In particular, I'd like to get some feedback on the argument I develop in Section 2. Most people don't seem to take modal possible-world sentences very seriously, but, if they take non-modal possible-world sentences seriously, I think they should. My main reason for thinking so is that, if basic modal sentences (e.g. ‘It is not possible that there are talking donkeys’)) are correctly analyzed as non-modal possible world sentences (i.e. ‘At no possible world, there are talking donkeys’)) (and incidentally I think they are not), then complex modal sentences (e.g. ‘It is possible that it is not possible that there are talking donkeys’)) should be analyzed as modal possible-world sentences (i.e. ‘It is possible that, at no possible world, there are talking donkeys’).

In my argument, I focus on that example and argue that, if 'It is not possible that there are talking donkeys’ is true if and only if there is no possible world at which there are talking donkeys, then ‘It is possible that it is not possible that there are talking donkeys’ is true if and only if it is possible that there is no possible world at which there are talking donkeys.

The argument for, if ‘It is possible that it is not possible that there are talking donkeys’ is true, then it is possible that there is no possible world at which there are talking donkeys goes like this.

- ‘It is possible that it is not possible that there are talking donkeys’ is true. (A)
- [It is necessary that] 'It is not possible that there are talking donkeys’ is true if and only if there is no possible world at which there are talking donkeys. (A)
- For all p, ‘It is possible that p’ is true if and only if it is possible that ‘p’ is true. (A)
- For all p and q, if [it is necessary that] p if and only if q, then it is possible that p if and only if it is possible that q. (A)
- It is possible that ‘It is not possible that there are talking donkeys’ is true. (from 1 and 4)
- It is possible that there is no possible world at which there are talking donkeys. (from 2, 3 and 5)

- It is possible that, at no possible world, there are talking donkeys. (A)
- [It is necessary that] 'It is not possible that there are talking donkeys’ is true if and only if at no possible world, there are talking donkeys. (A)
- For all p and q, if [it is necessary that] p if and only if q, then it is possible that p if and only if it is possible that q. (A)
- For all p, ‘It is possible that p’ is true if and only if it is possible that ‘p’ is true. (A)
- It is possible that 'It is not possible that there are talking donkeys’ is true. (1 and 4).
- 'It is possible that it is not possible that there are talking donkeys’ is true. (2, 3 and 5).

What is the difference between believing that there are possible worlds and believing that modal claims are made true by irreducibly modal features of the actual world? Why couldn't someone say that these features of yours are what possible worlds are composed of? That seems like a not totally unreasonable way to describe the view of David Armstrong, that possible worlds are composed out of universals that are part of the actual world. Perhaps there is something blocking such an interpretation of your features (indeed, perhaps you think Armstrong's universals can't really be used the way he uses them), but I'm very curious as to what it would be.

ReplyDeleteAaron,

ReplyDeleteThere's a lot to say but I don't want to sidetrack the discussion from the main topic of the post. So I'm going to make just to a few quick comments (maybe I'm going to talk about this a bit more some other time).

To your first question--one enormous difference between my view and Armstrong's is that he seem to deny that there are irreducibly modal features of the actual world (and in particular he clearly denies that the fundamental, natural properties are powers).

To your second question--of course, one

couldsay that possible worlds are composed of irreducibly modal features of the actual world (although, as I just said, this is not what Armstrongdoessay), but what role would possible world have in this picture where modal features of the actual world are already doing all the modal truthmaking?Finally, even conceding that Armstrong thinks that "combinatorial" possible worlds are truthmakers for modal sentences and that these possible worlds are possible combinations of features of the actual world, it would not seem to follow that the features of the actual world themselves are the truthmakers of modal truths. So, for example, if it is true that this (green) apple could have been red and what makes it true is that there are "combinatorial" possible worlds in which the state of affairs

this apple's being redobtains, it does not follow that [This apple being red] is made true by the features of the actual world that make up that possible state of affairs (e.g. by this apple and bybeing red) or that it is made true by any actual state of affairs involving either this apple orbeing red.I know this is quick and quite condensed, but I hope it clarifies.

