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One-boxing and objective consequentialism

I've been reading about objective consequentialism lately. It's interesting how pervasive and natural the use of counterfactuals is in this context: what an agent ought to do, people say, is whichever available act would lead to the best outcome (if it were chosen). Nobody thinks that an agent ought to choose whichever act will lead to the best outcome (if it is chosen). The reason is clear: the indicative conditional is information-relative, but the 'ought' of objective consequentialism is not supposed to be information-relative. (That's the point of objective consequentialism.) The 'ought' of objective consequentialism is supposed to take into account all facts, known and unknown. But while it makes perfect sense to ask what would happen under condition C given the totality of facts @, even if @ does not imply C, it arguably makes no sense to ask what will happen under condition C given @, if @ does not imply C.

The probability that if A then B

It has often been pointed out that the probability of an indicative conditional 'if A then B' seems to equal the corresponding conditional probability P(B/A). Similarly, the probability of a subjunctive conditional 'if A were the case then B would be the case' seems to equal the corresponding subjunctive conditional probability P(B//A). Trying to come up with a semantics of conditionals that validates these equalities proves tricky. Nonetheless, people keep trying, buying into all sorts of crazy ideas to make the equalities come out true.

Spelling out a Dutch Book argument

Dutch Book arguments are often used to justify various epistemic norms – in particular, that credences should obey the probability axioms and that they should evolve by condionalization. Roughly speaking, the argument is that if someone were to violate these norms, then they would be prepared to accept bets which amount to a guaranteed loss, and that seems irrational.

But it's hard to spell out how exactly the argument is meant to go. In fact, I'm not aware of any satisfactory statement. Here's my attempt.

Imaginary Foundations

My paper "Imaginary Foundations" has been accepted at Ergo (after rejections from Phil Review, Mind, Phil Studies, PPR, Nous, AJP, and Phil Imprint). The paper has been in the making since 2005, and I'm quite fond of it.

The question I address is simple: how should we model the impact of perceptual experience on rational belief? That is, consider a particular type of experience – individuated either by its phenomenology (what it's like to have the experience) or by its physical features (excitation of receptor cells, or whatever). How should an agent's beliefs change in response to this type of experience?

Simplicity and indifference

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?

Strengthening the prejacent

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.

Long-run arguments for maximizing expected utility

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?

What i and -i could not be

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.)

Might counterfactuals

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'.

A few links

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.

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