Here is the take-home final for those of you taking that.
Here is a model solution for homework 4.
Here is today’s handout.
Here is yesterday’s handout.
Here are our readings for our next meeting:
- SEP entry on fine-tuning, Section 1-4
- Kotzen, “Selection Biases in Likelihood Arguments,” in BJPS 2012
Here is the fourth set of homeworks, due on Fri Dec 1. See below for readings for our next meeting.
Today’s handout is here.
As announced in class, we’ll post the 4th homework prompt in the next few days.
Here are the model solutions for Homework 3.
Our next meeting is in two weeks (after Thanksgiving break). The readings are:
- Roger White, “Evidential Symmetry and Mushy Credence,” in Oxford Studies in Epistemology, Volume 3 (2010)
- Mahtani’s entry in OHFE
Handout from yesterday is available in the paper form we distributed only.
Readings for next week:
- Joyce, “A Nonpragmatic Vindication of Probabilism,” Philosophy of Science 65 (1998)
- Titelbaum book, Ch 10
Here is today’s handout on decision theory.
Readings for next week:
Here is homework 3. ( Now due Nov 10.)
Today’s handout: Jeffrey Conditionalizing.
Here is Matt’s Excel Spreadsheet.
Readings for next week: Thoma’s entry in OHFE
We’ll post Homework 3, which will be due next Friday Nov 3, soon. Note that the due date of that homework is also the last day to have a topic for a final paper approved by us; if you don’t do so you’ll default to the required take-home final exam instead.
Readings for next week:
- Field, “A Note on Jeffrey Conditionalization,” Philosophy of Science 45 (1978)
- Garber, “Field and Jeffrey Conditionalization,” Philosophy of Science 47 (1980)
- Weisberg, “Commutativity or Holism?” in BJPS 2009
Model answers to second homework are here.
Here is today’s handout.
For next week read:
Remember your second homeworks are due on Friday.
Fixed some errors in the presentation of Homework problem 23 (but it doesn’t change the problem’s solution).
Here are the second set of AGM notes:
I also added a hint to Homework problem 23.
In addition to providing a hint on Homework 23, I’ve come to think that the assertion about
Our reading for next week is:
We posted the full second homework set, which is due on Fri Oct 13 (end of next week). Some of the problems are about material that’s on the AGM handout we gave out last week (and linked to below), but haven’t yet talked through. We’ll start with that stuff in this week’s meeting.
Here are the handouts from today’s meeting:
- Webnotes on Orders
- Webnotes Introducing Non-Monotonic Consequence and AGM
The first was already linked last week, but there you have it again so you don’t have to scroll if you’re looking for it. The second we talked through through the “Revision in AGM” section, and started to talk about Contraction. Definitely review the material we’ve already discussed, and bring any questions about it to our next meeting (or ask us beforehand, as you like). You’ll probably also want to read through the rest of the handout at least once before that meeting, so that when we go through it in class you’ll have some anchor points.
Our reading for next time are these two:
- Sturgeon, The Rational Mind, Ch 6; the link contains both Ch. 5 and 6, but we were asking you to read Ch. 5 for this week, and Ch. 6 for next week’s meeting
- Lin, selection from Open Handbook chapter, read only §2 and §5
As we said in class, we’re aiming to release the full next set of homework problems by the end of this week/start of next week. Check back here.
Also feel free to approach us if you’ve got any questions on the first homework. Jim is the one who wrote the feedback on these submissions, but we’re both happy to discuss them with you.
One more thought about preorders vs partial orders. If we want to order people by what their height is, so that Cal ⊏ Logan would say that Cal is less tall than Logan, then presumably we’re going to want to allow ties. Individuals can belong to the same position in the ordering, without being the same people. (Perhaps Cal and Yan are equally tall.) So for such an ordering, we would not want anti-symmetry. It would be only a preorder, not a partial order. (Then there’s the further question of whether the order is connected/total, which in this case it presumably is. This would plausibly be a total preorder.)
On the other hand, suppose we don’t order people by their height, but instead order people’s heights — where these are thought of as numbers or some other kind of abstract mathematical quantity. Presumably here we would want anti-symmetry. If Cal and Yan are equally tall, then Cal’s height would be the very same quantity as Yan’s height. Here we wouldn’t want to allow for the possibility of numerically distinct heights occupying the same position in the ordering.
