Abstract
The question as to whether some objects are possible worlds that have an initial segment in common, i.e. so that their fusion is a temporal tree whose branches are possible worlds, arises both for those who hold that our universe has the structure of a temporal tree and for those who hold that what there is includes concrete universes of every possible variety. The notion of “possible world” employed in the question is seen to be the notion of an object of a kind such that objects of that kind play a certain theoretical role. Lewis’s discussion of the question is thereby clarified but is nevertheless inadequate; his negative answer is correct but even from his combinatorialist viewpoint the rationale he provides for this answer is misguided. I explain why the combinatorialist advocate of concrete plenitude should hold that no object is a tree of possible worlds. Then I explain that for a different reason the nomic essentialist advocate of concrete plenitude should hold this much too.
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Notes
My concern is with metaphysically possible worlds. In what follows “possible” “necessary,” “modal” and their cognates should be disambiguated so as to express concepts pertaining to metaphysical modality except where explicit indication is given to the contrary.
MacFarlane (2008).
Cf. Belnap (2003, pp. 34–35).
For the latter case see Belnap and Müller (2010, Sect. 3) and Belnap (1992, 2003, Sect. 2); for the former case see Lewis (1986, p. 206). Lewis (2004, footnote 13) seems to change his mind, since he appears to say that in the absence of “worldwide branching,” by which he presumably means branching over a simultaneity hyperplane, it is inappropriate to speak of “branching.” I am grateful to an anonymous referee for likewise pressing the suggestion that in the relativistic case talk of a “tree” is somewhat misleading if “branching” occurs at points rather than hyperplanes. I agree that the image of a tree does not serve as an adequate representation of branching in the relativistic case. In fact, for two reasons it is a bit misleading in even the non-relativistic case. Firstly, three dimensions are needed to represent the branching of even a one-dimensional space (one dimension representing the one-dimensional space; a second dimension representing time; and the third dimension representing the “logical space” into which the two dimensional branches spread out). Secondly, an object that occupies three dimensions so as to represent branching across a one-dimensional hyperplane of absolute simultaneity doesn’t look much like a tree: for example, since the initial maximal segment being represented is only two-dimensional, ideally the “trunk” of the object would have to be as thin as possible—a sheet of paper would be getting there but e.g. a pencil would be highly misleading.
The term “possible world” may also be parsed attributively as “possibly a world.” See Williamson (2000) for discussion of the distinction between the attributive and the predicative parsing of “possible X.”
Outside of philosophy we often speak of “the/our world” and “the/our universe” interchangeably. No doubt it is because “world” has the ordinary sense “universe” that Lewis feels free to begin On the Plurality of Worlds by making detailed claims about the “world we live in”—claims that are true of our universe—before claiming at the beginning of the third paragraph that “our world is but one world among many.” He makes these claims without giving any explanation of what “world” means and without suggesting that he is employing the term in any technical sense. It isn’t plausible to suppose that in so doing he is addressing only those few philosophers who are familiar with the technical sense in which “(possible) world” has come to be used.
This view is not universally shared. For example, Bricker (2001, p. 28) writes “given the range of disagreement over what sort of thing “possible world” refers to, those who assert that possible worlds exist cannot be taken to share a single view.” But the point that such people can, and should, be taken to share a single view was already made by van Inwagen (1986, pp. 192–193). Van Inwagen holds that “the” concept “possible world” is the concept of an object that plays a certain theoretical role, and that the dispute between those such as Plantinga (1976) who hold that “possible worlds” are abstract and those such as Lewis (1986) who hold that “possible worlds” are concrete is a dispute about the natures of the objects that play this role. The theoretical role van Inwagen sketches is much narrower than the one I have identified, however. Note too that van Inwagen does not recognize the ambiguity of “possible world” that I have identified. For the ersatzist view that possible worlds are sets of sentences, see Roy (1995).
Bricker (2001) argues against Lewis (1986) that a plural quantifier should be used in the equivalence schema. In effect, Yablo (1999) argues against Lewis (1986) that the set W should be taken to include not just universes but (at least some of) their proper parts too. The arguments of Bricker and Yablo are discussed in Sect. 3.2 below.
I follow Lewis (1986, p. 62) in taking a relation to be “external” iff it does not supervene on duplicates of its relata but does supervene on duplicates of the respective fusions of its relata. I am grateful to the anonymous referee who pointed out that some hold that the notion “external relation” is prior to that of the notion “duplicate” and is therefore not defined by this equivalence.
