Worlds are Pluralities

ABSTRACT

I propose an account of possible worlds. According to the account, possible worlds are pluralities of sentences in an extremely large language. This account avoids a problem, relating to the total number of possible worlds, that other accounts face. And it has several additional benefits.

KEYWORDS:

• possible worlds
• pluralities
• cardinality
• linguistic ersatzism

Acknowledgements

Thanks to Ethan Brauer, Phillip Bricker, Ashley Chay, Melissa Fusco, Martin Glazier, Christopher Menzel, Dee Payton, Alexandru Radulescu, Jonathan Schaffer, Ted Sider, Qiantong Wu, two anonymous referees, and the audience at the 2021 Central APA, for much helpful feedback and discussion.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

1 Similarly, as Boolos [1984] argues, pluralities are not entities. A plurality is not a new piece of ontology, that is, over and above the entities which it contains. Think of pluralities as the referents of plural terms.

2 Many different plural languages—corresponding to many different logical systems—could be used to formulate the plural account of possible worlds. To keep things simple, I will not explore those differences here. My preferred approach to plurals is the approach in [Linnebo 2003; 2013].

3 All of this is compatible with the view, to which I subscribe, that in many cases both the set containing some items and the plurality of those items exist. Just as there is a set of all natural numbers, for instance, there is a plurality of all natural numbers too.

4 Another important difference is that the unrestricted comprehension principle for pluralities, unlike the unrestricted comprehension principle for sets, avoids Russell’s paradox. The reason, in rough outline, is this: in plural logic, the formal regimentation of the sentence ‘there is a plurality consisting of all and only those entities which are not p-members of themselves’—this is the sentence whose formal analogue in set theory generates the paradox—is ungrammatical. So the unrestricted comprehension principle for pluralities does not imply the formal regimentation of that sentence.

5 To see why, just consider all sentences of the form ‘Set $x$ is self-identical’, where $x$ is a set which names itself. Since the number of such $x$ is greater than every cardinal, the number of such sentences is greater than every cardinal too.

6 In fact, the resources of plural logic can also be used to define something that corresponds to the standard set-theoretic definitions of cardinalities. The former definition, which follows a construction similar to the one given in footnote 7, could also be interpreted as a way of quantifying size. So ultimately, there are two formal constructions that could be taken to quantify how big various collections are: one based on pluralities, and one based on sets. In this paper, I assume that only the set-theoretic construction provides a genuine, joint-carving account of what size is; other constructions, based on pluralities, do not. This assumption might not be necessary for my purposes here. But it also might be; it is hard to tell. My worry—which may or may not be well-founded—is that without this assumption, my account might face a variant of the problem that I raise in sec. 3. Given this assumption, however, the risk of that seems low: since the formal construction based on pluralities does not capture what size really is, according to the assumption in question, no version of the problem in sec. 3 arises for the account of possible worlds to come.

7 Here is how. Say that $x$ is a pair plurality just in case $x$ is a plurality such that each p-member of $x$ is a pair. The domain of a pair plurality $x$ is the plurality $\mathrm{dom}(x)$ such that u is a p-member of $\mathrm{dom}(x)$ if and only if for some $v$, $⟨u,v⟩$ is a p-member of $x$ (here and in what follows, ‘$u$’ and ‘$v$’ are singular terms while ‘$x$’ is a plural term). The range of a pair plurality $x$ is the plurality $\mathrm{ran}(x)$ such that $v$ is a p-member of $\mathrm{ran}(x)$ if and only if for some $u$, $⟨u,v⟩$ is a p-member of $x$. A plurality $x$ is functional just in case $x$ is a pair plurality such that for all p-members $⟨u,v⟩$ and $⟨u,v^{\prime }⟩$ of $x$, $v$ is $v^{\prime }$. A functional plurality $x$ is one-to-one just in case for all p-members $⟨u,v⟩$ and $⟨u^{\prime },v⟩$ of $x$, $u$ is $u^{\prime }$. Finally, the language $\mathcal{L}$ is absolutely infinite just in case there exists a one-to-one functional plurality $x$ such that $\mathrm{dom}(x)$ is the plurality of all $\mathcal{L}$-sentences and $\mathrm{ran}(x)$ is the plurality of all cardinal numbers.

