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Possible worlds and non-principal ultrafilters

It is natural to think of a possible world as something like an extremely specific story or theory. Unlike an ordinary story or theory, a possible world leaves no question open. If we identify a theory with a set of propositions, a possible world could be defined as a theory T which is

  1. maximally specific: T contains either P or ~P, for every proposition P;
  2. consistent: T does not contain P and ~P, for any proposition P;
  3. closed under conjunction and logical consequence: if T contains both P and Q, then it contains their conjunction P & Q, and if T contains P, and P entails Q, then T contains Q.

It is often useful to go in the other direction and identify propositions with sets of possible worlds. We can then analyse entailment as the subset relation, negation as complement and conjunction as intersection. Of course, we may not want to say that a world is a (non-empty) set of (consistent) propositions and also that a consistent proposition is a non-empty set of worlds, since these sets should eventually bottom out. But that doesn't seem very problematic, and it is easily fixed as long as there is a simple 1-1 correspondence between worlds and logically closed, consistent and maximally specific theories. In particular, one might suspect that on the present definitions, every logically closed, consistent and maximally specific theory uniquely corresponds to a possible world, namely the sole member of the intersection of the theory's members.

Poor one-boxers

Imagine you're a hedonist who doesn't care about other people, nor about your past or your distant future. All you care about is how much money you can spend today. Fortunately, you're on a pension that pays either $100 or $1000 every day, plus an optional bonus. How much you get is determined as follows. Every morning, a psychologist shows up to study your brain. Then he puts two boxes in front of you, one opaque, the other transparent. You can choose to take either both boxes or only the opaque one. The transparent box contains a $10 bill. The opaque box contains nothing if the psychologist has predicted that you will take both boxes; if he has predicted that you will take one box, it contains $100. The psychologist's predictions are about 99% accurate. The content of your boxes is your bonus payment. In addition, you get your ordinary payment, which is either $100 or $1000 depending on how many boxes you took the previous day: if you took both, you now get $1000, otherwise $100. The ordinary payment is given to you before the psychologist studies your brain, so by the time you choose between the two boxes, you already know how much you received. What do you do?

Travel plans

I will probably be in Germany from about mid May until the end of June this year.

Two new papers

One: Variations on a Montagovian Theme.
Two: Belief Dynamics across Fission.

As always, comments are much appreciated.

Self-locating belief and diachronic Dutch Books

If beliefs are modeled by a probability distribution over centered worlds, belief update cannot work simply by conditionalisation. How then does it work? The most popular answer in philosophy goes as follows.

Let P an agent's credence function at time t1, P' the credence function at t2, and E the evidence received at t2. Since E is a centered proposition, it can be true at multiple points within a world. Suppose, however, that the agent assigns probability 0 to worlds at which E is true more than once. Then to compute P', first conditionalise P on the uncentered fragment of E -- i.e. the strongest uncentered proposition entailed by E. This rules out all worlds at which E is true nowhere. Second, move the center of each remaining world to the (unique) point at which E is true.

Assessing the evidence differently

Alice is randomly selected from her population to be tested for a rare genetic disorder that affects about one in 10,000 people. The test is accurate 99 percent of the time, both among subjects that have the disorder and among subjects that don't. Alice's test comes back positive.

Call the information in the previous paragraph E, and suppose it's all you know about the situation. How confident are you that Alice has the disorder?

Letting our subjective probabilities be guided by the stated frequencies, we can use Bayes' Theorem to figure out that P(disorder | positive) = P(positive | disorder) * P(disorder) / (P(positive | disorder) * P(disorder) + P(positive | ~disorder) * P(~disorder)) = 0.99 * 0.0001 / (0.99 * 0.0001 + 0.01 * 0.9999) = 0.0098. Assume then that your degree of belief is about 0.01.

Conditional probabilities and Humphreys' Paradox

Expressions like 'P(A/B)', or 'the probability of A given B', seem to be used in various different ways. On one usage, P(A/B) equals P(AB)/P(B), at least if P(B) > 0. Call this the ratio usage. Simple versions of the ratio usage define P(A/B) as P(AB)/P(B), and so entail that P(A/B) is undefined whenever P(B)=0. But I would like to admit views into the family on which P(A/B) is taken as a primitive binary probability, governed by something like the Popper-Renyi conditions.

Multi-indexing and the intransparancy of truth

One might suggest that for any English sentence S, 'S is true' has the same meaning as S. Assuming compositionality, it would follow that the two are intersubstitutable in every context. But they are not.

First of all, they are not intersubstitutable in attitude reports and speech reports. I don't think this is very problematic because such reports are partly quotational, and of course expressions with the same meaning aren't always intersubstitutable inside quote marks. But 'S is true' and S are also not intersubstitutable in simple intensional contexts, as witnessed by examples like

Paradoxes for "expresses the proposition"

There are familiar semantic paradoxes for "truth" and "reference", such as the Liar paradox and Berry's paradox. I would have thought that there should be similar paradoxes for "expression", i.e. for the relation between a sentence S and the proposition expressed by S. A quick duckduckgo search didn't come up with anything. Pointers?

Here is a Liar-style one I came up with myself. Assume propositions are sets of worlds (which is the case I'm interested in). Consider the sentence

E: E expresses the empty set.

If E is true, then the proposition it expresses contains the actual world, in which case E doesn't express the empty set. So E can't be true. Since we've just proved not-E from no empirical assumptions, ~E expresses the set of all worlds. Hence E expresses the empty set. So E is true. Contradiction.

One more

Yet another paper on counterpart-theoretic semantics: Generalising Kripke Semantics for Quantified Modal Logics. This one is a bit more technical than the others. I use a broadly counterpart-theoretic model theory to construct completeness proofs for very basic quantified modal logics, such as the combination of positive free logic and K. I also play around with adding an object-language substitution operator. There are some unfinished sections at the end, but since I haven't been working on this since January, I thought I might as well upload the current version. All the proofs are spelled out in detail, which makes the whole thing ridiculously long.

I'm not much of a logician, so I'd be very interested to hear if this looks like it is worth pursuing any further.

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