# Pleonastic possible worlds

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DOI: 10.1007/s11098-012-9857-z

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- Steinberg, A. Philos Stud (2013) 164: 767. doi:10.1007/s11098-012-9857-z

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## Abstract

The role of possible worlds in philosophy is hard to overestimate. Nevertheless, their nature and existence is very controversial. This is particularly serious, since their standard applications depend on there being sufficiently many of them. The paper develops an account of possible worlds on which it is particularly easy to believe in their existence: an account of possible worlds as pleonastic entities. Pleonastic entities are entities whose existence can be validly inferred from statements that neither refer to nor quantify over them as a matter of conceptual necessity. Definitions are proposed that ensure that this is the case for possible worlds.

### Keywords

ModalityPossible worlds, account ofPleonastic entitiesSchiffer, StephenThe role of possible worlds in philosophy is hard to overestimate. Much, if not all, of philosophy is concerned with discovering truths with modal import. Once such modal truths are at issue, philosophers find it hard not to slide into talking about what is the case at this, some or every possible world. But even when we employ ordinary modal idioms—like the modal auxiliaries ‘might’, ‘could’ and ‘must’, the modal sentential operators ‘it is possible that’ and ‘it is necessary that’ or the counterfactual conditional—we use sentences whose standard semantics is given in terms of possible worlds. Moreover, important philosophical concepts are often elucidated with the help of possible worlds, as for instance the concepts of supervenience, determinism and essential property. Lastly, some philosophers have hoped to reduce certain abstract entities—like propositions or properties—to sets of possible worlds or their inhabitants. In short, it is hard to do philosophy without relying on possible worlds.

At the same time, the nature and existence of possible worlds is anything but uncontroversial. Often when philosophers merely find it convenient to employ talk of possible worlds they accompany such talk with a disclaimer that they are not committed to a particular conception of possible worlds or even to the view that there really are any. And even those philosopher who accept that there are possible worlds disagree significantly on what sort of things they are supposed to be. Views range from maximal mereological sums of spatio-temporally related concrete entities (Lewis 1986), maximal properties of the universe (Stalnaker 2003, 2010), maximal possible states of affairs (Plantinga 1974) to maximal consistent sets of propositions (Adams 1974). Further, some of their proponents think that possible worlds have fundamental explanatory work to do (Lewis 1986; Stalnaker 2003), while others are happy to accept that possible worlds should themselves be explained in terms of what is modally the case (Plantinga 1976, p. 144, 1985, p. 89).

This disagreement about their nature and existence is all the more serious since possible worlds can only straightforwardly fulfil their roles in semantics and clarification if there are enough of them. Consider for instance the anti-realist about possible worlds who claims that there is at most one possible world, the actual world. Anti-realists should maintain that the standard semantics for possibility sentences is false. According to the standard semantics, ‘McCain could have won the 2008 elections’ is true just in case there is a possible world at which McCain wins the 2008 elections. Since McCain didn’t win but could have, the standard semantics for possibility sentences is false if the anti-realist is right. Likewise, anti-realists should deny the standard explication of the notion of an essential property. According to the standard explication, *P* is an essential property of *x* just in case *x* has *P* at every world at which *x* exists. Since things have properties that are not essential to them, the standard explication of an essential property is false if anti-realism is correct.

Whether or not we should be anti-realists about possible worlds depends at least partially on what we take possible worlds to be. If we take other possible worlds to be mereological sums spatio-temporally isolated from the actual world, and we have reason to believe that nothing is spatio-temporally isolated from the actual world, we should be anti-realists.
^{1} On other accounts realism about possible worlds may be more easily defensible. This paper develops an account of possible worlds on which it is particularly easy to be a realist—an account of possible worlds as what Stephen Schiffer (2003, chap. 2) calls *pleonastic entities*. Roughly, they are the kinds of things of which it is a conceptual truth that their existence can be validly inferred from truths that do not mention them at all. One advantage of this account of possible worlds is that it is but a special case of a plausible account of abstract objects in general. Another advantage is that a pleonastic account of possible worlds allows a satisfactory description of our epistemic access to them. This is spelled out in Sect. 1. Section 2 gives a brief overview of the pleonastic account of abstract objects in general. Section 3 applies the account to possible worlds.

Let me begin with a remark on the scope of this paper. In order to keep things manageable I will be exclusively concerned with possible worlds as they have to be for a semantics of *sentential* modal logics. That is, I will not worry about the domain of worlds or about questions of *de re* modality—e.g. the question of how to evaluate sentences with modal open subsentences, and, in particular, whether for their evaluation other *possibilia* are needed than just possible *worlds*. This is an issue that will have to be dealt with once the pleonastic account has been set up. But this is a topic for another paper.

## 1 Motivating pleonastic possible worlds

^{2}The canonical way of coming to know about them is via certain transformations that take one from sentences in which these entities are not referred to or quantified over to sentences in which they are. Before we entered the philosophy class room we knew many things modal. We knew that McCain might have won the 2008 presidential elections if only he had chosen a different running mate, we knew that it was impossible for him to win and lose the 2008 presidential elections at the same time and that it was necessary that either someone won or no one did. But we did not know about the existence or otherwise of possible worlds.

^{3}We did not know that there is a possible world at which McCain chooses a different running mate and wins. We did not know that there is no possible world at which he wins and loses the 2008 presidential elections. And we did not know that at all possible worlds, either someone wins the elections or no one does. This changed at the very latest when we took our first course in metaphysics. There we learned to move from modal sentences to their possible worlds counterparts. In particular, we learned to make transitions according to the following schemata:

*formally*valid. However, the idea suggests itself that the move may be

*conceptually*valid, i.e. that it is truth-preserving as a matter of conceptual necessity. Since the transition is conceptually valid, it may be thought, reasoning according to (P) and (N) is knowledge preserving. Alternatively, we may say that (P) and (N) are elliptical for

*modus ponens*. The claim that (P) and (N) are conceptually valid corresponds to the claim that the second premiss in (P*) and (N*) is a conceptual truth, a truth knowable—and typically known—simply on the basis of mastery of the relevant concepts. Since (P*) and (N*) are formally valid, reasoning according to (P*) and (N*) is knowledge preserving.