8 Note that Lagadonian may also include higher-order vocabulary. And Lagadonian allows for alien properties: for on my preferred view, properties can exist even if nothing instantiates them.

9 Superpluralities also facilitate the combination of (i) the Plural Account of worlds, and (ii) the account of groups in [Wilhelm 2020].

10 Likewise for philosophical theories that are formulated in terms of relations between (i) possible worlds, and (ii) entities like sentences, propositions, objects, agents, and so on. Such relations are usually assumed to obtain between entities only: they relate one entity—namely, a possible world—to another. But the corresponding philosophical theories can be reformulated so that they posit relations that obtain between pluralities and entities instead. The resulting relations can then obtain between (i) possible worlds, understood in accord with the Plural Account, and (ii) entities like sentences, propositions, objects, agents, and so on.

11 Principles akin to these are discussed in various places throughout the literature (see Hawthorne and Uzquiano [2011] and Bricker [2020]).

12 $2^{\mu }$ is the cardinality of the power set of $\mu$.

13 Analogous considerations suggest that it is better to endorse Occupation than to endorse its negation.

14 One might object that empirical science does not need cardinalities higher than sets of reals. I agree that this may be true for empirical science in the actual world. But even if this is actually true, it is possibly false: it seems at least metaphysically possible, in other words, that empirical science will posit spacetimes of higher cardinalities. And while Spacetime allows for that, its negation does not.

15 Infinitary languages are sometimes written in the form ${\mathcal{L}}_{\chi ,\lambda }$, where (i) $\chi$ and $\lambda$ are cardinals, (ii) $\chi$ bounds the number of formulas which can be conjoined ‘at once’, and (iii) $\lambda$ bounds the number of variables which can be quantified over ‘at once’ [Dickmann 1975].’ In this paper, I do not use that notation. My arguments apply to any language which has the two features mentioned above, regardless of whether that language is infinitary in the sense just described.

16 If both ${\kappa }_{\mathcal{L}}$ and ${\lambda }_{\mathcal{L}}$ are finite, then ${\kappa }_{\mathcal{L}}\cdot {\lambda }_{\mathcal{L}}$ is whichever finite number is equal to their product. If either ${\kappa }_{\mathcal{L}}$ or ${\lambda }_{\mathcal{L}}$ is infinite, then by the axiom of choice, ${\kappa }_{\mathcal{L}}\cdot {\lambda }_{\mathcal{L}}$ is whichever of ${\kappa }_{\mathcal{L}}$ or ${\lambda }_{\mathcal{L}}$ is largest.

17 In order to avoid this conclusion, one might adopt a potentialist conception of sets (thanks to an anonymous reviewer for pointing this out). Roughly put, according to the potentialist conception, sets are constructed in an iterative hierarchy: at each stage in the construction, new sets can be constructed using the sets which were constructed at previous stages. So there is no completed totality of all sets; rather, at each stage in the construction, many sets exist potentially, in that they would exist if they were constructed, but they do not actually exist at the stage in question. Now consider a Lagadonian language—call it Potentialized Lagadonian—which is relativized to stages: in particular, the constants of this language include all the sets which exist at the stage in question, but no sets which merely potentially exist. Arguably, a version of linguistic ersatzism which takes possible worlds to be maximal, consistent sets of Potentialized Lagadonian sentences—call this view Potentialized Ersatzism—validates both Spacetime and Occupation. Potentialized ersatzism may validate Spacetime because at each stage, potentialized ersatzism implies that possibly, more sets exist; and for each cardinal number μ, those merely possible sets could be used to construct a possible world out of Potentialized Lagadonian sentences whose spacetime has cardinality μ. And potentialized ersatzism may validate Occupation for similar reasons. Whether or not potentialized ersatzism really does validate Spacetime and Occupation, of course, depends on the details of the potentialist conception of sets in question; and it is beyond the scope of this paper to fully explore that here. Perhaps there is a way of formulating potentialized ersatzism which validates both Spacetime and Occupation, and which also satisfies several other relevant desiderata. But very briefly, it is worth pointing out two issues that may arise in the attempt to formulate an attractive version of potentialized ersatzism. First, because the potentialist conception of sets is often formalized using modal notions [Linnebo 2013], potentialized ersatzism might be viciously circular: the truth conditions for the modal operators used to state its potentialist conception of sets might presuppose the very worlds that potentialized ersatzism is used to analyze. Second, if possible worlds are identified with maximal, consistent sets of sentences in Potentialized Lagadonian, then each possible world may always be merely ‘potentially’ completed: for a set of sentences may be maximal and consistent—and so be a possible world, according to this version of linguistic ersatzism—at one stage, but non-maximal at later stages, as it will not contain sentences featuring the later stages’ sets as constants.