The term “suitable” is crucial: the relation x bears to y iff x is not identical to y is an external relation; so dropping “suitable” would have the undesirable consequence that it is analytically true that there is at most one universe. There are two ways of substantiating “suitable” that are worthy of note. One is to take spatiotemporal relatedness as a paradigm; an external relation can then be deemed “suitable” iff it is at least analogous to spatiotemporal relatedness. Another is to invoke the notion “natural” so as to hold that a relation is suitable iff it is natural (i.e. the presumption being that the relation of non-identity is not natural). Either way it is uncontroversial that spatiotemporal relations are suitable. For further discussion see Bricker’s (1996) dispute with Lewis (1986, pp. 69–78) regarding the relations with respect to which a “world” should be defined as being unified and maximal.
For the more liberal notion of “local” unifiedness, and for the proof that locally unified “worlds” cannot overlap (as two “worlds” that are branches of a temporal tree do), see respectively Sect. 2 and footnote 10 of Bricker (1996). Lewis (1986, p. 72) is inclined to require that “worlds” be globally unified but he also considers a liberalisation of this requirement; the liberalisation he considers falls short of local unifiedness, however. Bricker (1993, 1996, 2001) argues that no more than local unifiedness is required because spacetime is only locally unified if General Relativity is true.
When Lewis (1986, pp. 206–209) distinguishes “branching of worlds” from “branching within worlds” he should be interpreted as meaning “modal locus” by “world.” So a temporal tree the branches of which are modal loci exemplifies what he calls “branching of worlds,” while a modal locus that is a temporal tree of mere histories exemplifies what he calls “branching within worlds.” Curiously, Belnap and Müller (2010, fn. 4) interpret Saunders and Wallace’s (2008) use of “world” as “history” and then complain that “following Lewis, [they] obliterate the crucial distinction between histories and worlds.” Whatever the merits of this complaint against Saunders and Wallace it is wrongly directed at Lewis: Lewis does not obliterate this distinction; on the contrary, in distinguishing “branching within worlds” from “branching of worlds” he enjoins us to attend to it.
As I read him, Lewis (1986) is guided to a hypothesis as to which objects in the concrete plenitude are “worlds” in the technical sense of “modal loci” by his awareness of which objects in the concrete plenitude are “worlds” in the ordinary sense of “universes”; and since the resulting hypothesis that the modal loci are exactly the universes is seen by him to pass many tests he sticks to it. He suspends this hypothesis occasionally so as to discuss the merits of principles with which it is incompatible; in particular, he suspends it when he considers (on pp. 206–209) the independent merits of the supposition that certain “worlds” (i.e. modal loci) are branches of temporal trees.
Unfortunately, this opposition is hampered by the fact that “world” means “universe” in the ordinary sense and no other term for modal locus has become common usage. In the absence of an alternative term for the technical sense, to deny that the universes are the modal loci on the grounds that some modal loci are not universes is to hold that some worlds are not worlds (in the ordinary sense). Philosophers who are made nervous by this sort of thing wouldn’t have batted a eyelid had terms such as “universe” and “modal locus” been used all along to respectively express the two concepts unambiguously and no such term as “world” ever been used to express the concepts ambiguously. Bricker (2001) is one philosopher who is made nervous by it. In effect, he argues that Lewis’s hypothesis that the “worlds” (i.e. the modal loci) are exactly the universes is untenable. He recognizes that the problem he identifies could be solved by identifying the “worlds” not with the universes but with the universes and the fusions of universes. But he (p. 45) rejects this revision on the terminological ground that it would conflict with established “philosophical usage” (at least in the “realist” tradition) whereby “worlds have been essentially unified, and they have ... not overlapped ... extensively” and pursues an alternative solution involving plural quantification. Had the terms “modal locus” and “universe” been in play all along, however, Bricker would have felt no pressure whatsoever on the modification of Lewis’s theory he rejects: in effect, his view is that Lewis holds that the modal loci are exactly the universes but a better theory holds that the modal loci are the universes and the fusions of universes. (A word of warning. Bricker postulates maximally unified objects of two kinds: the metaphysical (which he calls “possible worlds”) and the “physical” (which he calls “universes”). In my view, his practice of withholding the term “universe” from the maximally unified objects that he deems “metaphysical” is unjustified and misleading; since Bricker (1996, p. 231) does not conceive the notion “particle” in such a way that no “metaphysical” object is a negatively charged particle he should not conceive the notion “universe” in such a way that no “metaphysical” object is a universe.)
Regarding the first ground, see especially Belnap and Green (1994), Belnap et al. (2001), Belnap and Müller (2010), and MacFarlane (2003); MacFarlane (2008). Regarding the second ground, for exposition see Albert and Loewer (1988) and Lewis (2004); for more recent developments see Saunders et al. (2010).