On both of these proposals knowing that there is a possible world at which McCain wins is, in important respects, like knowing that Kant is a bachelor. Although we do not have any direct possible world or bachelorhood detecting faculties, we have the means to know something—that it is possible that McCain wins and that Kant is unmarried and a man—on the basis of which we can validly *infer* that the former is the case. Further, this inference is backed by a conceptual truth: that there is a possible world at which McCain wins if it is possible that he wins and that Kant is a bachelor if both unmarried and a man respectively. That this is a good model generally of our knowledge of abstract entities is the key tenet of the pleonastic account of *abstracta* due to Stephen Schiffer.^{4}

There is, however, another answer to the question of why transitions like (P) and (N) should be knowledge preserving that one can extract from David Lewis’s writings on possible worlds. In order to further motivate the pleonastic account of possible worlds, I will explain why I think that an answer in the spirit of Lewis is unsatisfactory.

[I]f we are prepared to expand our existential beliefs for the sake of theoretical unity, and if thereby we come to believe the truth, then we attain knowledge. (Lewis 1986, p. 109)

Suppose Lewis is right and their usefulness in building philosophical and semantic theories gives us knowledge conferring reasons to believe that there are possible worlds. This does not yet explain how we can come to know that there is this or that possible world—that there is a possible world at which McCain wins, say—*on the basis* of our knowledge of the corresponding simple modal statements—that McCain might have won. One particular use of possible worlds might help here, though. For, possible worlds are supposed to figure in an illuminating *semantics* for ordinary modal sentences. In particular, the conclusions of (P) and (N) are supposed to give the semantics of the corresponding premisses. Thus, we might think that their use in the semantics of the corresponding statements gives us epistemic access to the existence of possible worlds. Roughly, this may work as follows: we know that McCain might have won. By semantic ascent we can thereby come to know that ‘McCain might have won’ is true. Since (we know that) our best semantics for modal sentences tells us that sentences of the form ‘*a* might have *F*-ed’ are true just in case there is a possible world at which *a**F*-s, we can gain knowledge that there is a world at which McCain wins.^{5}

Although this may be a reasonably attractive answer in the case of possible worlds, it is not an answer that easily generalises. The case of possible worlds appears to be very similar to that of other platonic entities. Thus, we should not accept the Lewisian answer in the case of possible worlds either.

^{6}

*via*the obvious transformations, expressible with the following sentences:

- (1)
Socrates belongs to the set of all philosophers;

- (2)
Socrates falls under the concept

*philosopher*; - (3)
The proposition that Socrates is a philosopher is true;

- (4)
There is such a fact as the fact that Socrates is a philosopher;

- (5)
The state of affairs of Socrates’ being a philosopher obtains.

Given the striking similarity of our canonical way of coming to know about possible worlds and the other abstract objects cited, it is at least prima facie desirable to have a unified account of the legitimacy of our epistemic access to them. One such unified account is a pleonastic account of the relevant objects that holds that the transitions are materially valid as a matter of the relevant concepts. On the other hand, the Lewisian idea that the transitions recorded in (P) and (N) are knowledge preserving for reasons of theoretical utility cannot plausibly apply to all of the cases mentioned here. Note that the transformations that result in sentences (1)–(5) are all based on the very same sentence, namely ‘Socrates is a philosopher’. Suppose someone knows that Socrates is a philosopher and, further, that ‘Socrates is a philosopher’ is true. He infers from this that Socrates has the property of being a philosopher, that Socrates belongs to the set of all philosophers, and so forth. For these inferences to be knowledge preserving in the way that (P) and (N) are on the current proposal, our best semantics for simple predications would have to tell us that sentences of the form ‘*a* is *F*’ are true just in case *a* has the property of being *F*, *and* just in case *a* belongs to the set of all *F*s, and so forth. But it would be very surprising, to say the least, if our best semantic theory for simple predications would avail itself of this vast array of apparently redundant theoretical machinery. Further, even though many philosophers endorse semantics for simple predications in terms of *one* of the kind of things in questions, no one endorses such a semantics in terms of *all* of them. But then, hardly anyone will be attracted by the view that our knowledge of all the abstract objects cited is due to their use in the best semantics of the sentences in question.^{7} Thus, it is at least prima facie desirable not to ascribe the legitimacy of our epistemic access to possible worlds to such an inference either.^{8}

A pleonastic account of abstract entities in general, and possible worlds in particular, on the other hand, is independently plausible and can do justice to the similarity of our epistemic access to possible worlds and other abstract entities. The next section gives a brief sketch of the outlines of a pleonastic account.

## 2 Pleonasticism

The main tenet of a pleonastic account of *F*s is that *F*s are the kinds of things that are subject to so-called *something-from-nothing transformations*. Something-from-nothing transformations are inferences from statements that neither mention nor quantify over *F*s to statements that logically entail the existence of *F*s, which are valid as a matter of conceptual necessity.^{9} The obvious questions towards such a view are (a) which concepts are responsible for the validity of something-from-nothing transformations, (b) how do they pull it off, and (c) what distinguishes them from clearly empty concepts attempting similar feats. This section sketches answers to these questions.