So whether you say you’ve got a preorder or a partial order will depend on what it is you take yourself to be ordering. The underlying facts about who is how tall remain the same, but if we’re thinking of people ordered by their heights — it’s an ordering of people — the structure is going to look different than if we’re thinking of people’s heights themselves, as separate abstract quantities, being ordered.
Also repeating one thing that came out in class discussion today: in ordinary (or ordinary philosophical) talk, saying that two things are “incomparable” might suggest it’s meaningless or a category mistake to compare them. Thus if I say that Marc is taller than Alex, you’d complain I said something false, but if I say that Marc is taller than middle C, you might want to characterize what I’ve done differently. You might say it’s not even false, or that even if it’s false, it’s worse than false. These ordinary suggestions are not part of the use of “incomparable” in formal literature. In formal literature, to say that a and b are incomparable by a relation ⊑ just means that a ⊑ b and b ⊑ a are both false.
In the homeworks, when asked to prove certain things, some of you gave line-by-line natural deduction style proofs. That format is OK, but usually won’t be expected. If you give a proof of that sort, though, we will be expecting it to meet the correct standards for a formal proof. There tended to be some sloppiness in the proofs that were presented in that way, and we want to be sure to identify and address any confusions you may there be manifesting. Those who gave more informal proofs may have had an easier time, because they weren’t being as explicit. So if the moves they were making were objectively licensed, but they had some background confusion hovering around, we may well not have noticed the confusion because the proofs weren’t as explicit.
One kind of mistake a few of you made was assuming that the domains of some relations had a given number of elements in it (such as more than one, or more than zero). Or assuming that the domain definitely would have an a, b, and c in it such that aRb and bRc. These assumptions could well be false. Your arguments need to be hypothetical on if it’s the case that there are elements such that so-and-so…
Another kind of mistake a few of you made came up for example in the proof that (Α ⨯ Β) ∩ (Α ⨯ Γ) ⊆ Α ⨯ (Β ∩ Γ). You’d say Assume
Good so far. Then you’d say e ∈ (Α ⨯ Β) ∩ (Α ⨯ Γ); then by definition of ∩, e ∈ Α ⨯ Β and e ∈ Α ⨯ Γ.By the definition of
Okay. But you’d continue ⨯, the first tells us that e = some (a,f) such that a ∈ Α and f ∈ Β.By the definition of
Here something dangerous has happened. You’re reusing the variables ⨯, the second tells us that e = some (a,f) such that a ∈ Α and f ∈ Β.a and f but you haven’t established that you’re entitled to assume they’re the same a and f as in the previous step. It’s correct that they will be the same, but you haven’t established it. This sets off a warning bell, because there might come another context where you make such a move and the assumption that the variables co-refer would be incorrect. (I’ll give an example below.)
A safer way to proceed here would be to say instead something like this: By the definition of
Then the proof could proceed as before.⨯, the second tells us that e = some (a′,f′) such that a′ ∈ Α and f′ ∈ Β.Since both pairs are = e, we have that f = f′.
In more-abbreviated, less explicit presentations of the proof, like many of you gave and the model answer also gives, this kind of mistake doesn’t really have space to manifest itself. I’m not saying this to encourage you to give less explicit proofs. I’m just explaining why we might be calling you out on it whereas you see other proofs that seem to just gloss the issue over go unremarked. Our primary pedagogical interest here is not in what you write down, but to make sure that you understand what’s going on. If there might be a fuzzy bit of your thinking here, that doesn’t get you into trouble with this proof but might do so elsewhere, this is a good opportunity to root that fuzziness out.
I promised an example where this kind of issue might be more problematic. This is taken from a math textbook’s report of an actual student homework. The student was trying to prove that a composition of onto functions is onto. That conclusion is correct, but the student’s “proof” is fallacious. The student wrote:
Suppose
f: X → Yandg: Y → Zare both onto. Then by definition offbeing onto,∀y ∈ Y. ∃x ∈ X. f(x) = y; and by definition ofgbeing onto,∀z ∈ Z. ∃y ∈ Y. g(y) = z. So(g ∘ f)(x) = g(f(x)) = g(y) = z, and thusg ∘ fis onto.