Thanks for the effort, but I guess I have to wait for the future longer discussion. I strongly suspect that the distinction is merely terminological (I tend to agree with the old positivists that that's almost always the case with metaphysics), and your explanation here would only have a chance to relieve that worry if I understood what you mean by "powers." Sadly, being told that Armstrong universals are not powers is not enough for me to be clear on what powers are.

ReplyDeleteHi Gabriele,

ReplyDeleteI've only flicked through the paper quickly, so apologies if you answer this directly!

I'm wondering: What's the difference between saying "There are possible universes" and saying "The accessibility relation between worlds is not S5"?

Here's a situation describable within the Lewisian framework. There are multiple S5-like clusters, i.e. modal regions where accessibility is reflexive, symmetric and transitive. Certain clusters can see other clusters---by which I mean simply that all members of cluster 1 can see cluster 2. But we make no assumptions at all about the general accessibility relations across clusters---so we don't assume that all members of cluster 2 can see cluster 1, etc.

The "clusters" reduce, I take it, to your possible universes? But what I've described was just a particular set-up of accessibility relations in the Lewisian framework.

By-the-by: I think this might undercut your argument in section 2. (In K, possibly possibly p does not reduce to possibly p.) To show that you're maker some assumptions regarding the accessibility relation, consider e.g. principle (4), which you use in your first numbered argument in this post:

(p = q) -> (poss[p] = poss[q])

This fails in K: you need to assume that accessibility is symmetric.

Actually thinking about it more, principle (4) is *really* weird, and fails even in S5. Countermodel:

ReplyDeleteWorld 0: ~p, ~q

World 1: ~p, q

Let World 0 be actual. Then p isn't possible, q is possible, and p = q is true in the actual world.

Hi Tim,

ReplyDeleteThanks for your comments.

Sorry! Obviously, the principle should have read box(p = q) -> (diamond[p] = diamond[q]) (I made the correction in the post and I should state more explicitly in the paper that the argument goes through only if the possible-world analysis is a strict biconditional, but I assume all analyses are).

As for your suggestion about accessibility relations, I'm not sure how it would work, but if it works in such a way that every world bears of fails to bear accessibility relations of different order to other worlds so as to mimic my hierarchy of possible universes of different orders that may be equivalent to my proposal, but, contrary to what you suggest, this is obviously a different analysis of modality from the standard possible-world analysis, as it assumes that there is not only one accessibility relation but multiple accessibility relations and that 'it is possible that it is possible that p' is true if and only if p is the case at a possible world 2-accessible from the actual world).

Moreover, as far as I can see, your picture does not seem to have any advantage over mine. Does it? In fact, it only seems to muddle things more (as if they weren't complicated enough on mine ;-)).

By the way, if the hierarchy of possible universes has the right structure, K would be the complete logical system.

I hope this is sufficiently clear despite the conciseness.

I don't claim this as "my picture". It's Lewis'. It doesn't need two different notions of accessibility, just the one. We identify as a *universe* any set of worlds S such that:

ReplyDelete(1) accessibility restricted to S is an equivalence relation

(2) accessibility restricted to any strict superset of S is not an equivalence relation.

That's what I meant by an S5-like cluster. The advantage is not reinventing the modal wheel.

Interesting paper. To follow up on Tim's comments:

ReplyDeleteWhat's wrong with using accessibility relations? ("It's possible that it's not possible that there are talking donkeys" goes into "There's a world x R-related to the actual world such that there's a world y R-related to that x such that in y there are talking donkeys.") Is the worry that this isn't non-modal?

But what's really the difference between your account and using accessibility relations?

If I understand Tim correctly, he shows how we can construct a universe model from a Kripke-model by taking clusters. We can now say that a cluster S is n+1 accessible from a cluster S' iff there is a cluster S'' such that S'' is n-accessible from S' and S is 1-accessible from S''. So that makes it come out true that possibly possibly p is true iff if p is true at a 2-accessible world.

But we can go the other way too. For take a universe model M. We construct a Kripke-model M' by taking as our worlds the worlds which occur at any level in M. The accessibility relation in M' is taken to be 1-accessibility. M' and M satisfy the same formulae.

Tim,

ReplyDeleteI can't really see how this picture would be in any way equivalent to the one I propose.

