Tuesday, June 12, 2007

a lewisian response to salmon

for the purposes of this post, let:
<> be the possibility daimond
[] be the necessity box
E be the existential quantifier
A be the universal quantifier

Let me run over Salmon's pen argument again. Suppose it's metaphysically possible for the pen to have originated from 5 blobs of material different than it actually did. We have the actual world @, world w1 at which the pen is from 4 blobs of material different, and w2 at which the pen is 4 blobs different than at w2. Salmon argues as follows:

1) w3 is an impossible world which legitimately exists
2) therefore Ex(x is an impossible world which legitimately exists)

Thus impossible worlds. Relative possibility always sort of bugged me. What's possility if not possibility simpliciter? What does possibly possible mean? Something like it could be possible? Well, the pen argument makes things a little clearer (at least it lit something up in my head).
Take Nathan Salmon. It's impossible for him to be a visa credit-card account. However, it's not impossible for anything to be a visa credit-card account(assuming we're realists about such things). This is possibility/impossibility relative to an object. Now imagine a world with no banks, no credit cards, nothing that could conceivably be a credit card account. It would be true that (N).
(N) ~Ex<>(x is a credit card account)
Unless you believe the barcan formula to be valid, this would not imply (N*)
(N*) ~<>Ex(x is a credit card account)
If (N*) were true, then credic card accounts would be truely impossible objects. As it stands they're only relative impossible objects. Everything's pretty simple so far.
Let @ be a proper name designating the actual world. Let 2x and 3x be complex predicates representing the maximal ways world 2 and world 3 are. Now (W) is true.
(W) ~<>3@
But (W) is silent with respect to (3).
(3) ~<>Ex3x
In fact, (3) is obviously false, since w2 can possibly be the way w3 is. Now, a truely impossible world would fit into be predicated by (I).
(I) Ix iff ~<>ExIx
That is, something is impossible if it's not possible for anything that exists to be that way.

So, as far as Lewis goes, he can respond to Salmon using this machinery. He can claim that he only rejects impossible worlds that satisfy Ix (someday I'll try rephrasing this so I don't quantify over impossibelia). This isn't the logical metaphysical distinction, this is the relative possibility, absolute possibility distinction. He admits the existence of W3, though he may make such moves and denying W3 is a counter-part of @ (this would be denying that @ is possibly the way w3 is).

So what about Salmon's pen? It still has the modal property of not being able to have come from 5 blobs different material. It also has the counter-factual modal property that if it had come from 4 blobs different material, it could have come from 8 blobs different material(as the material it actually came from).

That's my rant. Anyone well-versed in Salmon want to respond? Personally, I'm worried that (I) is trivially un-satisfiable (Lewis might be ok with that), although I'm unsure of how else to represent impossibility simpliciter.


Chris Tillman said...

I'm not really getting the reply here. Salmon holds that there are ways this world could not have been. He takes his table example to show this. Here is the objection:

First we will state a principle concerning origins that we will assume is true:

Table T could have come from slightly different wood from the wood it actually came from, but T could not have come from any wood that is too different.

Let's define an arbitrary cutoff. It does not matter where we make the cutoff since we could re-run the objection using any plausible sharpening of the principle. So let's suppose the cutoff is 5. So here is our principle:

(E) It is possible for T to have originated from 5 (or less) cellulose molecules different hunk of wood compared to the hunk it originated from and it is not possible for T to have originated from more than 5 molecules different hunk from the hunk it originated from.

Now let's describe some worlds:

w1: T originates from hunk H. Assume that w1 is actual.
w2: T originates from H/5, where 'H/5' denotes a particular hunk that is 5 cellulose molecules different from H.
w3: T originates from H/10.

Suppose everything else about w1-3 is just the same insofar as consistency permits.

Now the argument can be stated as follows:

1. If (E) is true, then w2 is a way that w1 can be and w3 is a way that w2 can be and w3 is not a way that w1 can be.

2. If w2 is a way that w1 can be and w3 is a way that w2 can be and w3 is not a way that w1 can be, then w3 is impossible relative to w1.

3. (E) is true.

4. So w3 is impossible relative to w1.

5. If (4), then w3 is impossible.

(Justification: assume (4). Given our assumption that w1 is actual, (4) implies that w3 is actually impossible. Since 'actually P' implies 'P', it follows that (5) is true.)

6. So w3 is impossible.

Note the conclusion is that w3 is impossible simpliciter, not merely that w3 is impossible relative to w1. What it means here to be impossible simpliciter is for something to be incompatible with the way the world is plus the laws of metaphysics (in particular, (E)).

Note, too, that by being more careful we do not need to quantify over impossibilia in order to run the argument. We can put things in a way such that it's clear that the argument shows that a way T could not have been is such that it originated from H/10, though it's possibly possible that T is that way.

