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.
A might counterfactual is a statement of the form 'if so-and-so were
the case then such-and-such might be the case'. I used to think that
there are different kinds of might counterfactuals: that sometimes
the 'might' takes scope over the entire conditional, and other times
it does not.
For example, suppose we have an indeterministic coin that we don't
toss. In this context, I'd say (1) is true and (2) is false.
(1) If I had tossed the coin it might have landed heads.
(2) If I had tossed the coin it would have landed heads.
These intuitions are controversial. But if they are correct, then the
might counterfactual (1) can't express that the corresponding would
counterfactual is epistemically possible. For we know that the would
counterfactual is false. That is, the 'might' here doesn't scope over
the conditional. Rather, the might counterfactual (1) seems to express
the dual of the would counterfactual (2), as Lewis suggested in
Counterfactuals: 'if A then might B' seems to be equivalent to
'not: if A then would not-B'.
I stumbled across a few interesting free books in the last few days.
1. Tony Roy has a 1051 page introduction
to logic on his homepage, which slowly and evenly proceeds from
formalising ordinary-language arguments all the way to proving
Gödel's second incompleteness theorem. All entirely mainstream
and classical, but it looks nicely presented, with lots of exercises.
2. Ariel Rubinstein has made his six
books available online (in exchange for some personal
information): Bargaining and Markets, A Course in Game
Theory, Modeling Bounded Rationality, Lecture Notes in
Microeconomics, Economic Fables, and the intriguing
Economics and Language, which applies tools from economics to
the study of meaning.
Is 'can' information-sensitive in an interesting way, like 'ought'?
An example of uninteresting information-sensitivity is (1):
(1) If you can lift this backpack, then you can also lift that bag.
Informally speaking, the if-clause takes wide scope in (1). The
truth-value of the consequent 'you can lift that bag' varies from
world to world, and the if-clause directs us to evaluate the statement
at worlds where the antecedent is true.
Here's a more interesting case.
(2a) Fred can dance if nobody's watching.
(2a) Fred can't dance if people are watching.
(More simply: Fred can dance, but not if people are watching.)
Many accounts of deontic modals that have been developed in response
to the miners puzzle have a flaw that I think hasn't been pointed out
yet: they falsely predict that you ought to rescue all the miners.
The miners puzzle goes as follows.
Ten miners are trapped in a shaft and threatened by
rising water. You don't know whether the miners are in shaft A or
in shaft B. You can block the water from entering one shaft, but you
can't block both. If you block the correct shaft, all ten will
survive. If you block the wrong shaft, all of them will die. If you
do nothing, one miner will die.
Let's assume that the right choice in your state of uncertainty is to
do nothing. In that sense, then, (1) is true.
There's something odd about how people usually discuss iterated
prisoner dilemmas (and other such games).
Let's say you and I each have two options: "cooperate" and
"defect". If we both cooperate, we get $10 each; if we both defect, we
get $5 each; if only one of us cooperates, the cooperator gets $0 and
the defector $15.
This game might be called a monetary prisoner dilemma, because
it has the structure of a prisoner dilemma if utility is measured by
monetary payoff. But that's not how utility is usually
According to the "revealed preference" account of orthodox
economics, the utility an agent assigns to an outcome is determined by
her choice dispositions. Thus if you wouldn't choose to defect in the
above game (and your choice can't be explained away as a slip), then
the game isn't a true prisoner dilemma. A true prisoner dilemma
is a game in which defection dominates, even though mutual cooperation
has greater utility for both players than mutual defection.
Suppose you prefer $105 today to $100 tomorrow. You also prefer $105 in 11 days to $100 in 10 days. During the next 10
days, your basic preferences don't change, so that at the end of that
period (on day 10), you still prefer $105 now (on day 10) to $100 the
next day. Your future self then disagrees with your earlier self about
whether it's better to get $105 on day 10 or $100 on day 11.
In economics jargon, your preferences are called time
inconsistent. Time inconsistency is supposed to be a failure of