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Inadmissible evidence in Bayesian Confirmation Theory

Suppose we are testing statistical models of some physical process -- a certain type of coin toss, say. One of the models in question holds that the probability of heads on each toss is 1/2; another holds that the probability is 1/4. We set up a long run of trials and observe about 50 percent heads. One would hope that this confirms the model according to which the probability of heads is 1/2 over the alternative.

(Subjective) Bayesian confirmation theory says that some evidence E supports some hypothesis H for some agent to the extent that the agent's rational credence C in the hypothesis is increased by the evidence, so that C(H/E) > C(H). We can now verify that observation of 500 heads strongly confirms that the coin is fair, as follows.

Conditional expressions

Most programming languages have conditional operators that combine a (boolean) condition and two singular terms into a singular term. For example, in Python the expression

'hi' if 2 < 7 else 'hello'

is a singular term whose value is the string 'hi' (because 2 < 7). In general, the expression

x if p else y

denotes x in case p is true and otherwise y. So, for example,

Evidentialism and time-slice epistemology

Time-slice epistemology is the idea that epistemic norms are history-independent: whether an agent at a time satisfies an epistemic norm is always determined by the agent's state at that time, irrespective of the agent's earlier states.

One motivation for time-slice epistemology is a kind of internalism, the intuition that agents should not be epistemically constrained by things that are not "accessible" at the relevant time. Plausibly, an agent's earlier beliefs are not always accessible in the relevant sense. If yesterday you learned that yew berries are poisonous but since then forgot that piece of information, it seems odd to demand that your current beliefs and actions should nevertheless be constrained by the lost information.

The broken duplication machine

Fred has bought a duplication machine at a discount from a series in which 50 percent of all machines are broken. If Fred's machine works, it will turn Fred into two identical copies of himself, one emerging on the left, the other on the right. If Fred's machine is broken, he will emerge unchanged and unduplicated either on the left or on the right, but he can't predict where. Fred enters his machine, briefly loses consciousness and then finds himself emerge on the left. In fact, his machine is broken and no duplication event has occurred, but Fred's experiences do not reveal this to him.

Evidentialism, conservatism, skepticism

An evil scientist might have built a brain in vat that has all the experiences you currently have. On the basis of your experiences, you cannot rule out being that brain in a vat. But you can rule out being that scientist. In fact, being that scientist is not a skeptical scenario at all. For example, if the scientist in question suspects that she is a scientist building a brain in a vat, then that would not constitute a skeptical attitude.

Preference and the Principal Principle

Decision theoretic representation theorems show that one can read off an agent's probability and utility functions from their preferences, provided the latter satisfy certain minimal rationality constraints. More substantive rationality constraints should therefore translate into further constraints on preference. What do these constraints look like?

Here are a few steps towards an answer for one particular constraint: a simple form of the Principal Principle. The Principle states that if cr is a rational credence function and ch=p is the hypothesis that p is the chance function, then for any E in the domain of p,

Magic, worlds, numbers, and sets

In On the Plurality or Worlds, Lewis argues that any account of what possible worlds are should explain why possible worlds represent what they represent. I am never quite sure what to make of this point. On the one hand, I have sympathy for the response that possible worlds are ways things might be; they are not things that somehow need to encode or represent how things might be. On the other hand, I can (dimly) see Lewis's point: if we have in our ontology an entity called 'the possibility that there are talking donkeys', surely the entity must have certain features that make it deserve that name. In other words, there should be an answer to the question why this particular entity X, rather than that other entity Y, is the possibility that there are talking donkeys.

The meaning of 'London'

Noam Chomsky's New Horizons in the Study of Language and Mind contains a famous passage about London.

Referring to London, we can be talking about a location or area, people who sometimes live there, the air above it (but not too high), buildings, institutions, etc., in various combinations (as in 'London is so unhappy, ugly, and polluted that it should be destroyed and rebuilt 100 miles away', still being the same city). Such terms as 'London' are used to talk about the actual world, but there neither are nor are believed to be things-in-the-world with the properties of the intricate modes of reference that a city name encapsulates. (p.37)

I don't know what Chomsky is trying to say here, but there is something in the vicinity of his remark that strikes me as true and important. The point is that the reference of 'London' is a complex and subtle matter that is completely obscured when we say that 'London' refers to London.

Please, don't put your papers on acedemia.edu

Every now and then, I come across a link to a paper on academia.edu that looks interesting. I myself don't have an account on academia.edu, and I don't want one. This means that in order to look at the paper, I have to go through the following process.

  1. I click "Download (pdf)".
  2. I get confronted with the message: "You must be logged in to download". I can choose to "connect" with Facebook or Google or create an account manually.
  3. I choose the third option, since I don't want academia.edu to access my Google profile (and I don't have a Facebook account).
  4. Now I have to fill in a form asking for "First Name", "Last Name", "Email" and "Password". I enter random expletives in all the fields because I don't want an academia account, I just want to see the bloody paper.
  5. After submitting that form, I get asked whether I have coauthored a paper in a peer-reviewed journal. I choose "No", fearing that otherwise I'll have to answer more questions about those papers.
  6. Next I'm asked to upload my papers. I don't want to upload any papers, so I click "Skip this step".
  7. Next I have to fill in my university affiliation: "University", "URL", "Department", "Position". I enter random expletives.
  8. Next comes a form where I have to enter my "Research Interests". I enter some expletives. (Turns out my expletives are a popular research interest, shared with 32 others.)
  9. Next I'm told again to "connect" with Facebook, even though I already chose not to at the start. I click "I don't have a Facebook account".
  10. Now, finally, I am presented with a link to the paper I wanted to have a look at.

As you can imagine, I rarely go through all that hassle. Usually I look around if I can find the paper somewhere else and give up if I can't.

Centred propositions and objective epistemology

Given some evidence E and some proposition P, we can ask to what extent E supports P, and thus to what extent an agent should believe P if their only relevant evidence is E. The question may not always have a precise answer, but there are both intuitive and theoretical reasons to assume that the question is meaningful – that there is a kind of (imprecise) "evidential probability" conferred by evidence on propositions. That's why it makes sense to say, for example, that one should proportion one's beliefs to one's evidence.

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