Wolfgang Schwarz

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.

Two papers on counterpart semantics

I've thought a bit about counterpart-theoretic semantics last year, both for natural language and for quantified modal logic. Here's a paper in which I present my preferred version of this framework as applied to natural language: Counterpart Theory and the Paradox of Occasional Identity. Apart from the semantics itself, my main claim is that the advantages of counterpart semantics do not require the metaphysics of "counterpart theory".

Here is another paper which covers related grounds, but from a more logical point of view: How Things are Elsewhere: Adventures in Counterpart Semantics. Comments on either paper are very welcome.

Online Papers Feed and Source

I've just replaced the Online Papers in Philosophy Feed by a newer version. Let me know if you run into any problems with that. (You may also consider switching to a feed from PhilPapers.)

Have I mentioned that the source code for the scripts that generate the feed is on github? Well, now I have.

(While I'm in the swing of mentioning, I might as well also mention (i) that my paper on updating self-locating beliefs is forthcoming in Phil Studies, (ii) that I won't be at the AAP this year, although I will be at various other events, like here, here and there, and (iii) that Holly and I are not "in a relationship" any more. In case you wondered about any of these.)