In metaphysics, "Humeans" are people who believe that truths
about laws of nature, counterfactuals, dispositions and the like
(truths about what must or would be the case) are in
some sense reducible to non-modal truths (about what is the
case).
One way to be a Humean is to deny that there are any laws
of natures, non-trivial counterfactuals, etc.: if there are no modal
truths, then trivially all modal truths are reducible to non-modal
truths. On this account, there are no "necessary connections between
distinct existences": eating arsenic might in fact be followed by
death, but it could just as well be followed by hiccups or anything
else.
This paper (recently
featured on the
physics arXiv blog) argues that if the universe never comes to an
end, then the universe will probably come to an end within the next 5 billion
years. The reasoning, as far as I can tell, goes roughly like
this.
First, define the probability of an event of type A given an event
of type B as the total number of A events over the number of B
events. If the universe is infinite, then the total number of A events
and B events will often be infinite. But infinity over infinity isn't
well-defined. So to have well-defined probabilities, the relevant
counts of A and B events must be restricted, e.g. to a finite initial
segment of the universe.
OK. We're back in Canberra. I've also finished the completeness proof
that I've been working on for the last few months. More on that soon. In the
meantime, here are some pictures from this year's bike
trip through the Alps.
.
Suppose you add to the language of first-order logic a sentence operator L for which you stipulate that all instances of
(L(p -> q) & Lp) -> Lq
are valid and that validity is closed under prefixing L's:
if p is valid, then so is Lp.
For example, L could be the modal operator 'necessarily', or it could
mean the same as '
'. If it means the same as
'
', then
Last week I accepted an offer for a post-doc at the ANU, starting in September. I will be working with Al Hajek on "the objects of probability". Should be great.
Extensional contexts are usually defined as positions in a
sentence at which co-refering terms can be substituted without
affecting the truth-value of the sentence. So 'Cicero' occupies an
extensional position in 'Cicero denounced Catiline', but not in
'Philip said that Cicero denounced Catiline'. One might think that a
term t occupies an extensional position in A(t) if and only if all
instances of the following schema are true:
(LL) x=y -> A(x) <-> A(y).
'x=y' is true iff 'x' and 'y' co-refer, and 'A(x) <-> A(y)' is true
iff 'A(x)' and 'A(y)' have the same truth-value. So to say that all
instances of (LL) are true is to say that
->
Two rather different things sometimes seem to go under the name
"norms of assertion", and it might be useful to keep them
apart. Often, e.g. by Williamson, norms of assertion are characterised
as constitutive norms of a particular speech act. Roughly, a
constitutive norm for an activity X is a norm you must obey, or try to
obey, in order to partake in activity X. The rules of chess are a
paradigm example: to play chess, you have to move the pieces in a
particular way across the board. The other kind of "norm of assertion"
would be a genuine social norm that is normally in force when
people make an assertion.
Suppose tonight you will fission into two persons. One of your
successors will wake up Mars and one on Venus. There are then two
possibilities for how things might be for you tomorrow: you
might wake up on Mars, and you might wake up on Venus. These are
distinct centered possibilities that do not correspond to distinct
uncentered possibilties. There is just one possibility for the
world, but two possibilities for you. Indeed, the two possibilities
are two actualities: you will wake up on Mars, and you will
wake up on Venus. It is tempting to go further and say that there are also two
possibilities for you now. I want to discuss three quite
different reasons for making this move.
In today's installment we take a look at the "imaging analysis" of subjunctive conditional probability. We will find that the analysis is fairly empty, and therefore fairly safe. In particular, it seems invulnerable to a worry that Robbie Williams recently raised in a comment on his blog. Let's begin with an example.
What if the government hadn't bailed out the banks? Some
of them would almost certainly have gone bankrupt, and other
companies would probably have followed.
Here we have some sort of conditional probabilities: "if A, then probably/almost certainly C". But they aren't ordinary conditional
probabilities of the kind that go in the ratio formula, P(A/B) =
P(AB)/P(B). I do not believe that if the government actually
didn't bail out the banks (but only made everyone believe it did),
then some of the banks went bankrupt. That is, my ordinary
conditional probability in the bankruptcies given that there was no
bailout is fairly low. Nevertheless, I believe that if the government
hadn't bailed out the banks, some of them would probably have
gone bankrupt. My subjunctive conditional probability in the
bankruptcies given no-bailout is high.
One of the grave threats to the development of mankind in general,
and philosophy in particular, is the assumption that the objects of
propositional attitudes can be expressed by that-clauses. The
assumption is often smuggled in via a definition, e.g. when propositions
are defined as things that are 1) objects of attitudes and 2)
expressed by that-clauses. No effort is made to show that anything
satisfies both (1) and (2) -- let alone that the things that satisfy (1)
coincide with the things that satisfy (2).