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:

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.

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:

(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?


Blogger Dan said...

I agree with Chelsea that a fallacy of relevance isn't entirely counter-intuitive. However, they're not particularly useful either. However, if on his way to countering logical omnicience, closure under entailment and the barcan formulas, he manages to take falacies of relevance down as well I'll shed no tears for them.
As for question 1, the semantics of priests language defeats any such relitivism. Tx (x is a teapot) iff delta(x) (the thing that x refers to) is in the extention of teapothood in world w. Should x refer to something different in two cases of the statement Tx, you would have to state it with two variables, and thus you would have Tx and Ty. In other words, if we keep using the same variables we have to assume they're reffering to the same thing every time.
Priest mentiones the idea of a "fuzzy" formula, as you termed it. A predicate, for example, would be fuzzy if the extension and co-extension were not exclusive. That is, there would be some n-tuples in D that were both in the extension and co-extension (then by definition, be both true and false). Also, if the extension and co-extension were not exhaustive of D (i.e., there were some n-tuples in D that were in neither) than those would be neither true or false with respect to the predicate. Priest treats matrices the same way as predicates in non-logical worlds (that is, he treats them as atomic). Their truth value is thus simply based on the extension and co-extension you wish to assign to them, it can be as fuzzy as you like! Take a matrix C where the extension = co-extension = D(sup n). That matrix would be both true and false with any combination of terms applied to it.
As an end note, I'd like to pose a question. I'm worried that by showing Q → Q is false in some world, priest shows that Q → Q itself is not a logical truth. That is more counter-intuitive than having p → (Q → Q) as a logical truth. As priest says "At possible worlds, for A → B to be true, we still require that every closed world where A holds, B holds". I suppose the answer to that would be, in a world where ~(Q → Q), Q is true, false, neither or both (the matrix of (Q → Q) doesn't seem to be directly dependant on the truth value of Q). In any case, that would not threaten its status as a logical truth in logical worlds. If Q holds in w, then it holds in w, so what if w is a logically inconsistent world in may fail as well.

11:06 PM  

Post a Comment

<< Home