The main proponent of pleonastic accounts regarding a variety of abstract entities like properties and propositions, Stephen Schiffer, does not go out of his way to answer questions (a) and (b). According to him, the concept responsible seems to be always the *sortal* concept of an *F*—the concept of a property, a proposition, a possible world and so forth. These sortal concepts manage to ensure the validity of the pertinent instances partly because it is part of their underived conceptual role that we are willing to thus infer.^{10} Presumably, the sortal concept of a property, for instance, is as if it were implicitly defined by the stipulation that instances of (H) and (L) are to be valid. However, Schiffer does not provide any details, and the topic of how exactly implicit definitions work, if at all, is a difficult one. To avoid these difficulties, I will explore the possibility of giving *explicit* definitions that would ensure the validity of something-from-nothing transformations. In this I will follow the work of Benjamin Schnieder who has proposed explicit definitions for property concepts that ensure the validity of something-from-nothing transformations.^{11} In a nutshell, the idea is that we can define *individual* property concepts like that expressed by ‘the property of being a philosopher’ by saying that it applies to whatever every philosopher must exemplify and every non-philosopher lack. Schematically, the proposal is the following^{12}:

**Definition schema: individual property concepts**

*x*= the property of being \(F\leftrightarrow_{\rm df.}\)

- (i)
\(\square \forall y\;(y\;\hbox{is}\;F\rightarrow y\;\hbox{has}\;x)\\\;\&\)

- (ii)
\(\square\forall y\;(\neg y\;\hbox{is}\;F\rightarrow y\;\hbox{lacks}\;x)\)

*show*that the definition is adequate. In the context of standard non-modal first order logic, this means showing that there is exactly one object that falls under the definiens. For instance, when we propose to define the singular term ‘\(\varnothing\)’ as follows:

- (6)
\(x=\varnothing \leftrightarrow_{\rm df.}x\;\hbox{is a set}\;\&\;\forall y\;(y \notin x);\)

However, this is not how we should think of the definition of the concept of the property of being a philosopher and the definitions of the individual property concepts in general, according to the pleonastic account of properties. For, there is no way *independent* of the individual property concepts of coming to know that there is this or that property or that there are properties at all. That is, there is no way of *first* coming to know that there is something that must be had by all philosophers and lacked by everything else, and *then* stipulating that this is the property of being a philosopher.^{13} Consequently, we cannot show that our definitions are acceptable in the usual way.

Intuitively, the difference between pleonastic properties and mere scam pleonastic entities like the god of the ontological proof—who is omniscient, omnipotentsuch a purported definition must be not merely consistent but

conservative: it must not introduce fresh commitments which (i) are expressible in the language as it was prior to the introduction of its definiendum and which (ii) concern the previously recognized ontology of concepts, objects, and functions, etc., whatever in detail they may be. (Hale and Wright 2000, p. 302)

*and*whose existence can be inferred by conceptual analysis—is that the former’s existence has no untoward consequences for the world as we knew it before their concepts were introduced, while the latter would have substantial consequences, and, thus, cannot fall under a pleonastic concept.

Schiffer spells this intuitive difference out in terms of conservative theory extensions. Roughly, in the case of pleonastic entities whose canonical individual concepts have explicit definitions: adding their definition stating biconditionals together with existence claims to any theory does not result in a theory that has new consequences statable in the original theory.^{14}

Although I think that it is on the right track, I mention this strand of the account only to put it aside. A detailed discussion of conservativeness deserves a much thorougher discussion than this paper can provide. My focus here is a proposal for a definition of the relevant possible worlds concepts that would ensure that possible worlds are pleonastic entities. Although I hope that it will become apparent that the possible world concepts, as defined, meet the conservative extension constraint if the property concepts do, I will have to leave things with this conditional claim and will now turn to possible worlds.

## 3 Pleonastic possible worlds

In this section, the main section of the paper, I proceed as follows. First, I motivate two principles about the individuation and characteristic properties of possibilities in Sect. 3.1. In Sect. 3.2, I propose definitions of individual proto-possibility concepts and the sortal concept of a proto-possibility, and show that the things that fall under these concepts satisfy the principles. In Sect. 3.3, I show how to get from proto-possibilities to possibilities and, finally, to possible worlds.

In order to make life easier while setting things up I start out assuming that S5 is the correct modal logic. In conjunction with the focus of this paper on possible worlds as they have to be for *sentential* modal logics, this allows me to ignore iterations of modal operators. This is so since in sentential S5 any formula with a subformula in the scope of more than one modal operator is equivalent to one in which no iterations of modal operators occur.^{15} Sect. 3.4 shows that the simplifying assumption that S5 is correct is not essential to the present account of possible worlds.

### 3.1 Principles about possibilities

What happens in math class when we are introduced to Kripke’s miniature possible worlds orAn analogy from school—in fact, it is not merely an analogy—will help to clarify my view [on possible worlds]. Two ordinary dice (call them die A and die B) are thrown, displaying two numbers face up. For each die, there are six possible results. Hence there are 36 possible states of the pair of dice, as far as the numbers shown face-up are concerned […]. We all learned in school how to compute the probabilities of various events (assuming equiprobability of the states). For example, since there are just two states—(die A, 5; die B, 6) and (die A, 6; die B, 5)—that yield a total throw of 11, the probability of throwing 11 is 2/36 = 1/18.

Now in doing these school exercises in probability, we were in fact introduced at a tender age to a set of (miniature) ‘possible worlds’. […] ‘Possible worlds’ are little more than the miniworlds of school probability blown large. (Kripke 1980, pp. 16ff.)

*possibilities*, as I will call them?

^{16}Well, we seem to learn to make certain simple transformations. From

- (7)
It’s possible that die A shows a 5 and die B shows a 6;

- (8)
There is the possibility that die A shows a 5 and die B shows a 6.

- (9)
It is possible that die A shows a 6 and die B shows a 5;

- (10)
There is the possibility that die A shows a 6 and die B shows a 5.

*two*possibilities out of 36 relevant to the case. Second, according to both possibilities and no other out of the 36, the total number shown by the dice must be 11. After all, we want to say that there are two possibilities

*according to which the total throw is 11*, and 34 others according to which the total throw isn’t 11. The following principles about possibilities ensure this. Let’s write ‘\(x\,\Vdash\,p\)’ for ‘according to

*x*,

*p*’ or ‘at

*x*,

*p*’ (I will typically say that

*x*forces that

*p*). Then the second point is secured if, speaking schematically, the possibility that

*p*forces whatever that

*p*entails

^{17}and vice versa, i.e.

