Philos Stud DOI 10.1007/s11098-012-9998-0

What are impossible worlds? Barak Krakauer

© Springer Science+Business Media B.V. 2012

Abstract In this paper, I argue for a particular conception of impossible worlds. Possible worlds, as traditionally understood, can be used in the analysis of propositions, the content of belief, the truth of counterfactuals, and so on. Yet possible worlds are not capable of differentiating propositions that are necessarily equivalent, making sense of the beliefs of agents who are not ideally rational, or giving truth values to counterfactuals with necessarily false antecedents. The addition of impossible worlds addresses these issues. The kinds of impossible worlds capable of performing this task are not mysterious sui generis entities, but sets of structured propositions that are themselves constructed out of possible worlds and relations. I also respond to a worry that these impossible worlds are unable to represent claims about the shape of modal space itself. Keywords

Possible worlds · Impossible worlds · Modality · Metaphysics

It is well-known that the framework of possible worlds suffers from certain difficulties if it is to be used as an analysis of particular philosophical concepts. Possible worlds are coarse-grained, in the sense that the space of possible worlds models only what is logically or metaphysically possible. Thus analyses that make use of possible worlds face problems if they are to differentiate content that is logically or metaphysically equivalent. The propositions \nine is prime[ and \I am a married bachelor[ are both impossible, but they are distinct. I could, for example, believe one without believing the other; different things might be the case if one of these propositions were true, but not the other. While one may attempt to

B. Krakauer (&) University of California, Santa Cruz, USA e-mail: [email protected]; [email protected]

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somehow differentiate this content within the space of possible worlds1, a more straightforward solution would be to adopt a system that allows us to model content in a more fine-grained fashion. Thus the goal of this project is to develop an account of impossible worlds of a certain kind, and show how these worlds allow us to give better analyses of philosophical concepts than we would be able to do with possible worlds alone. In Sect. 1, I will briefly discuss some of the limitations of possible worlds and potential applications of impossible worlds. In Sect. 2, I will develop an account of impossible worlds that is both ontologically innocent and robust enough to do the work required of these worlds. In Sect. 3, I respond to a worry that these worlds are unable to represent certain claims that are made about the space of worlds itself.

1 Uses of impossible worlds Why believe in impossible worlds? One kind of argument aims to show that we are already committed to the existence of impossible worlds. In his (1973), Lewis argues that possible worlds are simply ways things could have been. We all believe that there are ways things could have been, and possible worlds are merely ways things could have been, so we all believe that there possible worlds. Yet we also all believe there are ways things couldn’t have been, as pointed out by Naylor in her (1986). Since Lewis does not accept the existence of impossible worlds, and thus cannot use them to represent the ways things could not have been, he cannot comfortably endorse his original argument; indeed, he seems to switch to an argument based on the usefulness of possible worlds in his (1986). Nonetheless, we could also easily construe Naylor’s modus tollens as a modus ponens. Since there are obviously ways things could not have been, there are obviously impossible worlds.2 We might also think that we are committed to impossible worlds because their existence is entailed by other theoretical commitments. As King argues in (2007a), depending on what we take possible worlds to be, we should accept the existence of impossible worlds as well. If possible worlds are complete and consistent sets of sentences or propositions or states of affairs, for example, there is no principled reason why we shouldn’t also believe in incomplete or inconsistent sets of sentences, propositions, or states of affairs. The argument I want to defend here, however, does not rely on how we are to understand “ways,” nor does it rely on our commitment to sets and propositions. Rather, I hold, with Lewis, that possible worlds earn their ontological keep by the use they can be put to in most fields of philosophy, playing an important role in explicating concepts in logic, metaphysics, epistemology, and ethics. Yet the work done by possible worlds could be better done with the addition of impossible worlds. Consider, for example, the kind of work Lewis puts possible worlds to in his (1986). Lewis holds that possible worlds are extremely useful in explaining 1

See, for example, Lewis (1986, Ch. 1) or Stalnaker (1987).

2

This argument is made in more detail by Vander Laan in his (1997).

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modality, closeness, content, and properties. If we consider each of these applications in turn, we can see that our understanding of each of these families of concepts can be improved with the addition of impossible worlds.3 1.1 Modality The most obvious application of possible worlds is to explain modality. By quantifying over possible worlds, we can develop a much better understanding of what it is for something to be possible, necessary, or actual. We can understand the sense in which the laws of physics are necessary, and thus locutions such as ‘It is not possible to travel faster than the speed of light’ as restricted quantifications over possible worlds. That is, to say that ‘It is not possible to travel faster than the speed of light’ is to say that in all nomically accessible worlds (i.e., worlds which have the same laws of physics that our world has), nothing travels faster than the speed of light. Of course, there are possible worlds that have different laws of physics, and at these worlds, some things travel faster than light; nonetheless, these are not the worlds we were quantifying over. If we say, ‘The coin could land tails,’ we usually mean that there are nomically accessible possible worlds where the coin does land tails. Similar stories can be told for other flavors of necessity: some worlds share our history, some worlds share facts about what is practical, some worlds correspond to our moral or legal obligations, and so on; words like ‘must’ and ‘can’ and ‘possible’ correspond to restricted quantification over such possible worlds. Epistemic possibility, however, presents a challenge. How are we to make sense of sentences such as ‘Goldbach’s conjecture might be false,’ or ‘relevance logic could be correct?’ Assuming that Goldbach’s conjecture is true and that the proper analysis of the entailment relation is classical, these are mere epistemic possibilities: for all we know, there could be some very large even number that is not the sum of two primes, or perhaps the conclusion of some formal argument is always “relevant” to its premises, as formalized in a system such as that of Anderson and Belnap (1975). Yet there is (presumably) no possible world where Goldbach’s conjecture is false, and no possible world where relevance logic is correct. These claims, then, seem to be existential quantification over worlds, in the same pattern as ‘The coin could land tails,’ yet there are no possible worlds capable of representing the truth of these claims. Impossible worlds would provide an attractive and straightforward way of understanding claims about epistemic possibility. A logically impossible proposition could still be epistemically possible if we can quantify over impossible worlds. The claim that Goldbach’s conjecture could be false, then, should be understood as ‘At some epistemically accessible world, Goldbach’s conjecture is false.’4 3

This kind of argument is similar to that of Yagisawa (1987), though I reject his account of extended modal realism.

