Intentional Converse Barcan Formula

Proof: (I'll use 'U' for universal quantifier and 'aBUxAa' for a Bs that all x A x and (Rden. a under s,B) for the accessibility relation for B with respect to the denotation of 'a' relative to s.)

We want to show that aBUxAx entails UxaBAx in Priest. So let a be any term, x be any variable, and let A be any formula. Let P be any class of Priest models and let M be a member of P. We want to show that for every assignment s for M, and every world w in M, if aBUxAx, then UxaBAx. Let s be an assignment for M. Let b be a world in M. We now want to show that if aBUxAx, then UxaBAx in b in M,s. So assume that aBUxAx is true in b in M,s. We now want to show that UxaBAx is true in b in M,s. UxaBAx is true in b in M,s iff for all d in D, aBAx is true in b in M,s(x/d). So let d* be a member of D. We wish to show that aBAx is true in b in M,s(x/d*). aBAx is true in b in M,s(x/d*) iff for all w* in C such that b(Rden. a under s,B)w*, Ax is true in w* under s(x/d*). Let c be a member of C such that b(Rden. a under s,B)c. We wish to show that Ax is true in c in M under s(x/d*). Recall our assumption that aBUxAx is true in b in M,s. This is so iff for all w* in C such that b(Rden. a under s,B)w*, UxAx is true in w* under s(x/d*). Since b(Rden. a under s,B)c, we have that UxAx is true in c under s. But UxAx is true in c under s iff for all d in D, Ax is true in b under M,s(x/d). So Ax is true in C in M,s(x/d*), since d* is a member of D. But this is what we wanted to show. So we are done.

Two Proofs of Logical Omniscience in Priest

Direct Proof: (I'll use 'A' for formulae and 'B' for intentional operators. 'Rden.a in s,B' denotes the accessibility relation for intentional operator B with respect to the denotation of 'a' with respect to s.)

Let P be any class of Priest models and let M be a member of P. We want to show that for every assignment function s for M, and every world w in M, if A is a theorem of Priest, then so is aBA. Let s be an assignment for M. Let b be a world in M. We now wish to show that if A is a theorem of Priest, so is aBA. So assume that A is a theorem of Priest. We now wish to show that aBA is true at b in M,s. According to Priest, aBA is true at b in M,s iff for all w* in C such that b(Rden.a in s,B)w*, A is true in w* under s. So let c be a world such that b(Rden.a in s,B)c. We now wish to show that A is true at c in M,s. But by our assumption, A is a theorem of Priest. It follows that A is true at c in M,s. So aBA is true at b in M,s. So if A is a theorem of Priest, so is aBA. That is what we wished to show, so we are done.

Contrapositive Proof (Priest's Proof):

We want to show that if A is a theorem of Priest, then so is aBA. We will prove this indirectly by supposing that aBA is not a theorem of Priest and proving, on that assumption, that A is not a theorem of Priest. So suppose aBA is not a theorem of Priest. Then there is a model M of Priest, and assignment function s, and a possible world b in M such that aBA is false in b under M,s. According to Priest, aBA is false at b in M,s iff for some w* in C such that b(Rden.a in s,B)w*, A is false in w*. So let c be a world such that b(Rden.a in s,B)c. A is false at c in M,s. According to our definitions, A is a theorem of Priest iff A is trhe wiht respect to every model of Priest and every assignment of free variables. But this does not hold for A since A is false in c under M,s. So A is not a theorem of Priest. So we are done.

Notation

I got a question concerning notation for evaluating multiple variables with respect to an assignment. In case others were unsure about what to do, I thought I would post the standard notation. g(x/d1) is the function just like g except it assigns d1 to 'x'. g(x/d1)(y/d2) is the function just like g except that it assigns d1 to 'x' and d2 to 'y'. So, for example, when working on problem 4, you will need to consider g(x/d) for all d in D. In order to do that, you'll need to consider g(x/d)(y/d) for some d in D. In M, there are four options here: g(x/d1)(y/d1), g(x/d1)(y/d2), g(x/d2)(y/d1), and g(x/d2)(y/d2). Hopefully this will make sense when you get to problem 4. If not, comment on this post.

I also received a question about how to think about the terms in FOPL. It is helpful to think that individual constants behave like proper names in English and variables behave like indexicals or demonstratives. So here are two ways of saying the same thing:

Albert is fat.
That is fat. (or 'He is fat'.)