**P1**

∀*q* (the possibility that \(p\,\Vdash\,q \leftrightarrow\square(p\rightarrow q)).\)^{18}

- 1.
die A shows a 5.

- 2.
die B shows a 6.

- 3.
there are at least two dice.

- 4.
the total number shown by die A and die B is 11.

The first point about counting possibilities can be secured if, again schematically, the possibility that *p*_{1} is the same as the possibility that *p*_{2} just in case whatever is forced by the former is forced by the latter and vice versa, i.e.

**P2**

the possibility that *p*_{1} = the possibility that \(p_{2}\leftrightarrow\forall q\) (the possibility that \(p_{1}\,\Vdash\,q \leftrightarrow\) the possibility that \(p_{2}\,\Vdash\,q\))

According to (P2), the possibility that die A shows a 5 and die B a 6 is not the possibility that die A shows a 6 and die B a 5, since, e.g., although according to both the total throw is 11, the former but not the latter forces that die A shows a 5. Moreover, (P2) ensures that there are *only* 36 possibilities that differ at most in which number is shown by which die (and by whatever that entails) according to them. For instance, (P2) rules that the possibility that die A shows a 5 and die B a 6 just is the possibility that die B shows a 6 and die A a 5, since it is necessary that die A shows a 5 and die B a 6 just in case die B shows a 6 and die A a 5. Consequently, the former forces whatever the latter forces and vice versa.

It should be stressed that two things have to be kept apart. First, there is the question of how we know that it is possible that die A shows a 5 and die B a 6. At least in the use of ‘possible’ relevant in the school example, this is an empirical question. Suppose the teacher brought two dice, A and B, but B is a trick die: none of its sides displays a 6 (say two of them display a 1 instead). Then it is just not possible that die A shows a 5 and die B a 6 after the throw. But the question of whether there is the possibility that die A shows a 5 and die B a 6, *given that it is possible that die A shows a 5 and die B a 6* is not an empirical question. Rather, it is a question that can be answered (in the affirmative) by any student who has mastered the possibility concepts taught by the math teacher, without inspection of the dice. Likewise, the question of whether it is just as likely that die A shows a 5 and die B a 6 as it is that die A shows a 6 and die B a 5 is an empirical question. Obviously, if B is a trick die of the sort described, the answer to the question is ‘no’. But even if the sides show the usual numbers, the dice may be loaded so as to favour landing on a particular side. If B has added weight on the side showing a 2, for instance, it will be more likely that A shows a 6 and B a 5 than that A shows a 5 and B a 6. Again, this is an empirical question. But the question of whether the two possibilities are equiprobable *given that it is just as likely that A shows a 5 and B a 6 as that A shows a 6 and B a 5* is not an empirical question. Rather, it is a question that can be answered (in the affirmative) by any student who has mastered the possibility concepts (and the concept of equiprobability) taught by the math teacher, without inspection of the dice.

### 3.2 Proto-possibilities

^{19}of individual

*proto*-possibility concepts:

**Df. Schema: Individual Proto-Possibilities (Df-IPP)***x*= the proto-possibility that \(p\leftrightarrow_{\rm df.} \square \forall q\, (x\,\Vdash\,q \leftrightarrow \square(p\rightarrow q))\)

^{20}:

**Definition Proto-Possibility***x*is a proto-possibility \(\leftrightarrow_{\rm df.} \exists p\) (*x*= the proto-possibility that*p*)

- *
\( \forall x\, (x={\mathcal{P}}p_{1}\leftrightarrow \square\forall q\, (x\,\Vdash\,q \leftrightarrow \square(p_{1}\rightarrow q)\ensuremath)).\)

The proofs of (P1) and the left-to-right direction of (P2) are trivial:

**Proof of (P1)**

**Proof of** (\({\bf P2}\rightarrow\))

- **
\( \square \forall q \,({\mathcal{P}} p_{1}\,\Vdash\,q \leftrightarrow \square(p_{1} \rightarrow q)).\)

- ***
\( \square \forall q \,({\mathcal{P}} p_{2} \,\Vdash\,q \leftrightarrow \square(p_{2} \rightarrow q)). \)

**Proof of** (\({\bf P2}\leftarrow\))^{21}

*in fact*force the same things. First, to get from line 2 to line 3, we had to assume the S5 theorem that \((\square\varphi \rightarrow\square\psi)\rightarrow\square(\square\varphi \leftrightarrow\square\psi)\). Since this paper proceeds on the assumption that S5 is correct, this is not a problem. We also need to assume, to get from line 3 to line 4, that for the sentential quantifier the Barcan Formula holds, i.e.

\({\bf BF}\text{-}\forall \it{q}\quad \;\;\forall \it{q}\,\square\,\varphi\,\rightarrow\, \square\forall \it{q}\, \varphi \)

*identity criteria*hold for proto-possibilities.

^{22}One form uses canonical designators for them—singular terms of the form ‘\(\mathcal{P}p\)’—on the left-hand side while avoiding proto-possibility vocabulary on its right-hand side:

**IC**_{1}\(\forall p_{1}, p_{2} \,(\mathcal{P}p_{1}= \mathcal{P} p_{2} \leftrightarrow \square (p_{1}\leftrightarrow p_{2})).\)

**IC**_{2}\(\forall x, y \,((x\hbox{ is a proto-possibility } \&\, y\,\hbox{is a proto-possibility}) \rightarrow (x=y \leftrightarrow \forall q \,(x\,\Vdash\,q \leftrightarrow y\,\Vdash\,q))).\)

_{2}), there are no two proto-possibilities that force exactly the same things. Since the proto-possibility that

*p*

_{1}forces the same things as the proto-possibility that

*p*

_{2}just in case it is necessary that (

*p*

_{1}iff

*p*

_{2}), (IC

_{1}) is true as well. Note that these identity criteria ensure that proto-possibilities are just as fine-grained as they need to be in order for them to record modal differences: there are no two modally equivalent proto-possibilities.

