# Merely Possible Possible Worlds

# Merely Possible Possible Worlds

# Abstract and Keywords

This chapter first sketches a minimal theory of propositions—one that ascribes to propositions just the structure that anyone who is willing to talk of propositions at all must ascribe to them. It extends the minimal theory by adding some assumptions about the modal properties of propositions and possibilities, and then sketches a general model of logical space that makes room for merely possible possibilities. Next, it considers the relation between models and the reality that they purport to model and the extent to which our theory of propositions and possibilities provides a realistic semantics. Finally, it responds to some arguments against the thesis that propositions may exist contingently.

*Keywords:*
propositions, minimal theory, modal properties, possibilities

E. J. Lowe, in a general discussion of ontology, makes the following remark, in passing:

Many abstract objects—such as numbers, propositions and some sets—appear to be necessary beings in the sense that they exist “in every possible world.”… Indeed, possible worlds themselves, conceived of as abstracta—for instance as maximal consistent sets of propositions—surely exist “in every possible world.”

^{1}

I suggested in chapter 1 that this is false. Possible worlds, in the sense in which it is reasonable to believe that there are many of them—the sense in which they are “conceived of as abstracta”—are contingent objects. It does not take a very sophisticated argument to make at least a prima facie case for this claim. It seems plausible to assume, first, that there are some propositions—singular propositions—that are object-dependent in the sense that the proposition would not exist if the individual did not. It also seems plausible to assume that there are some objects that exist only contingently and that there are singular propositions about those objects. These assumptions obviously imply that there are propositions that exist only contingently, and if possible worlds are maximal consistent propositions, or maximal consistent sets of propositions, it implies (p.23) that there are possible worlds (or possible world-states) that exist only contingently. But despite the apparently compelling argument for this thesis, there are reasons to resist it, and it has been resisted. The thesis has some surprising and counterintuitive consequences, and there are some intuitively compelling arguments on the other side that we will have to consider. And if we accept the thesis, we need to consider its effect on a semantics for modal notions. My main aim in this chapter is to argue that we can reconcile the contingent existence of propositions with orthodox possible-worlds semantics, though the way of doing so that I will propose makes some concessions to the points made by Alan McMichael that I discussed briefly at the end of chapter 1. Our overall theory will, in a sense, involve a retreat, both from extensionality and from realism about possible states of the world, but it is a tactical retreat that, I will argue, preserves the virtues of both realism and extensionality.

Here is my plan for this chapter: I will first sketch a minimal theory of propositions—one that ascribes to propositions just the structure that anyone who is willing to talk of propositions at all must ascribe to them. In section 2 I will extend the minimal theory by adding some assumptions about the modal properties of propositions and possibilities, and in section 3 I will sketch a general model of logical space that makes room for merely possible possibilities. Sections 4 and 5 focus on the relation between models and the reality that they purport to model and on the extent to which our theory of propositions and possibilities provides a realistic semantics. In section 6 I will respond to some arguments against the thesis that propositions may exist contingently.

# 1. A Minimal Theory of Propositions

Propositions are the contents of speech and thought, and they are the objects to which modal properties like necessity and possibility are ascribed. There are many theories about what the things are that play these roles, but all will agree about a certain structure
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of relations between propositions: they may be compatible or incompatible with each other, one may entail another, two may be contraries or contradictories, or necessarily equivalent. They are also objects that are true or false. Many of the properties of and relations between propositions are interdefinable. A minimal theory of propositions can make do with just two primitive properties: a property of *consistency* applied to sets of propositions, and a property of *truth* applied to propositions. Before stating the postulates of the theory, I will define four additional properties in terms of consistency that will be useful for stating the postulates:

(D1) A set of propositions Γ is

maximal consistentiff it is consistent, and for every propositionx, if ΓU{x} is consistent, thenx∈ Γ.(D2) Two sets of propositions Γ

_{1}and Γ_{2}areequivalentiff for every set of propositions Δ, Γ_{1}UΔ is consistent if and only if Γ_{2}UΔ is consistent.(D3) A set of propositions Γ

entailsa propositionxiff ΓU{x} is equivalent to Γ.(D4) Two propositions

xandyarecontradictoriesiff they meet the following two conditions: (a) {x,y} is inconsistent, and (b) for every consistent set Γ, either ΓU{x} is consistent or ΓU{y} is consistent.

Individual propositions are said to be consistent or equivalent (respectively) when their unit sets are consistent or equivalent, and an individual proposition *x* entails a proposition *y* iff its unit set {*x*} entails *y*.

The postulates of the minimal theory are as follows:^{2}

(P1) Every subset of a consistent set is consistent.

(p.25) (P2) The set of all true propositions is maximal consistent.

(P3) Every proposition has a contradictory.

(P4) For every set of propositions Γ, there is a proposition

xsuch that Γ is equivalent to {x}.(P5) Every consistent set of propositions is a subset of a maximal consistent set.

(P6) Equivalent propositions are identical.

I take it that (P1) is unproblematic and needs no comment. (P2) is equivalent to the assumption that every proposition is either true or false. There may be applications of the notion of proposition for which this postulate might be denied (for example, a relativist semantics or a noncognitivist theory in which there are possibilities distinguished in thought such that there is no absolute fact of the matter which of them is actual), but I am going to ignore this complication. (P3) and (P4) are closure conditions on propositions. If propositions are something like truth conditions, then (P3) is just the assumption that if there is a certain truth condition, then there is also a condition that that condition *not* be satisfied, and (P4) is the assumption that for any set of different truth conditions, there is the condition that *all* of them be satisfied. It is not assumed that we necessarily have the resources to express all of these conditions—just that they exist.

