GA 1101. Formal Epistemology

GA 1101. Advanced Introduction to Formal Epistemology

Course Description

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.

Meetings and Preparation

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.)

Announcements

Nov 15

The reading for next week is Chapter 6 of Sider's book. Updated: You may find it helpful to read some of Chapter 2 as background: you could skip section 2.5, and skim sections 2.8 and 2.9 of that chapter. In Chapter 6 itself, you can skim section 6.6. We will be doing some of the homeworks from Sider's Chapter 6, but they won't be due this week. Still if you like you can get started on them. I'll ask you to do a certain number of the (unstarred) problems in his Exercise 6.2, 6.3, 6.5, 6.6, 6.6, 6.7, 6.8, 6.9, 6.10. (Sider answers the starred problems in the back of the book.) You'll have the option to choose which problems to work on, so you don't need to worry about solving some of these problems in advance, and then me assigning different ones.

Nov 12

The readings for this week are this Stanford Encyclopedia article and this Wikipedia page. There's no homework due this week.

Oct 31

I fixed the homework solution in light of our discussion in class.

As I said in class, next week we'll finish up our discussion of Harman, then we'll discuss the chapters 3-4 of Christensen, and then we'll discuss some subset of these readings. (I'll nominate one or two for you to focus on by Sunday, but they're all worthwhile.)

Oct 28

For Tuesday, we'll focus on the Foley article (linked below), and chapters 3-4 of Christensen (the whole book is linked here), and Hawthorne's contribution to

this debtate. ~~I may post more/refine this list on Sunday, but this will be a good start for your reading.~~

Oct 24

Here are answers for this week's homework.

Oct 18

Here is the homework for next week.

Here is the large pool of readings for next week. As I said, I'll propose a narrowed version of this to focus on in a few days; but I'm making all the reading available in case you want to browse. ~~I expect we will include at least the Foley reading in the narrower list.~~ Update: we'll read Foley for Oct 31. For Oct 24, read Christensen Chapter 2, and Friedman2013.

And here is the large pool of readings for the following week.

Oct 10

Here are readings for next week, Oct 17: Strevens (Chapter 11), the discussion of the "Old Evidence" and "New Theories" problems in Easwaran #2, Earman Ch. 5 and this paper by Roger White on "prediction versus accommodation."

Oct 9

Again late posting readings: sorry! For this week, it'd be good to re-read and continue in Strevens (Chapters 5-10, especially Ch. 8). Here are two further articles, that present the material in more depth: Earman Ch. 3 and Howson and Urbach Ch. 4.

The following week, I plan for us to talk about The Old Evidence Problem, and the epistemic differences between a theory predicting some new evidence, and the theory being able to accommodate evidence that's already familiar. The week after, I plan for us to talk about the relation between belief and credence. The week after, I plan for us to talk about how plausible it is that epistemic rationality requires our beliefs to be consistent and deductively closed. After that, we'll move to other (non-probabilistic) formalisms for modeling evidence and learning.

Oct 1

Sorry to be so late posting readings. I'll ask you to read one new article for this week's meeting, Ch 5 from Christensen 2004. I also encourage you to re-read and carefully consider the first three sections of Hajek's article, which was assigned for last week. (The later sections in his article concern arguments we're not going to have time to study.)

That Christensen chapter summarizes arguments he develops in the three papers I posted as optional reading back on Sept 15. If you want to read more, I think the 1996 paper is especially useful.

At this point in the seminar, you should be comfortable manipulating the probability calculus and solving problems like the homework you did for week 3. Some of you have previous experience with this, others not. For those of you who aren't yet fully comfortable with this, I'll be happy to answer questions and talk you through some more problems outside of class. But the only way you'll master the material is by investing more time, practice, and attenton into the expository readings I assigned in past weeks. You should especially focus on Chapters 6 and 8 of Skyrms, from p. 45 to middle p. 61 of Resnik, and my notes for week 3. Here is some additional reading covering more or less the same ground: Ch. 2 of Howson and Urbach.

You should also make sure you get to the point in your understanding where you're comfortable following discussions of what bets an agent with certain credences would/should assess as fair. If you're not there yet, then again I encourage you to spend more time reading the expositions assigned for earlier meetings, such as pp. 69-78 of Resnik. Here are some additional readings covering more or less the same ground: Ch. 4 of Gillies, selections from Ch. 2 of Earman, and Appendix 1 from Adams.

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 (this is especially good); this Routledge encylopedia piece (added); and here's something on Wikipedia.

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: {x² | x ∈ ℕ and x even}
singleton sets, empty set
subsets and powersets
notation for unions and intersections: A ∪ B; ⋃ {A, B}; ⋃_i=1ⁿ a_n; ⋃_{a ∈ A} Pow(a)
notation like Σ_i=1ⁿ i² or Σ_{i ∈ A} i²
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.