The first may be symbolized in a language like FOPL as 'Fa'; the second as 'Fx'. You can think of the assignment function as something like a context; with respect to different contexts, the second displayed sentence above will say different things. But (ignoring the fact that many people have the same name) the first displayed sentence says the same thing with respect to any context. So a "model" of English would determine the meaning of the first displayed sentence all by itself. But it would not determine the second; for that, we need a context (assignment).

If this last bit is more confusing than helpful, then just ignore it.

Sample Proofs

Sample proofs for Problem Set 1 are with the hardcopies of the readings on the 4th floor of University College.

Logic Fonts

In case you would like to type the problem set solutions in Word, it may help to download some math fonts that contain many of the symbols we've been employing. The fonts are available here.

Problem Set 1

For problem set 1, please show whether or not the following wffs are true with respect to model M and assignment g:

1. Gab
2. (crazy)Ux(Fx)
3. (crazy)Gx(Gxx)
4. (crazy)Ux(crazy)Gy(Gxy)

M = [D, delta], where
D = {d1,d2}, and
delta is the function such that:

a. delta(a) = d1,
delta(b) = d2,
and for all other individual constants alpha, delta(alpha) = d1.

b. delta(F) = {d1},
delta(G) = {[d1,d1], [d1,d2], [d2,d1]}

g is the function such that:
g(x) = d1,
g(y) = d2,
and for all other variables alpha, g(alpha) = d1.

Feel free to post questions or comments, but do not post your answers! Email them to me or print them off and slide them under my office door by 5pm Sunday.

Note: I'm using square brackets '[' and ']' for ordered pairs since Blogger takes the corner braces as html and strips them out.

Comment Paper 1

I've received a few questions about the nature of the comment paper. Let me try to address them. First, don't worry if you don't understand all or even most of the material in Chapters 1 and 2. The point of reading them now is to begin to familiarize yourself with Priest's ideas. You are not at all expected to be an expert or even fluent. Fluency (or at least comprehension) is the goal of presenting the logical material in class. Many things in the first two chapters we haven't had a chance to discuss. So don't worry if they are opaque to you. But it is worth emphasizing that there are several points discussed in the first two chapters that do not require technical sophistication to grasp. For your first comment paper, focus on one of these points and don't worry about the symbol nightmare. Good strategies for writing comment papers involve either (1) clarification: show how some point made by Priest can be made more precise; (2) criticism: criticize some aspect of the material in Priest; (3) question: if you have a question that came up in the reading, state that question and consider possible answers to it; (4) comment: say something relevant about a point in Priest that is neither a clarification, a criticism, or question.

Any of (1-4) and combinations of (1-4) are good ideas for comment papers. If you have further questions or concerns about comment papers, comment on this post.

I Probably Shouldn't Be Doing This But . . .

I think it's important that students are aware of this resource. Here is an example of what it can do for you.

Announcement

I forgot to mention in class that there will be a "meet and greet" to which you are all invited on Thursday, Sept. 14, 2:30-4:00 in the Senior Common Room, 200 University College. One may wish to come in order to meet department staff and each other, imbibe refreshments, or consume nibbles both hot, cold, and vegetarian.

In case you need a contact person (whether or not there is no contact person such that you need him/her), you may contact Laurie or Janelle at 453 University College, 4-6878.

I hope you all can come!

For Monday, Sept 18

Please read as much as you can of chapters 1 and 2 of Priest's book. Try to formulate specific questions about what, if anything, you don't understand in those chapters. For your comment paper, please write a short (1-2) page paper on some aspect of the readings.

Additional readings for those who are interested are linked from the previous post. Hard copies will be available shortly in the cubbies (or whatever they are) on the 4th floor of University College.

Feel free to comment on this post with any questions or concerns that arise from class or from the readings or to create your own post.

Required and Suggested Readings for Week 2

All readings other than those from Priest's book are available for photocopying in the philosophy department (4th floor of University College). A number of them are also available online. (In many cases they are only accessible from a campus computer.) Among them: Ben Caplan's "Existence", Nathan Salmon's "Existence", Nathan Salmon's "How to Become a Millian Heir", and Gottlob Frege's "On Sense and Reference".

(nb: I often have trouble accessing papers by Caplan. If you have difficulty, right click on the link and select "Save Target As" to download the paper.)

(nb2: The Salmon papers require JSTOR access. You should have access to them from any campus computer. The Caplan paper, and I think the Frege paper, should be accessible from any computer connected to the internet.)