Do Modal Claims Imply the Existence of Possible Worlds?

by Michael Huemer

I. An ontological argument for the existence of possible worlds?

David Lewis propounds an extraordinary theory in On the Plurality of Worlds. He concedes both that his view is highly counter-intuitive and that it therefore incurs a heavy burden of proof (133-5). What kind of argument does he give on behalf of the plurality of worlds? Essentially he gives four instances of one form of argument: he offers analyses of philosophically interesting concepts in terms of possible worlds; he then concludes that we should believe in possible worlds. I'm going to develop an objection to Lewis' general approach. I say to his general approach because I will not go into details having to do with how well Lewis' proposed analyses fit with the usage of the terms he tries to analyze, or anything of that sort -- let's grant for the sake of argument that there is an adequate fit there. I'm going to argue that regardless, Lewis' sort of argument can not be valid. The intuitive idea I have in mind is comparable to what is probably the normal intuitive reaction to the Ontological Argument for the existence of God. One feels that the argument can not be correct because the sort of premise it uses -- the mere definition of a certain term -- is unsuited to the kind of conclusion it ends up with -- viz. a conclusion as to the real existence of a certain being. Continuing, it's hard to bring oneself to think that the non-existence of something would be contradictory. What if there had just been nothing at all in the world -- surely that's internally consistent? Given that contradiction is a relation -- that is, there must be two things (at least) contradicting each other (as when I assert P and ¬P) -- it's hard to imagine where the contradiction could be found in a state of nothingness.

Well, by analogy, I think that the sort of premises David Lewis uses -- the analyses of certain concepts, or the fact that such analyses can be given -- seems unsuited to the kind of conclusion he draws -- positing the existence of uncountable infinities different worlds. I don't see how this sort of premise could, in principle, support this kind of conclusion. This will be my argument, in outline:

I'm only going to deal with Lewis' first two arguments, though, wherein he offers analyses of modality and counter-factual conditionals. I'm going to set aside his analyses of properties and belief contents, to keep things simple.

II. The notion of analyticity, and that the existence of possible worlds is not analytic

I am not unaware that the distinction between analytic and synthetic truths is subject to controversy, but I can not get into that debate here. Here I will take the distinction for granted. I take an analytic truth to be a sentence or proposition that is true by definition, whose denial is contradictory, and which can be known just through attention to the meanings of terms (I take all those to be at least materially equivalent). I take it that sentences like "All quadrilaterals have four sides" and propositions of the form, If A then A, are paradigm cases of analytic truths. And a 'synthetic' truth is, of course, any one that is not analytic.

I also consider existential claims -- that is, assertions that certain things exist -- to be paradigm synthetic claims. In defense of the claim that existential propositions are generally synthetic, I have little to add to this passage from Kant:

If, in an identical proposition, I reject the predicate while retaining the subject, contradiction results; and I therefore say that the former belongs necessarily to the latter. But if we reject subject and predicate alike, there is no contradiction; for nothing is then left that can be contradicted. To posit a triangle, and yet to reject its three angles, is self-contradictory; but there is no contradiction in rejecting the triangle together with its three angles. The same holds true of the concept of an absolutely necessary being. If its existence is rejected, we reject the thing itself with all its predicates; and no question of contradiction can then arise. There is nothing outside it that would then be contradicted, since the necessity of the thing is not supposed to be derived from anything external; nor is there anything internal that would be contradicted, since in rejecting the thing itself we have at the same time rejected all its internal properties. (B622-3)

I am not going to try to argue positively any further for this point because I think existential claims are already widely accepted as paradigmatically synthetic (by those who accept the distinction) and because I doubt in any case that Lewis or anyone else would want to say that the existence of possible worlds in Lewis' sense is an analytic truth.

But one might think that this is a counter-example to my general claim⁽¹⁾: Knowing the meaning of the statement "I exist" is enough to know it is true; so it must be analytic, right? No. As much as I covet the status of necessary existent, I might not have existed; in fact I didn't exist in 1950. So it cannot be analytic that I exist. I know that I exist by introspection, not by the analysis of language.