For one thing, the account I develop in the paper would hold that the accessibility relation that holds among actually possible first-order universes (the standard possible worlds) cannot be the same accessibility relation that holds among them and any possibly possible first-order universes (possible worlds in the actual Lewisian pluriverse and other possible pluriverses (if there are any)) (or even better it would hold that the same accessibility relation holds between any two nth-order universes within a certain (n+1)th-order universe but not between them and any nth-order universe in some other (n+1)th-order universe)

But, if we stick to the different accessibility relations between different possible universes proposal, the difference between the "first-order" and the "second order" accessibility relations is not that one is an equivalence relation and the other is not (they both are equivalence relations). The difference is that they have different extensions (the n-th order accessibility relation holds between any two possible nth-order universes within the same possible n+1th order universe but does not hold between any two nth-order universes within different n+1th order universes).

In any case, one of the things the account I develop is trying to do is to provide an analysis of modal possible-world sentences such as 'There could have been only one possible world' and I don't see how what you call "Lewis'" picture would do that (and as far as I can see, what you call Lewis proposal is not Lewis' proposal to deal with this problem, which is the one I'm mainly concerned with).

Jon: that's exactly what I have in mind. Thank you.

ReplyDeleteGabriele: not sure what to say any more. Maybe Jon can help me out!

Tim,

ReplyDeleteYou could help by explaining:

(1) how the accessiblity relations labeled 'x is 1-accessbile from y', 'x is 2-accessbile from y', etc. are not different relations as I claim @2:35 and Jon implies @12:02 but you seem to deny in your comment @6:16 (but rereading your comment I suspect you misunderstood my point while I thought you denied it(?))

(2) how the 'x is 2-accessible from y' relation is not an equivalence on my picture as you claim @6:16. On my picture, any world is 2-accessible from itself. For any worlds x and y, if x is 2-accessible from y, then y is 2-accessible from x. And, for any worlds x, y and z, if x is 2-accessible from y, y is 2-accessible from z, then x is 2-accessible from z.

(3) how you plan to deal with what I call modal possible-world sentences. Is it analogously to the way I do by quantifying over clusters? 'There could have been only one possible world' is true if and only if there is a cluster of mutually 1-accessible that are 2-accessible from the actual world at which there is only one possible world? but what about the truth-conditions of 'There could have been no possible world'?).

One last point. I don't think I'm claiming anywhere to have invented anything or that my picture is incompatible with possible world

semantics, I'm only claiming that the best way to makes sense of modal possible-world sentences and complex modal sentences is to reject the standard possible world analysis, which uses one accessibility relation or no accessibility relation at all, in favour of one that uses different accessibility relations (this is why I claim that the possible-universe analysis is not a different analysis of modality but a variant of it). Moreover, note that restricted accessibility relations, when used at all by standard possible world analyses, they are used to make sense of relative modalities not of absolute modalities, as I would be proposing to do.Gabriele: we seem to have different ideas about what would be a good analysis. Tim's point and mine is that your construction is equivalent to the standard analysis using Kripke models. (Equivalent in the sense that any universe-model can be transformed into a Kripke-model making true the same formulae; and vice versa.) Obviously, the structure of the universe-model is rather different, but the claim isn't that the structures are isomorphic. Do you disagree with the equivalence claim? (Perhaps there's an operator on the universe-structures which we cannot "carry over" to the Kripke models?)

ReplyDeleteIf you agree with the equivalence claim, what's the problem with using Kripke models?

About the sentence 'There could have been only one possible world': why is there a problem analyzing this? Why doesn't 'There is an R-accessible world at which the sentence 'there is at most one possible world' is true' work?

Jon: Thanks again! You have expressed very well precisely the concern I have here.

ReplyDeleteGabriele: since Jon is doing better than me at expressing this, I'm going to hand over to him. But, in brief:

(1) Because they're merely restrictions on the single accessibility relation.

(2) Then I misunderstood you in the first place, and I'm sorry for this.

(3) Jon has given the answer I would also want to give:

(A) There could have been only one possible world.

becomes

(B) There is a world, W, such that: (i) W is accessible from the actual world and (ii) no world is accessible from W.

