Lewisian realism and modal reduction

I’ve posted a draft of a new paper that, among other things, defends Lewis against the charge that he needs to employ primitive modality in order for his modal realism to be successful, thus undermining his claims to reduction. One thing I argue is that Lewis’s objectors fail to adequately distinguish two tasks: giving an account of what possibility is, and giving an account of the extent of possibility. The tasks are crucially different and, in my opinion, neither requires for its success the meeting of the other. In particular, an account of what possibility is can stay silent on the extent of what is possible.

So consider the Lycan/Shalkowski objection that Lewis needs a modal understanding of ‘world’ to ensure that there is the correct correspondence between worlds and possibilities, necessary for the material adequacy of Lewis’s account of possibility as truth at a world. Lycan says that Lewis needs ‘world’ to mean ‘possible world’ to rule out the inclusion of impossible worlds in Lewis’s ontology. Shalkowski says Lewis needs the notion of a world to be modal to ensure that the space of worlds is complete: that there are no worlds missing.

I think that’s wrong. What ensures that there are no impossible worlds is Lewis’s account of what possibility is. To be possible just is to be true at a world, so there’s simply no question of there being an impossible world for Lewis. Whatever worlds there happen to be, those worlds will all be possible and none of them impossible, because that’s just what possibility is. Similarly, there’s no question of there being a world missing – of there being a possible circumstance with no corresponding world. But what accomplishes this is not a modal understanding of ‘world’ but, again, Lewis’s account of what possibility is.

It just falls out from Lewis’s analysis that there’s no impossible world, and no possible circumstance unrepresented by a world. Now, here’s what doesn’t fall out from the analysis: that there’s no world with a round square as a part, or that there’s a world with a talking donkey as a part. But contra what Lycan and Shalkowski think, this doesn’t mean that Lewis’s analysis leaves it open that there are impossible worlds or not worlds enough for possibility. If it turns out that there’s a world containing round squares then this is not for it to turn out that there’s an impossible world, according to Lewis’s analysis – it’s for it to turn out that round squares are possible after all! Likewise, mutatis mutandis, if it turns out that there’s no world containing a talking donkey.

Now, Lycan and Shalkowski might complain that any analysis of modality that says that round squares are possible and talking donkeys impossible is not acceptable. Well maybe that’s right. But Lewis’s analysis of course doesn’t say this: it just doesn’t settle that round square are impossible or talking donkeys possible. But that’s fine: the account of what possibility is needn’t settle these claims about the extent of possibility. To demand that Lewis’s analysis settle these facts is to demand too much of analysis: it’s to confuse the two tasks that should be kept separate.

You might think that we need to be able to acquire warrant for thinking that there are no worlds with round squares and that there are worlds with talking donkeys if Lewis’s analysis is to be warranted in the first place. Well, again, that’s fine: Lewis has given us an argument for thinking that the space of worlds is like this. (Namely, that the posit that it is so is theoretically beneficial.) But it’s nothing about the meaning of ‘world’ or the nature of worlds that settles that the space of worlds is so, and nor need it be, since an account of what possibility is needn’t entail an account of the extent of possibility.

I think a similar thing is going on in Divers and Melia’s objection to Lewisian realism. Their argument is as follows. They assume that it’s possible for there to be alien natural properties, and so Lewis’s principle of recombination doesn’t give us a complete account of what worlds there are. Now, it seems that if there could be alien natural properties, there should be no finite bound on the number of possible alien natural properties out there. It seems ad hoc to say there are exactly 17, or a billion, alien natural properties in the multiverse; and so it seems that if we accept the possibility of alien properties in the first place, we should hold that for any finite natural number n, there are at least n alien properties to be found across the space of worlds. But once this is granted, argue Divers and Melia, there is no way to give in non-modal terms a complete account of what worlds there are. For we can’t just say that there are infinitely many alien natural properties spread across the worlds; or that for any finite n there is a world where n distinct alien natural properties are instantiated. Why not? Well, to satisfy those tenets there has to be, across the space of worlds, a denumerable sequence of alien natural properties P1, P2, . . ., Pn. Now, let S be the set of all the worlds that there are. S satisfies both those tenets, of course; but so does the set S* which is the subset of S containing all the members of S except those worlds where, say, P1 is instantiated. Because with P1 missing, there are still of course infinitely many alien properties left; so any tenet you laid down to tell you that there were infinitely many alien natural properties out there in the space of worlds won’t be able to discriminate between it being P1, P2, . . ., Pn that exist across the worlds or merely P2, . . ., Pn that exist. And so there is no tenet you can lay down that will completely yield all the worlds that there are. Unless, of course, we say something like ‘All the possible alien natural properties are instantiated somewhere across the space worlds’. And so the only way to completely say what worlds there are is to invoke primitive modality.