(P5) might be denied on the grounds that propositions might be “gunky”: one might think that for every proposition *x*, no matter how specific, there are always further propositions that are incompatible with each other, but each is compatible with *x*. But while I don’t want to exclude the possibility that there are domains of propositions for which (P5) fails, I will restrict attention to domains of propositions for which it holds. I suggested in chapter 1 that one should think of possible worlds as cells of a partition of logical space rather than as points in the space—partition cells that
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are fine-grained enough to settle all issues at hand. Equivalently, one can think of the domain of propositions as all of the propositions that concern the relevant subject matter. On this more deflationary and less metaphysical conception, the closure conditions (postulates (P3) and (P4)) are still reasonable assumptions as constraints on the set of propositions that concern a subject matter. And the problems concerning contingently existing propositions that are our concern will still arise.

Many accounts of what propositions are (for example, a Russellian theory of propositions or a Fregean account of Thoughts) would reject our last postulate, (P6), allowing for distinct propositions that are necessarily equivalent. But this postulate still belongs to a theory of propositions that is appropriately called “minimal” for the following reason: however propositions are individuated, all who are willing to talk of propositions at all should agree that propositions, as they understand them, *have* truth conditions and so that the equivalence relation in terms of which (P6) is stated is well defined. Our theory of coarse-grained propositions is minimal in the sense that it characterizes an entity that all theorists of propositions can agree about, even if they want to allow, in various different ways, for more fine-grained objects that determine propositions in this coarse-grained sense. For purposes of modal semantics, we need a notion of proposition that is the right grain to ensure that our semantics is compositional, and a notion of proposition satisfying (P6) seems to meet this condition, at least if we are not trying to represent intentional mental states, even if further distinctions may be needed for other purposes.

We can represent the minimal theory of propositions pictorially with a representation of logical space. Propositions are ways of dividing the space—they can be represented by subspaces, with compatible propositions represented by overlapping subspaces, and entailment represented by inclusion. The totality of propositions determines a maximally fine partition of logical space, with
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the partition cells corresponding to maximal consistent propositions. Every proposition will correspond to a set of these maximal consistent propositions, and every such set will determine a unique proposition. This one-to-one correspondence was the ground for the identification of propositions with sets of possible worlds, or world-states, with world-states identified with maximal propositions, but this identification becomes problematic when we recognize that propositions themselves may exist only contingently.^{3} If we allow for this possibility, we must confront the question (to use familiar, perhaps metaphorical, jargon) of the identification of propositions across possible worlds, as well as some methodological questions about what we are doing when we use modal concepts to talk about the framework that we want to use to analyze and represent modal concepts.

# 2. The Modal Properties of Propositions

If we allow for contingently existing propositions and for the possibility of propositions that do not in fact exist, then there may be propositions that are *maximal consistent* in the sense defined but that are only contingently maximal and thus have the *potential* to be further refined. In chapter 1, I used a version of Kripke’s dice example to give a toy illustration of this: two indiscernible dice are thrown; one lands 6, the other 5. Had that happened, there would
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have been a different possibility in which the one that landed 6 landed 5, and the one that landed 5 landed 6. Given that the dice are merely possible, no propositions that exist in the actual world can distinguish these two possible possibilities. There is, therefore, a maximal possibility that, had it been realized, would not have been maximal. Or consider a maximal proposition that entails that Saul Kripke had seven sons and that his seventh son was a lawyer. If this proposition were true, there would then exist singular propositions about that seventh son, including the proposition that he was a lawyer and the proposition that he was not a lawyer but a plumber. The first of these merely possible propositions (that he was a lawyer) would be true in *w*, even though it is not *actually* entailed by *w*, since it—the proposition—does not actually exist. The second of these singular propositions (that he was not a lawyer but a plumber) would be merely possibly true. There would also be a singular proposition about this seventh son that he did not exist—a proposition that would be false in the maximal counterfactual situation we are considering but (it seems reasonable to believe) merely contingently false. It would therefore be true with respect to certain possible worlds that would be counterfactual worlds if Kripke had had a seventh son. Among the possible worlds in which that singular proposition would be true is our actual world, at least assuming that in fact Kripke does not have seven sons. So there could have been a proposition that does not in fact exist but that if it had existed would have been true *with respect to the actual world*.

I used the term “possible possibilities” above, which is loose talk. There are no merely possible possibilities, of course, just as there are no merely possible people. What does exist is the possibility that there are possibilities, and propositions, that do not in fact exist, and the existence of these possibilities implies the existence of general propositions about propositions, for example, the (false) proposition that there exist singular propositions that witness to a particular (false) existential proposition. (These correspond to the (p.29) higher-order properties of containment properties that I discussed in chapter 1.) The problem is how to model the complex structure that relates propositions, including propositions about propositions, to each other.

Our minimal theory of propositions made no claims about the modal status of propositions—about their essential and accidental properties or about the relation between the propositions there are and the possibility of there being others. We might add two modal assumptions to the minimal theory:

(P7) There exists a proposition that

*necessarily*entails all propositions.(P8) For any set of propositions Γ, if Γ is consistent, then

*necessarily*if Γ exists, then Γ is consistent.

The first of these modal postulates is guided by the idea that even though different possible situations may provide different resources with which to partition logical space, it is the same logical space that we partition, whatever possibility is realized. Our minimal theory already implied that there exists a (unique) proposition that is entailed by all propositions. The additional modal principle (P7) adds that this proposition *necessarily* has the property of entailing all propositions (and so necessarily exists). The second modal principle implies that all the basic propositional properties and relations (entailment, consistency, incompatibility, etc.) are essential properties and relations: for example, it follows from (P8) that for any propositions *x* and *y*, if *x* entails *y* then it is necessary that if *x* and *y* exist, then *x* entails *y*.

