Many people think that talking about possible worlds is useful in philosophy. A good number of those people think that talking about impossible worlds is also useful. In most cases, talking about impossible worlds as well as possible worlds is innocuous. On most of our views about what worlds are, impossible worlds are no more ontologically problematic than possible worlds: sets of propositions all of which can’t be true together are no more mysterious than sets of propositions all of which can be true together; if talk of possible worlds is a merely pragmatically useful fiction, talk of impossible worlds can be such without any further mystery. And so if you hold some such view about worlds, the question as to whether we should talk about impossible worlds depends solely on whether to do so is useful – there is no metaphysical problem in doing so.

Not so, seemingly, if you are a Lewisian realist about worlds. For Lewis, a world at which there are blue swans is a world with blue swans as parts, and so a world with round squares is a world with round squares as parts. And so, to believe in the latter world is to believe in round squares. And this *is* to raise a metaphysical problem, for now one must admit into one’s ontology objects which could not exist. In brief, impossible worlds for Lewis are problematic because of how he thinks worlds represent: they represent something being the case by being that way, whereas his opponents think worlds represent in some indirect manner, by describing things to be that way, or picturing them to be that way, or etc. Impossible worlds are not metaphysically mysterious on the latter views because there is no metaphysical puzzle in there being a description of something that couldn’t exist, or a picture of something that couldn’t exist; but they are a metaphysical puzzle for Lewis, because there is a metaphysical puzzle in there being something that couldn’t exist.

Nonetheless, some think this is a price worth paying: they like Lewis’s account of possibilia but are impressed by the arguments for the need for impossibilia, so want to extend Lewis’s ontology to include impossible worlds. I’ve heard this move a few times in conversation, but the one person I know of who has defended it in print is Ira Kiourti. (Yagisawa defends a view similar to Lewisian realism with impossible worlds, but with some crucial differences.) Now, there are some big, and familiar, problems with believing in genuine impossible worlds that they each try to deal with, but I am yet to be convinced can be solved. (See my critical study of Yagisawa.) But I want to raise a problem for Lewisian realism with impossible worlds that I haven’t seen discussed and which I don’t even know how one would start to answer.

I can see how Lewisian realism with impossible worlds is supposed to deal with impossibilities like ‘There is a round square’ or ‘Frank is taller than Jim and Jim is taller than Frank’. I just need to believe in impossible objects – a round square and a man that is both taller and shorter than some other man – and then I can believe in worlds composed in part of such objects. Now, personally I can’t conceive of such objects – but so what? If I’ve got good reason to believe in them, I can postulate them. But I don’t see how we are meant to account for an impossibility like ‘2+2=5’. For Lewis, ‘2+2=4’ is necessary not because there’s a number system that is a part of each world and which behaves the same way at each world; rather it’s necessary that 2+2=4 because the numbers are *not* part of any world – they stand beyond the realm of the concreta, and so varying what happens from one portion of concrete reality to another cannot result in variation as to whether 2+2 is 4. And so, since contingency *just is*, for Lewis, variation across certain portions of concrete reality – namely, the worlds, which are just big concrete objects – there is simply no room for contingency with respect to the mathematical truths. Necessary truths about the realm of concreta are necessary because each relevant portion of concrete reality is a certain way: that is, *no matter what* variation you get across these concrete portions of reality, things are that way. But necessary truths about the realm of pure abstracta are necessary because they have nothing to do with how concrete reality is: so *irrespective* of what variation you get across these concrete portions of reality, things are that way. (See my paper on Lewis and reduction, esp. footnotes 3-5, and the corresponding discussion in the text.)