4

Of course, impossible worlds aren’t the only way of understanding epistemic possibility, though they probably are the simplest. Other accounts face various problems. Perhaps the leading contender for understanding epistemic possibility is two-dimensional semantics (see, for example (Chalmers 2004), according to which we can make sense of some epistemic possibilities by analyzing the primary intensions of the propositions expressed. This strategy is helpful in understanding some epistemic

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1.2 Closeness Possible worlds are also required to make sense of closeness. An account of closeness can be used to make sense of counterfactuals: when we say something like, ‘If I had missed the bus, then I would be late for class,’ I plausibly mean something like ‘In the closest situation where I miss the bus, I am late for class.’ The best way to make sense of such a ‘closest situation’ is to think of possible worlds ordered by some relation of overall similarity, as in (Lewis 1973). A counterfactual is true when the consequent is true in the closest (i.e., most similar overall) possible world (or worlds) where the antecedent is true. But it seems that there can be counterfactuals with impossible antecedents. Consider ‘If nine were prime, then it would not be divisible by three.’ Impossible worlds allow us to retain the structure of Lewis’s account of counterfactuals: we can say that this sentence is true because, in the nearest (impossible) world (or worlds) where nine is prime, it is not divisible by three.5 1.3 Belief content Possible worlds are also useful in discussing the content of propositional attitudes such as belief. We can represent what some agent believes as the set of possible worlds that she thinks might be actual. We can say that some possible worlds are doxastically accessible to some agent: these are the worlds that the agent thinks might be the actual world. An agent thinks some proposition might be true iff that proposition is true in some of her doxastically accessible worlds; she believes some proposition iff that proposition is true in every doxastically accessible world.6 We then represent how an agent makes inferences by updating the set of doxastically accessible worlds with the acquisition of new information: as an agent learns something about the world, she narrows the set of worlds she takes to be actual.7 This picture, however, makes several unreasonable assumptions about the rational behavior of agents. Agents are represented as being logically omniscient and perfect reasoners. They believe all logical truths, since logical truths are true at every possible world, and therefore true at every doxastically accessible world. The agents described in this model believe no contradictions, since contradictions are not true at any possible world, and therefore not true at any doxastically accessible world. Finally, these agents believe the logical entailments of all their beliefs, since if P entails Q, and P is true at every doxastically accessible world, then Q is true at every doxastically accessible world as well. Assuming we want these agents to be more realistic in these respects, we should look for ways to relax these assumptions. Footnote 4 continued possibilities; two dimensions semantics is helpful in understanding how we could conceive of water as not being H2O. It is less clear, however, how two dimensional semantics would help us understand how Goldbach’s conjecture or how some non-classical logic could be true. 5

This proposal is discussed in (Nolan 1997), (Vander Laan 2007), and others.

6

See, for example, (Hintikka 1962). The model presented is a simplification of Lewis’s scheme, which makes better sense of belief de se.

7

See, for example, (Stalnaker 1987).

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Realistic agents must be permitted to believe the impossible, fail to believe the necessary, and fail to believe some things that follow from their other beliefs.8 Impossible worlds provide the most straightforward remedy for this problem. There are impossible worlds at which contradictions are true, impossible worlds at which not all tautologies are true, and impossible worlds at which some propositions are true without their (classical) entailments. It would, therefore, be easy to represent agents who are not logically perfect if we allow the doxastically accessible worlds to include impossible worlds. If an agent believes in some contradiction, then all of her doxastically accessible worlds are impossible worlds that contain some contradiction. If she fails to believe in some logical truth, then some of her doxastically accessible worlds are impossible worlds that lack this logical truth. If an agent fails to believe some proposition Q that is entailed by some proposition P that she believes, then P is true in all of her doxastically accessible worlds, though some of her doxastically accessible worlds are impossible worlds where P is true but Q is not. 1.4 Propositions Possible worlds are also useful in providing an analysis of propositions. Possible worlds also allow us to analyze propositions as functions from possible worlds to truth values (or, equivalently, as sets of possible worlds). The proposition 〈grass is green〉 is the set of all worlds according to which grass is green. There are, however, intuitively distinct propositions that pick out the same set of possible worlds: 〈all bachelors are unmarried〉 and 〈2 + 2 = 4〉 pick out the same set of worlds, viz. the set of all possible worlds. Impossible worlds could differentiate these necessarily equivalent propositions or properties: there are, after all, impossible worlds where bachelors are married, and impossible worlds where 2 + 2 does not equal 4. At some impossible worlds, both of these are true; at other impossible worlds, one of these propositions is true, but not the other. Therefore, the set of all worlds (possible and impossible) where bachelors are unmarried is not the same set as the set of all worlds where 2 + 2 = 4.9 To be sure, the defender of a possible worlds account of properties and propositions has tools at her disposal to represent these entities in a more finegrained fashion. For example, one could use possible worlds to describe structured propositions, which allow us to individuate necessarily coextensive properties and 8

Some might cede these points, and hold merely that the formal model described here is merely an epistemic ideal; the model describes perfect reasoners, and thus describes a kind of normative goal for reasoning. It is certainly useful to keep the Hintikka model as a kind of ideal for rationality, but being able to represent imperfect agents will allow us to model more than just ideal rationality. If we can describe the behavior of realistic agents, we can use it to make predictions about how people will behave and update beliefs, for example. Furthermore, it may be useful to have a theory of how we ought to update our beliefs, should we believe contradictions or fail to believe tautologies.

