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.)

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.

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.

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.

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.

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.

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 the last four months I wrote a draft of a possible textbook on
decision theory. Here it is.

I've used these notes as basis for my honours/MSc course "Belief,
Desire, and Rational Choice". They're tailored to my usage, but they
might be useful to others as well.

The decision-theoretic concept of preference is linked to the concepts
of subjective probability and utility by the expected utility
principle:

(EUP) A rational agent prefers X to Y iff the expected
utility of X exceeds the expected utility of Y.

Economists usually take preference to be the more basic concept and
interpret the EUP as an implicit definition of the agent's utilities
(and sometimes also her probabilities).

According to a popular picture, some beliefs are justified by "seemings": under
certain conditions, if it seems to you that P, then you are justified
to believe that P, without the assistance of other beliefs. So
seemings provide a kind of foundation for belief, albeit a fallible
kind of foundation.