These additional principles may seem plausible enough, but is it legitimate for us to help ourselves to the notion of necessity in stating principles of the theory of propositions that will be the basis of our semantics for a modal language? I have disclaimed any pretension to be reducing modal to nonmodal concepts—I even claimed that it is a virtue of a theory that it avoids such a reduction. But I (p.30) also promised a vindication of possible-worlds semantics, a semantic theory that is presumed to get its explanatory power in part from its extensionality. As McMichael argued, “if we have to give up the extensionality of the possible-worlds approach, we might as well do without it.” But I hope to provide an interpretation of the orthodox semantics that retains the virtues of extensionality while also making use of primitive modality in the theory of propositions.

Kit Fine, following Arthur Prior, defended a thesis that he called “modalism”: “The ordinary modal idioms (necessarily, possibly) are primitive.”^{4} Fine takes modalism to be incompatible with the possible-worlds “analysis” of modal concepts, while acknowledging that possible-worlds semantics may have its uses if it is not regarded as providing an analysis. If by “analysis” one means an eliminative reduction, then I think most possible-worlds theorists (David Lewis aside) will agree with modalism, but one may still hold that possible-worlds semantics provides a genuine explanation, in some sense, of the meanings of modal expressions. The problem is to clarify the sense in which an explanation, short of reduction, is still an explanation. Consider the analogy with the semantics for first-order quantification theory: there are quantifiers in the semantical metalanguage, and of course the semantic “analysis” of the quantifiers provides no reduction of the concepts of existence and universality to anything more basic. But the theory still gives us a compositional semantics that sharpens and clarifies the structure of quantificational discourse—the ways that quantifiers interact with each other, as well as with names, predicates, and other logical operators. The project of reconciling possible-worlds semantics with intuitively plausible metaphysical commitments faces some special problems not present in the case of extensional quantification theory, but the fact that we need to use modal concepts to explain our primitives is not itself a problem.

# (p.31) 3. The Model of Logical Space

I will have more to say about primitive modality later in this chapter, but let me first elaborate a bit on our picture of logical space, partitioned by the maximal propositions: the way the space is partitioned is a contingent matter, depending on the resources available in the actual world—the world in which our theory of propositions is being stated. If one of the maximal propositions that is in fact false were to have been realized, there would have been different resources available, and logical space (the same logical space) would have been partitioned in a different way—perhaps more finely in some respects and more coarsely in others. (And when I say that different resources would have been *available*, I mean not just that those who are partitioning logical space would have had access to different resources; I mean that different resources would have existed.) We can model this conception of logical space with a set of points and a function taking each point to an equivalence relation that provides the maximal partition that *would* divide logical space, were that point to represent the possible state of the world that is realized. Now we know what the cells of our basic partition (for the actual world) represent: maximal consistent propositions. But what do the *points* represent? Specifically, what is the difference in what is represented by two distinct points within the same partition cell? The answer is that they all have exactly the same representational significance, but we need many of them in order to represent the way in which maximal propositions have the potential to be further refined. This potential is reflected in higher-order propositions about propositions and in the iterated modal propositions that were the basis of McMichael’s challenge to what he called “atomistic actualism.” Intuitively, one may think of the points as representations of possibilities, one of which *would* be maximal if the partition cell of which they are members had been realized. Which one of them would have been realized? There is no fact of
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the matter about that, since each point within an equivalence class has exactly the same representational significance (in the actual world) as every other point within its equivalence class. But we need more than one such point in order to represent the different possibilities that *would* exist if that possibility were realized. To return to our toy example of the dice, we want to represent the fact that if our dice had existed, there would have been *two* possibilities in which one die landed 6 and the other 5, since had there been two such dice that landed 6 and 5, there would have been a distinct possibility in which they landed the other way. But since they don’t in fact exist and are characterized generically as indiscernible, there is, in the actual world, only one maximal property (one 6, the other 5) to represent these two distinct possible possibilities.

Since a maximal consistent proposition will, by definition, decide *all* (actual) propositions, including the higher-order ones about what kinds of propositions would exist if that maximal proposition were realized, our framework must impose structural constraints on the family of equivalence relations to ensure that each of the points within any maximal equivalence class will decide all the higher-order propositions in exactly the same way. The technical details of these structural constraints are spelled out in appendix A.

# 4. Models and the Reality They Model

When we come to doing our modal semantics, it will be the points, not the maximal equivalence classes of points, that play the role of possible worlds. But now the attentive reader may be inclined to cry foul, complaining that she has been subjected to a bait and switch. I started by asking what possible worlds are and answered that they are properties that a universe might have—maximal properties or, equivalently (I proposed), maximal consistent propositions. I also promised a vindication of orthodox possible-worlds semantics, but now I am saying that the entities that are the “possible
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worlds” in the models of our orthodox semantics are different from the maximal propositions that I began by identifying with possible worlds. While we might be persuaded to be realists about maximal propositions (the critic complains), it is another matter to be realists about these points that our models must use to make sense of iterated modal claims. Have we simply replaced the unacceptable nonactual possibles with *so-called* nonactual possibles (to echo McMichael), things such as numbers or sets that are *not* nonactual possibles but that are suitable to model them?

This complaint is fair enough, but I did warn you that there would be some concessions to the critics of actualist possible-worlds semantics and a tactical retreat from a fully realistic interpretation of that framework. And while the interpretation of the semantics that I want to give is perhaps not straightforwardly realistic, I will argue that it can answer the criticisms that McMichael makes.