I think Divers and Melia’s argument that the Lewisian is not going to be able to give a complete account of the space of worlds in non-modal terms is pretty convincing. But unlike them, I see no reason to think this casts doubt on the reductive ambitions of the theory. Why should we demand that the Lewisian be able to give a complete non-modal account of what worlds there are? Given the Lewisian analysis, that’s to demand a non-modal account of the space of possibilities. But why should we demand this? To say what it is to be possible is one thing, to say what is possible another. Maybe no complete account of the space of possibility can be given: that should lead us only to epistemic humility, not to abandon a reductive account of what it is to be possible.

The paper goes into these issues in more detail, as well as making some methodological remarks about how to assess whether something can appropriately be included in a reductive basis. Comments on any of it would be welcome.

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63 responses to “Lewisian realism and modal reduction

  1. Here's another way to put that. You hold the following claim: "there [are] no semantic guarantees that 'Actually, there are no Fs' unless there is a semantic guarantee that 'there are no Fs' is true".I'm thinking that in the relevant sense, Louis thinks there is a semantic guarantee that there are no island universes. Speaking with the usual restriction on the quantifier to actualia, that's guaranteed to be true.Of course, there are island universes. But now I've forced the context where I'm speaking unrestrictedly. But if I can still use that quantifier within the scope of actually, in this context there's no longer a semantic guarantee that there aren't actually any island universes.So I think Louis can agree with your principle on either disambiguation: it only looks like he can't by shifting context.

  2. I can see how Louis can hold onto principle (A). So if that's what you're pushing, I'll spot it. What I can't see is how he can hold onto (A) and (N) together. Maybe that's a stable position, but my point was simply that both strike me as pretty plausible constraints on the interpretation of actuality and necessity. So maybe what I should've said is that there can be semantic guarantees that actually p only if there are semantic guarantees that necessarily p. That's what (!) seems to say. I'm totally winging this, incidentally

  3. Right, good, I think we're in agreement now. Your claim that there can be a semantic guarantee that actually p only if there's a semantic guarantee that necessarily p is similar to my worry that every possible being can truly say that everything's s-t connected despite the possibility of things failing to be s-t connected. In each case, the worry is that possible circumstances shouldn't be ruled out as being actual by semantic fiat. Right?