Again, perhaps it is analytic that there is a successor of 2? If so, the debate over Platonism in mathematics is more easily resolved than I have been led to believe. No, the proposition that the number 3 exists, in the same sense as we say cats exist, I exist, and Mount Everest exists, is so far from being analytic that it is highly implausible. It may be (as I think) analytic that 3 is the successor of 2, but to conclude then that there exists a successor of 2, requires also the premise that 3 exists. This latter, if it even makes sense, is at least a substantive claim, and no triviality. And while I concede there may be a sense in which abstract objects can be said to exist, I don't think they exist in just as full-blooded a manner as I do. This seems like a fair point to make, since Lewis holds that possible worlds exist in exactly the same manner as we exist and the things we see around us and occasionally bump into exist (2-3), and that possible worlds are concrete existents⁽²⁾; and since it is particularly plausible that what exists in that manner is no analytic question.

This is certainly not the place to digress into philosophy of mathematics or the problem of universals. I believe it is sufficient to observe that Lewis' thesis of the plurality of worlds is not intended as, and is not, an analytic truth. If it can be shown that Lewis' arguments would commit him to saying it is an analytic truth, then his arguments must be mistaken. This is all I seek to establish in this section.

III. What is Lewis' argument?

I'm torn between two alternatives in interpreting Lewis. One interpretation gives him a plausible premise that is irrelevant to his conclusion; the other gives him a stronger and less probable premise that entails his conclusion.

Let's just consider how the analysis of modality is supposed to support the plurality-of-worlds thesis. (The other proffered analyses presumably work the same way.) On any interpretation, I think Lewis' first premise is that we are often correct in making modal claims (e.g. "Possibly p", "Necessarily q"). He says he does not "make any case against a hard-line actualism that rejects any sort of quantification over possibilities" but that he would not favor such a view (viii). So one alternative to Lewis' theory would be the view that modal discourse is all nonsense or delusory. In that case, presumably, the possibility of analyzing it in terms of possible worlds would be no support for the reality of possible worlds. (I do not claim that this view is credible.)

The second premise is that we can in some sense analyze such discourse in terms of possible worlds -- in Lewis' words, "we can take the diamond and the box as quantifiers ... over possible worlds." (9; emphasis mine) And with this analysis, "modality turns into quantification: possibly there are blue swans iff, for some world W, at W there are blue swans." (5; emphasis mine) Perhaps the reader will think I am making too much of Lewis' choice of words, but why does he speak of 'taking' modal operators as quantifiers over possible worlds, as an action open to us, rather than just saying the operators are quantifiers over possible worlds? If the analysis he suggests is true, then don't we have to take the box and diamond as quantifiers over possible worlds?

Elsewhere he is more forthright:

I myself, of course, do think that modal operators are quantifiers over possible worlds ... I do not just think that the indices of frames 'may be regarded as' possible worlds. (20)

This is a significant issue: what is supposed to be the status of the possible-worlds analysis of modality, i.e. of the statement

Possibly there are blue swans iff, for some world W, there are blue swans at W.

and its cousins? From the above quotation from Lewis (p. 20), and from common sense, it is natural to assume that it is supposed to be a true (and not merely useful or aesthetic) analysis of the expression on the left hand side of the biconditional. It is also natural to assume that the purpose of an 'analysis' is to explain the meaning of a certain expression in terms of simpler or more readily understood ones, so that, for instance,

x is a bachelor iff x is an unmarried man.

would be a true analysis of "bachelor". And the reason we want to have analyses of philosophical concepts around is so that we may more firmly grasp the analyzed concepts.

If these assumptions were correct, then Lewis would have a straightforward deduction of the existence of possible worlds:

If our modal discourse is discourse about possible worlds, and our modal discourse is not just the expression of delusions on our part, then there must be possible worlds to make it true.