I have no particular thoughts on the truth of (B). But I'm interested in how you'd deal with:

(C) There could have been only one possible universe.

Tim and Jon,

ReplyDeleteI briefly rehash a basic argument against your interpretation of Tim's (A) in Section 3 of the paper. A much more carefully developed argument for the same conclusion can be found in John Diver's (1999) (the argument has a few precursors though). If you are not convinced by what I say in the paper and what Diver's says there, I'm afraid there is little I can say here to persuade you, but I'd be interested to hear why you are not convinced by the argument.

In a nutshell, the basic argument goes like this. Lewis tells us, e.g., that there are many possible worlds beside the actual world, but, say, could have there been only one world? Or, to pick another example, could the accessibility relations among the possible worlds been different from what they are?

Just like questions about how the actual world could have been are not settled just by looking at how the actual world is, so by parity of reasoning, it would seem that questions about how the "actual" pluriverse (i.e. the pluriverse at which the actual world is located) could have ave been are not settled by how the actual pluriverse is.

Note that this is a point about semantics not metaphysics. As a matter of fact the "actual" pluriverse may be the only possible pluriverse and both of the questions above should be answered negatively. But in order for the questions above to even make sense we would have to specify under what conditions they should be answered positively or negatively.

Tim,

Your (C) only makes sense if you specify the order of the universe on my proposal.

(C*) There could have been only one possible n-th order universe

would be

(D) At the actually possible n+2th order universe there is a possible n+1th order universe that contain one and only one possible nth order universe.

As for

(E) There could have been no possible worlds

(which I claimed standard possible-world analyses cannot provide truth-conditions for) would be true on a possible universe analysis if and only if:

(F) At the actually possible third-order, there is a second-order universe that contains no first order universes.

Gabriele,

ReplyDeleteFor what it's worth, here's my reason for not accepting the argument against Tim's/my reading of (A):

There are two different quantifiers over worlds. One is the quantifier over worlds in the meta-language in which we're giving our interpretation of the modal object-language. Then there's the quantifier over worlds as it occurs in the modal object-language, e.g., in "Possibly, there are two different possible worlds".

Suppose I grant that (in the sense of the meta-language quantifier) that at each world there is only one world. Does anything follow about how we should interpret the object-language quantifier over worlds? No. We can give that quantifier any interpretation we like. In particular, we could have it range over all worlds ranged over by the meta-language quantifier; or over those worlds which are accessible. Taking (A) we would get something like this:

(A') There is a world w such that in the domain, D, assigned to w there is at most one world.

If I'm right about this, Tim's analysis of (A) is a bit quick. We cannot take for granted that we have an interpretation of 'for some possible world' as this occurs in the object language. Besides, on Tim's analysis (A) implies that is possibly necessary that 0=1. (Any world which cannot access any other world makes the absurd necessary.)

Jon,

ReplyDeleteFirst of all, I don't think that as far as Lewis is concerned it's a matter of object-language (e.g. ordinary English) versus meta-language (e.g. "Counterpartese") as much as a question of quantifier restriction. So, what I am claiming is that, contrary to what Lewis seems to believe, the quantifiers in, say, 'There is a possible world at which there are blue swans' are still not totally unrestricted. This time however are not restricted to the actual world but to the "actual" pluriverse--i.e. the pluriverse Lewis professed to believe in. Modal sentences about the actual pluriverse, on my proposal, should be interpreted as further removing the restriction and as being about a higher order pluriverse containing the actual pluriverse and any other possible pluriverses (if there are any). The crucial point is that even if the actual pluriverse is the only possible pluriverse, modal sentences about it are not analysable as non-modal sentences about it but as sentences about a higher-order pluriverse that contains only the actual pluriverse. (In analogy with what would be the case if standard possible-world analyses were correct but there was only one possible world--the actual world).

Second, even if the distinction between object-language and metalanguage was relevant here, I don't see how it would help your case. Because what I would be claiming is that the truth conditions for modal sentences in the meta-language should be given in terms of (non-modal) sentences in a meta-meta-language. So, I don's see why the interpretation of the quantifiers in the object-language would matter at all. The modal sentences I am interested would be sentences of the meta-language obtained by seemingly unobjectionable modal principles such as modal ubiquity (every truth is either necessary or contingent).