The main point of Salmon's argument is not to argue that there are ways things could not have been. Such a conclusion can be supported in a more direct and less contentious way. His main point is to argue that the accessibility relations between worlds that correspond to metaphysical possibility are intransitive. (The countermodel is just that w1 sees w2 and w2 sees w3 but w1 does not see w3.) Given Kripke's results, intransitivity of accessibility is equivalent to a rejection of S4, which says that if P is possibly possible, then P is possible.

Lewis's thought, by contrast, seems to be that Salmon's argument establishes that w3 is among the worlds in logical space. Thus it is possible. Thus the argument is a reductio of (E). Salmon responds to this objection at length in the passages we read from "The Logic of What Might Have Been:" http://links.jstor.org/sici?sici=0031-8108(198901)98%3A1%3C3%3ATLOWMH%3E2.0.CO%3B2-7

Dan said...

>Justification: assume (4). Given >our assumption that w1 is actual, >(4) implies that w3 is actually >impossible. Since 'actually P' >implies 'P', it follows that (5) >is true

I suppose it's this justification that the response is meant to attack. For Lewish 'actually' is indexical, and it was suggested (p 63-64 Divers) that the inference from actually P to P may not be valid under Lewis. Thus, as I stated in the post, there may be ways that THIS world could not be that are not impossible simpliciter.
The intuition pump here is that if something could be F, then being F is not impossible simpliciter. And it seems that W2 could've be like W3 (however you cash out PW, that has to remain true at least in a count-factual sense. Indeed, Salmon's argument depends on it).
So clearly, this response is a denial of (5) in your (Salmon's) argument.
It seems to me we have a choice of denying either
(1) actually P -> P
(2) Ex<>Fx -> F is not impossible simpliciter

Dan said...

That Salmon link didn't work for me

Chris Tillman said...

As stated your (2) does not make sense. And Lewis need not and should not deny that '@P' implies 'P'. In fact, he can't. On his view, if it is true that @P, then the actual world is a member of the set of propositions expressed by 'P'. But that's sufficient for the truth of 'P'. The direction that Lewis seemingly must deny is that 'P' implies '@P' though it's not even clear to me that he has to deny this. (I think it depends on what it takes on Lewis's view for something to be true at a world.)

And Salmon certainly can agree that if something (actual) could be F then being F is not impossible simpliciter. What Salmon denies is that if it could be the case that x could be F, then x could be F. But that's different. Let me put it another way: Salmon denies that, speaking unrestrictedly, if x could be F relative to the way something or other is, then x could be F simpliciter.

I also think the indexicality of 'actual' is playing no role here whatsoever. It's true, on Salmon's view, that 'w3 could actually obtain' is true when evaluated at w2. But this does not imply that it is true that w3 could actually obtain. Compare: 'I have long hair' is true when evaluated at Dan, but it is not true that I have long hair. It's very important not to confuse object language with metalanguage, especially when dealing with indexicals. I'm not sure you're making this mistake but I thought it would be worth mentioning.

Chris Tillman said...

The link was to Salmon's article on JSTOR. You need to be on a computer with JSTOR access to get to it. If you google scholar the title of the article, the JSTOR link should be the first hit.

Dan said...

>And Salmon certainly can agree >that if something (actual) could >be F then being F is not >impossible simpliciter. What >Salmon denies is that if it could >be the case that x could be F, >then x could be F.

But Lewis could go as far as saying if something non-actual could be F, then being F is not impossible simpliciter (since being non-actual doesn't in any way make you ontologically inferior, or even different, on his view). As in there's something really out there that could be F! But on a more sane metaphysics, I take Salmon's point.
Perhaps this points to an even wierder aspect of Lewis's metaphysics. For lewis, the assumption (E) in the first note would be interpreted all wierd like so:
(EL) T has a counter-part that originated from 5 (or less) cellulose molecules that are not counter-parts of T's cellulose molecules. And T does not have a counter-part that originated from more than 5 cellulose molecules that are not counter-parts of T's cellulose molecules.
It's hard to see what sort of counter-part relation is significant here, or how this would fit into salmon's argument, or whether or not it's plausible in the first place.

Chris Tillman said...

But if Salmon's argument is sound, it's false that there's really something out there that could be F. The point is logical, not metaphysical. Someone who accepted Lewis's account of worlds could just as well accept the soundness of Salmon's argument.

Sure, Lewis could hold that if <><>. . .<>P, for any iterations of '<>', then P is not impossible simpliciter. But this just seems like an empty stipulation. It's still the case that if Salmon's argument is sound, then a way the world cannot be is such that P. So ways worlds could be according to which P is the case are not possible ways for the world to be. That seems sufficient for them to be impossible, whether or not there is an attenuated, technical sense in which they are not impossible simpliciter. It seems too much to require that in order for w to be impossible simpliciter, it must be a way no world could be. For trivially, w is that way.