In their 2018 paper, 'Living on the Edge', Ginger Schultheis issues a powerful challenge to epistemic permissivism about credences, the view that there are bodies of evidence in response to which there are a number of different credence functions it would be rational to adopt. The heart of the argument is the claim that a certain sort of situation is impossible. Schultheis thinks that all motivations for permissivism must render situations of this sort possible. Therefore, permissivism must be false, or at least these motivations for it must be wrong.

Here's the situation, where we write $R_E$ for the set of credence functions that it is rational to have when your total evidence is $E$.

- Our agent's total evidence is $E$.
- There is $c$ in $R_E$ that our agent knows is a rational response to $E$.
- There is $c'$ in $R_E$ that our agent does not know is a rational response to $E$.

Schultheis claims that the permissivist must take this to be possible, whereas in fact it is impossible. Here are a couple of specific examples that the permissivist will typically take to be possible.

Example 1: we might have a situation in which the credences it is rational to assign to a proposition $X$ in response to evidence $E$ form the interval $[0.4, 0.7]$. But we might not be sure of quite the extent of the interval. For all we know, it might be $[0.41, 0.7]$ or $[0.39, 0.71]$. Or it might be $[0.4, 0.7]$. So we are sure that $0.5$ is a rational credence in $X$, but we're not sure whether $0.4$ is a rational credence in $X$. In this case, $c(X) = 0.5$ and $c'(X) = 0.4$.

Example 2: you know that Probablism is a rational requirement on credence functions, and you know that satisfying the Principle of Indifference is rationally permitted, but you don't know whether or not it is also rationally required. In this case, $c$ is the uniform distribution required by the Principle of Indifference, but $c'$ is any other probability function.

Schultheis then appeals to a principle called *Weak Rationality Dominance*. We say that one credence function $c$ *rationally dominates* another $c'$ if $c$ is rational in all worlds in which $c'$ is rational, and also rational in some worlds in which $c'$ is not rational. Weak Rationality Dominance says that it is irrational to adopt a rationally dominated credence function. The important consequence of this for Schultheis' argument is that, if you know that $c$ is rational, but you don't know whether $c'$ is, then $c'$ is irrational. As a result, in our example above, $c'$ is not rational, contrary to what the permissivist claims, because it is rationally dominated by $c$. So permissivism must be false.

If Weak Rationality Dominance is correct, then, it follows that the permissivist must say that, for any body of evidence $E$ and set $R_E$ of rational responses, the agent with evidence $E$ either *must know of each* credence function in $R_E$ that it is in $R_E$, or they *must not know of any* credence function in $R_E$ that it is in $R_E$. If they *know of some* credence functions in $R_E$ that they are in $R_E$ and *not know of others* in $R_E$ that they are in $R_E$, then they clash with Weak Rationality Dominance. But, whatever your reason for being a permissivist, it seems very likely that it will entail situations in which there are some credence functions that are rational responses to your evidence and that you know are such responses, while you are unsure about other credence functions that are, in fact, rational responses whether or not they are, in fact, rational responses. This is Schultheis' challenge.

I'd like to explore a response to Schultheis' argument that takes issue with Weak Rationality Dominance (WRD). I'll spell out the objection in general to begin with, and then see how it plays out for a specific motivation for permissivism, namely, the Jamesian motivation I sketched in this previous blogpost.

One worry about WRD is that it seems to entail a deference principle of exactly the sort that I objected to in this blogpost. According to such deference principles, for certain agents in certain situations, if they learn of a credence function that it is rational, they should adopt it. For instance, Ben Levinstein claims that, if you are certain that you are irrational, and you learn that $c$ is rational, then you should adopt $c$ -- or at least you should have the conditional credences that would lead you to do this if you were to apply conditionalization. We might slightly strengthen Levinstein's version of the deference principle as follows: if you are unsure whether you are rational or not, and you learn that $c$ is rational, then you should adopt $c$. WRD entails this deference principle. After all, suppose you have credence function $c'$, and you are unsure whether or not it is rational. And suppose you learn that $c$ is rational (and don't thereby learn that $c'$ is as well). Then, according to Schultheis' principle, you are irrational if you stick with $c'$.

