Friday, May 25, 2007

Does Lewis Reject K?

Lewis's concrete modal realism endorses the view that (speaking unrestrictedly) there exists a plurality of possible worlds. By a principle of plenitude, we have that every way something could be is a way that something is (speaking unrestrictedly). Given that a way this world could be is such that it has blue swans, Lewis's view logically entails that (speaking unrestrictedly) there are blue swans. (We might have to pack a lot into the statement of Lewis's view in order to get logical entailment, but I think we can do that in a way Lewis would accept.)

So let's call the relevant theses of Lewisian Modal Realism 'LMR', for short. Given the foregoing point, LMR faces a dilemma. Call the claim that blue swans exist 'BS'. One the basis of the foregoing, we have:

1. LMR --> BS (where '-->' is the arrow of logical entailment)

But by necessitation (N), we have:

2. [](LMR --> BS)

But by K, (3) follows from (2):

3. []LMR --> []BS

But Lewis does not take LMR to be merely contingently true; he aims to give a metaphyiscal account of all of reality (as opposed to what's "merely" actually the case or "merely" what is possible with respect to what is actually the case). So Lewis should accept (4):

4. []LMR

But now it follows from the foregoing that (5) is the case:

5. []BS

But (5) seems to be manifestly false. So either (4), K, or N have to go, or we need to explain how to live with (5).

(This form of the dilemma is from Parsons; Parsons is responding to a proposed solution to this problem from Divers. The presentation of the problem in Divers occurs in his book Possible Worlds but I understand it has an earlier source. I think Parsons' and Divers' proposed solutions are no good and I won't discuss them further in this post.)

I think the responses by a proponent of LMR may differ depending on whether that proponent accepts QML or counterpart theory (CT). Here is what I take to be the best reply for a proponent of QML and LMR:

(5) is true. Since nothing in the semantics of '[]' interacts with the unrestricted quantifier in BS, and since identity is necessary, (5) is a nearly trivial consequence of BS, and BS is entailed by LMR. But we should take care to note that (5) does not commit us to the following:

Domain Existence Principle (DEP): For all individuals x, features F, and domains d, if 'x is F' under the intended interpretation is true at d, then x is a member of/part of/among d.

Consider the spatial case: It's plausible that a proposition that is true at any place is true at every place. (I'm assuming that sentences under intended interpretations are sentences associated with propositions.) So 'Joshua has a head' expresses something that is true at my living room, but, unfortunately, Joshua and his head are not located within (are not members of/are not parts of/are not among the plurality of things in) my living room. So DEP is false.

A more controversial instance of DEP is MEP:

Modal Existence Principle (MEP): For all individuals x, features F, and worlds w, if 'x is F' under the intended interpretation is true at w, then x is a member of/part of/among w.

I think the proponent of LMR should reject (MEP) as well. Lewis does as much in "Postscript to CT and QML": he says that something can be in a world by being either a part of that world, by having a part that is part of that world, or by "belonging to the least restricted domain that is normally--modal metaphysics being deemed abnormal--appropriate in evaluating the truth in that world of quantifications." (Postscript pp. 40 in Phil Papers I.)

So properly understood, (5) amounts to the claim that blue swans are a part of the totality of what is. And that is no shocker, given LMR.

What I take to be the second best response for proponents of LMR and QML basically endorses the first response but adds something about semantic context-sensitivity of quantifier expressions. Since I believe this view is false, I do not believe it is part of the best response. But it also certainly would be part of the story for Lewis, were he to convert to QML and retain LMR.

But what should Lewis say? Lewis was no fan of QML. But we can begin by noting that Lewis recognized that 'Everything actual necessarily exists' comes out true even if something actual lacks a counterpart in another world. Since nothing is special about actuality in obtaining the result, we can obtain a parallel result for w: blue swans are a part of w. So given this, and that Lewis rejects (MEP), one response for Lewis would be just to accept the replies I suggested on behalf of the proponent of QML.

But I don't think Lewis's more mature view would accept that everything actual exists necessarily. In Plurality, Lewis says:

"According to what I said, . . . Humphrey satisfies 'necessarily x exists' and fails to satisfy 'possibly x does not exist' iff he has no counterpart at any world W who does not exist at W. but what can it mean to say that the counterpart is 'at W' if not that, at W, the counterpart exists? So it seems that Humphrey does satisfy 'necessarily x exists' and doesn't satisfy 'possibly x does not exist'. That is wrong. For all his virtues, still it really will not do to elevate Humphrey to the ranks of the Necessary Beings."

