Philosophers often explain what could be the case in terms of what is, in fact, the case at one possible world or another. They may differ in what they take possible worlds to be or in their gloss of what is for something to be the case at a possible world. Still, they stand united by the threat of paradox. A family of paradoxes akin to the set-theoretic antinomies seem to allow one to derive a contradiction from apparently plausible principles. (...) Some of them concern the interaction between propositions and worlds, and they appear to afford the means to map classes of propositions into propositions – or, likewise, classes of worlds into worlds – in a one-to-one fashion that leads to contradiction. Yet another family of paradoxes threaten the view that whatever could exist does, in fact, exist, which is in line with modal realism, for example. This article aims to survey and identify the source of each family of paradoxes as well as to outline some responses to them. (shrink)

We look at recent accounts of the indefinite extensibility of the concept set and compare them with a certain linguistic model of indefinite extensibility. We suggest that the linguistic model has much to recommend over alternative accounts of indefinite extensibility, and we defend it against three prima facie objections.

This note is to show that a well-known point about David Lewis’s (1986) modal realism applies to Timothy Williamson’s (1998; 2002) theory of necessary existents as well.1 Each theory, together with certain “recombination” principles, generates individuals too numerous to form a set. The simplest version of the argument comes from Daniel Nolan (1996).2 Assume the following recombination principle: for each cardinal number, ν, it’s possible that there exist ν nonsets. Then given Lewis’s modal realism it follows that there can be (...) no set of all (that is, Absolutely All) the nonsets. For suppose for reductio that there were such a set, A; let ν be A’s cardinality; and let µ be any cardinal number larger than ν. By the recombination principle, it’s possible that there exist µ nonsets; by modal realism, there exists a possible world containing, as parts, µ nonsets; each of these nonsets is a member of A. (shrink)

This paper discusses the principle of recombination for possible worlds. It argues that arguments against unrestricted recombination offered by Forrest and Armstrong and by David Lewis fail, but a related argument is a challenge, and recommends that we accept an unrestricted principle of recombination and the conclusion that possible worlds form a proper class.

The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the (...) existence of wide sets. Drawing upon Cantor’s notion of the absolute infinite, the paper argues that the modifications are warranted and preserve a robust iterative conception of set. The resulting theory is proved consistent relative to ZFC + “there exists an inaccessible cardinal number.”. (shrink)

Say that some things form a set just in case there is a set whose members are precisely the things in question. For instance, all the inhabitants of New York form a set. So do all the stars in the universe. And so do all the natural numbers. Under what conditions do some things form a set?

We argue that certain modal questions raise serious problems for a modal metaphysics on which we are permitted to quantify unrestrictedly over all possibilia. In particular, we argue that, on reasonable assumptions, both David Lewis's modal realism and Timothy Williamson's necessitism are saddled with the remarkable conclusion that there is some cardinal number of the form ℵα such that there could not be more than ℵα-many angels in existence. In the last section, we make use of similar ideas to draw (...) a moral for a recent debate in meta-ontology. (shrink)

We often speak as if there are merely possible people—for example, when we make such claims as that most possible people are never going to be born. Yet most metaphysicians deny that anything is both possibly a person and never born. Since our unreflective talk of merely possible people serves to draw non-trivial distinctions, these metaphysicians owe us some paraphrase by which we can draw those distinctions without committing ourselves to there being merely possible people. We show that such paraphrases (...) are unavailable if we limit ourselves to the expressive resources of even highly infinitary first-order modal languages. We then argue that such paraphrases are available in higher-order modal languages only given certain strong assumptions concerning the metaphysics of properties. We then consider alternative paraphrase strategies, and argue that none of them are tenable. If talk of merely possible people cannot be paraphrased, then it must be taken at face value, in which case it is necessary what individuals there are. Therefore, if it is contingent what individuals there are, then the demands of paraphrase place tight constraints on the metaphysics of properties: either (i) it is necessary what properties there are, or (ii) necessarily equivalent properties are identical, and having properties does not entail even possibly being anything at all. (shrink)

Are there, in addition to the various actual objects that make up the world, various possible objects? Are there merely possible people, for example, or merely possible electrons, or even merely possible kinds? We certainly talk as if there were such things. Given a particular sperm and egg, I may wonder whether that particular child which would result from their union would have blue eyes. But if the sperm and egg are never in fact brought together, then there is no (...) actual object that my thought is about. Or again, in the semantics for modal logic we presuppose an ontology of possibilia twice over. For first, we countenance various possible worlds, in addition to the actual world; and second, each of these worlds is taken to be endowed with its own domain of objects. These will be the actual objects of the world in question, but they need not be actual simpliciter, i.e., actual objects of our world. What are we to make of such discourse? There are four options: (i) the discourse is taken to be unintelligible; (ii) it is taken to be intelligible but nonfactual, i.e. as not in the business of stating facts; (iii) it is taken to be factual but reducible to discourse involving no reference to possibilia; (iv) it is taken to be both factual and irreducible. These options range from a full-blooded form of actualism at one extreme to a full-blooded form of possibilism at the other. The two intermediate positions are possibilist in that they accept the intelligibility of possibilist discourse but actualist in that they attempt to dispense with its prima facie commitment to possibilia. All four positions have found advocates in the literature. Quine, in his less irenic moments, favours option (i); Forbes advocates option (ii), at least for certain parts of possibilist discourse; many philosophers, including Adams and myself, opt for (iii); while Lewis and Stalnaker have endorsed versions of (iv), that differ in how full-blooded they take the possible objects to be.. (shrink)

Consider two standard quantified modal languages A and P whose vocabularies comprise the identity predicate and the existence predicate, each endowed with a standard S5 Kripke semantics where the models have a distinguished actual world, which differ only in that the quantifiers of A are actualist while those of P are possibilist. Is it possible to enrich these languages in the same manner, in a non-trivial way, so that the two resulting languages are equally expressive-i.e., so that for each sentence (...) of one language there is a sentence of the other language such that given any model, the former sentence is true at the actual world of the model iff the latter is? Forbes (1989) shows that this can be done by adding to both languages a pair of sentential operators called Vlach-operators, and imposing a syntactic restriction on their occurrences in formulas. As Forbes himself recognizes, this restriction is somewhat artificial. The first result I establish in this paper is that one gets sameness of expressivity by introducing infinitely many distinct pairs of indexed Vlach-operators. I then study the effect of adding to our enriched modal languages a rigid actuality operator. Finally, I discuss another means of enriching both languages which makes them expressively equivalent, one that exploits devices introduced in Peacocke (1978). Forbes himself mentions that option but does not prove that the resulting languages are equally expressive. I do, and I also compare the Peacockian and the Vlachian methods. In due course, I introduce an alternative notion of expressivity and I compare the Peacockian and the Vlachian languages in terms of that other notion. (shrink)

The doctrine that whatever could exist does exist leads to a proliferation of possibly concrete objects given certain principles of recombination. If, for example, there could have been a large infinite number of concrete objects, then there is at least the same number of possibly concrete objects in existence. And further cardinality considerations point to a tension between the preceding doctrine and the Cantorian conception of the absolutely infinite. This paper develops a parallel problem for a variety of possible worlds (...) accounts of modality which eschew the commitment to a plethora of possibly concrete objects. Moreover, the difficulty is importantly different from more familiar threats of paradox exemplified by certain descendants of Russell's paradox of propositions and Kaplan's paradox. (shrink)