### 3.3 From proto-possibilities to possible worlds

Not all proto-possibilities are possibilities, and many proto-possibilities are not possible worlds. However, if the present suggestion is on the right track, *all but one* of the proto-possibilities are possibilities, and those possibilities with a certain maximality property—those that are ‘blown large’—are possible worlds. First, there is exactly one proto-possibility that is an *im*possibility. It’s the proto-possibility that forces *everything*. Consider, for instance, the proto-possibility concept expressed by ‘the proto-possibility that snow is white and not white’. According to (Df-IPP), \(\mathcal{P}\) (snow is white and not white) forces whatever is entailed by the proposition that snow is white and not white, i.e. \(\mathcal{P}\) (snow is white and not white) \(\Vdash\,p\) iff \(\square\) (snow is white and not white \(\rightarrow p\)). Since it is impossible that snow is white and not white, \(\square\) (snow is white and not white \(\rightarrow p\)), for any *p*. Thus, \(\mathcal{P}\) (snow is white and not white) \(\Vdash\,p\), for any *p*. Since there are no two proto-possibilities that force exactly the same things, and the same argument works for all and only proto-possibilities that are proto-possibilities that *q*, when it is impossible that *q*, there is exactly one such proto-possibility.

**Possibility**_{1}\(\mathcal{P}p\) is a possibility \(\leftrightarrow \lozenge p\)

**Possibility**_{2}*x*is a possibility \(\leftrightarrow_{\rm df.}\) (*x*is a proto-possibility & ∃*p*(\(x \not\Vdash\,p\)))

*Possibility*

_{2}(together with (Df-IPP) and the general definition of a proto-possibility) entails

*Possibility*

_{1}. Suppose \(\mathcal{P}p\) is a possibility. Then, according to

*Possibility*

_{2}, it does not force everything. That is, there is some

*q*such that \(\lozenge (p \& \neg q).\) But then, also ◊

*p*. Thus, if \(\mathcal{P}p\) is a possibility, ◊

*p*. Suppose now that \(\mathcal{P}p\) is not a possibility. Since \(\mathcal{P}p\) is a proto-possibility, \(\mathcal{P}p\) must force everything if it is not a possibility, according to

*Possibility*

_{2}. Since it forces everything, it forces \(\neg p\). But \(\square(p\rightarrow \neg p)\) only if \(\neg\lozenge p\). Thus, if \(\lozenge p, \mathcal{P}p\) is a possibility.

Not every possibility is what philosophers would call a possible world. Possible worlds, as Kripke says, are possibilities blown large. Intuitively, they decide every question that is to be decided. Many possibilities, on the other hand, don’t. For instance, the possibility that Socrates is a famous philosopher forces whatever is entailed by Socrates’ being a famous philosopher. For instance, it forces that Socrates is a philosopher, that there is at least one famous philosopher, that Socrates is a famous philosopher or a carpenter, and so on. Since \(\mathcal{P}\)(Socrates is a famous philosopher) is a possibility, it does not also force that Socrates is not a philosopher, that there is no famous philosopher, that Socrates is neither a famous philosopher nor a carpenter. But these are not the only things it does not force. For instance, it neither forces that Obama is president in 2014 nor that Obama is not president then. Hence, it does not decide the question of whether Obama is president in 2014. Thus, it is not what philosophers would call a possible world.

Of course, \(\mathcal{P}\)(Socrates is a famous philosopher & Obama is president in 2014) decides that question (while there are many questions it does not decide either). Moreover, \(\mathcal{P}\)(Socrates is a famous philosopher) and \(\mathcal{P}\)(Socrates is a famous philosopher & Obama is president in 2014) stand in an interesting relation: the latter forces everything the former does. Let’s say, following Humberstone (1981), that the latter *refines* the former, which I will abbreviate by writing ‘\(\geqslant\)’. That is,

**Refinement**

\(x \,\geqslant\,y \leftrightarrow_{\rm df.} \forall p\,(y\,\Vdash\,p \rightarrow x\,\Vdash\,p)\)

If \(x \,\geqslant\,y\) and \(y\not\geqslant\, x, x\)*properly refines**y* (*x* > *y*). That is,

**Proper Refinement**

\(x > y \leftrightarrow_{\rm df.} x\, \geqslant\, y \,\&\, \exists p\,(x\,\Vdash\,p \,\&\, y \not\Vdash\,p)\) Thus, not only does \(\mathcal{P}\)(Socrates is a famous philosopher & Obama is president in 2014) refine *P*(Socrates is a famous philosopher), it also properly refines it since the former forces that Obama is president in 2014, while the latter does not.

*impossible*proto-possibility properly refines any possibility. But there may be possibilities that are not properly refined by any

*possibility*. These deserve the title of a possible world. That is

**Df. Possible World***x*is a possible world \(\leftrightarrow_{\rm df.}\)

- (i)
*x*is a possibility & - (ii)
\(\neg\exists y\) (

*y*is a possibility &*y*>*x*)

*q*, say. Then \(\mathcal{P}p \not\Vdash\,q\) and \(\mathcal Pp\not\Vdash \neg q\). So, \(\lozenge (p \,\&\, \neg q)\) and ◊(

*p*&

*q*). But then \(\mathcal{P}(p \,\&\, q) >\mathcal{P}p\), even though \(\mathcal{P}(p \,\&\, q)\) is a possibility. Thus, if some possibility leaves a question undecided, it is not a possible world, according to our definition. On the other hand, suppose that \(\mathcal{P} p_{1}\) leaves no question undecided, i.e. \(\forall q \,(\mathcal{P}p_{1}\Vdash\,q \vee \mathcal{P} p_{1}\,\Vdash\,\neg q)\). Now suppose that there is a proto-possibility that properly refines \(\mathcal{P} p_{1},\mathcal{P}p_{2}\). Since \(\mathcal{P}p_{2} > \mathcal{P}p_{1}\) and \(\forall q \,(\mathcal{P}p_{1}\Vdash\,q \vee\mathcal{P}p_{1}\Vdash \neg q),\exists q\,(\mathcal{P}p_{2}\Vdash\,q \,\&\, \mathcal{P} p_{2}\,\Vdash\,\neg q)\). But then \(\mathcal{P} p_{2}\) is not a possibility, since \(\exists q\,(\square(p_{2}\rightarrow q) \,\&\, \square(p_{2}\rightarrow \neg q))\) only if \(\neg\lozenge p_{2}\). Thus, if a possibility leaves no question undecided, it is a possible world, according to our definition.

