Tuesday, January 09, 2007

Dummett on modality

[A word of warning, this is not the clearest post.] In a paper called "Could there be Unicorns?" Dummett goes into Kripke's argument in Naming and Necessity that there could not be unicorns. Dummett presents three ways of understanding Kripke's argument and on two of those ways, the argument doesn't work. I didn't understand that part of the paper (nor do I remember it well), so I won't talk about that. In the conclusion Dummett doesn't sound terribly convinced of it either. What is more interesting is some of the background discussion. First, Dummett talks about the revival of modality among philosophers. As the story goes, modal concepts like possible worlds fell into ill-repute among philosophers in the first half of the 20th century, due in large part to arguments by Quine. Modal notions were "spooky" intensional notions and so modal logic was a "spooky" intensional logic. The only good notion was an extensional one, so modal notions were right out. Modal logic and possible worlds were revived, and made palatable, in the 60s and 70s by Kripke's giving a completeness proof and a semantics for modal logic in addition to his Naming and Necessity lectures. Dummett thinks that this story does not get things right. Whatever the reasons for modal logic becoming popular among philosophers again, he thinks that Kripke's work is not relevant. One of the big advances was that the Leibnizian idea of possible worlds could be extended to weaker logics by relativizing the accessibility. Also, Kripke's big proof was for K. However, K is not the logic that most philosophers use. They use an unrestricted accessibility relation, which is what is used in S5. This leads Dummett to say that philosophers don't put any structure into their accessibility relation (which is sort of odd since it is fairly structured, in an equivalence relation kind of way). I think what he means is made clearer when he says that in S5, there is nothing special about the actual world. In general in K or S4, the actual world is special in virtue of what is accessible from it and what is accessible to. At least, that is what I think he means. So, since Kripke's work didn't do anything to make S5 any clearer than it already was, philosophers shouldn't use it as a justification for using possible worlds talk in the sense of S5. This is not to say that S5 is obscure. Dummett seems to think that Leibniz and others made unrestricted modality and possible worlds somewhat respectable, although not completely clear.

The other bit of background that was neat was Dummett's attempt to make sense of the relative accessibility relation. He says that no one has made philosophical sense of relative accessibility. Nuel Belnap seems to agree with him. Mathematically it is clear and elegant, but it is philosophically a bit flaccid. What is Dummett's proposal? While he doesn't explain exactly what the relative accessibility relation could be, he tries to give an example that would motivate having an accessibility relation that is not an equivalence relation. He does this by casting things in terms of states of affairs. Dummett's example motivates rejecting symmetry. A state of affairs S is accessible if a state of affairs T which is presupposed or required by S is in the world under consideration. Suppose we are in a world w that has T and S and is related to a world u that has S that is related to a world v that has neither. The relation is transitive so w is related to v. However, v is not related to w since v does not have S. It could still be related to u, but it is necessary for a world to have S to be related to a world with T. This seemed
somewhat convincing, although he admits the difficulties in extending this to something that would motivate abandoning transitivity.

Finally, Dummett expresses some worries about possible worlds talk in general. To quote, "I believe that the use by philosophers of possible-worlds semantics has done, on balance, more harm than good." In regards to his arguments in the article, Sir Michael says they "may possibly not be watertight".

4 comments:

Aidan said...

"This leads Dummett to say that philosophers don't put any structure into their accessibility relation (which is sort of odd since it is fairly structured, in an equivalence relation kind of way)."

Can't we just ignore accessibility relations in S5? Could that be what Dummett means? (0r is that just the proof systems I'm familiar with?)

Shawn said...

Hi Aidan,
I think you're right on the Dummett interpretation. Since all worlds are accessible from all others, we can ignore the accessibility relation. It isn't relativized at all. This relation is in effect what some philosophers use when they say "accessible world" and then go on to consider all possible worlds.

Dummett's turn of phrase struck me as an odd way to put it, but it does make sense.

1Z said...

It's "Sir Michael" not "Sir Dummett".

Shawn said...

My mistake on the honorifics. I'll adjust that.