9

Impossible worlds can also be used to distinguish necessarily coextensive properties. We might think that triangularity and trilaterality are distinct properties, even though every three-angled figure is also a three-sided figure across all possible worlds. Impossible worlds allow us to represent figures that are triangular but not trilateral, or vice versa; thus, the two properties pick out distinct sets of figures across worlds.

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propositions. Though there are many accounts of this kind of structured entity,10 we can illustrate this strategy with the system Lewis sketches in (1986, pp 56–59). The project is to augment the possible worlds account with some kind of structure that mirrors the structure of the language used to express the proposition. On an unstructured account of propositions, for some proposition P, the propositions P and ¬¬P pick out the same set of possible worlds (viz, the set of all worlds where P is true). A structured account of propositions could represent negation as a relation N that holds between unstructured propositions and the sets of worlds where that proposition does not hold. The structured proposition ¬P could then be represented as 〈N, P〉, while the structured proposition ¬¬P could be represented as 〈N, 〈N, P〉〉. On the structured account of propositions, P and ¬¬P are distinct propositions, since they pick out different ‘tuples.11 If structured propositions give us a fine-grained notion of properties, why do we need impossible worlds? If we can come up with some entity that enables us to make the distinctions we need to make between properties and propositions, why not simply make use of them and not take on the additional ontological burden of impossible worlds? Because we can do more with worlds than we can with structured propositions alone. It is, of course, useful to give some perspicuous account of propositions in the course of metaphysics. But we cannot simply point to some class of set-theoretic structures that identify and differentiate content in the right way and think that our work is done. Our account of propositions must differentiate distinct propositions, but it must also do more: it must also pick out the objects that are able to play the role that is required of them by modality, counterfactuals, the content of propositional attitudes, and so on. Structured propositions alone cannot do this work. As seen above, worlds are required to give an account of modality, content, and closeness. It is worlds that stand in the similarity relations that underwrite an account of counterfactuals; it is worlds that can be used to model how agents (ideal or otherwise) reason and update their beliefs; it is worlds that represent what a believer takes to be true. Structured propositions can give us fine-grained entities to individuate certain properties and propositions, but they cannot be put to the same use that worlds are. An adequate account of propositions is one on which we can put them to the kinds of uses argued above. Propositions are not idle pieces of metaphysics, but

10

See, for example, King (2007b) and others.

11

It must be conceded that structured propositions will not solve every problem of hyperintensionality. One could believe one, but not the other, of two necessarily equivalent and structurally isomorphic structured propositions. Whether or not this is a problem for the defender of the structured proposition account depends on whether there is some plausible alternative treatment for such cases. For example, if the equivalent and isomorphic propositions in question are 〈London is pretty〉 and 〈Londres est jolie〉 , then it is plausible that the reason that Pierre would assent to one but not the other are metalinguistic in nature. Certainly, if Pierre realizes that the two sentences are equivalent (or even that ‘London’ and ‘Londres’ are names for the same city), then he would retract his acceptance of one of these propositions. To be sure, this account is threatened if all hyperintensional content can be treated metalinguistically, but this does not seem to be the case. Pierre’s confusion about ‘London’ and ‘Londres’ is not akin to how some logicians understand non-classical entailment relations, and it would border on absurdity to hold that nobody grasps the meanings of their terms, since everyone falls short of ideal rationality.

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rather objects that have modal properties, stand in entailment relations, counterfactually entail other propositions, are the proper targets of certain propositional attitudes, and so on. If an impossible proposition is merely some ordered set, we do not have any handle on how to understand epistemic possibility, counterpossibles, or belief in the impossible. To say that if A were true, B would be true, for example, is to say (roughly) that in the nearest world where A is true, B is true. It is unclear what to make of these truth conditions if A and B are merely sets: there is no theory on the table of the comparative nearness of sets. Of course, one could attempt to develop such a theory, but the considerations that are generally in play with respect to the nearness of worlds (e.g., matching regions of matters of fact or preserving laws) are more naturally applied to worlds than they are to sets. If we have impossible worlds, however, these questions of how to understand the work done by these propositions become tractable, since worlds provide the kind of framework that allows us to meaningfully talk about modality, entailment, counterfactuals, belief content, and the like. 1.5 Impossible worlds and triviality There is at least one serious worry about these uses of impossible worlds. The reason that possible worlds play such a central role in describing modality, closeness, belief content, properties, and propositions is that it allows us to reduce these notions to collections of possible scenarios. If impossible worlds are to play a role in our treatment of these concepts, we risk trivializing our analyses. For example, we could use impossible worlds to distinguish the proposition 〈nine is prime〉 from 〈some bachelor is married〉, since there is some impossible world where 〈nine is prime〉 is true, but not 〈some bachelor is married〉; the sets of worlds are not the same. However, it seems that we will also be forced to distinguish the propositions 〈some bachelor is married〉 and 〈some bachelor is wedded〉 since, at least depending how the worlds are specified, there will be impossible worlds where one is true, but not the other. Intuitively, however, these are the same proposition; they do not differ with respect to what is meant. Similarly, our models of belief should not license outlandish inferences, and any realistic agent should be expected to make certain simple inferences, even if she fails to make more complex ones. Formal epistemology would be trivialized if the structure provided no norm for how agents should reason, and provided no predictions for how agents likely will reason. If the space of impossible worlds lacks the inferential structure of the space of possible worlds, then it is unclear what sense we can make of how agents update their beliefs. The worry, then, is that impossible worlds lack the kind of logical structure that makes possible worlds an attractive tool. There are at least two ways of modifying our account of impossible worlds in such a way that they individuate content in a way that is fine-grained but not trivial. One proposal requires that we restrict which impossible worlds are used when giving our accounts of modality, properties, propositions, beliefs, and so on. We determine which impossible worlds are relevant to our analysis by considering the worlds that are relevantly nearby to the actual world. Yagisawa, in his (1987),