McMichael had two reasons for finding a nonrealist strategy unsatisfactory. First, for any semantics that “contains nonrealistic elements, the problem will arise of distinguishing what aspects of the semantics are of genuine significance and what aspects are purely artificial. We will want a method for ‘factoring out’ the artificial aspects. But a nonrealistic semantics coupled with a method of ‘factoring out’ is just a realistic semantics.”^{5} I think this is exactly right, but McMichael’s own discussion points to the strategy for answering his challenge. He discusses the analogy with a relational theory of space. According to the relationist, there are really no such things as spatial locations—there are just spatial relations between things. But the best way to model the structure of spatial relations is with a (mathematical) space, made up of points (*so-called* spatial locations). One “factors out” the artifacts of the model—separates them from the realistic claims of the theory—by adding to the theory an equivalence relation between spatial models. Equivalent models
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are those that differ in artificial ways but that agree in the realistic claims they make about the spatial relations between things. There are different versions of this kind of relational theory, made precise with different equivalence relations. One might say that all and only permutations of spatial points that preserve distance relations between points are equivalent representations.^{6} So a possible world in which everything, throughout history, is three feet to the north, or rotated forty-five degrees on a certain axis, from the way things are in the actual world is just a conventional redescription of the actual world. Or one might say that only *ratios* of distances need to be preserved. Adolph Grünbaum once discussed the verifiability and intelligibility of what he called the “universal nocturnal expansion hypothesis” that everything should instantaneously double in size. (Why this has to happen at night, I am not sure.)^{7} The question is one of the identification of spatial location, and spatial properties and relations, across possible worlds. Could I fix the reference of a spatial location (for example, the place of the center of mass of the sun, at a specified time) and then stipulate that a certain counterfactual situation should be one where some specified kind of event takes place *there*? The reference-fixing act presupposes that there exist locations reference to which can be fixed.

There are familiar questions of the identification of locations across time, as well as across worlds (or, equivalently, about the relations between spatial and temporal structures). Since the basic laws of Newtonian physics are invariant in all inertial frames, one might accept Newtonian physics while being a Galilean relativist, holding that there is no such thing as absolute velocity, only velocity relative to an inertial frame. An equivalence relation, defined by a class of permutation relations on spatio-temporal points, makes (p.35) the notion of an inertial frame, and this relativistic thesis, precise. The Galilean relativist is, in a sense, anti-realist about spatial locations, but he can use the same basic framework for representing his theory as the Newtonian absolutist, and he can, as McMichael puts it, “factor out” the conventional, or anti-realist, aspects of his physics from the part about which he claims there is a fact of the matter. If we can be as precise as the Galilean relativist in stating the relevant equivalence relation, I think we will have given an adequate interpretation of our modal semantics.

I should add that there is more than just an analogy here. One may think of abstract spaces in general as representational devices for modeling properties and relations.^{8} The application to *physical* space and time is just the most prominent application. We will have more to say about space and spatial relations in the next chapter.

Another analogy that is relevant is utility theory: real numbers are used to specify utility values, which are intended to reflect in a systematic way the motivational states that dispose a rational agent to act in a range of different possible circumstances. But the numbers themselves are conventional: any positive linear transformation of a utility function is a representation of exactly the same motivational state. Even with fundamental measurement—quantities such as length and mass—there is a conventional element in the use of real numbers to represent the quantities, but the relevant equivalence relation is stricter, requiring the preservation of ratios, and not just ratios of intervals. In general, if one is trying to model a purely relational structure, the strategy of “factoring out” the artifacts of the model with an equivalence relation is a familiar one. According to an actualist, the facts about iterated modalities, at least those involving the possibility of things that do not in fact exist, are the kind of purely relational facts that should be modeled in this way.

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As McMichael says about the spatial analogy, “the problem of distinguishing . . . invariant features from artificial ones is just as important as finding a coordinate system that ‘works.’ ”^{9} The family of equivalence relations that is part of the modal theory of propositions I have sketched (and that is spelled out in more detail in the appendix) aims to make this distinction precise. If McMichael is right that “a nonrealistic semantics coupled with a method of ‘factoring out’ is just a realistic semantics,” then our interpretation of the orthodox possible-worlds semantics should count as a realistic one.

But is McMichael right about this? Is this indirect way of giving a semantics for a modal theory sufficient? The critic might say: “What I want to know is, what is there *really*, according to your theory?” The answer is: there are individuals—actual ones only—though what individuals there are is a substantive question, mostly empirical, and our abstract theory remains neutral about most of that. There are also properties, propositions, and relations (again, actual ones only). About them we have more to say, though we still remain largely neutral about what properties and relations there are (for example, about whether there are absolute spatial locations, or whether there are irreducible mental properties). But there are (according to our theory) not only properties of individuals^{10} but also higher-order properties and relations: properties of properties, relations between properties, and relations between properties and other things. I talked about some of these higher-order properties and relations in the discussion of containment properties in chapter 1: there was, for example, a correspondence relation between generic and specific properties. There will be a correspondence relation like this between existential propositions and their witnesses. Salient among the higher-order properties and relations are
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*propositional functions*. A function, in general, is a special case of a relation, so a propositional function is a kind of relation between an individual and a proposition. And there are also functions from individuals to propositional functions, functions from individuals to functions from individuals to propositional functions, and on up from there. (See appendix B for a digression on propositional functions and a way to do the modal semantics that does not involve quantification, in the metalanguage, over merely possible individuals.)

So there are properties and relations. We can say, in the context of Kripke models, how properties, relations, and propositional functions are modeled: properties, for example, are modeled as functions from possible worlds (points) to subsets of the domain of that world. But what *are* properties, really, in themselves? As I said in chapter 1, properties are to be understood in terms of the way the world would have to be for them to be instantiated. The concept of a property is a basic concept, not reducible to something else, but it is not an isolated concept: we can say a lot, not just about particular properties (by elaborating, say, about what a thing must be like to be a donkey, to be yellow, or to be soluble) but also about how properties and relations are related to each other—about what their role is in a complex relational structure. How do we do that? By using models, such as standard Kripke models, together with a family of equivalence relations on the points in the model that spell out what features of the models are artifacts and what parts are features of the reality being modeled.

