Wolfgang Schwarz

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

Preferring the less reliable method

Compare the following two ways of responding to the weather report's "probability of rain" announcement.

Good: Upon hearing that the probability of rain is x, you come to believe to degree x that it will rain.

Bad: Upon hearing that the probability of rain is x, you become certain that it will rain if x > 0.5, otherwise certain that it won't rain.

The Bad process seems bad, not just because it may lead to bad decisions. It seems epistemically bad to respond to a "70% probability of rain" announcement by becoming absolutely certain that it will rain. The resulting attitude would be unjustified and irrational.

Williamson on modal knowledge

Apropos Williamson. The following question came up last year when we discussed The Philosophy of Philosophy in Canberra. I thought it had a sensible answer that we just couldn't figure out, but then Dorothy Edgington raised the same question at the recent phloxshop workshop in Berlin, and even though there were quite a few Williamsonians present, there was no agreement on what the answer is, and the proposals didn't sound very convincing.

The question is simply how, on Williamson's account, we can have knowledge of substantial metaphysical necessities, e.g. of the fact that gold necessarily has atomic number 79. Williamson explains that when we counterfactually imagine gold having atomic number 78 (knowing that it has number 79), we will "generate a contradiction", because we hold "such constitutive facts [as atomic number] fixed" (p.164). But the distinction between constitutive and not-constitutive facts can hardly be analysed as the distinction between whatever we happen to hold fixed and the rest, given Williamson's commitment to strong mind-independence of metaphysical modality. So what justifies our holding fixed the atomic number?