### 3.4 Iterations

Let me end this section by briefly indicating why the simplifying assumption that we may neglect iterations of modal operators is relevant to the current proposal and how the proposal may be varied so that the simplifying assumption can be dropped. Suppose that the characteristic S4 axiom is invalid, so that, e.g.,

Intuitively, what we want in this case is that, although no possibility accessible^{23} from the actual possibility forces that *p*_{1}, there is some possibility that is accessible from a possibility accessible from the actual possibility which forces that *p*_{1}. However, given (Df-IPP) and the first conjunct of (S4), the only proto-possibility that forces that *p*_{1} is the proto-possibility that forces everything, the impossibility. Thus, when accessibility enters the picture, the current proposal yields too few possibilities. Indeed, it only yields those possibilities accessible from possibilities that are actual since what a possibility forces depends on what propositions are *in fact* entailed by others.

The idea of how to deal with this complication is simple: we must not only consider which propositions are *in fact* entailed by which, but also which propositions are possibly entailed by which, which are possibly possibly entailed by which and so forth. We can then start with the actual possibility and build outwards. In order to do so, we will have to exploit the *modal* information possibilities themselves contain. Where the original proto-possibility that *p* forced whatever is entailed by the proposition that *p*, the new proto-possibilities that *p* will force whatever is entailed by the proposition that *p*, *according to some other possibility*. Let me elaborate.

*actual*proto-possibility’ or ‘@’

^{24}:

**Df. The Actual Proto-Possibility (Df-@)**\(x=@\leftrightarrow_{\rm df.} \square \forall p \,(x\,\Vdash\,p \leftrightarrow \mathcal{A} p)\).

^{25}@ is also a possibility since it does not force everything. Moreover, given that \(\forall p\,(\mathcal{A}p\vee \mathcal{A}\neg p)\), @ is even a possible world. Now, @ forces many things non-modal: that Socrates is a philosopher, that Socrates is not a carpenter, perhaps that Obama is president in 2014. But it also forces modal truths. For instance, since it is actually possible that Obama is president in 2014, @ forces that it is possible that Obama be president in 2014. Since it is actually necessary that Obama is or is not president in 2014, @ \(\Vdash\,\square\) (Obama is or is not president in 2014), and so forth.

**Df. Schema: Individual @-Proto-Possibilities (Df-@IPP)***x*= the @-proto-possibility that \(p \leftrightarrow_{\rm df.} \square \forall q \,(x\,\Vdash\,q\leftrightarrow @\,\Vdash\,\square(p \rightarrow q))\)

*p*—\(\mathcal{P}_{@} p\), for short—must force whatever is entailed by the proposition that

*p*,

*according to @*. It is easy to see that all and only our original proto-possibilities are @-proto-possibilities: for any of the former there is exactly one of the latter that forces the very same things and

*vice versa*. For, consider \(\mathcal{P}p\), the proto-possibility that

*p*as originally defined. \(\mathcal{P}p\, \Vdash\,q\) just in case \(\square(p\rightarrow q\)). If \(\square(p \rightarrow q), \mathcal{A} \square(p \rightarrow q)\). So, by (Df-@), \(@\,{\Vdash}\,\square(p \rightarrow q)\). Thus, by (Df-@IPP), \(\mathcal{P}_{@} p\) forces that

*q*. On the other hand, if \({\mathcal{P}}{p}{\not\Vdash}\,q\), then \(\neg \square(p \rightarrow q)\). So, \(\neg\mathcal{A}\square(p \rightarrow q)\), and, by (Df-@), \({@} {\not\Vdash} \square(p \rightarrow q)\). Thus, by (Df-@IPP), \(\mathcal{P}_{@} p \not\Vdash\,q\). The other direction is similar.

*their*modal information to get further proto-possibilities and so on. The general definition schema can be put as follows:

^{26}

**Df. Schema: Individual w-Proto-possibilities (Df-wIPP)**If

*w*is a proto-possibility, then*x*= the*w*-proto-possibility that \(p \leftrightarrow_{\rm df.} \square \forall q\,(x\,\Vdash\,q\leftrightarrow w\,\Vdash \square(p \rightarrow q))\)

*p*

_{1}. Let’s call one of them \({\mathfrak{p}}\). Since \({\mathfrak{p}}\,\Vdash\,\lozenge p_{1}\), there will also be a possible \({\mathfrak{p}}\)-proto-possibility which forces that

*p*

_{1}, and, thus, a possibility which forces that

*p*

_{1}. The latter will just not be accessible from @.

Let me end this section by commenting on a distinctive feature of the current proposal. According to it, in a sense, all possibilities are centered around the actual possibility. Picturesquely, there are only possibilities reachable, perhaps via intermediate steps, from @—there are no pockets of absolutely unreachable possibilities. Less picturesquely, all possibilities are related to the actual possibility by the transitive closure of the accessibility relation.

This may be thought to be a limitation of the present account since there are certainly *formal models* of modal sentential logic that represent things to be otherwise. This has nothing in particular to do with dropping the simplifying assumption that S5 is correct. There are also formal S5 models in which accessibility is an equivalence relation but the set of all possible worlds is partitioned into more than one equivalence class.