In characterizing possible worlds-states as a kind of property, I emphasized that they are therefore not (or at least not essentially) representations. But of course we can theorize about properties, propositions, and possible states of the world only by representing them (just as an astronomer can theorize about planets and stars only by representing them and their properties). Unlike the possible states of the world—maximal ways a world might be—that
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are being modeled in our semantic theory, the points (the “possible worlds”) in a Kripke model *are* representations—representations of the properties that I claim possible states of the world are. The whole Kripke model represents not just these properties but also a structure of relations between these properties (the possible states of the world) and between them and other things. The points themselves are not properties—they are points in an abstract space that are being used to represent possible states of the world. So I won’t even complain if you call the points in a Kripke model “ersatz possible worlds” or better, “ersatz possible *states* of the world,” since it is ways a world might be that they represent.

Aviv Hoffmann, in a commentary on the paper that this chapter draws on,^{11} argues that I have not really defended the thesis that propositions and possible world-states are contingent entities, but rather the different thesis that the things I call propositions and possible worlds are only contingently propositions and possible worlds. According to Hoffmann, what I have shown, at best, is that being a proposition may be a contingent property but that on my account, the things themselves that are contingently propositions are still necessary existents. The argument is based on a principle of modal set theory, which should be uncontroversial: a set exists in a given possible world if and only if all of its members exist at that world. Hoffmann argues that since for each point in logical space it is necessary that there exists some proposition containing that point, it follows that all the points, and so all the subsets of the set of all points, necessarily exist. His conclusion is that while I can deny that certain subsets are propositions, relative to certain possible worlds, I cannot deny that they exist, relative to that world. But I think Hoffmann’s argument conflates the propositions and possible states of the world with the sets of points of a space that represent them. The points are elements of an abstract space.
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Whether the space itself, with its points, is a necessary existent or not is an independent question. Even if mathematical spaces of the right structure existed only contingently, one could still use them to represent propositions that entail the nonexistence of the space itself. On the other hand, if the space and all of its points are necessary existents, that does not prevent us from using them to represent contingently existing things. (We cannot, for example, infer from the fact that Babe Ruth is represented by the number 3 that the Babe himself is therefore an eternal and necessary being.)

# 5. Nonrealistic Semantics

I have argued that our overall semantic theory is, in the sense that matters, a realistic one, while acknowledging that there is a sense in which we are not realists about the possible worlds, and possible individuals of the Kripke models used to represent the ways things might be. I have tried to meet, in a precise way, McMichael’s challenge to “factor out” the nonrealistic elements of the theory, and if I have succeeded, then McMichael’s more general concern about a nonrealist semantics should not apply to the account I am promoting. But let me consider that concern and distinguish more explicitly the account I want to defend from one that simply uses a domain of “so-called” possibilia as surrogates for what might exist but does not.

McMichael concedes that a nonrealistic semantics that uses some arbitrary surrogates for possibilia might be sufficient if our project is only to do semantics for modal *logic*. He suggests that a nonrealistic semantics might give an adequate account of the notions of validity and satisfiability that are appropriate for modal language, but “what we are ultimately interested in,” he says, “is to give truth conditions for some nice modal fragment of a natural language,” and a Kripke semantics, by itself, “does not supply conditions for *truth* [as contrasted with conditions for *validity*].”
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If what we are interested in is *truth* conditions, he claims, then we need to answer questions about “the number and relationships of nonactual possible individuals.”^{12}

I don’t think the distinction between natural language semantics and semantics for the formal language of modal logic is relevant to the issue McMichael is raising. In a robustly realistic theory, the languages in which we formulate our philosophical commitments might be regimented formal languages rather than fragments of natural language. The issue he is raising, I think, is about how to understand the models that are used to interpret the language. I take the point to be something like this: an *intended* model, in a robustly realist semantics, is not a *representation* of the subject matter of one’s theory; it is the subject matter itself. If I formulate a theory that I wish to defend—if I claim that my theory is true—then in the intended model for my theory, the domain will be the things I claim to be talking about and not substitutes for them. In the intended model for a theory about DNA molecules, for example, the domain (which is part of the model) contains DNA molecules themselves and not models (perhaps made out of Tinkertoys) of them.

Unintended models, as McMichael suggests, have their place. Suppose I have a friend who has an elaborate theory about elves, sprites, fairies, and leprechauns. I don’t accept his theory and in fact would have a hard time specifying, in a precise way, the *truth* conditions for his theory—saying exactly what the world would have to be like for his theory to be true. But if my task is just to assess the validity of his reasoning when he expounds his theory and draws conclusions from its basic principles, I will have an easier time. I might regiment his theory in a first-order (extensional) language and specify a model for it, perhaps using a set of natural numbers for the domain (to be the *so-called* elves, sprites, fairies, and
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leprechauns). My model is not my attempt to specify my friend’s *intended* model, but if my interpretation makes the sentences he uses to state the theory come out true in the model, that might be good enough for my purpose, which is just to assess his reasoning. I will not, however, have provided a semantics for my friend’s theory that is realist, in any sense.

