Pleonastic possible worlds

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.

Appendix

The proof to the effect that principle (P2) can be derived from the proposed definitions relies on the sentential Barcan formula:²⁷

• ${\mathbf{BF-}} {\forall q} \quad \forall {q}\; \square\;\varphi \,\rightarrow\,\square {\forall}{q\,}\varphi .$

I call the principle sentential Barcan since it is a variant of the notorious standard Barcan formula, differing from it only in that it uses a sentential instead of a nominal quantifier. Standard Barcan conflicts with fairly basic everyday modal intuitions. It implies, for instance, that there couldn’t have been things that do not actually exist. This raises the worry that the—at least initial—implausibility of standard Barcan may rub off on sentential Barcan.²⁸

This worry can be substantiated.²⁹ For, it seems that we can generate a counterexample to sentential Barcan out of a counterexample to standard Barcan in the following way: take any predicate that makes standard Barcan false. For definiteness, suppose ‘is something that actually exists’ (‘is an actual existent’, for short) is such a case. Let us use square brackets ‘[…]’ to abbreviate ‘the proposition that …’. Now consider the instance of $(\hbox{BF-}\forall {q})$ in which ‘$\varphi$’ is substituted by ‘[q] is a singular proposition $\rightarrow[q]$ is about an actual existent’:

BF

$\forall q \square$ ([q] is a singular proposition $\rightarrow[q]$ is about an actual existent) $\rightarrow\square\forall q$ ([q] is a singular proposition $\rightarrow[q]$ is about an actual existent).³⁰

We might think that (BF) is false, and, thus, a counterexample to sentential Barcan, given plausible principles about singular propositions. For, suppose singular propositions are essentially singular and existentially dependent on the things they are about.³¹ Since, we assumed, there might have been things that do not actually exist, there might have been singular propositions—singular propositions about such mere possibilia—that do not actually exist. Consider, for instance, the merely possible proposition P that says about a merely possible die d ₁ that it shows a 6.³² If P had existed, P would have been (i) singular and (ii) only about something, d ₁, that does not actually exist. Thus, the consequent of (BF) is false. On the other hand, all actual singular propositions are about actual existents, and, thus, they are necessarily such that, if they are singular, they are about something that actually exists. So (), the antecedent of (BF) is true, and (BF) is a counterexample to $\hbox{BF-}\forall {q}$.

If the reasoning of the last paragraph is correct, I lose an essential resource for showing that my definitions entail principle (P2) (which says that no two proto-possibilities force the same). But, one might think, this is good news, since, despite initial appearances, (P2) is false as well. For, consider a merely possible die d ₂ distinct from d ₁. $\mathcal{P}$ (d ₁ shows a 6) and $\mathcal{P}$ (d ₂ shows a 6) are different proto-possibilities since they could have forced different things. For instance, had d ₁ existed, the former but not the latter would have forced that d ₁ showed a 6. But in the actual world, no singular propositions about the two merely possible dice are available to distinguish the two proto-possibilities. So (), $\mathcal{P}$ (d ₁ shows a 6) and $\mathcal{P}$ (d ₂ shows a 6) actually force the same, despite the fact that they are not identical, contra (P2).

I see four possible reactions to the problem: first, we might deny the principles about singular propositions that fuel the argument. In particular, we might deny that singular propositions can only exist when the things they are about exist.³³ Second, we might bite the bullet and accept standard Barcan as well. If sentential Barcan is as plausible as I think it is, and if the argument from standard to sentential Barcan is valid, this would provide at least some reason to accept standard Barcan, despite initial appearances.³⁴ Third, we might just concede the point, and look for a substitute that can do the work of (P2) instead. A candidate that comes to mind is (P2)’s modal strengthening: $(\hbox{P2}_\square)$. $(\hbox{P2}_\square)$ leaves open that there may be proto-possibilities that happen to force the same, but insists that no two of them are such that they necessarily force the same. It is easy to see that the modified principle also follows from the proposed definitions, and that a proof of this fact does not need to rely on sentential Barcan.

However, a second look at the arguments against sentential Barcan and (P2) reveals that they neither force the acceptance of controversial metaphysical principles nor the modification of the argument in the main text. For, they presuppose a very tight connection between sentential quantification on the one hand and the existence of propositions (amongst which are singular propositions) on the other. Roughly, they presuppose that the ‘range’ of the sentential quantifier varies from possible world to possible world with the propositions that exist at these worlds. In particular, the argument against (P2) presupposes that in order for

• (11) $\forall p \,(\mathcal{P}\, (d_1\hbox{ shows a 6})\,\Vdash\,p\leftrightarrow\mathcal{P}\,(d_2\hbox{ shows a 6})\Vdash\,p)$

to be true, the corresponding nominal quantification about propositions has to be true:

• (12) $\forall x\, (x\hbox{ is a proposition}\rightarrow(\mathcal{P}\,(d_1\hbox{ shows a 6})\hbox{ forces }x\leftrightarrow\mathcal{P}\,(d_2\hbox{ shows a 6})\hbox{ forces }x))$

Otherwise, the transition marked by the parenthesised lightning bolt in the argument against (P2) is not legitimate, for the argument asks us to accept the falsity of (11) on the basis of considerations only directly relevant to the falsity of (12). ³⁵ But even though the paper sometimes uses proposition talk in its informal formulations, this is just for expositional purposes: it is not committed to strict intersubstitutability. In particular, considerations about the existence conditions of propositions such as the ones above may just be taken to show that there is no such close link. This means, of course, that a certain ontic interpretation of sentential quantification is ruled out, namely an interpretation that associates the sentential quantifier with a set of propositions as its range. But such an interpretation is far from obligatory. In effect, it tries to understand sentential quantification in terms of nominal quantification. An attractive alternative position maintains that such an understanding is impossible: just like nominal quantification, sentential quantification is conceptually basic and cannot be understood in more fundamental terms. ³⁶