A survey of basic methods and results in formal epistemology, in probabilistic, discrete, and modal forms.

We'll begin the semester discussing Bayesian/probabilistic models of belief. I aim to shift direction around 33%-50% through the semester to talk about models of belief in terms of sets of sentences/propositions, the most prominent of these being the AGM model. Then we'll shift direction to talk about modal representations of belief and knowledge, in the tradition of von Wright and Hintikka.

If time permits, we'll finish by talking about some issues at the interface between formal and informal epistemology, such as what is the relation between degrees of confidence and "categorical" or all-or-nothing attitudes like belief and agnosticism/suspended judgment; when and why it's helpful to regard beliefs as deductively closed and always consistent; and what it means for something to be evidence for a hypothesis. (Some of these questions will have come up for discussion during the earlier parts of the course, too.)

We'll draw on some literature in decision theory, but that will not be our focus.

The seminar meets on Tuesdays from 10 AM - 12 noon in the NYU Philosophy Department (5 Washington Place), on the 2nd floor.

This seminar is aimed at graduate students in philosophy, or others with comparable preparation. Faculty, post-docs, students from other philosophy departments, and the like, are all welcome to participate.

If you're interested in taking or auditing the seminar, it'd help if you emailed me at jim.pryor@nyu.edu ahead of time so I have some sense of the group before the first meeting. Also, I can then include you on announcements I may send out by email. (Most announcements will only be posted below.)

- Sept 19
- Here are answers to this week's homework. There is no homework for next week. Readings for next week are already posted below; I may also post some more optional reading later.
- Sept 15
- Last week we were talking about different "interpretations of probability," that is, different (alleged) phenomena that one might model using the mathematical framework of probabilities. If you want to read more about this, look at Ch. 7 of Skyrms; pp. 61-80 of Resnik; this SEP article, and here's something on Wikipedia. If you want more, I can supply it.
- The readings for our next meeting are just these notes.
- The exercises for our next meeting are here. They may take you a fair amount of time. If you get stuck, don't be discouraged. The homeworks help us to figure out what's easier and what's harder for the group to catch on to. Come to class with questions.
- For the meeting after, we're going to discuss Dutch Book arguments in favor of having credences that obey the probability calculus, and updating only by conditionalizing. The readings for that will be:
**(added)**Skyrms pp. 137-143,**(added)**Resnik pp. 69-78, (reread) pp. 19-22 of Strevens, read from bottom of third page to end of Easwaran #1, read Hajek 2008. Extra reading (optional for now): Christensen 1991, 1996, 2001. - Sept 12
- Here are my answers to the set theory homeworks. Readings and exercises for next week to be posted later.
- Sept 6
**Readings for week 2**: Chapters 6 and 8 of Skyrms; from bottom of p. 6 to top of p. 17 and from p. 45 to middle p. 61 of Resnik; the first three pages of Easwaran #1 and the first two pages and the section on Bayesian Epistemology of Easwaran #2; and Chapters 1-6 of Strevens' Notes on Bayesian Confirmation Theory. That's a good amount of reading, but there is a lot of overlap in what they discuss, and we'll take as much time as we need to get on top of it.- Sept 5
- If you didn't get an email announcing this website, but want to be put on a mailing list with occasional updates about this seminar, let me know. I'll post a number of readings on this site; you'll be prompted for a username and password to retrieve these. I'll email these to members of the seminar.
- In class today, I distributed this handout on set theory, and this associated homework exercise. Everyone taking the class for credit should do that homework and submit to me by the start of our next meeting (either electronically or you can hand-write your answers if you want). Those auditing the class don't need to do the homework, but unless you're thoroughly familiar with this material (as I know some of you are), it could be a useful exercise.
The packet I distributed in class also had a second homework attached to it, covering some of the same material. You don't have to complete that homework (the answers are already provided), but I recommend looking it over and asking about anything that's unclear.

Here's another handout covering much of the same material. There are homework questions interspersed through that handout, but I'm not expecting you to complete them.

In class I mentioned that Cantor proved that for any set (even infinite sets), the cardinality of its powerset was always larger than that of the set itself. Here's a snippet explaining how that proof goes.

Here's a list/reminder of the notions we discussed in class today. Most of them are explained in some of the handouts above; I've provided links where they aren't:

- sets and set-builder notation: {x
^{2}| x ∈ ℕ and x even} - singleton sets, empty set
- subsets and powersets
- notation for unions and intersections: A ∪ B; ⋃ {A, B}; ⋃
_{i=1}^{n}a_{n}; ⋃_{a ∈ A}Pow(a) - notation like Σ
_{i=1}^{n}i^{2}or Σ_{i ∈ A}i^{2} - the notion of a set's complement
- the size or cardinality of a set, written |A| (although elsewhere in our reading the vertical bars will mean "absolute value")
- countable vs uncountable infinities
- the notions of a binary operation being associative and/or commutative
- the notions of a function being injective/surjective/bijective
- the notions of a relation being reflexive/symmetric/transitive
- permutations: bijections or "shufflings" from a set to itself, including the identity function
- combinations: how many subsets of size k are there from a base set of size n?

Here are some notions it'll also be good for you to be acquainted with, but which we didn't discuss in class. Some of them are addressed in the above handouts:

- partitions and equivalence relations
- inverse function
- composition of functions and relations
- the notion of an interval: this is a set of real numbers between two endpoints. If the set includes the upper endpoint, it's called a closed (on the top) interval, and if not it's called an open (on the top) interval; similarly for the lower endpoint. So for example, {x∈R | 0≤x and x≤1} is closed on both sides. This interval is commonly written as [0,1]. {x∈R | 0≤x and x<1} is closed on the bottom, and open on the top. It's described as "half-open". This interval is commonly written as [0,1). {x∈R | 0<x and x<1} is open on both sides. This interval is commonly written as (0,1).

Notice the unfortunate coincidence between the notation for the open interval (0,1) and (one way of writing) the ordered pair (0,1). Those are very different things. For explicitness, I'll write (0..1) when I want to specify the interval, rather than the more standard notation of just (0,1). The notation [0,1] also has some uses other than the one described here. - the notion of a "supremum"/least upper bound/join, and the notion of a "infimum"/greatest lower bound/meet. I'm not sure whether you'll need to master the exact meaning of these notions; if you do we'll talk about them later. But you may come across them in some reading. As a first pass, you can think of the first notion as usually amounting to a
*union*, and the second notion as usually amounting to an*intersection*. That is, the join of two sets A and B (written A ∨ B) will usually be their union A ∪ B, and so on. - Another notion you may come across is the question of whether a set is convex. This is primarily applied to sets of points on a line, or in a plane, or in a 3-space, and so on. Suppose x and z belong to the set A. We could then ask whether the point directly halfway between them is also in A, and whether the point 1/3 of the way from x to z is in A, and so on. In general, if (0..1) is the open interval between 0 and 1, we can ask whether ∀x,z∈A ∀y∈(0..1) (the point y of the way between x and z is in A). When that's true, then the set A counts as convex. See Wikipedia for some diagrams.

- sets and set-builder notation: {x