Monday, September 25, 2006

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.

Friday, September 22, 2006

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.

Thursday, September 21, 2006

Sample Proofs

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

Wednesday, September 20, 2006

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.

Monday, September 18, 2006

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.

Sunday, September 17, 2006

A World of Cabbages and Kings.

In section 1.5 Priest maps out a solution, or rather two sub-solutions, that are meant to take care of the problem of ‘fallacies of relevance’. He notes at the start of the section that, “As things stand so far, Q→Q is true at all worlds. Hence, P→(Q→Q) is a logical truth. That is, given the semantics there are ‘fallacies of relevance’: logical truths of the form A→B where A and B share no propositional parameter. This is counter-intuitive.”(15) As a preliminary point, I am not fully behind Priest’s assumption that this is counter-intuitive. Or rather, why we should expect A and B to share a propositional parameter at all? Question 1 below, further explains this point.

The remaining portion of the section attempts to cleanse our semantics from this fallacy by the “employment of a distinctive kind of world”.(15) He introduces the normal and non-normal worlds. Normal worlds are logically possible and non-normal are logically impossible. That is, non-normal worlds are ones in which the laws of logic do not follow the ‘actual’ logical laws. The goal here (I do believe…) is to introduce and arrange a world within the class of closed worlds, C, in which Q→Q fails. And therefore show that: “…given such impossible worlds, P may hold at one of them where Q→Q fails. At possible worlds, for A→B to be true, we still require that at every (closed) world where A holds B holds. Hence P→(Q→Q) is not a logical truth.”(15-16)

Priest presents two ways of accomplishing this task. We either take the RM way or the ‘other’ way. It’s the ‘other’ way that presents some questions for me. So, I will very loosely sketch out in point form the section about matrices in impossible worlds, from Priest’s own exposition found on pages 16 to 18. Then follow up with a few questions.


The matrix of a formula:

The term t occurs in the formula A(t)

t is free if it contains no occurrence of a free variable that is bound in A(t)
→ so ∫x is free in P∫x but not free in GxP∫x

Conditions for a matrix:
All free terms are variables
No free variable has multiple occurrences
All free variables that occur in it, x1, …, xn, are the least variables greater than all the variables bound in the formula

We then assign the appropriate denotation to the matrix of the formula which occur in the form of A→B.

The truth conditions for conditionals stay the same as those for the set of possible worlds.

And, any matrix of the form A→B is treated as atomic, with assignments of extensions and co-extensions.

So in the end we can assign these ‘atomic formulas’ with the appropriate extension so that Q→Q fails at an impossible world.


Now on to the questions:

1.
When the statement or theorem Q→Q is made, are the propositions Q and Q really the same thing? Is Q really implying itself, or something slightly different than itself? Should or do they really share the same parameter? Take for example the sentence:

If x is a tea pot then x is a tea pot.

When this is taken in the context of a statement, one follows the other. Think of the person making this statement as uttering it in ‘real time’ pointing to a tea pot. They point to the tea pot, make the statement “If x is a tea pot…”, either they choose to keep pointing or re-point to the tea pot, and continue with the sentence, “…then x is a tea pot.” The first, ‘x is a tea pot’, is the question begging tea pot (the antecedent) and the second is the answering / confirming tea pot (the consequent). Perhaps this is a bit of a cheat. It is not the addition of the ‘if’ and the ‘then’ that makes the difference in the ‘x is a tea pot’. At least it is not supposed to be. But, if this is the case (perhaps not for this particular reason) that Q and Q are not the exact same thing, then this fallacy of relevance is not a problem from the get go.

2.
If I understand this correctly, one of the benefits to the impossible world matrices is to take A→B as an atomic formula. By building up a matrix of free variables into a formula, the falsity of A→B as a whole can be obtained at a particular world. The first question that comes to mind is whether such a matrix is at all possible to construct. As the formula is laid out, I don’t see any reason to suppose that a single matrix could not include multiple free variables (as long as they are not identical). If this is the case, then is it not possible to have contradictory variables? You could essentially end up with a formula that has a ‘fuzzy’ condition (which I am all for!) rather than a true / false one. The second question has to do with the main ‘goal’ at hand. The stated intent was to postulate a world in which Q→Q fails to hold, so that P→(Q→Q) is not a logical truth within the closed sphere of worlds. In other words we want:

P → (Q→Q)
(holds) (doesn’t hold)

So that we may show a contradiction. But what Priest seems to illustrate with impossible worlds, is that instead of the above, we get something like this:

{P→(Q→Q)}
(doesn’t hold)

Is this a correct assessment of what is going on? And if so, is this a fair treatment of the relevance fallacy at hand?

Comment Paper on Chapter 1 and 2 of Toward Non-Being

I hope this is where the comment papers are supposed to go on the blog:

Priest builds up his semantics in chapters 1 and 2 of his book, Towards Non-Being, and makes a number of assumptions along the way. Hopefully by tying some of these assumptions together with his semantics and the material in class, Priest’s motivations and the power behind his interpretations will be made more clear. Priest makes the assumption that Noneism is correct (the view that not all objects exist) and that when we quantify over all or some things, this quantification includes objects at all worlds, regardless of their existential status at the actual world. He even goes as far as to introduce new quantifiers so as to not confuse the reader (yes, he did an excellent job there).

Therefore it seems a good fit to attempt to apply this semantics to interpreting some of the paradoxical problems of the kind discussed during the first class.

The problem of negative existentials is likely from where the motivation for Noneism originally came, because it is easy to ascribe the property of non-existing to a non-existent object, as we cannot fail to refer when making mention of it. The sentence ‘Santa Claus does not exist’ is true because, at least when written at the actual world, the object referenced by ‘Santa’ does not exist. This object may have existence at other worlds (possible or not), but that does not matter, as ‘Santa’ is a rigid designator that picks out an object with multiple identities, one for each world. This solution works because Priest would reject the ontological assumption, that for two objects that stand in a certain relation, the relata must exist. Similar solutions hold for the problems raised by satisfaction, empty thoughts, necessary existents and compositionality when non-existent objects are brought in (these being more of the issues brought up in class).

The handling of fictional objects is simple as well, since that for any fiction (consistent or inconsistent), the stories, characters and events will be realized by some possible or impossible world, to which we can refer while discussing those things. For the sentence ‘I believe that Sherlock Holmes was a detective’, we can interpret it as meaning that for the world* where Conan Doyle’s story is realized, the person to which Sherlock refers is a detective, one that doesn’t exist at the actual world.

An intentional context where an exported object is indeterminate can be handled in a similar manner. Consider the familiar sentence ‘I want to own a sloop’. If my interpretation of Priest is correct, he would say that it is true only if for all worlds where sloop ownership is realized by my counterpart/identity at that world, my desire would in fact be satisfied. If at some world, I had a rundown sloop and was not satisfied, then it is not true to say that I want to own a sloop, I must want to own a non-rundown sloop instead. Taking it further, even in cases where all sloops are tragically destroyed at the actual world, it has no effect on the non-actual worlds that do the work of determining the truth of the sentence. Therefore, under Priest’s interpretation, there is no problem with exportation because sloops are exported at every world where my desire is realized.

This should hopefully clear up more of the sorts of problems this semantics is intended to solve.

* Would a fiction be realized by one world, or multiple? There should only be one ‘correct’ interpretation of a fiction, though I suppose something like an ambiguity would fork one interpretation into multiple, both equally correct. But this is beside the point I guess.

Wednesday, September 13, 2006

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.

Monday, September 11, 2006

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.

Wednesday, September 06, 2006

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.)