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suggests one such framework for doing this. Yagisawa suggests that, in the same way that some spheres of worlds contain those worlds that share our physical (or logical) truths, other spheres of worlds contain the worlds that correspond to our analytic truths. Consider some world “analytically familiar” to the actual world iff it shares the analytic truths of the actual world. The notion of analyticity in play is not analyzed, but in all of the analytically familiar worlds, vixens are female foxes, bachelors are unmarried men, and so on; it is not analytic, however, that trilateral things are triangular, or that nine is prime.12 Our purposes determine which sphere of worlds is relevant to the analysis at hand. If we are attempting to model the behavior of perfectly rational epistemic agents, for example, the model will only include the sphere of worlds that are logically possible. If we are analyzing the propositional content of some agent, then our model will only contain those worlds that are analytically familiar to the actual world. This will rule out worlds where, for example, bachelors are not unmarried men. Consider an additional assumption: there is a unique, “best” way to analyze propositions, such that (for example), ‘bachelor’ is analyzed as ‘unmarried man,’ and not vice versa. A world whose propositions are completely decomposed in this fashion is analytically basic. A proposition, then, can be identified as the set of analytically basic worlds where that proposition is true, or a function from semantically basic worlds to truth values. Thus, propositions such as 〈some bachelor is married〉 and 〈some bachelor is wedded〉 will pick out the same set of analytically basic worlds, since these two propositions will decompose in the same way; furthermore, these are not the same worlds picked out by 〈nine is prime〉, since that proposition has a different decomposition. Such an approach could be defended if acceptable notions of analyticity and propositional decomposition could be worked out. Since I am not particularly sanguine about either possibility, it may be better to sketch another kind of approach to the problem. Another proposal may avoid these issues. On such a view, the impossible worlds are sorted into equivalence classes according to the semantic content of their propositions: two worlds are in the same set iff they agree with respect to what is represented as true at those worlds. Call two worlds that agree with respect to what is represented semantically equivalent. While I have no analysis of what it is for two propositions to mean the same thing13, it is clear that we have some robust intuitions about when two propositions are equivalent. Propositions such as 〈some bachelor is married〉 and 〈nine is prime〉 have different semantic content, while 〈some bachelor is married〉 and 〈some bachelor is wedded〉 do not. Two worlds w1 and w2 are semantically equivalent iff for every proposition P that is true in w1, either (a) P is true in w2, or (b) a proposition that means the same thing as P is true in w2. Two semantically equivalent worlds, then, will always agree with respect to whether some bachelors is married, even if one world represents this as 〈some bachelor is married〉 and the other represents it as 〈some bachelor is wedded〉; these same worlds, however, could disagree with respect to whether nine is prime. It is these 12

In so far as I understand the notion of analyticity, I do not share these judgments.

13

An immediate worry might be that worlds will no longer be useful in determining the semantic content of utterances, if some notion of semantic content is built into the structure of the space of worlds.

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classes of semantically equivalent worlds that are used in the analysis of propositions. A proposition, strictly speaking, is not a set of worlds at which that proposition is true, but rather a set of set of worlds: a proposition is identified with the set of sets of worlds that are semantically equivalent to every world at which that proposition is true. According to this view of propositions, 〈some bachelor is married〉 and 〈some bachelor is wedded〉 may pick out different sets of worlds, but they will pick out the same collection of semantically equivalent worlds, since no two sets of semantically equivalent worlds can differ only with respect to the truth of these propositions. Of course, neither of these proposals are complete as they stand. One must ultimately say more about analyticity and propositional decomposition, or more about what it means for two worlds to be semantically equivalent. Yet these problems are not intractable, especially for the latter proposal. As long as a response to the triviality objection along one of these lines can be given, we need not let this worry prevent us from giving an account of impossible worlds that is capable of performing the tasks described above. 1.6 Worlds as models and worlds as analysis There is one other kind of objection to discuss before beginning the project in earnest. One may worry that any attempt to explicate belief content, propositions, and the like in terms of worlds—possible or impossible—is misguided from the start. As Plantinga worries in his (1987), Lewis’s project might better be thought of as giving some kind of model for notions such as propositions, rather than an analysis, since sets of worlds simply cannot play the role required of propositions. There is room for debate about how we might think of the contents of belief, but it seems that whatever our theory of belief content may be, our beliefs are not sets: sets do not make claims, and so cannot be accepted or rejected. Similarly, whatever propositions are, it seems clear that they must be bearers of truth, but sets of worlds (or functions from worlds to truth values) do not have truth values. These set theoretic constructions may be useful in the course of modeling the contents of belief or propositions, but they are simply not the kinds of things that can be identified with beliefs or propositions. If the best that a worlds-based approach to content can do is to provide a model, then one can consider this project an effort to provide a better model. The possible worlds model is popular and fruitful, but could be improved if it were made suitably fine-grained. Making sense of how this model can be extended is, by itself, a worthwhile project. In this fashion, we may hold that propositions are ultimately to be analyzed as classes of sentences, thoughts, or ideas in the mind of God, but still find use for the model of propositions as sets of worlds in formal epistemology, semantics, economics, artificial intelligence, and so on. One might still have reason to be hopeful that the more ambitious project is still possible: one may still hold that propositions really are sets of worlds of some sort. Of course, sets of worlds are not the kinds of things that are true or false in the ordinary sense, or are believed or disbelieved by people, but we need not let that

