I'd like to try and standardize Fine's argument against "proxy reduction."
Fine is concerned with certain sorts of (actualist) attempts to make sense of possibilist discourse. One option for the actualist is to make sense of possibilist discourse by employing “proxies,” in something like the following manner:
“With each possible x is associated another entity x’, acceptable to the actualist, and any statement Φ(a,b,…) about the possibles a,b,…is then understood in terms of a corresponding statement Φ’(a’,b’,…) about the associated entities a’,b’,…”
According to Fine the most natural way of thinking about the relationship between the entities in the first set and those in the second is in terms of the identity relation. And if that is the assumption, he argues, then the following argument can be presented against any form of actualism employing this form of proxy reduction:
Where Mx: x has the (modal) property of possibly-being-the-world
Rxy: x ‘goes proxy’ for y
w: some possible world w
r: any actualistically acceptable proxy: i.e: a maximal consistent set of propositions or states of affairs; a maximal structural property; or a ‘way a world could be.’
(1) □∀x ∀y [(x = y)-->(Fx-->Fy)]
(2) □∀x ∀y [~(Fx-->Fy)-->~(x=y)] (1) CONTRA
(3) □∀x ∀y [~(~Fx v Fy)-->~(x=y)] (2) IMPL
(4) □∀x ∀y [(Fx & ~Fy)--> ~ (x=y)] (3) DEM & DN
(5) ∀x ∀y [~(x = y)--> ~Rxy]
(6) Mw
(7) ~Mr
(8) Mw & ~Mr (5&6)
(9) ~(w=r) (4,7)
(10)~(r=w) (9)
(11) ~Rrw (5,10)
This seems to be faithful to the English argument given by Fine. I *believe* it is valid as presented. If anybody would comment and let me know if and where I am going off-course with this, it would be appreciated.
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7 comments:
There's some weirdness in your dictionary that's keeping things from being more straightforward than they could be. I'd recommend forgetting about going proxy. I'm not sure anyone knows exactly what that is. Here are some options that I do think make sense: (i) Version v of AR attempts to provide a model of modal reality (in the sense B&Z mention in connection with Lewis's ontological reductions), or (2) Version v of AR gives a metaphysical account of the nature of possible worlds.
I take it Fine's target is the latter. (And generalizations of it to individuals, etc.) His objection is that such identifications must fail, since analysans and analysandums fail to share all properties. So they are not identical (insert whatever entity version v of AR purports to identify worlds with).
The main weirdness with the dictionary comes from the singular terms, I think. So you're allowed two types in QML: individual constants (names) and variables. You can restrict the variables to certain sorts of things or you can effect the restrictions you want by employing appropriate predicates. Examples of the latter:
Suppose we want to say that all worlds are states of affairs. Using unrestricted variables (i.e., variables that the assignment function is allowed to choose anything in the domain as the thing assigned as the value of that variable) we might say this:
Ax (Wx --> Sx)
where 'A' is understood as universal quantification, 'W' expresses the one-place property of being a world, and 'S' expresses the one-place property of being a state of affairs.
Note the variable is unrestricted in the above sense, but we implemented a restriction in the conditional; the conditional applies to everything, but the consequent's applying to something only matters if the thing in question is a world.
This is the standard way to effect restrictions in QML.
A less standard way, but one that can be done if we're more explicit about the particular language we're using, would be to employ two sorts of variables. So for example we can let 'x', 'y', 'z', with or without subscripts, be variables that range over any object that is not a world, while we reserve variables 'w', with or without subscripts, that can be assigned to any member of the domain that is a world and only members of the domain that are worlds. Then we'd write 'every world is a state of affairs' simply as:
Aw(Sw)
using the same interpretation for 'A' and 'S' as before.
It seems like it would be easier to go with the latter option. Can I do the following?
Let the variable 'w' be assigned to any and only those objects in the domain that are possible worlds, and let 's' be assigned to any and only those objects in the domain that are maximal-consistent-states-of-affairs, and let P express the one-place property of possibly-being-the-world. If this is ok, then we can write the (relevant bits of the) argument as:
AwPw
As~Pw
AwAs (Pw &~Ps)
*This would obviously have to be cleaned up to generate the conclusion Fine wants. But I'm just wondering if this would solve the dictionary problem.
Thanks
Would the following work? The move from 1. to 2. is omitted.
Let ‘w’ be a variable that is assigned to only those objects in the domain that are possible worlds, and let ‘s’ be a variable that is assigned to only those objects in the domain that are maximal states of affairs. Let ‘P’ stand for the one place predicate of possibly-being-the-world.
1. □∀x∀y∀F [(x = y)->(Fx->Fy)]
2. □∀x∀y∀F [(Fx & ~Fy)->~(x=y)]
3. ∀wPw
4. ∀s~Ps
5. ∀w∀s(Pw & ~Ps)3,4
6. ∀w∀s~(w=s)
Here's the argument in propositional form.
1. A possible world is possibly the world.
2. No maximal proposition(MP), property(P), state-of-affairs(SA), or way-things-could-be(W) is possibly the world.
3. 1&2
4. (1&2)->there is a possibility for a possible world that is not a possiblity for MP, P, SA, W; namely, possibly-being-the-world.
5. If there is a possibility for a possible world that is not a possiblity for MP, P, SA, W, then a possible world is not MP, P, SA, W.
6. A possible world is not a maximal proposition, maximal property, state-of-affairs, or way-things-could-be.(3-5)
Do you want your conclusion to be that there exists a possible world that fails to be blah or that all things are such that if they are possible worlds then they are not blahs? Reformulate the conclusion to make your intention clear and make the necessary corresponding adjustments to the penultimate premise.
I think a proponent of AR will be able to accept the conclusion of this argument (under either interpretation Chris suggests). The latter interpretation I believe is the intended one. Fine seems to define a possible world as something that is possibly the world. Under that conception and AR proponent may indeed admit that whatever they take as a pw is not a pw in this sense. However, they do not need it to be. They need only admit that there is one possible world in the Kit Fine sense (the actual world), and that this world could have:
PR) had a different state of affairs obtain
BR) had different things be true of it
NR) instantiated a different maximal property
CR) been arranged differently
To claim that there exists something which is possibly the actual world (which in fact isn't) would be an identity nightmare. If the actual world is @, the claim might look something like this:
Ex~(x=@)&<>(x=@)
And the proponent of such a claim would have to sit in the corner (unless they're a counterpart theorist).
I agree with Dan and Chris that the most plausible interpretation of the view that Fine is attacking is (2). But (2) leaves it open as to whether a sentence like
(i) A possible world is a maximal proposition.
employs the 'is' of identity or the 'is' of predication. To say that the view targeted by Fine gives a metaphysical account of the "nature" of possible worlds suggests the latter. But, from what Fine says in the paper, this isn't what he's up to. He's concerned with the best way to translate Lewis-talk into actualist-friendly talk. Any time Lewis says that there exists a possible, non-actual entity @, the actualist interprets this as a statement about some actually-existing entity @', and argues that @=@'.
Dan's second point also seems right: one could avoid the objection by switching to a view that doesn't have the ontological committments that Fine's target has.
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