In that case, while I can add to my ontology round squares if I wish, and hence believe in a big concrete object with a round square as a part, and thereby have a world that represents the impossible circumstance of there being a round square, I don’t see even what weird metaphysical move to make to get a Lewisian world that represents 2+2 as being 5. Worlds don’t have numbers as parts: they are sums of concrete individuals; and if we give that up, we don’t have something worth calling an extension of Lewisian realism (Yagisawa gives this up, which is why I’m ignoring his view here). A Lewisian world nonetheless *represents* that there are numbers, because numbers exist from the standpoint of that world. And while the concrete objects that exist in one Lewisian world are never the same as the concrete objects that exist at another world, the numbers that exist from the *standpoint* of each world are just the same. And that’s why mathematical truths are necessary, for Lewis: because a world represents some mathematical claim as being the case just because the numbers represented as existing from the standpoint of that world *are* as the claim says they are. And since the same numbers exist from the standpoint of any two worlds, no two worlds differ in what mathematical claims they represent as being the case. Given Lewis’s account of what a world is and how they represent something to be the case, there is simply no room for variation across worlds in what mathematical claims are represented as true: hence there is no room for contingency in mathematical claims, but nor is there room for a world that represents some impossible mathematical claim as true, *no matter what *we think about the extent of what concrete worlds there are. So the Lewisian simply can’t extend her ontology to admit worlds that account for every impossible situation.

Dear Professor Cameron,

I think you are onto something importantly right. However, as it stands, it seems Lewis could offer two replies. First, since we’re dealing with impossible worlds, he might deny that all parts of such worlds are concrete. Perhaps “having abstract parts” is part of what makes an impossible world impossible. (Even so, one rejoinder might be to insist that some impossible worlds are such that 2+2=5 AND that they have no abstract parts. That also could be an impossible state-of-affairs, but that too could be what defines an impossible world as such.)

Second, I’m unsure whether Lewis is committed to worlds having all concrete parts, even in the case of possible worlds. In _Plurality_, he writes “As for the parts of worlds, certainly some of them are concrete, such as the other-worldly donkeys and protons and puddles and stars. But if universals or tropes are non-spatiotemporal parts of ordinary particulars that in turn are parts of worlds, then here we have abstractions that are parts of worlds” (p. 86).

Still, I may be inclined to deny, as van Inwagen does, the intelligibility of “abstract parts.” (We sometimes say things like “algebra is a part of mathematics,” but that could be dismissed as loose talk.) So even though there may be two Lewisian replies to your claim, your point still seems defensible in the end.

On the second point: that’s why I talked only about ‘pure’ abstracta, like numbers. It’s these that Lewis can’t admit as parts of worlds. I agree that if there are abstracta that depend on concreta, like tropes or universals, Lewis can admit that those are parts of worlds.

On your first point. I agree that something along those lines might be an option a theorist could take, but I stand by my claim that it would too serious a departure for it still to be legitimately thought of as an extension of Lewisian realism. Here’s one way of seeing that: you can’t just say ‘Okay, let abstract objects be parts of some worlds’ and leave it at that. One now needs a totally different definition of ‘world’: a world can’t now be a sum of spatio-temporally related things, because pure abstracta aren’t s-t related to anything. So what IS the new definition of ‘world’? And remember that we can’t invoke modality for it to be anything like Lewisian realism. (In effect that’s what Yagisawa does, which is why he doesn’t face this issue: he in effect has a primitive worldmate relation.)

I think you’re right that the abstract-parts view would be too much of a departure from Lewisian realism. Nonetheless, it seems we must take the worldmate-relation as primitive regardless, once we introduce impossible worlds. Argument: If the relation is not primitive, there will be some true sentence of the form “x and y are worldmates iff Rxy,” for some appropriately selected predicate ‘R’. Suppose we also take your earlier suggestion that impossible worlds can (at the very least) be represented by sets of inconsistent statements. Then, without further provisos, there will be impossible worlds represented partly by statements of the form “x and y are worldmates and ~Rxy.” But if that holds in some impossible world, then “R” cannot define the worldmate-relation across both possible and impossible worlds. Yet since “R” is arbitrary, the worldmate-relation here is undefinable…

Not sure I buy that. Why can’t we hold Lewis’s definition: x and y are worldmates iff they are spatio-temporally related, and then say that the impossibility of a and b being worldmates without being s-t related is realised at the world where a and b both are and aren’t s-t related?

And in any case, if your problem is a problem, why does having a primitive worldmate relation help? We still have to account for the impossible world where two things are worldmates but don’t stand in the worldmate relation.