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worry us too much. Plantinga is right to point out that propositions do not bear truth in the same way that sentences do; that should not be in dispute. A proposition (qua set of worlds) is true at some world iff that world is a member of the set of worlds that is identified with that proposition; a proposition (qua set of worlds) is believed by S iff it is true at all of S’s doxastically accessible worlds.14 A set of worlds clearly does not have a truth value in the way that a sentence does, but there is no reason why truth, in its varied ordinary language guises, cannot be profitably reduced to this notion of set membership. Similarly, a set of worlds makes no claims, but there is no reason why a propositional attitude such as belief cannot be understood as a claim about which worlds some agent takes to be actual. I do not dispute the intuition that sets do not have truth values and do not make claims; the claim is rather that our ordinary notions of truth and the like are a motley crew, and that adopting a somewhat more technical notion of truth at a world allows us to better understand the notions of truth, properties, propositions, and belief.

2 The ontology of impossible worlds 2.1 Ersatz impossible worlds It is plausible to think of impossible worlds as ersatz constructions of some sort. On pain of contradiction, they cannot be concrete.15 Perhaps they are mere fictions of some sort,16 but even if they are, it would be important to better characterize this fiction if it is to do any explanatory work. If impossible worlds are constructions, from what are they constructed? Brogaard and Salerno, in their (2008), argue that they are constructions out of sentences of some world-making language. This kind of impossible world is intuitively plausible, since we use language to describe impossible scenarios in a way that cannot be done with concrete worlds or (perhaps) pictures.17 A linguistic ersatzist account of impossible worlds inherits many of the difficulties faced by the linguistic ersatzist program in general. Most importantly, it is not clear that a world-making language would have the resources required to describe all the impossible worlds. The linguistic ersatzist faces embarrassing questions about how the appropriate world-making language represents content related to alien individuals or alien properties. Furthermore, it is unclear how one could understand the meanings of the terms of the world-making languages without the kind of semantic tools that already rely on possible worlds: the referring terms and predicates of the world-making language cannot express a function from 14 This is not Lewis’s considered position. Ultimately for Lewis, the contents of belief (and other propositional attitudes) are properties, rather than propositions. Specifically, to believe in some proposition P is to self-locate one in logical space within the set of P-worlds. See Lewis (1978) and (1986). 15

See (Lewis 1986, fn 3).

16

See, for example (Rosen 1990).

17

See, for example, (Sorensen 2002).

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possible worlds or individuals at possible worlds, as they traditionally do. The ability of this language to express impossibilities would be constrained by its ability to name individuals and properties that do not exist in the actual world, as well as the grammar of the world-making language. An account of ersatz impossible worlds that does not rely on language would be more fruitful. 2.2 Worlds as simple set theoretic constructions A more promising account of impossible worlds is one in which impossible worlds are set-theoretic constructions of possible worlds. Construing impossible worlds in such a fashion has many advantages: most importantly, such worlds are as ontologically innocent as possible worlds, as the impossible worlds are merely constructions of possible worlds. Furthermore, if the possible worlds remain basic in a way that the impossible worlds are not, then we can continue to use the possible worlds to describe metaphysical possibility and necessity without threat of circularity. The challenge, then, is to determine how to best describe such worlds capable of playing the role required of them in Sect. 1. Rescher and Brandom, in their (1979), describe what they call schematic worlds, at which there are truth value gaps, and inconsistent worlds, at which there are truth value gluts. These worlds are created from the possible worlds. Consider two possible worlds, w1 and w2. Some proposition P is true at w1, and ¬P is true at w2. We can now describe a schematic world by a process they call schematization: w1 ∩ w2 describes a world such that, for any proposition ϕ, ϕ is true in the world iff ϕ is true in w1 and w2. \ / is true at w1 w2 iff / is true at w1 and / is true at w2 : Note that, in this schematic world, P is not true, since P is not true at w2 and ¬P is not true, since ¬P is not true at w1. We can describe an inconsistent world by a process they call superposition: w1 ∪ w2 describes a world such that, for any proposition ϕ, ϕ is true in the world iff ϕ is true in w1 or w2. [ / is true at w1 w2 iff / is true at w1 or / is true at w2 Note that, at this inconsistent world, P is true, since ¬P is true in w1, and ¬P is true, since ¬P is true in w2.18 Note that, even though P and ¬P can both be true in an inconsistent world, P ∧ ¬P will not be true at any world, since that conjunction is not true at any possible world; similarly, even though neither P nor ¬P can be true in a schematic world, P ∨ ¬P will always be true, since that disjunction is true in all possible worlds. Can Rescher–Brandom worlds do the work required of impossible worlds? These inconsistent and schematic worlds describe certain ways things couldn’t be, since they give us worlds where P and ¬P are both true, or where neither P nor ¬P is 18 Note that schematization and superposition can also be applied to the non-standard worlds, so a world can be both schematic and inconsistent, if we take the superposition of two schematic worlds, or the schematization of two inconsistent worlds.