Gabriele: You write that

ReplyDelete"[W]hat I would be claiming is that the truth conditions for m"I wodal sentences in the meta-language should be given in terms of (non-modal) sentences in a meta-meta-language. So, I don's see why the interpretation of the quantifiers in the object-language would matter at all"

I was probably unclear: the meta-langauge I'm envisaging is non-modal: no modal operators whatsoever. All the advanced modalizing goes on in the object langauge. Since there aren't any modal operators around in the meta-langauge, the question of advanced modalizing doesn't arise

at all.

Doesn't this just define the problem away?

Maybe, but it seems to me that you areforced to do the same thing, that is, take your meta-language to have quantifier over n-pluriverses for all n and have no modal operators. For if you had modal operators in you

meta-language we could do advanced modalizing about higher-order universes. And then the same problems arise all over again.

In your reply to Tim about his sentence

(C) There could have been only one universe

yousuggest that advanced modalizing with (higher-order) pluriverses is unproblematic. You hold thatquantification over universes only makes sense if we specify the order of the universe over which we're quantifying, right? If so there's no problem in seeing how we can get a non-modal paraphrase of a modalized

higher-order universe sentence.

However, we can introduce untyped quantification over universes by using a truth-predicate, and then we can do advanced modalizing about universes. We can, e.g., express:

(1) Possibly, there is a world which is not in any universe.

This goes into:

(1) Possibly, there is a world w such that for all n it is not true that

"w is in a n-th order pluriverse."

How are you going to treat examples like that?

Josh Parsons in "Against Advanced Modalizing", http://pukeko.otago.ac.nz/~jp30/papers/, discusses how the object/meta-language distinction can help with advanced modalizing.

Jon,

ReplyDeleteAlthough I don't see any problems with saying that a certain formal meta-language can be regimented so as not to have any modal operators, I don't understand how you plan to regiment natural language like this and as far as I understand the claims of the Lewisian modal realist makes are claims in English (although a variant of standard English whose quantifiers are not restricted to the actual world). Now, as far as I can see, if someone says that it is the case that p, it is always legitimate to ask whethrer this is so necessarily or contingently (it's the principle of modal ubiquity I was appealing to above). Of course, one can reject that principle but (1) that rejection seems to be rather implasuable and ad hoc to me and (2) Lewis himself seemed to think that possible worlds exist necessarily.

As for the sentence that is supposedly problematic on my proposal (which, let me say once again, as a hardcore actualist I do not endorse) i.e.

Possibly, there is a world that is not in any universe

it would be analyzed as

There is a first-order universe that is not in any nth-order universe for any n.

which, since any universe is in itself, is (necessarily) false.

Its possible that some of this is burdened with David Lewis' vocabulary.

ReplyDeleteDo we "dwell in possibility" or is contingency the nature of system might be a larger question than some specifics.

The dilemma with your first numerated list might be how "if-ey" # 2 is.

In other words, the entire account can be re-assessed in terms of coherency and contingency rather than coherency and variable calculus.

I assess the problem in terms of empirical facts. E.g. is there some way in which an empirical fact is so singular that it cannot exist in combination.

The truth is that combinations have some level of actuality out of combinations, or pragmatic reason, e.g. anomalies are uninteresting, not non-existant, because they are disfunctional.

However, singular evidences are only proven when they exist in one universe or world.

It then goes further to say that the existence of primary facts is a result of a sense-datum or contingent measurement which assures for example the following:

1. That subtleties are more likely to exist in a higher dimensional world

2. That where subtleties exist in high dimension, everything is more likely to exist

3. That where existence is rational, the additional likelihood is also rational.

This makes it impelling that there might be strong arguments (for God) except that quantification is still an advantage in the existence of a datum of substantiality.

What this suggests is that for example, quantity may be the basis for the substantiality of God, whereas subtlety might be the substantiality of the contingency of the worlds.

If subtlety is the contingency, it is too foolish to think that universes are "clear cut" at all; rather we assume differences because we are not dwelling in a higher-dimensional plane.

--Nathan Coppedge