This is what I think Lewis ought to be arguing, and this is a valid argument. However, remarks sprinkled throughout the book give me pause. Why does he deny that he has a conclusive argument (viii, 4) and instead talk about weighing the 'price' of his thesis against its 'benefits' (4-5, 135)? The argument I stated is, I think, conclusive, provided of course that we know the premises. Well, perhaps he thinks there is some doubt as to whether the premises are true. But what do benefits or gains have to do with the issue? At times it sounds as if Lewis is giving an entirely different sort of argument from the one I suggested, one which is purely pragmatic. According to the blurb (back cover), "Lewis argues that the philosophical utility of modal realism is a good reason for believing that it is true." Even if Lewis didn't write that, the book is filled with similar remarks about the 'benefits' of modal realism, and he cites the fact that "the hypothesis is serviceable" as the reason for believing in a plurality of worlds at the outset. (3) In what sense is the hypothesis useful? Lewis describes himself as "trying to improve the unity and economy of our total theory by providing resources that will afford analyses." (134; Cf. p. 4) This is something of a laconic description of the nature of the benefits Lewis conceives himself to be providing, but nowhere is he any more expansive.

Here's one interpretation of Lewis' point of view: We should believe in a plurality of worlds because something good will happen if we do. The good thing that will happen is we will be able to give more analyses of things. It's good to give analyses of things because doing so makes our theories of the world simpler, because it cuts down on the number of primitive concepts we have.

This argument is still not entirely clear. First, it's not obvious why it's good that our 'total theory' be simpler, and Lewis does not elaborate. Perhaps this is an aesthetic value, or a pragmatic one. If so, I would consider the argument invalid. I do not think the fact that good things will happen in this sense if we believe something, is any evidence of its truth. Or perhaps Lewis thinks the simplest theory, in general, is the most likely to be true. I will not argue this point, since Lewis doesn't, except to say it is not obvious to me, at least, that this principle has the same merit as applied to philosophy, as it does when applied in science.

Second, it is unclear that cutting down on the number of primitive concepts or terms we have to use makes our worldview simpler in the relevant sense. Wouldn't simplifying our worldview require cutting down on the concrete theoretical entities that we posit as existing? This is something that Lewis' record is less impressive on.

Third, it is unclear in what sense the giving of analyses cuts down on the number of primitive concepts we have. We have the concept of modality before we read Lewis' book. Hopefully, we still have it afterwards. A philosophical innocent might wonder how anything Lewis does could decrease the number of primitive concepts we have (even assuming we wanted this). If 'modality' is in fact a primitive concept, then nothing David Lewis can do can change that. Though he might be able to get us to falsely say and think it is not primitive, I assume we don't want that. I assume what we want is not "to reduce the diversity of notions we must accept as primitive" (4) by means of saying concepts of modality et al. are complex, even while in fact they are primitive; nor do we want to literally decrease the number of primitive concepts we have, by unlearning some concepts. Rather, I suppose the case is this: Lewis expects a priori that it is very unlikely that humans should have a large number of primitive concepts. For some reason, it is much more probable that we have only a few, and that other, seemingly primitive concepts are really definable in terms of those few.

The reader may worry that I am putting words into David Lewis' mouth, but I do not seek to make a straw man; I am trying to make out an argument that makes sense, on the basis of the elliptical remarks Lewis makes. I do not think he would literally think that the benefits to us of believing something would constitute reason to believe, in the sense of evidence, that it is true; and I do think Lewis means to be giving that sort of reason, that is, logical reasons for his modal realism. Let us even grant without argument that theories using fewer primitive concepts are generally more likely to be true. Now we are still left with the conclusion we stated earlier:

(Possibly p) iff (there is a world in which p).

is supposed to be a true analysis of "possibly", for surely it would not be considered an advantage of modal realism that it provide resources for more false analyses of philosophical terms.