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true.19 As we have seen, however, they do not give us worlds where contradictions are true. This means impossible worlds cannot describe some of the ways the world couldn’t be: for example, they don’t describe worlds where P ∧ ¬P is true, or worlds where P ∨ ¬P fails to be true. Rescher–Brandom worlds, then, would conflate all of these impossibilities; if propositions are represented by the set of all worlds where they are true, we would not be able to differentiate intuitively distinct contradictions, since they all pick out the set of no worlds. Similarly, Rescher– Brandom worlds would be unable to represent the beliefs of logically imperfect agents who believe contradictions or fail to believe tautologies. We could attempt to answer this problem by modifying the approach in such a way that the semantics given above hold only for atomic formulae, and changing our semantics for the evaluation of complex formulae.20 Roughly following the approach of (Restall 1997), we create non-standard worlds by superposing or schematizing the atomic formulae of possible worlds, which gives us the truth of atomic formulae, but not the complex formulae.21 Instead, we can define how the connectives work at worlds the following way22: :/ is true at w iff / is false at w / ^ w is true at w iff / is true at w and w is true at w: / _ w is true at w iff / is true at w or w is true at w: Thus according to such a system, there can be inconsistent worlds where P ∧ ¬P is true: in our original example, the superposed world w1 ∪ w2 describes such a case. Since P is true at w1 and ¬P is true at w2, P ∧ ¬P is true at w1 ∪ w2. Similarly, there can be worlds where P ∨ ¬P will not be true, such as the schematized world w1 ∩ w2. By making use of the Restall-inspired semantics, we are able to describe nonstandard worlds where logical falsehoods are true and worlds where tautologies are not true. This proposal still does not provide the kind of worlds required to represent all logical contradictions. Our logic is weak, but it will certainly have some theorems: 19 Are all schematic worlds “ways things couldn’t be?” Consider a world w that is the schematization of the actual world and some other world. Note that w does not contain any proposition that is not true at the actual world; it contains a proper subset of the truths of the actual world. It does not follow that w is actual, or even possible: w describes some part of the actual world, but not all of it. Unlike propositions, worlds are supposed to tell the “whole truth” about what goes on in the scenario described, and since w is incomplete, it cannot tell the whole truth about actuality; if the possible worlds are all complete, then w is not possible. 20

Note that we shift here to discussion of atomic and complex formulae, rather than propositions.

21

This is also similar to the approach of (Priest 1979). Restall and Priest only consider worlds that are inconsistent. The impossible worlds described here, however, can be inconsistent or schematic. See also (Berto 2009), whose impossible worlds are also built from sets of possible worlds. 22 It might seem more natural to define negation as w ⊨¬ϕ iff w ⊮ ϕ, but this rules out the possibility of P and ¬P being true at some world together. If we want P and ¬P to be true at some inconsistent world, then we cannot tie the truth of ¬P to the lack of truth of P. Rather, the model must ascribe falsity to some proposition independent of whether it has also ascribed truth to it. Another strategy to allow P and ¬P to be true at a world would be to make ¬P true when P is not true at some world w*. See, for example, (Dunn 1993).

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for example, either direction of double negation (¬¬ϕ ↔ ϕ) is valid in this framework. In all worlds as described above, ¬¬P is true whenever P is true; these impossible worlds do not describe some ways things could not be. We cannot respond to this problem simply by weakening the logic we have chosen, either. We could stipulate, for example, that the law of double negation is not a theorem, and include in our model worlds where some proposition P can be true without ¬¬P being true, and vice versa. But this does not address the central worry. As long as we provide some system of worlds that describes some entailment relation, the theorems associated with that entailment relation will be true in all worlds described by the system. If propositions are sets of worlds as described by this kind of system, then, they simply will not be fine-grained enough. Such a system will conflate the meanings of any two sentences that are true in the same set of worlds, given some system. If it seemed wrong that all propositions that are logically true in classical logic are indistinguishable, then it should seem wrong that all propositions that are logically true in some weaker system are indistinguishable. Thus if impossible worlds are to let us differentiate properties and propositions that are necessarily coextensive, the structures provided so far are simply not up to the task. If impossible worlds are to capture the phenomenon of epistemic possibility, these worlds simply do not go far enough. As we have seen, the law of double negation is valid in this system. A formula such as (¬¬P ⊃ P) will be true in all worlds, and thus we cannot make any sense of the notion that the failure of the law of double negation is epistemically possible in some scenario. Similarly, these impossible worlds will not solve the problem of logical omniscience. Since agents can now be doxastically related to Restall worlds, they can be represented as believing some logical falsehoods such as P ∧ ¬P. They cannot, however, be represented as believing other logical falsehoods, such as ¬ (¬¬P ⊃ P). The agents represented by this model will be logically omniscient with respect to the theorems of whatever logic is described by the worlds posited by the model: they will believe all of its logical truths, they will fail to believe anything that is logically impossible in the system, and they will accept all the (non-classical) entailments of their beliefs. A further problem with this kind of proposal is that it is not applicable to many of the applications that motivated our project. The kinds of impossible worlds discussed above are all ways of representing various kinds of formal impossibilities: Rescher–Brandom worlds give us structures where propositions such as P and ¬P can be both true or both false, and Restall worlds give us structures where propositions such as P ∧ ¬P are true and P ∨ ¬P are false. These worlds might be sufficient if our project is to provide a structure to represent certain logical falsehoods, but our ambitions are greater than this. We are not concerned merely with formal contradictions, but impossibilities in general. Consider propositions such as 〈nine is prime〉 or 〈some bachelor is married〉. Presumably, it is impossible that nine is prime, or that some bachelor is married, but we still want to be able to represent these propositions in a non-trivial way, if they are to serve as potential targets for belief (‘Bob believes that nine is prime’), antecedents of counterpossibles (‘If some bachelor were married, his wedding