Let us start by noting that, although there are such formal models, their inclusion of absolutely inaccessible possible worlds—worlds that are not even mediately accessible from the actual world—is gratuitous when the aim is to systematise modal facts. For, what is true at absolutely inaccessible worlds according to a model is completely irrelevant to what is true *simpliciter* according to the model. Any models that agree on what is true at all worlds related to the actual world by the transitive closure of accessibility agree on which sentences, including those with modal vocabulary, are true *full stop*. Thus, whenever there is a formal model that includes absolutely inaccessible worlds, there is another that makes true exactly the same sentences but does not include absolutely inaccessible worlds. Inaccessible worlds, if there were any, would be of no help in systematising modal facts.

For a related reason, the consequence that there are no absolutely inaccessible possibilities is not an idiosyncrasy of the present proposal but a consequence of any pleonastic conception of possibilities. It is part of the pleonastic conception of abstracta that which of them exist and what is true of them depends on what goes on with less problematic denizens of reality. Whether wisdom is exemplified by Socrates depends on what goes on with *Socrates*, viz. on whether Socrates is wise. Whether there is a possibility at which donkeys talk depends on what things are like modally, viz. on whether it is possible that donkeys talk, or possibly possible that donkeys talk, or possibly possibly possible that donkeys talk, and so on.

If there were absolutely inaccessible possibilities, their existence would not depend on what is modally the case. For, absolutely inaccessible possibilities would be possibilities that are neither accessible from the actual possibility, nor accessible from a possibility accessible from the actual possibility, nor accessible from a possibility accessible from a possibility accessible from the actual possibility, and so on *ad infinitum*. Let \({\mathfrak{p}}\) be such an alleged absolutely inaccessible possibility which forces that *p* and leaves every question whose answer is not entailed by the proposition that *p* undecided. Since \({\mathfrak{p}}\) is not accessible from \({@,\mathfrak{p}}\) forces that *p*, but @ does not force that it is possible that *p*. Since \(@ \not\Vdash\lozenge p\), \(\neg\mathcal{A}\lozenge p\), and, thus, \(\neg\lozenge p\). Consequently, the following is not the case: \({\mathfrak{p}}\) exists because it is possible that *p*. Since \({\mathfrak{p}}\) is not accessible from any world accessible from @, no world accessible from @ forces that it is possible that p. But then, \(@\not\Vdash \lozenge\lozenge p\). Thus, \(\neg\mathcal{A}\lozenge\lozenge p\). Consequently, the following is not the case: \({\mathfrak{p}}\) exists because it is possible that it is possible that *p*. By analogous arguments, the following is not the case: \({\mathfrak{p}}\) exists because it is possible that … *p*, where ‘…’ stands for any iterations of ‘it is possible that’. In short, \({\mathfrak{p}}\) does not exist because of what is modally the case. If possibilities are pleonastic, their existence is grounded in what is modally the case. Consequently, there are no absolutely inaccessible possibilities.

## 4 Conclusion

Possible worlds play an important role in philosophy. They figure in the standard semantics for the modal fragment of natural languages. And they are meant to help clarify important philosophical concepts. However, this does not mean that they belong to the basic furniture of reality. On the contrary, it may be held that their being in some sense derivative entities explains why possible worlds are able to fulfil the systematising role that makes them suitable for semantics and clarification. This paper offered an account of possible worlds that starts from this observation, namely an account of possible worlds as pleonastic entities—entities subject to conceptually valid something-from-nothing transformations.

The proposal developed in this paper explains how the validity of something-from-nothing transformations regarding possible worlds can be a conceptual matter. The transition from ‘it is possible that *p*’ to ‘there is a possible world at which *p*’ is backed by the conceptual truth that, if it is possible that *p*, there is a possible world at which *p*. This conceptual truth is a straightforward consequence of the definition of the pertinent individual proto-possibility concept together with the concepts of a possibility and a possible world, granted that refinement has to stop at some point. By spelling out the details, the present proposal thus goes some way towards a justification of the view that possible worlds are pleonastic entities.

Additionally, the view that possible worlds are pleonastic brings possible worlds in line with a plausible account of abstract objects in general. It explains our epistemic access to possible worlds in accordance with how we actually come to know about them. And it ensures that there are enough possible worlds to go around for a semantics of the modal fragment of natural languages and for the truth of the attempted clarifications of philosophical concepts. Pleonastic possible worlds seem to be just the things we want as an epistemically accessible tool in semantics, metaphysics, and virtually all other areas of philosophy.

I would have said ‘other *abstract* entities’ were it not for the fact that one of the major proponents of possible worlds, David Lewis, conceives of them as (mereological sums of) *concreta*. The present use of ‘platonic’ is borrowed from Yablo (2000, p. 198).

Of course, the specific timing does not matter. What matters is that there is a stage of conceptual sophistication at which one is in principle in a position to have pieces of ordinary modal knowledge while not being in a position to have knowledge of possible worlds. At that stage one lacks (i) all but the most general concepts under which possible worlds fall, and (ii) concepts for world-relativisation. Therefore, one is not in a position to know anything specifically about possible worlds, e.g. that there is a \(\underline{\hbox{possible world at which}}\) McCain wins.

On this proposal, reasoning according to (H) and (L) will at best be elliptical (since it leaves out the step with the explicitly semantic content). One might think that this already disqualifies it as a reconstruction of how we typically gain the knowledge in question. However, since homophonic semantic ascent may seem trivial, and is, thus, easy to miss, I would not want to put too much weight on this consideration.