Or perhaps Lewis would say he is not analyzing words or even concepts, but rather the objects of those concepts -- i.e., it is not the concept of possibility, but possibility itself which Lewis analyzes. Of course, if "possible" just means "true in some possible world", then it follows that in general something is possible iff it is true in some possible world; but not conversely. So let's suppose "possible" does not mean "true in some possible world" but nevertheless it turns out that all and only those propositions that are possible are true in some possible world. How would this surprising discovery constitute a simplification or increase in unity and economy of our understanding of the world? Well, perhaps possibility is truth in some possible world, notwithstanding that the corresponding expressions don't mean the same? I find it difficult to understand what this would mean, though, despite that I can verbally formulate the possibility. This would presumably not be a contingent identity, but nor is it analytic. Could it be discovered by scientific means (much as we discovered that water is H₂O)? It is clear that it has not been. The only method of investigation that has been used to arrive at the conclusion that

(Possibly p) iff (p is true in some possible world)

is armchair reflection. This is consistent with its being an analytic truth.

I suppose I cannot conceive of its being a non-analytic identity because I would imagine that the Fregean sense of "possible" is distinct from the sense attaching to "true in some possible world" (pursuant to the biconditional claim being synthetic), and consequently that these two senses must constitute two distinct properties that a proposition could have; and so possibility would not be the same as truth in some possible world.

But let us resist the digression into philosophy of language and turn to the textual evidence of Lewis' views. He does speak of reducing our primitive notions, and then of interpreting or analyzing expressions (5-14 passim). For instance he refers to "the interpretation of expressions that are not explicitly quantificational, but that reveal implicit quantification under analysis," (5-6), where the context makes clear he is talking about modal locutions. The text is fairly univocal in supporting our interpretation. One would not, for instance, refer to "Water is H₂O" as an 'analysis' of "water", as Lewis does refer to his own claims as 'analyses'.

The burden of this section has been just this: According to David Lewis,

(Possibly p) iff (p is true in some possible world).

and the other 'analyses' he offers, are analytic truths. That, after all, is what makes them analyses, and not merely large generalizations. This is not yet to say that "There are possible worlds" is analytic (that's for the next section); only that "Anything that is possible is true in some possible world" is analytic -- just as "All unicorns have horns" is analytic, irrespective of whether there are unicorns. The parallel here is instructive. "Unicorn" means -- that is, may be analyzed as -- "horse with a horn on its head." Therefore, analytically, all unicorns have horns. This does not mean that there are any unicorns, nor that there are any horns. However, if there are unicorns then it follows logically that there are horns. And if someone for some reason thought it was analytic that unicorns exist, then he would also have to admit the existence of horns as analytic.

By analogy, I say that Lewis' analysis of "possible" as "true in some possible world" makes it analytic that all possibilities are true in some possible world. This does not yet mean that there is anything that is possible, nor that there are any possible worlds. But if anything is possible, then it follows deductively that there are possible worlds (at least one). And if (as remains to be seen) it is analytic that anything is possible, then it must be analytic that there is a possible world.

Hopefully, I have set this up clearly enough. David Lewis' claim to analyze expressions such as "possibly p" and "if p then q" or to analyze the corresponding concepts, commits him to certain propositions' being not only true but analytically true.

IV. If Lewis' argument is correct, then the existence of possible worlds is analytic

I do not believe Lewis considers the existence of possible worlds to be an analytic truth. If we asked him, he would almost certainly say, "No, it's not analytic." Nevertheless, I argue that he is committed to its being analytic, whether he likes it or not. This is an unwelcome consequence of his theory, not a tenet that he asserts.

