s credence in $p$, after $R$ is conditioned on the proposition that $R$ is the rational credence function for you to hold. $$ C(p \mid R \text{ is rational } ) = R(p \mid R \text{ is rational }) $$

This formulation of New Rational Reflection says only that you should defer in this way to your currently rational credences. If we apply it to you after you’ve learnt which $e \in \mathscr{E}$ is true, then—since you’re certain that $D_e$ is the rational credence function for you to hold iff $\mathbf{T}e$ is true—New Rational Reflection (present) says that, for each $e, f \in \mathscr{E}$, \begin{aligned} D_f(p \mid D_e \text{ is rational }) &= D_e(p \mid D_e \text{ is rational }) \\\
D_f(p \mid \mathbf{T}e) &= D_e(p \mid \mathbf{T}e)
\end{aligned}

As with Rational Reflection, there’s no need to restrict the principle to your currently rational credences. If you should treat your current rational self as an expert, then so too should you treat your future rational self as an expert. So, in particular, before you learn which proposition in $\mathscr{E}$ is true, you should satisfy the following constraint: for each $e \in \mathscr{E}$, conditional on $D_e$ being the rational posterior credence function, your credence in $p$ should be $D_e(p \mid D_e \text{ will be rational })$. Since $D_e$ will be rational iff $e$ is your total evidence, this means that, conditional on $\mathbf{T}e$, your credence that $p$ should be $D_e(p \mid \mathbf{T}e)$.

New Rational Reflection (future) Conditional on $e$ being your total evidence, your credence that $p$ should be $D_e(p \mid \mathbf{T}e)$. \begin{aligned} C(p \mid D_e \text{ will be rational }) &= D_e(p \mid D_e \text{ is rational }) \\\
C(p \mid \mathbf{T}e) &= D_e(p \mid \mathbf{T}e) \end{aligned}

Henceforth, I’ll just call the conjunction of New Rational Reflection (present) and New Rational Reflection (future) ‘New Rational Reflection’.

Again, this deference principle is equivalent to a claim about updating. In this case, the claim is that you should be disposed to update with a Jeffrey shift on the partition $\mathbf{T}\mathscr{E} = \{ \mathbf{T}e, \mathbf{T}f, \mathbf{T}g, \dots \}$. What it is to be disposed to update with a Jeffrey shift on the partition $Q = \{ q_1, q_2, \dots, q_N \}$ is for the following to be true of your learning dispositions: for every $e \in \mathscr{E}$, there is some collection of weights $\lambda_1, \lambda_2, \dots, \lambda_N$ such that $\sum_i \lambda_i = 1$ and, for every propostion $p$, $$ D_e(p) = \sum_i C(p \mid q_i) \cdot \lambda_i $$ Or, equivalently: you are disposed to update with a Jeffrey shift on the partition $Q = \{ q_1, q_2, \dots, q_N \}$ iff, for every proposition $p$, each $e \in \mathscr{E}$, and each $q_i \in Q$, $D_e(p \mid q_i) = C(p \mid q_i)$ (if defined, of course).

Jeffrey Shift You are disposed to update with a Jeffrey shift on the partition $Q = { q_1, q_2, \dots, q_N }$ iff, for each $e \in \mathscr{E}$ and each $q_i \in Q$, $$ D_e(p \mid q_i) = C(p \mid q_i) $$

Therefore, to show that New Rational Reflection requires you to update with a Jeffrey shift on the partition $\mathbf{T}\mathscr{E} = \{\mathbf{T}e \mid e \in \mathscr{E} \}$, we would have to show that it requires that, for each $e, f \in \mathscr{E}$, $$ D_e(p \mid \mathbf{T}f) = C(p \mid \mathbf{T}f) $$

We can show this easily. New Rational Reflection (present) tells us that
$$ D_e(p \mid \mathbf{T}f) = D_f(p \mid \mathbf{T}f)
$$ And New Rational Reflection (future) tells us that $$ C(p \mid \mathbf{T}f) = D_f(p \mid \mathbf{T}f) $$ Putting these two identities together gives us that $$ D_e(p \mid \mathbf{T}f) = C(p \mid \mathbf{T}f) $$ So: New Rational Reflection entails that you should be disposed to update with a Jeffrey Shift on the partition $\mathbf{T}\mathscr{E}$.