Making transitions according to (H) and (L) will not always be safe. For, consider any true instance in which ‘*F*’ is replaced with ‘a non-self-exemplifier’, e.g. ‘the property of being green is a non-self-exemplifier’ (it’s true since, *pace* Plato, the property of being green is not itself green). This, together with the general validity of (H), seems to imply the existence of the Russell property of being a non-self-exemplifier. But the Russell property leads to paradox: if it existed it would exemplify itself just in case it didn’t exemplify itself. Similar comments apply to some of the transitions mentioned in the next paragraph. For the purposes of this paper, I will simply ignore the vexing issues surrounding the paradoxes.

Lewis endorses a different argument from theoretical utility in the case of sets (Lewis 1986, pp. 108ff.): very roughly, we can know that there are sets because they are so useful in mathematics. However, I can see no way of exploiting this observation to ground our epistemic access to the fact that Socrates belongs to the set of all philosophers *on the basis* of our knowledge that Socrates is a philosopher.

We could have run a more direct argument starting with a *de re* modal sentence, e.g. ‘Socrates might have been a carpenter’. From this, it seems, we can infer a truth about possible worlds—that there is a world at which Socrates is a carpenter—as well as truths about properties—that Socrates has the property of possibly being a carpenter—about sets—that Socrates belongs to the set of all possible carpenters—and so forth. Since these latter transformations are slightly more controversial than those from simple, non-modal, predications, I chose the argument in the main text.

Definitions of singular concepts may either take the form of equivalences or identities (cf. e.g. Suppes 1972, p. 17). Schnieder chooses the former. Alternatively, the following would have done as well:

**Definition Schema: Individual Property Concepts*** the property of being \(F =_{\rm df.}\;\hbox{the}\;x\;\hbox{such that}\;\square\;\forall y \,(y\;\hbox{is}\;F \rightarrow y\;\hbox{has}\;x)\;\&\; \square\;\forall y \,(\neg y\;\hbox{is}\;F\rightarrow y\;\hbox{lacks}\;x)\) However, given that the definitions are adequate, they provably yield the same results.

Of course, similar things may be said about the empty set on a pleonastic account of sets. What I said in the last paragraph simply assumes that there is a way of coming to know the set theoretical axioms before accepting (6) as a definition of the concept of the empty set.

In this I follow Humberstone (1981). Possibilities may only differ in name from *situations*, introduced into the philosophical and linguistic discussion by Barwise (1981) and Barwise and Perry (1983). Under the name ‘situations’, possibilities have come to some prominence since they promise to have a broad range of applications in natural language semantics. See, e.g., Kratzer (2010).

I use ‘entail’ in the sense of ‘strictly implies’ here and in what follows, i.e. that *p* entails that *q* just in case it is necessary that (*q* provided that *p*). Incidentally, talking about entailment relations ostensively between propositions makes things easier but is an inessential feature of my presentation. Everything I say could be reformulated using only the resources used in the official definitions to follow, in particular without appeal to propositions.

Quantifiers binding ‘*p*’ and ‘*q*’ are sentential quantifiers. I do not need to assume a specific account of sentential quantification for the purposes of this paper, except that they do not range over propositions. For more on sentential quantification see, e.g., Hugly and Sayward (1996, Part III) and compare the appendix to this paper below. Incidentally, for the sake of readability I suppress brackets where possible, assuming that ‘\(\Vdash\)’ binds more tightly than any of the logical operators, but less tightly than ‘the possibility that’.

Roughly, in order to test for conservativeness, we add, *inter alia*, existence assumptions concerning the new entities to theories. But the assumption that the possibility that *p* exists would, on any plausible account of possibilities, entail all by itself that it is possible that *p*. Thus, possibilities would fail the Conservative Extension test with flying colours.

This is strictly parallel to the case of properties. See Künne (2007, p. 347). Cp. also Künne (2006). A Neo-Fregean account of possible worlds—as developed, e.g., in Berkovski (2011)—would take these identity criteria as its starting point. However, although Pleonasticism and Neo-Fregeanism are similar in spirit, I take it to be a virtue of the present account that it can explain why such identity criteria hold instead of having to stipulate them.

**Accessibility***x*has access to \(y \leftrightarrow_{\rm df.} \forall p\,(y\,\Vdash\,p\rightarrow x\,\Vdash\,\lozenge p\)

‘\(\mathcal{A}\)’ is the actuality operator, to be read as ‘It is actually the case that’. For detailed discussion see, e.g., Crossley and Humberstone (1977).

It is the proto-possibility that *p*, where ‘*p*’ is a place-holder for the conjunction of all the atomic truths.

I use ‘*w* is a proto-possibility’ as short for ‘*w* = @∨ ∃ *w*′, *q* (*w* = the *w*′-possibility that *q*’ for the sake of readability.

Powerful defenses of the standard Barcan formula in the face of initial implausibility include Linsky and Zalta (1994) and Williamson (2002).

Thanks to an anonymous referee whose discussion is the basis of much of this appendix, including the potential counterexample to sentential Barcan discussed below.

One might be able to spell out the idea behind (BF) without reliance on talk about propositions. Since nothing in the discussion will hinge on this feature of my presentation I omit the details.

These assumptions follow, for instance, from a Russellian conception of singular propositions as complexes essentially made up of the things they are about.

In setting up the problem I pretend that we can refer to mere possibilia. I do so merely for ease of exposition.

For considerations about *second-order* Barcan that may lead one to accept standard Barcan see Williamson (forthcoming).

Analogous considerations apply to the justification of (BF)’s antecedent in the argument against sentential Barcan.

## Acknowledgments

Many thanks to Dorothy Edgington, Mark Kalderon, Stephan Krämer, Fraser MacBride, Robert Schwartzkopff, an anonymous referee for this journal as well as the other members of the Phlox research group Nick Haverkamp, Miguel Hoeltje, Benjamin Schnieder and Moritz Schulz for very helpful comments and discussion of the paper’s material at various stages of completion. I would also like to thank the participants of the Amsterdam Graduate Philosophy Conference 2010 and the 14th Annual Oxford Philosophy Graduate Conference, especially my commentators at these events Paul Dekker and Gabriel Uzquiano Cruz.