Do you see the mistake in this proof? Hint: how would the proof look if in the second sentence, instead of reusing the variable y, the student had used a new variable, say y′? Have they established that the y which is the output of f(x) and the y that when input to g yields z are the same y?
Model answers to the first homework are available here. We’re still reviewing your submissions, but so far everybody seems to be doing fairly well. Such mistakes as there are are minor. Will return your homeworks in class.
Some of you were emailing questions about the order webnotes. Here are some comments that may help. Consider an order like the one we call Example 6:
0 ⊏ 3 ⊏ 4 ⊏ 5 ⊏ ... ⊏ 1 ⊏ 2Two points. First, when we talk about the structure of this order, ignore anything specific about the identities of the different elements. Think of them as just anonymous dots. Any other order that has corresponding elements (also ignoring everything specific about their identity) in the same relative positions would count as having the same structure.
Second, this example is meant to be understood such that there is an infinitely ascending chain of numbers after 3, 4, 5, .... There is a number that comes immediately after 5, and then another that comes immediately after that, and so on forever. Then after all those numbers, there are two extra numbers, 1 and 2. All the other numbers come before them, but there is no number immediately before 1. For any number before 1, such as 9381, there will be other numbers between it and 1. Immediately after 1 comes 2, and then after 2 there are no numbers in the ordering. That is, 2 is a maximal element in this ordering.
Other orderings like Example 3 differ in that they have only one extra number coming after their infinitely ascending chain. The familiar ordering (Example 1) has no extra numbers coming after its infinitely ascending chain. Other examples differ in having two infinitely ascending chains, or infinitely descending chains, and so on. (By “infinitely ascending” I don’t mean anything about which numbers occupy which positions in the ordering. Only that for each number in the chain, there is another that comes immediately next in the ordering, and so on forever. By “infinitely descending”, I mean for each number in the chain, there is another that comes immediately before it in the ordering.)
First homeworks due by the end of the day this Friday. Please email your submission to Matt and Jim both.
Read for next week:
- Webnotes on Orders
- Sturgeon, The Rational Mind, Ch 5
Obviously class not meeting today with the second lockdown. Matt and Jim will figure out how to adjust the schedule, but the immediate upshots are:
- What we were going to discuss today, will be postponed until next week
- But we will still post the first homework here shortly, still to be due at the end of next week. The problems that are due next week only draw on things that we’ve already covered.
Here is the homework.
Here is the handout we discussed in class today, and the diagram that was on the back of the handout (and on the board).
Next week, there’s a little bit more setup material we’ll talk through, but most of the session will be devoted to talking through some puzzles and debates where the modal machinery for representing belief and knowledge are deployed. (Some of these puzzles are also discussed using other tools; it needn’t be that they have to be approached with these formalisms.) All of these readings are short, but the puzzles aren’t easy to solve. We recommend looking at them and thinking about them in the order we’ve listed them. The latter two puzzles draw on developments of the modal machinery that go beyond anything we’ve presented, but hopefully you’ve learned enough of the basics to think in general terms about how the puzzle might look through the lens of a modal treatment:
- Wikipedia on Moore’s Paradox
- Wikipedia on Church/Fitch Paradox
- Wikipedia on Surprise Exam/Hanging
- Fagin, Halpern, Moses, Vardi, Reasoning about Knowledge (1995), selection on muddy children
Next week, we’ll post the first set of homework problems, which will be due the end of the following week (Fri Sept 22).
Since the university was closed yesterday, we’ll continue next Wednesday, Sept 6, discussing the material we were scheduled to discuss this week. So no new readings. We’ve adjusted the syllabus, pushing everything a week later (and adjusting homework due dates). We dropped the unit at the end about prediction versus accommodation, which we regret, but that seemed the best fit overall for our course narrative and the shortened semester. We’ll be glad to discuss those issues and readings with anyone who wants to explore them outside of class.
Here are the handouts that Matt and Jim summarized in class today:
For next week’s meeting, read this chapter: Meyer, “Epistemic Logic,” from Blackwell Guide to Philosophical Logic (in section 9.5, skip all but the first two pages).
(updated) Note also that there’s a typo in LO5 on p. 191 of the Meyer article. As they make clear in the text, they mean to be discussing a thesis with two conjunctions (∧ or &) in it, rather than one with two disjunctions (∨), which is what’s printed. The thesis with two disjunctions in it is weaker and less interesting.