Wolfgang Schwarz

Frequentism and the end of time

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.

Update

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.

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Quick logic question

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

ANU

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.

Extensionality and Leibniz' Law

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

"Norms of assertion"

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.

Pre-fission possibilities

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.

Imaging, counterfactuals, and expected conditional chance

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.

Centering and self-ascription

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).

Semantic guilt

When reading technical material outside philosophy, I am often struck by the widespread use of non-rigid names and variables. A typical example goes like this. You introduce 'X' to stand for, say, the velocity of some object under investigation. When you want to say that at time t1, the velocity is 10 units, you put exactly this into symbols: 'at t1, X = 10'. If the velocity changes, we get a violation of the necessity of identity:

At t1, X = 10.
At t2, X = 20.

Or suppose you have a population of n objects with various velocities. Your statistics textbook will tell you that the variance of the velocity in the population is defined as