  4. Hi Rich, Hi Ross,I have been totally winging it from the beginning, and I still am.There are apparent counter-examples to (!). For instance, let P be "snow is white iff actually: snow is white." Then `actually P' is valid, but `Necessarily P' is not.One reaction is to suggest that (!) should be restricted to `actually'-free sentences. Another is to think that (!) (and, in particular, (N)) is just on the wrong track.Another apparent counter-example: Let P be "I am here". (In case you don't think that Kaplan's treatment gives the correct semantics for `I' and `here', let me stipulate that I am using an artificial language for which that semantics is correct.) Then `Actually, I am here' is valid, but `Necessarily I am here' is not.One reaction is to suggest that (!) should be restricted to indexical-free sentences. Another is to think that (!) (and, in particular, (N)) is just on the wrong track.Back in the 60's, modal logicians working in QML got really worried about the rule of necessitation. In particular, they thought that sentences of the form `(Ax)Fx => Fa' were valid, but not necessitatable; to see the problem, just let F be "exists". If they are right, this is a third kind of counter-example to (!) (and, in particular, to (N)). One reaction is just to change your notion of validity, e.g., moving to a free logic. Another reaction is to restrict the rule of necessitation. A final reaction is to abandon the rule of necessitation altogether. Either of the last two involves abandoning (N).Bottom line: I don't think (!), and, in particular (N), is anything like non-negotiable.Finally, when it is stipulated that "actually" is a quantifier restrictor, then we should expect failures of (!), and of (A) in particular. For instance, that silly "unhorsily" operator I stipulated will yield counter-examples to the analogous claim:($) `unhorsily P' is valid only if `necessarily P' is valid.A counter-example is provided if we let P be `there are no horses.' What is non-negotiable for an actuality operator? Three desiderata spring immediately to mind:(1) The schema "P iff actually P" is valid.(2) The schema "if actually P, then necessarily actually P" is valid.(3) The operator can be used to model the natural language locutions which seem to require it, e.g., "Your yacht could have been longer."For the record, it's not clear to me that Lewis's actuality operator delivers, especially on (1). But whatever problems it has on this score seem far removed from accommodating the possibility of island universes, or satisfying (!).- Louis

  5. Hi Louis. Whilst I agree that there are some delicacies with (A) and (N), I'm just not sure what to make of some of the cases. The Kaplan thing is very subtle: we only get apparent failures of (N) when we hold character fixed over models, and we don't get failures when we don't. So if we're talking about *narrowly logical* consequence, and that's the sense of validity we're interested in, (N) is still ok. And then there are also issues to do with 1-necessity and the like. (Incidentally: we get apparent failures of (N) for non-indexical cases: "There are thinkers" is LD-valid but its necessitation is not.) I know people make a big deal of the Kaplan stuff, but the cases aren't obvious to me unless we make some very substantial assumptions. And then there are issues to do with what's going on with names in QML which will have a bearing on the fate of (N). Then there are also issues to do with (2) within Lewis's setting. Then there are also issues to do with what the philosophical significance of actuality operators and QML more generally. I know some people *think* in QML, but some — Lewis? — have thought it a thoroughly truncated, expressively inadequate medium that gets in the way of the real business of translating between natural language and the langauge counterpart theory. So, yeah, the little argument I presented gets us into lots of issues that we're not in a position to answer. But let me back off to this: there is a prima facie in favour of the following: If P is metaphysically contingent, then our semantics shouldn't entail that any possible individual can know that P is false at their own world. The semantics you're backing forces us to reject that. That looks a bad-making feature of the semantics. Maybe it's negotiable — maybe everything is — but it's a cost nonetheless.

  6. Hello again,The prima facie virtue Rich identifies seems to be based on an argument like this:(1) P is metaphysically contingent, so P is true at some worlds and false at some worlds.(2) Let X be a world at which P is true, and Y be an individual at X.(3) Y can know P is false at his world, which is X.So P is false at X, which contradicts (2).However, Louis's semantics for 'actually' doesn't lead to any contradictions, anymore than the semantics for 'unhorsily' does, although Rich and Ross are right that it has a peculiar feature which Bricker's primitive actuality is designed to counter. But it's hard spelling out exactly what this peculiar feature is.We imagine an inhabitant of one of two island universes saying "there's only one universe" and think they must be wrong, but in fact they're not, because they are also part of a world which has only one universe, and that is the world they are talking about. The Anglophone inhabitants of island-universe worlds (and that includes us) just don't talk about the other islands.The problem arises because in order to accommodate worlds containing island universes without introducing extra ideology we say that worlds are arbitrary sums of universes, and this means speakers will be parts of more that one world. Our intuitions about actuality work on the assumption we are parts of only one world, though. This is why it sounds fair enough in (3) when I said "his world, which is X".One solution is to deny the possibility of island universes. Another is to say that when someone talks about actuality, or makes ordinary claims like "there are no talking donkeys", the quantifiers are restricted to the most natural world of which they are a part. This (presumably) will be the connected one. So although some worlds contain island universes with thinkers, these are not anybody's most natural world, which puts "there are no island universes", construed as an ordinary claim, roughly on a par with "there are no thinkers".One can even say that the most natural worlds are the connected ones and the whole pluriverse, and charity determines that in ordinary contexts we restrict to the connected one, and in extraordinary contexts we restrict to the pluriverse.The latter solution gives you the possibility of island universes, and while it does have odd results in the area under discussion, these are explained by our intuitions being based on the incorrect assumption that we're only parts of one world. Put this way, it sounds (to me) like a trade-off which is definitely negotiable.