I agree with McMichael on this general point but would argue that the relationship between a theory and its subject matter may be more complex than the simple story suggests. Models (in the model theorist’s sense of the term) may be used in different ways in one’s representation of the subject matter of a theory; in particular, models may be interposed between a language used to talk about some domain and the domain that the theory talks about. That, I want to suggest, is what is going on when one uses a model, plus an equivalence relation, to represent a structure of relations, as in a relational theory of space and time. A theorist who represents her theory of space and time in this way may be a realist about spatio-temporal relations and about the physical objects that exemplify such relations, even if she is not a realist about spatial locations themselves. As I have said, the merely possible individuals, and the points in logical space used in Kripke models as I am interpreting them, are like the spatial points in a relativist’s model of spatial structure. The intended subject matter of our modal theory consists of the actual individuals, the (actual) properties and relation that they might exemplify, and the (actual) higher-order properties and relations that might be exemplified by properties, relations, and propositions, as well as by individuals. About all these things, our theory can be resolutely realistic.

But while I agree that to be a realist, in the sense I want to be a realist, is to accept a commitment to the existence of the things in the domain of the intended model of one’s theory (allowing that models may also play a intermediate representational role), this is not necessarily to accept a commitment to the comprehensiveness (p.42) of one’s theory—to accept (to echo the words of Barwise and Perry quoted in chapter 1) that there is a unique intended model of super-reality that will provide us with a complete inventory of all the things, properties, and relations there are in the universe. I am inclined to be skeptical about the positive answer to the contentious philosophical question whether it makes sense to quantify over absolutely everything, but this is a separate question on which the kind of modal theory I am defending remains neutral.

If you press me on the question, how many points there should be in a model that represents the metaphysical possibilities, including the possibilities of things that might exist but don’t, I am inclined to answer, as many as you need to model the modal propositions that you want to model. The answer to the question, how many is that, will depend both on the expressive resources of the language one is using the models to interpret and on one’s metaphysical views about what is possible. The abstract semantic theory won’t answer those questions, but I hope at least that it will help clarify what is being asked.^{13}

# 6. Objections and Replies

I will conclude this chapter by looking at some of the problematic consequences of the thesis that propositions may exist merely contingently, but first let me remind you of the prima facie case for this thesis.

It is motivated by just two simple assumptions. The first is a doctrine Alvin Plantinga calls *Existentialism*: “a singular proposition is ontologically dependent on the individuals it is directly about.”^{14} The second is the claim that there are some things that exist only
(p.43)
contingently. The second of these assumptions seems to require little explanation or defense, although some philosophers have denied it, as we will see. It seems at least prima facie reasonable to take it to be a Moorean fact that people and ordinary physical objects are things that might not have existed. But what about the first assumption? There are different accounts of what propositions are that might motivate this thesis. If you think of a singular proposition as a kind of Russellian proposition, an ordered sequence containing the individual, along with properties and relations, as constituents, then it is natural to think that the existentialist thesis must be true, since it is natural to believe that sets and sequences are ontologically dependent on their elements. But even if one is presupposing, as I am, a coarse-grained conception according to which propositions are individuated by their truth conditions, it seems prima facie plausible to think that propositions about particular individuals are ontologically dependent on the individuals they are about. On the coarse-grained conception, propositions are truth conditions, and the truth condition for a singular proposition is a condition that the world must meet (for the proposition to be true) that essentially involves the individual that the proposition is about. It seems reasonable to believe that a condition that depends for its satisfaction on the way Socrates is requires, for its existence, the existence of Socrates. While our propositions are not complexes with properties and individuals as constituents, we retain the idea that propositions are built out of the materials we find in the actual world. Any actualist must accept this, but what materials one thinks there are will depend on one’s metaphysical and empirical beliefs. One can reconcile actualism with a rejection of object-dependence, if one is willing to make certain metaphysical commitments. One way to do this is to hold that there are qualitative conditions that are necessary and sufficient for the existence of a particular individual, so that singular propositions about actual or possible individuals are reducible to purely qualitative propositions. We will
(p.44)
discuss this strategy in chapter 3. Alvin Plantinga adopts a different strategy, which is to hold that for each individual, there is an individual essence, or haecceity, that exists independently of an individual that exemplifies that individual essence. So for Plantinga, while there may be no actually existing thing that would be Saul Kripke’s seventh son if he had seven sons, there do exist properties, probably infinitely many of them, that would suffice to individuate each of the possible individuals who, in each of the possible worlds in which Kripke had seven sons, would have been his seventh son. One might use an identity property to fix the reference of a term referring to an individual essence, for example, the property of being identical to Obama. But the assumption is that while reference may be fixed in this way, the individual essence itself is a property that exists independently of the object used to fix the reference. This may seem an extravagant metaphysical commitment, but Plantinga has an argument against the thesis of object-dependence, or existentialism, which commits him to it.

Plantinga’s argument has five premises:

P1. Possibly, Socrates does not exist.

P2. If P1, then the proposition

*Socrates does not exist*is possible.P3. If the proposition

*Socrates does not exist*is possible, then the proposition*Socrates does not exist*is possibly trueP4. Necessarily, if

*Socrates does not exist*had been true, then*Socrates does not exist*would have existed.P5. Necessarily, if

*Socrates does not exist*had been true, then Socrates would not have existed.^{15}

The conclusion drawn from these premises is as follows:

C. It is possible that both Socrates does not exist and the proposition

*Socrates does not exist*exists.

(p.45)
The first three premises obviously entail that *Socrates does not exist* is possibly true, and this, together with P4 and P5, entails C. The latter inference has the form:

This is an inference that is valid in any normal modal logic, so the argument as a whole is valid.