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would be well-attended’) and so on. But these propositions do not seem to be conjunctions of contraries. At least on the surface, they attribute a property to some object. It would be strange, then, to attempt to use schematic or inconsistent worlds to describe some scenario where nine is prime: just what would be conjoined or disjoined to make nine prime? To be sure, the proposition 〈some bachelor is married〉 could be complex in a way that is not evident from its syntax when expressed in English. Perhaps ‘bachelor’ is to be represented as ‘man’ and ‘not-married,’ and thus the proposition is analyzed as 〈some not-married man is married〉, which does seem to involve a conjunction of contraries.23 If this view is correct, all impossibilities reduce to contradictions under analysis; once we have such a formal contradiction, we can analyze it as the superposition of the possible worlds expressed by the contradiction’s conjuncts. At some point, however, it becomes unclear how to carry out this strategy. It is simply implausible that all impossibilities can be analyzed as contradictions. How can we tell a similar story for 〈water is XYZ〉? There does not seem to be any formal contradiction; even if water is necessarily H2O, this was an empirical, rather than logical or semantic discovery. If there is a formal contradiction contained in this proposition, it is harder to uncover, and it is certainly more difficult to argue for some specific formulation of this contradiction as the “correct” representation of the proposition. 2.3 Worlds as complex set-theoretic constructions If impossible worlds are to do the work required of them, they cannot be settheoretic constructions out of possible worlds, since the schematization or superposition of possible worlds does not allow us to express all the impossibilities required of a robust account of impossible worlds. The worlds would better represent the kind of content required if they were built from propositions, rather than worlds. 2.3.1 From possible worlds to structured propositions We have already seen one method of using possible worlds to build structured propositions in Sect. 1.4; we can continue to use this system for purposes of illustration, though not much hangs on our choice of how we build structured propositions. Unlike a theory of content paired with the kinds of worlds described by Rescher–Brandom or Restall, Lewis’s account of structured propositions is able to distinguish content at an appropriately fine level of grain: 〈P〉 and 〈N, 〈N, P〉〉, for example, are, after all, distinct ‘tuples, and therefore can be used to represent distinct beliefs. 23 Thinking about what propositions turn out to mean “under analysis” in this fashion raises a great many questions. Is there a unique analysis for every proposition? Are there semantic “atoms” to be found to analyze more complex components of propositions? The account of structured propositions discussed below avoids these questions. Nonetheless, whatever kinds of propositions one prefers and however one answers the above questions, one may yet wonder whether the kinds of worlds described by Rescher and Brandom describe the right kind of impossibilities.

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It is also clear how to represent informal impossibilities: 〈some bachelor is married〉 can be represented simply as the ‘tuple 〈E, 〈B, M〉〉, where E is a relation that holds between some property and the worlds where something instantiates that property; B is the unstructured property of being a bachelor, and M is the unstructured property of being married. Of course, our work is not done yet. For the reasons discussed above, we need impossible worlds if we are to make better sense of modality, closeness, and belief content. Nonetheless, structured propositions such as these are useful in order to describe and differentiate propositions in a more fine-grained way than would be otherwise available. Furthermore, structured propositions give us the tools necessary to construct the impossible worlds that are capable of giving complete accounts of epistemic possibility, the analysis of counterpossibles, and the contents of belief. 2.3.2 From structured propositions to impossible worlds Structured propositions are ordered sets composed of possible worlds, collections of entities at possible worlds, and relations between and among them. We can then form sets of these structured propositions to create worlds, which would be able to describe ways things could (or could not!) be. Call these worlds structured worlds. Structured worlds are built from structured propositions, rather than from worlds or superpositions or schematizations of worlds. A structured proposition is true at a structured world iff that proposition is a member of the set that composes the world. A structured world could be complete and consistent, but a structured world that is not complete and consistent is an impossible world. If some impossible world is described by the set {P, Q}, where P and Q are two unstructured propositions, then all that is true at this world are P and Q. It does not follow, for example, that ¬¬P is true, that P ∧ Q is true, or that P ⊃ P is true, since these propositions are not members of the set {P, Q}. There is little more to say about the behavior of structured worlds. Previous accounts described a logic of impossible worlds, giving truth conditions for the behavior of logical operators and connectives, but there can be no such logic given for structured worlds in general. This is because these impossible worlds are anarchic by design: any set of structured propositions is an impossible world. Thus there is no interesting logical structure to the space of these worlds. Any proposition could be true at some world if that proposition is a member of the set that describes the world; any other proposition could fail to be true at that world if that proposition fails to be a member of that set. Principles such as the law of double negation and modus ponens will not be valid in these worlds. Since there is no logic of structured worlds, they are flexible in a way that their competitors are not. Impossible worlds are capable of describing any way that the world could not be, since structured propositions have the resources to express content in a way that is arbitrarily fine-grained. Impossible worlds will therefore be capable of representing epistemic possibilities that possible worlds by themselves cannot.24 24 As mentioned above, it is likely that a separate account will have to be given for certain cases, such as the beliefs of agents who do not know the meanings of the terms they use to describe their beliefs.

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Furthermore, worlds such as these can give a formal epistemologist the structure required to represent agents that are not logically omniscient with respect to any logical system. Since anarchic impossible worlds are arbitrary fine-grained, there are worlds that can represent any all the ways that agents may be less than ideally rational. An agent can be doxastically related to worlds where any logical principle fails, or worlds that lack the (non-classical) entailments of their other beliefs. Finally, one can appeal to structured worlds as part of an account of counterpossibles. These impossible worlds, after all, are capable of representing the kind of impossible situations expressed by the antecedents of counterpossibles. Another ingredient is necessary as well: there must also be a similarity metric for impossible worlds so as to make sense of how one impossible world can be closer to the actual world than some other impossible world. Giving an account of the similarity of worlds is well beyond the scope of this paper, but it seems that a generalization of an account of trans-world similarity for possible worlds could be offered. If, for example, we think that a possible world is nearby to the extent that it holds fixed matters of fact and minimizes the occurrence of violations of (our) physical laws, we could say that a structured world is nearby to the extent that it holds fixed matters of fact and minimizes the occurrence of violations of (our) metaphysical and logical laws. If structured worlds are these kinds of set theoretic constructions, it is clear that we should accept them in our ontology. They are cheap: they are merely sets composed of things we already believe in. Furthermore, they are useful: if we can argue from the utility of possible worlds to their existence, then the same kind of case can be made for structured worlds. If we want to understand epistemic possibility, model the beliefs of rationally fallible agents, analyze propositions and properties in a more fine-grained fashion, give truth conditions to counterpossibles, and so on, structured impossible worlds offer the most straightforward way of doing so.