Hmm, would it be Lewisian enough to try something funky with counterpart relations? Step one: if there can be (say) a universal that exists in two worlds, we could fiddle with the counterpart relations so that it in w1 doesn’t count (according to counterpart relation R) as cross-world identical to w2. Step two: if that’s right for things that literally are parts of two worlds (e.g. universals), it should be right for things that aren’t literally parts of two worlds but count as existing ‘from the perspective’ of both (e.g. singletons of universals). And if

that’sright, it should be in principle possible to pull the same trick for pure sets. So maybe we can multiply possibilities here by multiplying counterpart relations; a(n im)possibility where 2+2=5 is just like a possibility where Lumpl isn’t Goliath: one where some pure set x is a counterpart of 4 under a “two-plus-two” relation and another pure set y is a counterpart of 5 under a “five” relation.Yes, this is all super sketchy, and no I don’t see how to make it less sketchy, and yes, maybe any attempt to make it work properly will collapse Lewisian Modal Realism into Dorrian Counterpairing Realism. But it seems to me like the most natural thing to try.

Parts of classes?

What about it?

Doesn’t it give a Lewis-acceptable model of how one might duck out of commitment to pure abstracta, and thus reduce your second problematic type of impossibility to the first?

Are you thinking of the appendix? It’s been a while – I’d have to re-read it and think through how that plays out.

I was thinking that given a very large (inaccessible cardinal sized) pluriverse of concrete mereological atoms and plural quantification we could treat mathematical claims like 2+2=4 in something like the way the in re arithmetical structuralist does, and 2+2=5 as realized by there being an impossible concrete such structure.

Hey, Ross–one thing that struck me about your construal of modal realism (I don’t know if you handle it somewhere, so sorry if it’s old hat) is that it might be bad news for Lewis’s argument against Boxes and Diamonds on OTPOW p10-12. “So [Humphrey] satisfies both ‘x is human’ at all worlds and ‘x does not exist’ at some worlds; so he satisfies both of them at some worlds; yet though he satisfies both conjuncts he doesn’t satisfy their conjunction. How can that be?”

Here’s how that can be: if “x is human” is indeed necessary, then the conceptual truth it rests on has the same status as mathematical necessity. Then Humphrey satisfies “x is human” and “x does not exist” *in different ways* at worlds where he satisfies both, and hence he does not in any one way satisfy their conjunction…

Yeah, Andy, that might be right. Interesting that you have to go that way, but yeah it sounds like a workable Lewis-friendly option.

Richard – sorry, I’m not following that very well.

Maybe you don’t need the pluriverse to be that size if there’s an impossible object that is big enough (and not big enough).

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Mathematics is not such a special case: For Lewis, logic is absolutely general and not dependent on any worldly features. So, in the Lewisian picture, mathematical and logical necessity, both, are world-invariant, their necessary status not arising in virtue of particular features of each world. This is the basis of the joke in Stalnaker’s ‘Impossibilities’ where ‘Louis’ replies to ‘Will’’s postulation of impossible worlds by challenging how we can begin to attribute logical properties to a world, or what it would be like to discover that some world is not deductively closed. So – Conjecture: whatever one says about how logical laws vary from world to world is applicable to the case of mathematical truths varying from world to world.

Moreover, if I understand this correctly, advocates of non-classical logics such as Priest, Mortensen, Routley have developed inconsistent number systems, which are based on non-classical logical systems. This again seems to suggest that if there is room for non-classical logics in a concrete pluriverse, ditto for alternative mathematical structures.

So, lets focus on the case of logical laws: I discuss variation in logical laws between worlds in chapter 5. One way to conceive of logical properties as anchored in a world is to see them as worldly structural properties – complex (second order) properties of some kind. The fact that we cannot imagine how a world might realise some such abstract structure is again beside the point. Chris Mortensen, in ‘Anything is Possible’, reminds us that just because some general truth can be presented in a totally abstract way this doesn’t imply it is without anchor in physical reality – think of, say, complex abstract equations in physics. All we want is for worlds to realise different such structural properties to get going. (We may want to deny that worlds have any such structural features at all, but then a case could be made that all such features arguably stand or fall together, at which point, arguably, the status of all kinds of necessities may be called into question.)