In the previous blogpost, I objected to Levinstein's deference principle, and others like it, because it relies on the assumption that all rational credence functions are better than all irrational credence functions. I think that's false. I think there are certain sorts of flaw that render you irrational, and lacking those flaws renders you rational. But lacking those flaws doesn't ensure that you're going to be better than someone who has those flaws. Consider, for instance, the extreme subjective Bayesian who justifies their position using an accuracy dominance argument of the sort pioneered by Jim Joyce. That is, they say that accuracy is the sole epistemic good for credence functions. And they say that non-probabilistic credence functions are irrational because, for any such credence function, there are probabilistic ones that accuracy dominate them; and all probabilistic credence functions are rational because, for any such credence function, there is no probabilistic one that accuracy dominates it. Now, suppose I have credence $0.91$ in $X$ and $0.1$ in $\overline{X}$. And suppose I am either sure that this is irrational, or I'm uncertain it is. I then learn that assigning credence $0.1$ to $X$ and $0.9$ to $\overline{X}$ is rational. What should I do? It isn't at all obvious to me that I should move from my credence function to the one I've learned is rational. After all, even from my slightly incoherent standpoint, it's possible to see that the rational one is going to be a lot less accurate than mine if $X$ is true, and I'm very confident that it is.

So I think that the rational deference principle is wrong, and therefore any version of WRD that entails it is also wrong. But perhaps there is a more restricted version of WRD that is right. And one that is nonetheless capable of sinking permissivism. Consider, for instance, a restricted version of WRD that applies only to agents who have no credence function --- that is, it applies to your initial choice of a credence function; it does not apply when you have a credence function and you are deciding whether to adopt a new one. This makes a difference. The problem with a version that applies when you already have a credence function $c'$ is that, even if it is irrational, it might nonetheless be better than the rational credence function $c$ in some situation, and it might be that $c'$ assigns a lot of credence to that situation. So it's hard to see how to motivate the move from $c'$ to $c$. However, in a situation in which you have no credence function, and you are unsure whether $c'$ is rational (even though it is) and you're certain that $c$ is rational (and indeed it is), WRD's demand that you should not pick $c'$ seems more reasonable. You occupy no point of view such that $c'$ is less of a depature from that point of view than $c$ is. You know only that $c$ lacks the flaws for sure, whereas $c'$ might have them. Better, then, to go for $c$, is it not? And if it is, this is enough to defeat permissivism.

I think it's not quite that simple. I noted above that Levinstein's deference principle relies on the assumption that all rational credence functions are better than all irrational credence functions. Schultheis' WRD seems to rely on something even stronger, namely, the assumption that all rational credence functions are equally good in all situations. For suppose they are not. You might then be unsure whether $c'$ is rational (though it is) and sure that $c$ is rational (and it is), but nonetheless rationally opt for $c'$ because you know that $c'$ has some good feature that you know $c$ lacks and you're willing to take the risk of having an irrational credence function in order to open the possibility of having that good feature.

Here's an example. You are unsure whether it is rational to assign $0.7$ to $X$ and $0.3$ to $\overline{X}$. It turns out that it is, but you don't know that. On the other hand, you do know that it is rational to assign 0.5 to each proposition. But the first assignment and the second are not equally good in all situations. The second has the same accuracy whether $X$ is true or false; the first, in constrast, is better than the first if $X$ is true and worse than the first if $X$ is false. The second does not open up the possibility of high accuracy that the first does; though, to compensate, it also precludes the possibility of low accuracy, which the first doesn't. Surveying the situation, you think that you will take the risk. You'll adopt the first, even though you aren't sure whether or not it is rational. And you'll do this because you want the possibility of being rational and having that higher accuracy. This seems a rational thing to do. So, it seems to me, WRD is false.