Plantinga takes premise P1 to be uncontroversial, not anticipating Williamson’s reason for rejecting his conclusion, and everyone will accept P5. But Plantinga notes that each of the other premises has been denied (by Larry Powers, Arthur Prior, and John Pollock, respectively), and he considers three different defenses of object-dependence that choose one of these premises to reject. This way of setting up the problem exaggerates the differences between the three responses to the argument that he considers, since (I will argue) there is an equivocation in the consequent of P2 (and the antecedent of P3), and the choice of which premise to reject depends on how that equivocation is resolved. The responses that Plantinga calls “Priorian existentialism,” which rejects P2 and “Powersian existentialism,” which rejects P3, are different only in that they resolve the equivocation in different ways. The third response that Plantinga considers, “Pollockian existentialism,” which rejects P4, also turns on the distinction between the two ways of understanding truth, though since I don’t think one can get to the second stage of the argument in any case, the rejection of P4 on one (less natural) interpretation is not necessary to defeat the argument.

To bring out the different interpretations of the clause that is the consequent of P2 and the antecedent of P3, let me introduce some notation. First, I will use ‘π’ as a term-forming operator on sentences. For any sentence ϕ, ‘πϕ’ will denote the proposition expressed by ϕ. So if S abbreviates the sentence “Socrates does not exist,” then ‘πS’ will denote the proposition that Socrates does not exist. Second, I will use the letter ‘T’ to be the monadic truth
(p.46)
predicate, applied to propositions. Third, I will use a binary predicate, ‘I’ relating propositions, for “entails” (as defined in our minimal theory of propositions). So ‘I*xy*’ says that proposition *x* entails proposition *y*. Fourth, I will use a variable-binding abstraction operator to form complex predicates. ‘$\widehat{x}\text{}(\text{F}x\text{}\vee \text{}\text{G}x)$’ will be a monadic predicate that will have, as its extension, the individuals that are in the extension of either F or G. For present purposes, the variables ‘x’ and ‘y’ will range over propositions generally, but the variable ‘*w’* will be restricted to possible worlds, understood as maximal propositions.

Plantinga’s premises P2 and P3 are stated in terms of a predicate of propositions, “is possible,” which might be defined in terms of truth or entailment in several different ways. Here are two definitions, which may not be equivalent.

The first predicate applies to propositions that are possibly true *in* some possible world, while the second applies to propositions that are true *of* or *entailed by* some possible world. The two definitions will be equivalent if all propositions exist necessarily, but not if some do not.

If one understands the predicate of possibility, as it occurs in the premises of Plantinga’s argument in the first way, then the defender of object-dependence should reject P2 but accept P3 (opting for the Powers response). On the other hand, if one understands the predicate in the second way, then the defender of object-dependence should accept P2 but reject P3 (opting for the Prior response). The singular proposition *Socrates does not exist* is a proposition that will be true *of* or *entailed by* only possible worlds in which that proposition does not exist. Since we are agreeing with Plantinga that nothing can be truly predicated of something that does not exist, the truth predicate will not apply to any proposition in a possible world in which that proposition does not exist.

(p.47) The Pollock response rejects P4. The most natural way to take P4 is to take the antecedent of the conditional at face value as an application of the predicate of truth to the proposition. But there may be some uses of the word “true” that should be understood as treating it like a redundant operator, its role being either rhetorical or to help mark a scope distinction. But given that there is a way to understand P5 so that it is true, it does not really help to find an interpretation according to which it is false. So I think it is the diagnosis in terms of the equivocation in P2 and P3 that shows where the argument fails.

I have argued that Plantinga’s argument can be resisted, even by those who accept its premises, but the thesis that some propositions and possible states of the world exist only contingently does have some discomfiting consequences. To start, I want to note that any account of propositions and possible worlds, the one I am defending or any alternative to it, that allows for contingently existing propositions will require the distinction that I appealed to in my discussion of Plantinga’s argument between what Kit Fine has called inner and outer truth: there will be propositions that are true *of* or *at* or *with respect to* a possible world, while not being true *in* that possible world. For a proposition to be true *in* a possible world is for it to have, in that world, the property of truth. For a proposition to be true *of* a possible world is for it to stand (in the actual world) in a certain relation (the entailment relation) to that possible world. That these two notions come apart follows from the following assumptions: (1) some propositions exist only contingently; (2) every proposition has a contradictory (this is a postulate of our minimal theory of propositions); and (3) necessarily, only existing things have properties, and in particular, only existing propositions have the property of truth. Here is the argument. By (1), there is a proposition *x* that exists only contingently, which means that there will be a possible world-state *w* that does not include *x* in its domain. But then by (2), its contradictory will also not be in the domain of *w*, and so by (3), neither *x* nor its contradictory
(p.48)
will have, in *w*, the property of truth. But world-states are, by definition, maximal, and so for any proposition, *w* will entail either the proposition or its contradictory. So since *w* will entail either *x* or the contradictory of *x*, there will be a proposition that is true *of* that world-state but not true *in* it.

But if there are cases of propositions that are true *of* or *at* a certain possible situation but not true *in* that situation, because they do not exist there, there will be violations of the necessitation of a simple truth schema—the schema for sentences of the form “ϕ if and only if it is true that ϕ.” In the notation we have been using, the schema and its necessitation are ϕ ↔ Tπϕ and □(ϕ↔Tπϕ)). In possible worlds *of* which ϕ is true but in which the proposition does not exist, Tπϕ will be false in virtue of the reference failure, in that possible world, of the term πϕ. Furthermore, there will be a divergence between the *entailment* relation and necessary truth preservation; that is, there will be counterexamples to (I*xy* ↔ □(T*x* → T*y*)). Proposition *x* might entail proposition *y*, even if *y* might not exist in a possible world in which *x* is true, in which case the equivalence will fail in the left-to-right direction. (An example: The proposition that no one is immortal entails the proposition that it is not the case that Obama is immortal. But if Obama had not existed, the proposition that he was [or that he was not] immortal would not exist, and so the proposition that it is not the case that he is immortal would not be true. But it might still be true, in such a counterfactual situation, that no one was immortal.)