  7. Hi All,Michael, I'm glad the suggestion I've offered the Lewisian doesn't turn out to entail a contradiction! Rich, you offer an argument for rejecting the suggestion, based on this idea: "If P is metaphysically contingent, then our semantics shouldn't entail that any possible individual can know that P is false at their own world." Is the P you have in mind is the sentence "there are island universes"? If so, the proposal does not run afoul of this claim. I can't know that "there are island universes" (as uttered by me here and now?) is false at my own world, because, as Michael notes, no world is uniquely my own. I can know that it is false at the minimal world containing me (if, that is, I can know the truth of the Lewisian theory under discussion), but that doesn't seem to me to be a problem. – LouisP.S. I'd like to ask you about your remarks on the Kaplan stuff when we see one another in Berlin.

  8. I'm confused a bit. every possible is individual can know that, 'actually, there are no island universes' is true, right?And I thought you thought that p iff actually p was non-negotiable?

  9. Rich,I can't speak for Louis, but on my suggestion it would work like this:Since I'm part of many worlds, "my world" has no unique satisfier, so saying "there are no other island universes in my world" has a failed definite description. The quantifier doesn't get restricted because the claim is an extraordinary claim about worlds, not an ordinary one about universes.But when I say "there are no island universes" or "actually there are no island universes" the quanitifers are restricted to my most natural world, in the former case by its being an ordinary claim, and in the latter by the actually operator.Both these are true, because my most natural world is the minimal one. This secures p iff actually p.

  10. Hi Michael. I don't get how your thing is meant to work. You want to say that there are various candidate semantic values for "my world". And you want to say that the most natural candidate is the minimal world. But then why doesn't naturalness break the tie and force the reference of "my world" to the minimal world? Naturalness is there, in part, to break ties when other things are equal. And by your own lights it breaks ties in the normal case of "there are no island universes". My headache is probably getting the better of me, though…

  11. Hi Rich,Remember how I had doubts that the Lewisian (either in Lewis's original theory, or in the souped-up version I've been suggesting) could endorse the validity of (*) P iff actually P? This is (part of) why. The Lewisian can say that every instance of (*) is true-at-@, but, I think, can't say that it is true simpliciter (with quantifiers in P on the LHS wide open). To see that this is not peculiar to the suggestion I offered, let P be "there are no talking donkeys". (There's some discussion between you and Ross on a related point a little farther up.)So: *I* think that (*) is non-negotiable. But the Lewisian, I think, has to say something a little subtler. Maybe Michael's suggestion is a start. Anyway, all this is quite independent of the dispute over the possibility of island universes.- Louis

  12. Rich,I was thinking there are no candidate values for "my world", because no world is uniquely mine, and a world would have to be my only world to be a candidate. In the case of "there are no island universes/talking donkeys" there are several candidates for the domain of quantification, and naturalness selects the most natural world of which I'm a part.Alternatively, we could view "my world" as an incomplete definite descripiton referring to the most salient of my worlds, which could plausibly be the minimal one, perhaps because of its naturalness. If we do look at it this way though, there's no guarantee that "Y's world" will refer to a world X just because Y is part of X. The problem in the argument I set out is still with (3).Either way "my world" is understood, thinkers will only be able to know that there are no island universes at their most natural world. Island universes with thinkers in are possible because of worlds which are nobody's most natural one.

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