The deontic logic of Desire as Belief
Assume that for any proposition A there is a proposition \(\Box A\) saying that A ought to be the case. One can imagine an agent – call him Frederic – whose only basic desire is that whatever ought to be the case is the case. As a result, he desires any proposition A in proportion to his belief that it ought to be the case:
\[\begin{equation*} (1)\qquad V(A) = Cr(\Box A). \end{equation*} \]Let w be a maximally specific proposition. Such a "world" settles all descriptive and all normative matters. In particular, w entails either \(\Box w\) or \(\neg \Box w\). Suppose w entails \(\Box w\). Does Frederick desire to live in such a world? Yes. On the assumption that w is actual, the entire world is as it ought to be. That's what Frederick wants. So he desires w.
Suppose Frederick desires not to live in any other kind of world – any world w that doesn't entail \(\Box w\). So we have:
\[\begin{equation*} (2)\qquad V(w) = \begin{cases} 1 & \text{ if $w$ entails $\Box w$}\\ 0 & \text{ otherwise}. \end{cases} \end{equation*} \]Define G as \(\{ w \mid V(w) = 1 \}\). From (1) and (2) and Jeffrey's (Jeffrey 1965) theory of V, we get
\[\begin{equation*} \qquad Cr(\Box A) = Cr(G/A). \end{equation*} \](Lewis 1988) points out that this equality between an unconditional and a conditional credence can't hold for all rational credence functions Cr. But it can hold for some particular credence function – for Frederick, say.
But what else would Frederick have to believe?
The characteristic axiom of standard deontic logic is that a proposition and its negation can't both be obligatory:
\[\begin{equation*} (D)\qquad \Box A \rightarrow \neg \Box \neg A. \end{equation*} \]Can Frederick be confident that this is true?
No. We can assume that Frederick gives positive credence to multiple worlds inside G. There must then be a proposition A that entails G for which 0 < Cr(A) < Cr(G). Since A entails G, we have
\[\begin{equation*} \qquad Cr(\Box A) = Cr(G/A) = 1. \end{equation*} \]And since \(Cr(G/\neg A) > 0\),
\[\begin{equation*} \qquad Cr(\Box \neg A) > 0. \end{equation*} \]But if \(Cr(\Box A) = 1\) and \(Cr(\Box \neg A) > 0\), then
\[\begin{equation*} \qquad Cr(\Box A \land \Box \neg A) > 0. \end{equation*} \]So Frederick can't accept (D).
The axiom (D) is somewhat controversial in deontic logic because it seems to rule out moral dilemmas. Some people think that there can be cases in which a proposition and its negation are both obligatory. But it's odd that our assumptions about Frederick entail that he must believe in moral dilemmas!
What went wrong?
There's something odd about (2). There are three kinds of worlds. Some worlds entail that they ought to be the case; some entail that they ought not to be the case; and some entail neither. In the third group, we find worlds whose norms are somewhat liberal: they don't require any specific world to obtain; they merely put constraints on worlds, and they satisfy those constraints. Everything that happens in such worlds conforms to the norms of these worlds. (2) assumes that Frederick doesn't want to live in such a world. This seems wrong. Maybe (2) should be replaced with (2'):
\[\begin{equation*} (2')\qquad V(w) = \begin{cases} 1 & \text{ if $w$ doesn't entail $\Box \neg w$}\\ 0 & \text{ otherwise}. \end{cases} \end{equation*} \]But this doesn't help. The exact same argument as above shows that Frederick can't accept (D).
What if we let Frederick distinguish all three kinds of worlds?
\[\begin{equation*} (2'')\qquad V(w) = \begin{cases} 1 & \text{ if $w$ entails $\Box w$}\\ -1 & \text{ if $w$ entails $\Box \neg w$}\\ 0 & \text{ otherwise}. \end{cases} \end{equation*} \]Then we get
\[\begin{equation*} \qquad Cr(\Box A) = Cr(G/A) - Cr(B/A), \end{equation*} \]where \(G = \{ w \mid V(w) = 1 \}\) and \(B = \{ w \mid V(w) = -1 \}\). It doesn't help. We can still run the exact same argument against (D).