These consequences might lead one to be skeptical of the true-in/true-at distinction. Timothy Williamson rejects it, as applied to propositions, though he acknowledges that it makes sense for sentences and other objects that express propositions.^{16} His claim is that the only way to understand “true-at” is as “true-in.” Since he also accepts object-dependence, or existentialism, he draws
(p.49)
the conclusion that everything exists necessarily and that nothing could exist except what does exist. I find his reasons for rejecting the distinction unpersuasive, but one of his worries points to the general issue we’ve met before about the extent to which the concepts of our theory are explanatory. He puts the problem as a worry about circularity. Suppose we define a possible world as a consistent and complete set of propositions, where a set Γ of propositions is consistent if and only if for any proposition *y*, if there is a valid argument from Γ to *y*, then there is not a valid argument from Γ to the contradictory of *y*, and Γ is complete if and only if for any *y*, there is a valid argument either from Γ to *y* or from Γ to the contradictory of *y*. (If we understand “validity” as entailment, then this is exactly what we have done in our minimal theory of propositions, so no problem so far.) But now he proposes that we explain the notion of validity (entailment) as *necessary* truth preservation. Possible worlds are thus explained in terms of the notion of validity, which is explained in terms of truth and necessity. But necessity is explained in terms of truth at (or with respect to) all possible worlds.

In our minimal theory of propositions, we did use a necessity operator in the metalanguage, and even though using this resource is clearly incompatible with a use of possible worlds to provide a reductive analysis of possible worlds, I argued that it did not make our semantics unacceptably circular (though I offered no general account of what makes a nonreductive theory explanatory). But I did not use our metalinguistic necessity operator to analyze the notion of entailment in the way that Williamson suggests. I took *consistency* as a primitive of the theory of propositions, though we could have begun with entailment and defined consistency in terms of it. A more austere theory of propositions might have just one primitive—a truth predicate—defining entailment (in the modal metalanguage) as follows: proposition *x* entails proposition *y* if and only if necessarily, if *x* is true, *y* is true. But as I have
(p.50)
noted, our account rejects this analysis, not because it makes the explanatory circle too tight (though this might be reason enough) but because it gets entailment wrong. There is a dialectical standoff here, since Williamson is right that true-of collapses to true-in, if we accept the more austere theory of propositions. I acknowledge that it seems prima facie plausible to identify entailment between propositions with the necessity of the (material) conditional that if one is true, so is the other. But if we acknowledge the possibility of contingently existing propositions, it is easy to understand why and how this identification may fail. Given the metaphysical implausibility of the ontological commitments we must undertake to avoid contingently existing proposition, it seems to me that giving up this identification is a small price to pay.

I don’t have an argument against the existence of the kind of individual essences for nonexistent things that Plantinga believes in, nor do I have an argument against the existence of a vast population of actual things that could have been living people and other material beings but actually are not and reside in some realm outside of space and time. The famous incredulous stare that David Lewis took to be the strongest argument against his modal realism is good enough for me, not only for Lewis’s modal realism but also for the metaphysical commitments of Plantinga and Williamson, which seem to me to have about the same degree of prima facie plausibility. My aim has been to show that one can at least give a coherent account of modality that allows for an expansive view of what is possible—one that accords with our pre-theoretic modal beliefs—without committing oneself to an excessively extravagant view of what actually exists. The modal framework is supposed to be neutral, allowing for the kinds of ontologies that Williamson and Plantinga endorse, as well as for those that accord more closely with common opinion. But if we succeed in rebutting some unsound arguments in support of such ontologies, I think the temptation to believe in them should go away.

(p.51) Still, if all we do, in our metaphysical argumentation, is rebut arguments for accepting certain metaphysical theses, there will remain a question about what it is that makes one or another of the available alternative metaphysical doctrines correct, or at least worthy of acceptance. I won’t have a general answer to this question, which continues to puzzle me, but in the next chapter I will consider some of the arguments for and against some contrasting views about the relation between individuals and their properties.

What Is Haecceitism?

## Notes:

Parts of this chapter were previously published as Stalnaker 2009. Thanks to Oxford University Press for permission to include them here.

(^{2})
These postulates differ from those given for a minimal theory or propositions in Stalnaker 1976. Earlier changes were in response to problems pointed out by Philip Bricker. Later changes were in response to problems pointed out to me by Damien Rochford.

(^{3})
Aviv Hoffmann has provided a decisive refutation of the thesis that possible worlds can be identified with sets of maximal propositions, on the assumption that there are object-dependent singular propositions about contingently existing individuals. The contingency would proliferate, implying that purely existential propositions, in fact all propositions, were object-dependent. In Stalnaker 1976, I compare the two orders of analysis and note that given certain independently plausible assumptions, a minimal theory of propositions will yield the conclusion that there is a one-to-one correspondence between propositions and sets of maximal consistent propositions. But the move from this correspondence to an identification of the propositions with the corresponding sets ignores the modal properties of propositions.

(^{6})
For simplicity of illustration, I am assuming the independence of spatial and temporal dimensions.

(^{7})
See Grünbaum 1964. I believe that *Playboy* magazine once took note of this hypothesis, wondering (based just on its name, in the title of an article) what the hypothesis might be and what these philosophers were up to.

(^{8})
See Hawthorne and Sider 2002 for an interesting discussion of abstract spaces used to represent properties and relations.

(^{10})
By “individual” here, I mean things that are not themselves properties, propositions, or relations.

(^{13})
Thanks to an anonymous reader for questions and comments that helped me clarify my response to McMichael’s worries about anti-realist interpretations of the modal semantic.

(^{16})
See Williamson 2001.