18 For more discussion of assumptions concerning the sizes of classes, and how those assumptions generate the same sorts of issues concerning size that standard set theory generates, see Uzquiano [2015] and Bricker [2020].

19 This provides a reason to favour the Plural Account over linguistic ersatz accounts that appeal to classes. For arguably, while the axioms of class theory can be used to generate a version of the upper bound problem for linguistic ersatz accounts of the latter sort, the axioms of plural logic cannot be used to generate a version of the upper bound problem for the Plural Account.

20 The argument is as follows. Let $S$ be any maximal, consistent set of propositions; note that given propositionalism, and given the existence of at least one possible world, such a set exists. Let $S{}^{\mathrm{\prime }}$ be the propositions in $\mathcal{P}$ which are not in $S$. Since $S$ is maximal and consistent, the following holds: for every proposition in $\mathcal{P}$, either that proposition is in $S$ or the negation of that proposition is in $S$ (but not both). Therefore, there is a bijection between $S$ and $S{}^{\mathrm{\prime }}$. It follows that $S{}^{\mathrm{\prime }}$ is a set. Since $\mathcal{P}$ is the union of $S$ and $S{}^{\mathrm{\prime }}$, $\mathcal{P}$ is a set too.

21 Lewis would not describe the upper bound problem as a problem, of course. For as mentioned above, Lewis more-or-less straightforwardly assumes an upper bound on how many worlds exist. There are, basically, two reasons why. First, without this assumption, Lewis’s recombination principle would imply Spacetime: for any cardinal $\mu$, possibly, spacetime has cardinality $\mu$. But it seems fishy, to Lewis, that a recombination principle about how spacetime might be occupied would have consequences for the possible size of spacetime itself [1986: 89]. Second, it provides Lewis with a way around an argument due to Forrest and Armstrong [1984]. As Forrest and Armstrong show, if the possible worlds are few enough to form a set, then Lewis must make that assumption: otherwise, Lewis’s recombination principle is inconsistent. But neither of these reasons—for making the assumption—strike me as compelling. It makes sense, to me, that a principle about how spacetime might be occupied would have consequences for the possible size of spacetime. And there is a more plausible way of avoiding the argument due to Forrest and Armstrong: deny that the possible worlds are few enough to form a set.

22 The upper bound problem is somewhat similar to, yet distinct from, other cardinality problems in the literature. It differs from Lewis’s original objection to lingustic ersatzism, which assumed that linguistic ersatzism constructs worlds using sentences in a countable language [1973: 90]. The upper bound problem also differs from other objections that have been raised to Lewis’s theory of possible worlds [Forrest and Armstrong 1984; Nolan 1996; Hawthorne and Uzquiano 2011]: while many of those objections depend on principles of recombination or transworld fusions, the upper bound problem does not.

23 The assumptions depend, in fairly sensitive ways, on the details of how consistency is defined. See [Dickmann 1975] for discussion of the notion of consistency in the context of massive languages.

24 Roy argues that worlds are sets of sentences in an extremely large language [1995: 220–5]. Though there is much to like about Roy’s account, it implies that for every world $w$, there are proper-class-sized many worlds $w{}^{\mathrm{\prime }}$ such that (i) exactly the same sentences are true at $w$ and $w{}^{\mathrm{\prime }}$, and (ii) $w$ and $w{}^{\mathrm{\prime }}$ are distinct. In other words, Roy’s account implies the existence of many ‘redundant’ worlds: the world-building language overgenerates how many worlds are needed, in order to represent all the possibilities that there are.

25 The basic idea of the Plural Account—drop sets for pluralities—can be extended to other accounts of possible worlds. Plural versions of combinatorialism, propositionalism, and modal realism, can be formulated using pluralities too.