### On the proposition that there are a plurality of worlds

David Lewis thinks that there are many concrete possible worlds (at least one for each de dicto possibility) and propositions are sets of those worlds. I disagree. I will present two dilemmas against this view.

It seems clear that the following is true:

1. Possibly, there are talking donkeys.

On Lewis's view, this means that there is a world and there are talking donkeys in that world. But, this implies that:

2. There are (quantifiers wide open) talking donkeys.

This is a proposition. So, according to Lewis, it must be a set of possible worlds. But which set is it? As I see it, there are only two options. If propositions are sets of worlds, then either (2) is the set of worlds that contain (within their worldly boundaries) talking donkeys, or (2) is the set of all worlds.

Consider the first option: that (2) is the set of worlds that contain (within their worldly boundaries) talking donkeys. This option can't be right. After all, (2) is true. Moreover, for any proposition p, if p is true and propositions are sets of worlds, then the actual world is a member of p. The actual world, though, is not a member of the set of worlds that contain talking donkeys. So, if propositions are sets of worlds, then (2) is not the set of worlds that contain talking donkeys (within their worldly boundaries).

The only other plausible option is that (2) is the set of all worlds. This seems plausible since (given that the quantifiers in (2) are wide open) it is true at any world. But, then (2) is the same as the proposition that arithmetic is incomplete. But, that is absurd. So, (2) is not the set of all worlds.

Even if you deny the absurdity of identifying (2) with the proposition that arithmetic is incomplete, there are still problems for the second option. Consider the following proposition:

3. There are (quantifiers wide open) cats.

Since the quantifiers in (3) are wide open, just like those in (2), then it seems we have no reason to treat them differently. No reason to say, for example, that (2) is a set of worlds that contain cats whereas (3) is the set of all worlds. So, we should treat them the same. If (2) is the set of all worlds, (as the second option indicates), then it must be that (3) is the set of all worlds too. So, if (2) is the set of all worlds, then (2) is identical to (3). But, clearly those are different propositions. I believe (3) yet I think that (2) is false.

To recap, if propositions are sets of worlds, then (3) is either the set of all worlds that contain talking donkeys or the set of absolutely all worlds. But, it is not the set of all worlds that contain talking donkeys and it is not the set of absolutely all worlds. So, propositions are not sets of worlds.

That's the end of the first dilemma against the Lewisian view. Here, then, is the second dilemma. Consider Lewis's modal realist thesis itself:

PW: There are (quantifiers wide open) many concrete possible worlds.

If (PW) is true and propositions are sets of worlds, then either (PW) is the set of worlds that contain (within their worldly boundaries) many concrete possible worlds or it is the set of all worlds. If (PW) is a set of worlds that contain (within their worldly boundaries) many concrete possible worlds, then it is the empty set (since no world contains many concrete possible worlds). But, if it is the empty set, then it is false (since on the sets-of-worlds view of propositions, a proposition is true iff the actual world is a member of the set that is that proposition). So, if (PW) is true, (PW) is not the set of possible worlds that contain many concrete possible worlds.

Perhaps (PW) is the set of all worlds. But, if that is the case, then (PW) is identical to the proposition that 2+2=4. But, it if it is identical to the proposition that 2+2=4, then I believe (PW) (since I believe that 2+2=4). But, I don't believe (PW). So, (PW) is not a set of all worlds.

It follows that either (PW) is not true or propositions are not sets of worlds. If propositions are sets of worlds, then (PW) is false. But, since the view that propositions are sets of worlds is true only if (PW) is true (the view makes sense only if (PW) is true), it follows that propositions are not sets of worlds.

## 4 Comments:

Regarding your first dilemma, I think it is clear that, on David Lewis's view, since (1) expresses a necessary truth, and there is only one necessary proposition (the set of all possible worlds), the proposition expressed by (1) is the set of all possible worlds. Likewise for (2) and (3).

So I think you are right to reject the first horn of the dilemma, but I think it is less clear that Lewis's view will fall prey to the objection you use against the second horn of the dilemma. I'm not sure but I think Lewis's paper "What Puzzling Pierre Does Not Believe" might shed some light on the sort of maneuvers he might employ to respond to that objection (see also Mike McGlone's Lewis on What Puzzling Pierre Does Not Believe).

The proposition one normally asserts by saying "there are cats" is the set of worlds at which "there are actually cats" comes out true, namely, the set of concrete possible worlds containing cats.

Similar considerations apply to the second dilemma. Any time you specify "quantifiers wide open" for Lewis, you will be identifying a proposition that is (on his view) either necessarily true (and thus, the set of all worlds) or necessarily false (and thus, the empty set). I'm not sure that the objection has more force when you demonstrate that (on Lewis's view) some metaphysical proposition winds up being the same proposition as a mathematical proposition, than the version of the objection which points out that I don't believe arbitrary mathematical truths in virtue of believing that 1+1=2.

That said, I'm definitely sympathetic to the objection you offer to the latter horn of each dilemma.

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Hi Joshua,

It's clear from

On the Plurality of Worldsthat Lewis (i) thinks that different sets satisfy the different roles associated with 'proposition', though all of them are sets containing possibilia in their transitive closure; (ii) sets of possible worlds satisfy the 'truth conditions' role; but (iii) structured intensions play something like the 'cognitive significance' role (1986: 55-59). So I imagine that Lewis would respond to both dilemmas by grasping the second horn, and argue that nothing absurd follows so long as 'proposition' is used univocally.But that response is boring. Here's a more exciting response: modify which types of possibilia get to be members of the sets that (for Lewis) are the propositions, and then modify the account of 'truth at a world' accordingly. As follows:

(P) Associated with any proposition

Pis a unique set of possibilia: propositions are sets of possible worldsand proper parts thereof.(TW) Proposition

Pis true at a possible world w iff either w orsome proper partof w is a member ofP.Now, (P) is something found in Lewis (1979): for him, the contents of propositional attitudes are properties, which for Lewis are sets of possibilia. (TW) seems to me the obvious way to modify the account of 'truth at a world' in light of (P).

How would this help the Lewisian? Here's the thought. We take the proposition 'there are [quantifier unrestricted] talking donkeys' to be the set of all talking donkeys (these represent worlds in which there are [quantifier restricted to the world of utterance] talking donkeys) and all possible worlds in which there are no [quantifier restricted to the world of utterance] talking donkeys. Likewise for the proposition 'there are [quantifier unrestricted] cats'.

So take, for example, the actual world: applying (TW) the first proposition (call it

P) is true at the actual world in virtue of the fact that the actual world itself is a member ofP. In contrast, second proposition (call itQ) is true at the actual world in virtue of the fact that some of the proper parts of the actual world (namely, the actual cats) are members ofQ.So (P) and (TW) allow both propositions to be true at the actual world. But clearly the two sets are distinct: so if the Lewisian can let these sets be the two propositions expressed by the respective sentences, then the Lewisian can deny that the two sentences express the same proposition.

Lewis,

Thanks for the comment. I found McGlone's paper very helpful.

Alex,

I like your idea. If, in giving a theory of propositions, we are want to identify propositions with entities that can play the appropriate role in our theories of propositional attitudes, then it seems like the view you suggest does a better job than the theory that propositions are merely sets of worlds.

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