[Ontologically, both logical and mathematical necessities can be pictured in the Lewisian framework as relations (higher order properties) holding between certain kinds of impure sets: in the one case we are dealing with propositions, i.e. sets of worlds, (sets that are not purely abstract) – and in the other, we are dealing with numbers, i.e. constructs out of the empty set, (again not purely abstract, given how tongue-in cheek Lewis is about the null-set: all it has to be is member-less. ‘A possum will do’. Indeed any concrete part of any world we are interested in). This picture, again, may allow for at least some minimal anchoring of mathematical entities in worlds.]

Beyond this, you may ask about impossibilities of the following kind: it is impossible that there is a world where 2+2=5 but which has no structural properties – so there must be such a world, albeit impossible. I would have to say that a world, which both does and does not have such properties, may have to suffice.

Thanks, Ira. I’m not exactly sure what the answer is, from that, but I’ll have a look at Ch.5 of your thesis in more detail!

Hi Ross,

The second bit of the answer is quick, yes. But the point I am making – which I take to be stated clearly enough – is that mathematics is not a special case. So whatever goes for logic goes for mathematics. This is key as you seem to think the question of mathematical truths uniquely problematic – i say not so. No need to go into the thesis for that. I didn’t explain in detail how logical structures can differ from world to world (a difficult issue John Divers raises), but we can talk about that if you like (that’s the bit I talk about in the thesis). :)

No, I wasn’t thinking maths was a special case. Just picking an example where the goings on concern abstracta rather than concreta

Ok. In that case, the only way forward with the general problem is to allow such mathematical and logical truths to be anchored in concrete reality in some way that allows them to vary. Structural properties seems to be the way to go (my only worry is a clash with humean supervenience here, but even Lewis does not take the thesis to be necessarily true). The fact that mathematical entities do not have to be considered pure abstracta if we ‘naturalise’ the null set as Lewis seems to suggest in parts of classes may further detract a little from the worry about purely abstract truth

(Btw. Thinking of the impossible as that which cannot possibly exist is an ambiguous statement if you are debating concrete impossible worlds. Arguably, just like the physically impossible cannot possibly exist given a particular set of physical laws (and not absolutely), the logically impossible cannot possibly exist given a particular set of logical laws, which, ex hypothesi, given the consideration of logically impossible worlds, are not taken to determine existence. You cannot go for impossible worlds and agree with Lewis on this matter of the status of logical laws.)

Cool. One thing both you and Andy are bringing to light though is that the best way for the impossible Lewisian to go is to embrace what Lewis says in PoC. I think that’s an interesting consequence; but I also personally am disappointed, because I like what Lewis says about worlds but don’t really like what he says about sets!

Interesting indeed. Ah, I tend to quite like this consequence. But I have to go back to examine it more carefully.

Maybe the deeper source of the issue that you’re raising is the non-recombinable nature of ‘magical’ selection relations. It’s tempting to allow magical set-theoretic such relations but forswear modal/semantic ones (perhaps on indispensability grounds). But this may not be stable if a suitably rich metaphysical account of the former makes trouble for the way you hoped to avoid commitment to the latter.

Well, I *do* like how intricately related every bit of Lewis’s system is to every other bit! But I would like to be able to do all the modal stuff without saying *crazy* things about sets! The empty set ain’t no possum!

Ha! And what a surprise that a possum will serve just as well!

I think that impossible worlds is an analogy too far for dealing with things that cannot exist like square wheels. I think the real problem here is that such things are literally non-sense, i.e. there is not (and cannot) be anything that they refer to. So to postulate a world in which such things do reside cannot therefore itself make sense, i.e. there cannot be something it refers to either. The difference between a possible world and nonsense is that it is possible that there could be something that is referred to in a possible world and it can be populated with individuals that the statements do refer to.

I feel your pain Matthew. Logic was a great gift to the world but in the hands of significant intellects it is liable to suck in time and vitality only to deposit its victims on high, barren platforms, scaffolded by non-sense and easily demolished with simple observation.

The Pseudo-scientists are all the time eating into the territory where we can still resonate but yet you continue to pick at the dry crumbs left in the intellectual hermitage.

Enjoy your swim with blue swans, I probably won’t see you there and nor will anyone else.