In fact, New Rational Reflection is equivalent to the claim that you should be disposed to update with a Jeffrey Shift on the partition $\mathbf{T}\mathscr{E}$. Assume that you are disposed to update with a Jeffrey shift on $\mathbf{T}\mathscr{E}$. Then, for any $e, f \in \mathscr{E}$, $$ (\star) \qquad \qquad D_e(p \mid \mathbf{T}f) = C(p \mid \mathbf{T}f) $$ If we let $f = e$ in ($\star$), then we get New Rational Relection (future): $$ C(p \mid \mathbf{T}e) = D_e(p \mid \mathbf{T}e) $$ If we instead swap $e$ and $f$ in ($\star$), we get: $$ D_f(p \mid \mathbf{T}e) = C(p \mid \mathbf{T}e) $$ And putting these two identities together gives us New Rational Reflection (present): $$ D_f(p \mid \mathbf{T}e) = D_e(p \mid \mathbf{T}e) $$ So: New Rational Reflection is equivalent to the claim that you should be disposed to update with a Jeffrey shift on $\mathbf{T}\mathscr{E}$.

In my paper Updating for Externalists, I pointed out that Schoenfield conditionalization (according to which $D_e(p)$ should be $C(p \mid \mathbf{T}e)$) requires a kind of immodesty that externalists should want to reject. For that rule requires you to always end up certain about what your total evidence is. Being certain about what your total evidence is means being certain about which credence function is rational. But externalists should want to say that you could be less than certain about what your total evidence is, and less than certain about which credence function is the rational one for you to adopt. Since Schoenfield conditionalization is equivalent to Rational Reflection, this was really just a re-hashing of Elga’s argument that externalists should reject Rational Reflection. And, in fact, the alternative update I recommended for externalists is closely related to Elga’s New Rational Reflection.

The alternative rule I recommended for externalists, called ‘externalist conditionalization’, says: $$ D_e(p) = \sum_f C(p \mid \mathbf{T}f) \cdot C(\mathbf{T}f \mid \mathbf{U}e) $$ (Here, I’m summing over the $f \in \mathscr{E}$.) This is a Jeffrey shift on the partition $\mathbf{T}\mathscr{E}$. That is, it is a rule of the form $$ D_e(p) = \sum_f C(p \mid \mathbf{T}f) \cdot \lambda_f $$ where, in the case of externalist conditionalization, $\lambda_f = C(\mathbf{T}f \mid \mathbf{U}e)$.

Another, equivalent, presentation of externalist conditionalization is this: your learning dispositions should be such that:

You are disposed to update with a Jeffrey shift on $\mathbf{T}\mathscr{E}$: that is, for every $e, f \in \mathscr{E}$, $$ D_e(p \mid \mathbf{T}f) = C(p \mid \mathbf{T}f) $$ and
For each $e, f \in \mathscr{E}$, upon learning that $e$, you are disposed to think $\mathbf{T}f$ is as likely as you currently think it is, conditional on your updating on $e$: $$ D_e(\mathbf{T}f) = C(\mathbf{T}f \mid \mathbf{U}e) $$

What we’ve just seen is that (1) is equivalent to New Rational Reflection. So a third, equivalent presentation of externalist conditionalization is this:

Externalist Conditionalization Your learning dispositions should be such that:

They satisfy New Rational Reflection; and
For each $e, f \in \mathscr{E}$, upon learning that $e$, you are disposed to think $\mathbf{T}f$ is as likely as you currently think it is, conditional on your updating on $e$: $$ D_e(\mathbf{T}f) = C(\mathbf{T}f \mid \mathbf{U}e) $$