More on the Swamping Problem for Reliabilism

In a previous post, I floated the possibility that we might use recent work in decision theory by Orri Stefánsson and Richard Bradley to solve the so-called Swamping Problem for veritism. In this post, I'll show that, in fact, this putative solution can't work.

According to the Swamping Problem, I value beliefs that are both justified and true more than I value beliefs that are true but unjustified; and, we might suppose, I value beliefs that are justified but false more than I value beliefs that are both unjustified and false. In other words, I care about the truth or falsity or my beliefs; but I also care about their justification. Now, suppose we take the view, which I defend in this earlier post, that a belief in a proposition is more justified the higher the objective probability of that proposition given the grounds for that belief. Thus, for instance, if I base my belief that there was a firecrest in front of me until a few seconds ago on the fact that I saw a flash of orange as the bird flew off, then my belief is more justified the higher the objective probability that it was a firecrest given that I saw a flash of orange. And, whether there really was a firecrest in front of me, the value of my belief increases as the objective probability that there was given I saw a flash of orange increases.

Let's translate this into Stefánsson and Bradley's version of Richard Jeffrey's decision theory. Here are the components:

• a Boolean algebra $F$
• a desirability function $V$, defined on $F$
• a credence function $c$, defined on $F$

The fundamental assumption of Jeffrey's framework is this:

Desirability For any partition $X_1$, ..., $X_n$, $$V(X) = \sum^n_{i=1} c(X_i | X)V(X\ \&\ X_i)$$ And, further, we assume Lewis' Principal Principle, where $C^x_X$ is the proposition that says that $X$ has objective probability $x$:

Principal Principle $$c(X_j | \bigwedge^n_{i=1} C^{x_i}_{X_i}) = x_i$$ Now, suppose I believe proposition $X$. Then, from what we said above, we can extract the following:

1. $V(X\ \&\ C^x_X)$ is a monotone increasing and non-constant function of $x$, for $0 \leq x \leq 1$
2. $V(X\ \&\ C^x_X)$ is a monotone increasing and non-constant function of $x$, for $0 \leq x \leq 1$
3. $V(X\ \&\ C^x_X) > V(\overline{X}\ \&\ C^x_X)$, for $0 \leq x \leq 1$.

Given this, the Swamping Problem usually proceeds by identifying a problem with (1) and (2) as follows. It begins by claiming that the principle that Stefánsson and Bradley, in another context, call Chance Neutrality is indeed a requirement of rationality:

Chance Neutrality $$V(X_j\ \&\ \bigwedge^n_{i=1} C^{x_i}_{X_i}) = V(X)$$ Or, equivalently:

Chance Neutrality$^*$ $$V(X_j\ \&\ \bigwedge^n_{i=1} C^{x_i}_{X_i}) = V(X_j\ \&\ \bigwedge^n_{i=1} C^{x'_i}_{X_i})$$ This says that the truth of $X$ swamps the chance of $X$ in determining the value of an outcome. With the truth of $X$ fixed, its chance of being true becomes irrelevant.

The Swamping Problem then continues by noting that, if (1) or (2) is true, then my desirability function violates Chance Neutrality. Therefore, it concludes, I am irrational.

However, as Stefánsson and Bradley show, Chance Neutrality is not a requirement of rationality. To do this, they consider a further putative principle, which they call Linearity:

Linearity $$V(\bigwedge^n_{i=1} C^{x_i}_{X_i}) = \sum^n_{i=1} x_iV(X_i)$$ Now, Stefánsson and Bradley show

Theorem Suppose Desirability and the Principal Principle. Then Chance Neutrality entails Linearity.

They then argue that, since Linearity is not a rational requirement, neither can Chance Neutrality be -- since the Principal Principle is a rational requirement, if Chance Neutrality were too, then Linearity would be; and Linearity is not because it is violated in cases of rational preference, such as in the Allais paradox.

Thus, the Swamping Problem in its original form fails. It relies on Chance Neutrality, but Chance Neutrality is not a requirement of rationality. Of course, if we could prove a sort of converse of Stefánsson and Bradley's result, and show that, in the presence of the Principal Principle, Linearity entails Chance Neutrality, then we could show that a value function satisfying (1) is irrational. But we can't prove that converse.

Nonetheless, there is still a problem. For we can show that, in the presence of Desirability and the Principal Principle, Linearity entails that there is no desirability function $V$ that satisfies (1). Of course, given that Linearity is not a requirement of rationality, this does not tell us very much at the moment. But it does when we realise that, while Linearity is not required by rationality, veritists who accept the reliabilist account of justification given above typically do have a desirability function that satisfies Linearity. After all, they value a justified belief because it is reliable -- that is, it has high objective expected epistemic value. That is, they value a belief at its expected epistemic value, which is precisely what Linearity says.

Theorem Suppose $X$ is a proposition in $F$. And suppose $V$ satisfies Desirability, Principal Principle, and Linearity. Then it is not possible that the following are all satisfied: 

• (Monotonicity) $V(X\ \&\ C^x_X)$ and $V(\overline{X}\ \&\ C^x_X)$ are both monotone increasing and non-constant functions of $x$ on $(0, 1)$;
• (Betweenness) There is $0 < x < 1$ such that $V(X) < V(X\ \&\ C^x_X)$.

Proof. We suppose Desirability, Principal Principle, and Linearity throughout. We proceed by reductio. We make the following abbreviations:

• $f(x) = V(X\ \&\ C^x_X)$
• $g(x) = V(\overline{X}\ \&\ C^x_X)$
• $F = V(X)$
• $G = V(\overline{X})$

By assumption, we have:

• (1f) $f$ is a monotone increasing and non-constant function on $(0, 1)$ (by Monotonicity);
• (1g) $g$ is a monotone increasing and non-constant function on $(0, 1)$ (by Monotonicity);
• (2) There is $0 < x < 1$ such that $F < f(x)$ (by Betweenness).

By Desirability, we have $$V(C^x_X) = c(X | C^x_X)V(X\ \&\ C^x_X) + c(\overline{X} | C^x_X) V(\overline{X}\ \&\ C^x_X)$$ By this and the Principal Principle, we have $$V(C^x_X)= x V(X\ \&\ C^x_X) + (1 - x)V(\overline{X}\ \&\ C^x_X)$$ So $V(C^x_X) = xf(x) + (1-x)g(x)$. By Linearity, we have $$V(C^x_X) = x V(X) + (1-x)V(\overline{X})$$ So $V(C^x_X) = xF + (1-x)G$. Thus, for all $0 \leq x \leq 1$, $$x V(X) + (1-x)V(\overline{X}) = x V(X\ \&\ C^x_X) + (1 - x)V(\overline{X}\ \&\ C^x_X)$$ That is,

• (3) $xF + (1-x)G = xf(x) + (1-x)g(x)$

Now, by (3), we have $$g(x) = \frac{x}{1-x}(F - f(x)) + G$$ for $0 \leq x < 1$. Now, by (1f) and (2), there are $x < y < 1$ such that $F < f(x) \leq f(y)$. Thus, $F - f(y) \leq F - f(x) < 0$. And so $$\frac{y}{1-y}(F-f(y)) + G < \frac{x}{1-x}(F-f(x)) + G < 0$$ And thus $g(y) < g(x)$. But this contradicts (1g). Thus, there can be no such pair of functions $f$, $g$. Thus, there can be no such $V$, as required. $\Box$