At the end of the last section, I argued that on Lewis' analysis, if it is analytic that anything is possible, then it has to be analytic that there are possible worlds. Lewis does not directly say this, but he does more or less explicitly say that "possible" means "true in a possible world"; which I think implies the former. The same holds about his other proffered analyses. E.g. he says that "If A then B" (where the conditional is counterfactual), means that in the nearest possible world in which A holds, B holds. So, on his analysis, if it is analytic that if A then B, for any A and B, then it will have to be analytic that in the nearest possible world in which A holds, B holds. And, as I think, it follows that, analytically, the nearest possible world in which A holds exists.

These inferences of mine suppose the following principle, which I assume without argument: If it is analytic that p, and p entails q, then it is analytic that q.

It now remains to see whether there are any analytic truths of the form "Possibly p" or "If A then B." This question can be answered in the affirmative.

A. It is analytic that something is possible

If anything is analytic, it is analytic that all bachelors are unmarried. Now "All bachelors are unmarried" entails "Possibly, all bachelors are unmarried." Therefore, analytically, it is possible that all bachelors are unmarried. According to Lewis, "It is possible that all bachelors are unmarried" means that there is a possible world at which it's true that all bachelors are unmarried. Therefore, analytically, there is a possible world in which all bachelors are unmarried. Therefore, analytically, there is a possible world. Thus Lewis' commitment to the analytic existence of possible worlds.

B. It is analytic that if A then B, for some A and B

Again, if anything is analytic, statements of the form If A then A are analytic. Thus, for instance, analytically, if there were ten planets there would be ten planets. (Equally well, if there were nine then there would be nine.) Now according to Lewis, "If there were 10 planets then there would be 10 planets" means that in the nearest possible world in which there are 10 planets, there are ten planets. Therefore, analytically, in the nearest possible world in which there are 10 planets, there are 10 planets. Now, on a reasonable analysis of definite descriptions, this latter entails that the nearest possible world in which there are ten planets exists; therefore, analytically, the nearest possible world in which there are ten planets exists. Hence, analytically, a possible world exists.

Incidentally, this last bit of reasoning enables us to establish analytically the existence of infinitely many possible worlds, and not just one. For we can establish analytically the existence of a possible world in which there are 11 planets, the nearest world in which there are 12 planets, etc.; and we can establish analytically that these are distinct, since, analytically, a world can't contain exactly 10 planets and also 11; nor can one contain exactly 10 planets and also 12; etc.

To anticipate an objection to what we have just said, suppose that there is no world in which there are 10 planets. Then what happens to a statement beginning "In the nearest world in which there are ten planets ..."? Generally, what is the truth-value of "If A then B" in the event that there is no possible world in which A holds? I have assumed that the statement is then automatically false, or at least not true. This is analogous to how "The King of France is bald" is automatically false, or at least not true, in the event that there is no King of France. That is, to make a claim about the King of France certainly can't be to speak the truth, if there is no King of France. Likewise, I should think, to make a claim about the nearest possible world in which A holds can't be true if there is no nearest world in which A holds.

But perhaps Lewis would feel differently. Perhaps he would say "If A then B" is stipulated to be vacuously true in case there is no world in which A holds. In that case, "If there were 10 planets there would be 10 planets" could still be analytically true without implying the existence of a world in which there are 10 planets. What it would then have to be taken to mean, is that there isn't a world in which there are 10 planets, closer than any other in which there are 10 planets, such that in that world there aren't ten planets. And there is no difficulty in taking this to be analytic.

I am willing to be generous on this matter, since I will win in any event. Both of the following are clearly analytic:

(C1) If there were ten planets, there would be ten planets.

and

(C2) Not: If there were ten planets, there wouldn't be ten planets.

Now Lewis has open three ways of dealing with conditionals that make implicit reference to possible worlds that don't exist. Suppose that there is no possible world in which there are ten planets. (I, of course, think this supposition is true, since I think there are no possible worlds whatsoever.) We suppose also, though, that "If A then B" does mean "In the nearest world in which A holds, B holds." Then there are three things we might say about sentences of the form "If there were ten planets, x", or "In the nearest world in which there are ten planets, x holds":

Now I say that on any of these suppositions, I can make my case, because Lewis could not preserve the truth of both C1 and C2.