"What I want to say, of course, is that Humphrey exists necessarily iff at every world he has some counterpart, which he doesn't; he has the possibility of not existing iff at some world he lacks a counterpart, which he does. It's all very well to say this; but the problem is to square it with my general account of the satisfaction of modal formulas."

". . . Shall we dump the method of counterparts? --That wouldn't help, because we can recreate the problem in a far more neutral framework. . ."

"What is the correct counterpart-theoretic interpretation of the modal formulas of the standard language of quantified modal logic? -- Who cares? We can make them mean whatever we like. We are their master. We needn't be faithful to the meanings we learned at mother's knee--because we didn't. If this language of boxes and diamonds proves to be a clumsy instrument for talking about matters of essence and potentiality, let it go hang. Use the resources of modal realism directly to say what it would mean for Humphrey to be essentially human, or to exist contingently." (pp. 10-13)

So let's take Lewis's advice and use the resources of modal realism directly: 'Necessarily blue swans exist' is false because at some worlds, nothing that is a part of those worlds is a member of the set of blue swans. (Things are complicated somewhat by not just considering an individual, but I don't think this affects any point I want to make substantially.)

But transworld individuals, like the fusion of all possible worlds (reality itself) is in any world by the account of being in a world given in "Postscript" and quoted above. So 'LMR is necessarily true' is true in CT. But this corresponds to the acceptance of the CT translation for (4) above, along with the denial of the CT translation for (5). So given (1), Lewis must reject N or K. I'm thinking if one has to go, it's K.

Lewis might not sweat this much. His dim view of QMP was registered in the above quote, and elsewhere in "Postscript" (pp. 45 in Phil Papers I) he says "if counterpart theory calls for the rejection of some popular modal principles, that needn't worry us." And K is (ahem!) popular. So maybe this is a question of how we proponents of QML should understand Lewis on this issue rather than the basis of some sort of crippling objection to him.

On the other hand, saying things in English that correspond to instances of K sound like ways to say something true in English. We can again follow Lewis and use the resources of LMR directly to say what they mean, but I don't see how we could say something that is compatible with (1), the CT translation of (4), and the CT translation of (5). So I guess I do think there's a problem for him here.

The only other clear option that I see is to cop out and say that when writing Plurality, in particular in the quoted section, his quantifiers were restricted so he said something true when he said that it just wouldn't do to elevate Humphrey to the status of the Necessary Beings. This, I think, would amount to adopting a version of the second strategy for proponents of LMR and QML.

(There are some other options, those disappointments Lewis mentions for a "friend of boxes and diamonds" on pp. 12 of Plurality. I am assuming he does not want any of those disappointments for himself and believes he can avoid them.)

Wednesday, May 02, 2007

The Vagueness of 'Very'

Since nothing's been posted around here recently, I thought I'd post something in an attempt to stimulate conversation. The topic of the post is the vagueness, or lack thereof, of 'very'. Joshua, Andrew, and I discussed this on the way to Syracuse last week, so I'd especially like to get the opinions of others. But Joshua and Andrew are also encouraged to register their thoughts, of course.

It is clear that many predicates of the form 'very F' are vague. For instance, there seem to be borderline cases of being very tall, very nice, and so forth. What is at issue is whether the vagueness of expressions of this form is due at all to the vagueness of 'very' or whether in all such cases the vagueness is due to the adjective 'F' to which 'very' attaches. Or, in other words, the question is whether 'very' is vague at all.

One way to show that 'very' is vague is to discover an adjective 'F' such that 'F' is not vague but 'very F' is vague. Since the vagueness of 'very F' could not then be due to the vagueness of 'F', it must be due to the vagueness of 'very'. But are there any such adjectives?

On the way to Syracuse, Joshua, Andrew, and I puzzled over this for a bit. We each came up with different candidates for an adjective meeting the above condition, but many of the proposed candidates were either plainly vague or not clearly nonvague. However, one of us then suggested 'late'. This seemed like a plausible candidate. It seemed to us that 'late' is not vague but 'very late' is vague.

I would like to know what the rest of you think about this case. I would also like to know whether any of you can come up with a spatial adjective 'F' that seems not to be vague but is such that 'very F' is vague. (One might expect that if there is a temporal adjective, like 'late', that has this feature, then there is a spatial adjective that has it as well.) In addition, I would like to note that some (for instance, Peter Unger) have thought that adjectives like 'flat' are not vague, claiming that a necessary condition on something's being flat is that nothing could be flatter than it. But presumably 'very flat' is vague. Is 'flat' another plausible example that can be used to show that 'very' is vague? (I'm inclined to think not, since I think that Unger was mistaken. But I'm wondering what the rest of you think.)