On the modal analysis of belief, 'S believes that p' is true iff p is
true at all possible worlds compatible with S's belief state. So
'believes' is a necessity modal. One might expect there to be a dual
possibility modal, a verb V such that 'S Vs that p' is true iff p is
true at *some* worlds compatible with S's belief state. But there
doesn't seem to be any such verb in English (or German). Why not?

What do we use if we want to say that something is compatible with
someone's beliefs? Suppose at some worlds compatible with Betty's
belief state, it is currently snowing. We could express this by "Betty
does not believe that it is not snowing". But (for some reason) that's
really hard to parse.

Gibbard's 1981 paper "Two recent theories of conditionals" contains
a famous passage about a poker game on a riverboat.

Sly Pete and Mr. Stone are playing poker on a Mississippi
riverboat. It is now up to Pete to call or fold. My henchman Zack sees
Stone's hand, which is quite good, and signals its content to Pete. My
henchman Jack sees both hands, and sees that Pete's hand is rather
low, so that Stone's is the winning hand. At this point, the room is
cleared. A few minutes later, Zack slips me a note which says "If Pete
called, he won," and Jack slips me a note which says "If Pete called,
he lost." I know that these notes both come from my trusted henchmen,
but do not know which of them sent which note. I conclude that Pete
folded.

One puzzle raised by this scenario is that it seems perfectly
appropriate for Zack and Jack to assert the relevant conditionals, and
neither Zack nor Jack has any false information. So it seems that the
conditionals should both be true. But then we'd have to deny that 'if
p then q' and 'if p then not-q' are contrary.

I've been reading about objective consequentialism lately. It's
interesting how pervasive and natural the use of counterfactuals is in
this context: what an agent ought to do, people say, is whichever
available act *would* lead to the best outcome (if it *were*
chosen). Nobody thinks that an agent ought to choose whichever act
*will* lead to the best outcome (if it *is* chosen). The
reason is clear: the indicative conditional is information-relative,
but the 'ought' of objective consequentialism is not supposed to be
information-relative. (That's the point of *objective*
consequentialism.) The 'ought' of objective consequentialism is
supposed to take into account all facts, known and unknown. But while
it makes perfect sense to ask what *would* happen under condition
C given the totality of facts @, even if @ does not imply C, it
arguably makes no sense to ask what *will* happen under condition
C given @, if @ does not imply C.

It has often been pointed out that the probability of an indicative
conditional 'if A then B' seems to equal the corresponding conditional
probability P(B/A). Similarly, the probability of a subjunctive
conditional 'if A were the case then B would be the case' seems to
equal the corresponding subjunctive conditional probability
P(B//A). Trying to come up with a semantics of conditionals that
validates these equalities proves tricky. Nonetheless, people keep
trying, buying into all sorts of crazy ideas to make the equalities
come out true.

I am puzzled about these efforts, for two reasons.

Dutch Book arguments are often used to justify various epistemic
norms – in particular, that credences should obey the
probability axioms and that they should evolve by
condionalization. Roughly speaking, the argument is that if someone
were to violate these norms, then they would be prepared to accept
bets which amount to a guaranteed loss, and that seems
irrational.

But it's hard to spell out how exactly the argument is meant to go. In
fact, I'm not aware of any satisfactory statement. Here's my
attempt.

For concreteness, I'll focus on the argument for probabilism,
but the case of conditionalization is similar.

My paper "Imaginary
Foundations" has been accepted at Ergo (after rejections from
Phil Review, Mind, Phil Studies, PPR, Nous, AJP, and Phil
Imprint). The paper has been in the making since 2005, and I'm quite
fond of it.

The question I address is simple: how should we model the impact of
perceptual experience on rational belief? That is, consider a
particular type of experience – individuated either by its
phenomenology (what it's like to have the experience) or by its
physical features (excitation of receptor cells, or whatever). How
should an agent's beliefs change in response to this type of
experience?

According to the Principle of Indifference, alternative
propositions that are similar in a certain respect should be given
equal prior probability. The tricky part is to explain what should
count as similarity here.

Van Fraassen's cube factory nicely illustrates the problem. A
factory produces cubes with side lengths between 0 and 2 cm, and
consequently with volumes between 0 and 8 cm^3. Given this
information, what is the probability that the next cube that will be
produced has a side length between 0 and 1 cm? Is it 1/2, because the
interval from 0 to 1 is half of the interval from 0 to 2? Or is it
1/8, because a side length of 1 cm means a volume of 1 cm^3, which is
1/8 of the range from 0 to 8?

Sometimes, when we say that someone can (or cannot, or must, or
must not) do P, we really mean that they can (cannot, must, must not)
do Q, where Q is logically stronger than P. By what linguistic
mechanism does this strengthening come about?

**Example 1**. My left arm is paralysed. 'I can't lift my (left)
arm any more', I tell my doctor. In fact, though, I *can* lift
the arm, in the way I can lift a cup: by grabbing it with the other
arm. When I say that I can't lift my left arm, I mean that I can't
lift the arm *actively*, using the muscles in the arm. I said
that I can't do P, but what I meant is that I can't do Q, where Q is
logically stronger than P.

Why maximize expected utility? One supporting consideration that is
occasionally mentioned (although rarely spelled out or properly
discussed) is that maximizing expected utility tends to produce
desirable results in the long run. More specifically, the claim is
something like this:

(*) If you always maximize *expected* utility, then over time
you're likely to maximize *actual* utility.

Since "utility" is (by definition) something you'd rather have more of
than less, (*) does look like a decent consideration in favour of
maximizing expected utility. But is (*) true?

According to realist structuralism, mathematics is the study of
structures. Structures are understood to be special kinds of complex
properties that can be instantiated by particulars together with
relations between these particulars. For example, the field of complex
numbers is assumed to be instantiated by any suitably large collection
of particulars in combination with four operations that satisfy certain
logical constraints. (The four operations correspond to addition,
subtraction, multiplication, and division.)

An allegedly attractive feature of realist structuralism is that it is
faithful to mathematical practice. Unlike various forms of
eliminativism or fictionalism, we can accept mathematical theorems as
literally true statements about an objective, mind-independent part of
reality. Unlike classical Platonism, we don't have to assume that
there is a special realm of abstract particulars. According to realist
structuralism, the number 2 is not a special particular, but a "place
in a structure". In fact, the number 2 figures in different
structures, and thus has different properties depending on whether we
do arithmetic, real analysis, or complex analysis.