Although I think this objection to WRD works, I think it's helpful to see how it might play out for a particular motivation for permissivism. Here's the motivation: Some credence functions offer the promise of great accuracy -- for instance, assigning 0.9 to $X$ and 0.1 to $\overline{X}$ will be very accurate if $X$ is true. However, those that do so also open the possibility of great inaccuracy -- if $X$ is false, the credence function just considered is very inaccurate. Other credence functions neither offer great accuracy nor risk great inaccuracy. For instance, assigning 0.5 to both $X$ and $\overline{X}$ guarantees the same inaccuracy whether or not $X$ is true. You might say that you are more risk-averse the lower is the maximum possible inaccuracy you are willing to risk. Thus, the options that are rational for you are those undominated options with maximum inaccuracy at most whatever the threshold is that you set. Now, suppose you use the Brier score to measure your inaccuracy -- so that the inaccuracy of the credence function $c(X) = p$ and $c(\overline{X}) = 1-p$ is $2(1-p)^2$ if $X$ is true and $2p^2$ if $X$ is false. And suppose you are willing to tolerate a maximum possible inaccuracy of $0.5$, which also gives you a mininum inaccuracy of $0.5$. In that case, only $c(X) = 0.5 = c(\overline{X})$ will be rational from the point of view of your risk attitudes --- since $2(1-0.5)^2 = 0.5 = 2(0.5^2)$. On the other hand, suppose you are willing to tolerate a maximum inaccuracy of $0.98$, which also gives you a minimum inaccuracy of $0.18$. In that case, any credence function $c$ with $0.3 \leq c(X) \leq 0.7$ and $c(\overline{X}) = 1-c(X)$ is rational from the point of view of your risk attitudes.

Now, suppose that you are in the sort of situation that Schultheis imagines. You are uncertain of the extent of the set $R_E$ of rational responses to your evidence $E$. On the account we're considering, this must be because you are uncertain of your own attitudes to epistemic risk. Let's say that the threshold of maximum inaccuracy that you're willing to tolerate is $0.98$, but you aren't certain of that --- you think it might be anything between $0.72$ and $1.28$. So you're sure that it's rational to assign anything between 0.4 and 0.6 to $X$, but unsure whether it's rational to assign $0.7$ to $X$ --- if your threshold turns out to be less than 0.98, then assigning $0.7$ to $X$ would be irrational, because it risks inaccuracy of $0.98$. In this situation, is it rational to assign $0.7$ to $X$? I think it is. Among the credence functions that you know for sure are rational, the ones that give you the lowest possible inaccuracy are the one that assigns 0.4 to $X$ and the one that assigns 0.6 to $X$. They have maximum inaccuracy of 0.72, and they open up the possibility of an inaccuracy of 0.32, which is lower than the lowest possible inaccuracy opened up by any others that you know to be rational. On the other hand, assigning 0.7 to $X$ opens up the possibility of an inaccuracy of 0.18, which is considerably lower. As a result, it doesn't seem irrational to assign 0.7 to $X$, even though you don't know whether it is rational from the point of view of your attitudes to risk, and you do know that assigning 0.6 is rational.

There is another possible response to Schultheis' challenge for those who like this sort of motivation for permissivism. You might simply say that, if your attitudes to risk are such that you will tolerate a maximum inaccuracy of at most $t$, then regardlesss of whether you know this fact, indeed regardless of your level of uncertainty about it, the rational credence functions are precisely those that have maximum inaccuracy of at most $t$. This sort of approach is familiar from expected utility theory. Suppose I have credences in $X$ and in $\overline{X}$. And suppose I face two options whose utility is determined by whether or not $X$ is true or false. Then, regardless of what I believe about my credences in $X$ and $\overline{X}$, I should choose whichever option maximises expected utility from the point of view of my actual credences. The point is this: if what it is rational for you to believe or to do is determined by some feature of you, whether it's your credences or your attitudes to risk, being uncertain about those features doesn't change what it is rational for you to do. This introduces a certain sort of externalism to our notion of rationality. There are features of ourselves -- our credences or our attitudes to risk -- that determine what it is rational for us to believe or do, which are nonetheless not luminous to us. But I think this is inevitable. Of course, we might might move up a level and create a version of expected utility theory that appeals not to our first-order credences but to our credences concerning those first-order credences -- perhaps you use the higher-order credences to define a higher-order expected value for the first-order expected utilities, and you maximize that. But it simply pushes the problem back a step. For your higher-order credences are no more luminous than your first-order ones. And to stop the regress, you must fix some level at which the credences at that level simply determine the expectation that rationality requires you to maximize, and any uncertainty concerning those does not affect rationality. And the same goes in this case. So, given this particular motivation for permissivism, which appeals to your attitudes to epistemic risk, it seems that there is another reason why WRD is false. If $c$ is in $R_E$, then it is rational for you, regardless of your epistemic attitude to its rationality.