3 Misrepresenting modal truths Some impossible worlds will contain some truth about modality that is incompatible with what is true at the worlds around it. An impossible world could, for example, represent the claim that all worlds accessible to it are worlds where some proposition P is true, even though P is false in some accessible worlds; an impossible world could represent the claim that, if A were true, then C would be true, even if the nearest A-worlds are not C-worlds. But this leads to an immediate worry: earlier, I had claimed that a structured proposition is true at a world iff it is a member of the set that comprises that world, but some of these structured propositions will make claims about modal space that are false from the perspective of the world in question. One could try to respond that such a mismatch between what a world claims about its modal neighborhood and what those modal facts actually are is impossible. Perhaps facts about accessibility and trans-world similarity somehow depend on what is true at the relevant worlds in such a way that a world cannot misrepresent

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the shape of modal space around it. If it is true at some world w that P is necessary, for example, then the accessibility relation in question must be one that only makes worlds where P is true accessible to w. It is unclear how to fill in the details of the story in a plausible way, and such an approach would not allow one to reduce modal claims to claims about the truth of propositions at worlds. Yet there is a still more serious problem, given how we understand structured worlds: some impossible worlds will make inconsistent modal claims. Consider, for example, some world w, where both □P and ¬□P are true. If the truth of these propositions determines how the accessibility relation is to be understood, then all worlds (relevantly) accessible to w will be worlds where P is true, and not all worlds (relevantly) accessible to w will be worlds where P is true; the accessibility relation will inherit the inconsistency of the impossible world. If our project is to allow for impossible worlds while keeping our logic sane and classical, then we cannot accept this result. Another way to ensure that worlds will not misrepresent modal space is to hold that structured worlds cannot represent modal truths. The propositions that make up a structured world would make claims about (say) the arrangement of matter, but not about what is necessary or possible, or what would occur in counterfactual situations; if legal, moral, and nomic propositions require some analysis in terms of what is true at other worlds, then such claims could not be represented within a structured world, either. There could be no such modal mismatch because structured worlds could not represent modal truths at all. Of course, one of the strengths of the structured worlds account is that it provides us with a straightforward way of giving robust content to modal propositions, allowing us to differentiate claims about what is necessary or possible, or to represent the modal beliefs of agents. If structured worlds cannot represent modal truths, then it is unclear how to differentiate propositions such as □P from □◇P, or how to represent the beliefs of agents that accept modal claims such as these. A more ambitious position is to hold that we can make sense of these problematic modal claims by recognizing that there are two separate notions of truth involved. Such an approach is not ad hoc, since there are other instances in which how a possible world represents itself differs from what is really true of that world. Bricker (2006) discusses this problem with respect to the property of actuality and possible worlds: each possible world represents itself as actual, yet (presumably) only one possible world is actual.25 It is true at each possible world that it is actual, but only true of one world that it is actual. A proposition such as □P may be true with respect to some world w because □P is a member of the set of propositions that compose w, or P may true at all worlds that are (relevantly) accessible to w. In the first kind of case, we can say that □P is true at w, and in the second kind of case, we can say that P is true of w. If these are indeed separate notions of truth, then there is no “mismatch” at all. Truth at a world is determined by set membership, as discussed above, while truth of a world is determined by the kind of semantic machinery we ultimately 25

This distinction is also needed to make sense of counterpart theory: consider some world w that contains a counterpart of me. The claim that w has me among its parts is true at w, but not of w. If possible worlds are sets of propositions (or sentences), then truth at a world and truth of a world will diverge in many more kinds of cases as well.

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endorse to make sense of claims about necessity, possibility, counterfactuals, and other modal notions.26 Which notion of truth to make use of depends on our purposes. Representing the content of beliefs, for example, requires a highly finegrained approach, which is best modeled in terms of the notion of truth at a world. If, for example, we are to represent agents that believe that some proposition is both necessary and not necessary, the notion of truth of a world will be inadequate, for reasons seen above. If, on the other hand, we are interested in determining what would follow from some impossibility, then we are more interested in the truth of notion. In attempting to determine what rules of inference would be true if intuitionistic logic were true, it would be necessary to see what is actually true in the space of worlds that are actually described by intuitionistic logic. As long as the notions of truth are distinct, there is no direct threat to our theory from the phenomenon of modal misrepresentation.

4 Conclusion In this paper, I have argued for a particular conception of impossible worlds. We should accept impossible worlds so construed because they are ontologically cheap and theoretically useful. We can use these structures to describe ways the world could not be. Moreover, we can use these worlds to make better sense of epistemic possibility, counterpossibles, belief, propositions, and properties. Structured impossible worlds are both adequate to the task of serving as the impossible worlds needed in our theory of counterpossibles, and are immune to several kinds of objections one might level against impossible worlds. They are robust enough to represent the kinds of content they would need to represent, they are capable of standing in similarity relations, and they are easily added to our ontology because they are merely set theoretic constructions of possible worlds. In short, this kind of impossible world should be accepted because it is serviceable and cheap. Acknowledgments This paper has benefitted from comments and suggestions by Phillip Bricker, Sam Cowling, Joseph Levine, Jonathan Schaffer, and an anonymous reviewer.

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