If we take alternative 1, then we have to say that C1 is false. Also, if we take alternative 3, we have to say that C1 is at least not true.

If we take alternative 2, then we get to say that C1 is true; however, we now have to deny that C2 is true. For, according to alternative 2, "If there were ten planets, there wouldn't be ten planets" is (vacuously) true. Since C2 is the contradictory of exactly that, C2 is false.

Now here is the point: I have just shown that if there were no possible worlds, then, logically, C1 and C2 could not both be true, on Lewis' analysis. So the fact that C1 and C2 are both true entails the existence of possible worlds. So the fact that C1 and C2 are both analytically true entails the analytic existence of possible worlds. In order to get C1 and C2 both being true, Lewis needs there to be a possible world in which there are ten planets. Of course, that's okay since he does think there is a possible world in which there are ten planets. But here's the problem: in order to get C1 and C2 both being analytic, Lewis would need it to be analytic that there is a possible world in which there are ten planets.

V. Conclusion

That was a complicated argument. Here it is, again, in brief:

I think Lewis would agree with this, and it should be clear in any case from §II.

This should be established by §III, because if (A) and (B) are not true, then modal realism only enables us to give false analyses of terms, and this could not be claimed as an advantage.

Examples of this were given in §IV.

This follows from 3, 4, and 5.

I have not offered any alternative analysis of modal notions or of the concept of counterfactual dependence (the relation denoted by "if ... then"), and I do not ever intend to. Each of these seems to me an eminent candidate for a primitive concept -- introspection on my part reveals no simpler constituent concepts -- and I cannot see the urgency behind the giving of analyses in any case. I do not feel confused about these notions, such that they might be cleared up for me by reading Lewis' book or by working hard on contriving other 'analyses' of them. But I will make some negative observations on the meanings of modal and counterfactual discourse, that I do think have the power to remove some confusion. My argument above does more than refute David Lewis' analyses of modal notions. It refutes any analyses along the same lines. The work of this final section is to spell out how broad "along the same lines" is.

All that was essential to the argument was that Lewis' analyses make modal discourse contain commitments to concrete existences. All assertions of concrete existence are both synthetic and (though we have not discussed this previously) contingent. But some modal assertions are analytic and necessary. Therefore, these modal statements contain no existential commitments. Since presumably modal locutions mean the same thing when they appear in analytic sentences as when they appear in synthetic ones (e.g. "If ... then" means the same thing in "If pigs flew, pigs would fly" as it does in "If pigs flew, air travel would be more dangerous"), modal locutions in general make no commitments to concrete existents.

This is a roundabout route to this intuitive point: modal discourse is intrinsically hypothetical in character. It is a mistake to try to analyze it out in purely categorical terms. Put another way, modal discourse (including counterfactual conditionals) adverts to merely possible objects and mere would-be situations. It is not about what exists at all; it is about what might have, but doesn't exist. It is a fundamental confusion, and, really, an attempt to expunge modality, to try to analyze talk about what might have been in terms of talk about what is.

Any propositions are able to be analytic and necessary only because of their merely hypothetical character, because they do not require anything to exist. Even "All bachelors are unmarried" has to be read hypothetically, in order to be analytically and necessarily true -- that is, it just means that if there were bachelors, they wouldn't be married. If we read it with existential import, then it would have to be contingent, synthetic, and empirical.

I do not claim any theoretical gains for my view, except to the extent that saying the truth is a theoretical gain. The way to see the merit of my point of view is not to count the number of primitives that it would commit you to, but rather to reflect on your own thoughts when you use modal idioms, and appreciate directly, introspectively, how closely they compare with what I have said.

Notes

1. So thinks Tim Maudlin, for instance.

2. He says he doesn't know what "concrete" means, but then it turns out that possible worlds are concrete according to all four of the interpretations of the word he suggests. (81-6)