Those who maintain that the semantics of tenses or historical modalities demands branching structures may reject Our Tree by maintaining the ersatzist view that in even the intended model the branches of the structure are abstract objects. Although ersatz branching models are popular amongst those who advocate the “A-theoretic” doctrine that tensed facts are fundamental, and, in particular amongst those who hold a no future variant of this doctrine i.e. according to which nothing concrete is later than the present, such ersatzism does not require A-theory. Just as the modal operators have been taken to engender modal loci that are not concrete, so too have the operators that express historical possibilities been taken to engender loci for historical modalities that are not concrete; in neither case is a B-theoretic rejection of Our Tree impugned. In effect, a B-theory that employs ersatz branching structures in the semantics of historical modalities while rejecting Our Tree is committed to what Belnap and Green (1994) call the “thin red line”; it holds that exactly one of the many ersatz futures veridically represents the (concrete) future (and so comprises the thin red line through the ersatz branching structure).
Just as the modal loci are the objects with respect to which the modal operators are “defined” so too are loci of other kinds objects with respect to which operators of other kinds are “defined.”
Wilson (2011, p. 371) shows qualified sympathy for this variant. See footnote 33 below for how his sympathy is qualified.
This claim about sets is to be understood in such a way as to be compatible with a reduction of sets to paradigmatically concrete objects.
Lewis (1986) holds that there are mountains that are non-actual; Bricker (2001) holds that there are mountains that are (merely) metaphysical; Meinongian advocates of Concrete Plenitude hold that there are mountains that are non-existent. Some hold that “is a mountain” is in effect ambiguous between “exemplifies the property of being a mountain” and “encodes the property of being a mountain.” I do not recognize this ambiguity; those who do should respect my express intention that my use of predicates be disambiguated in terms of exemplification, not encoding. Accordingly, advocates such as (Zalta (2006), Sect. 2) of the doctrine that every satisfiable predicate is such that some object encodes the property it expresses do not thereby advocate Concrete Plenitude; a golden mountain is a mountain irrespective of whether it is actual or exists, but an object that merely encodes the property of being a mountain is not a mountain.
(CP1) remains problematic even when its vagueness is put aside, and this is why it is only a rough first approximation to Concrete Plenitude. (CP1) is inconsistent: the possible variety of objects includes an object such that it is round and there are goldfish but also an object such that it is round and there are no goldfish; so the doctrine that there are concrete objects of every possible variety entails that there are goldfish and that there are no goldfish. Accordingly, Concrete Plenitude must be understood as the thesis that results when the term “possible” is restricted minimally to “suitably possible” so as to restore consistency. The problem of defining “suitably” is akin to the so-called “characterization problem” faced by Meinongians who would like to hold, if so doing weren’t inconsistent, that for every property \(F\) some object is completely characterized by the fact that it is \(F\).
Lewis (1986) is the pre-eminent advocate of Concrete Plenitude. Of course Lewis (1990) holds that Concrete Plenitude is an ontological thesis, and that to suppose otherwise is unintelligible. But as van Inwagen (1986, pp. 188–189) observes, “Meinongian” advocates of Concrete Plenitude, who he suspects are plentiful, deny that ontology is about what there is. Bricker (2001) advocates something of a compromise; with Lewis, the merely possible worlds exist; against Lewis, the universe of which we are a part is “physical” but every possible world is “metaphysical” and so there is no possible world to which our universe is identical. If the existence of our universe is thought of as a result of the “actualization” of one of the possible worlds, Bricker leaves it open, epistemically, as to whether more than one possible world has been actualized, i.e. so as to result in multiple “island” universes that are “physical.” Bricker’s view is not easily classified; one might say that he advocates Concrete Plenitude with respect to the “metaphysical” but rejects it with respect to the “physical.”
To say that combinatorialism is true is not to say that it is necessarily true; one might coherently maintain that although the actual fundamental properties are subject to combinatorial reshuffling there could have been fundamental properties that are not subject to combinatorial reshuffling. To my knowledge no one has ever exploited this distinction and I see no grounds to support so doing. I shall assume that if combinatorialism is true it is necessarily true.
Cf. Lewis (1986, p. 206).
This assertion relies on the fact that Concrete Plenitude is a common assumption with the reader. Combinatorialism alone does not entail Existential Splitting; nor does it do so when combined with the view that there are concrete modal loci. The reason is that it is compatible with a doctrine with which Existential Splitting is incompatible—namely, the doctrine that all fundamental properties are “locus bound” in the sense that no fundamental property is instantiated at two different modal loci. Combinatorialism is equivalent to an infinite conjunction of propositions that are each of the form “it is possible that ...” . As such it is neutral as to how exactly modalities are to be “defined” in terms of modal loci. In particular, in principle it could be reconciled with the doctrine that all fundamental properties are locus-bound by invoking a counterpart relation between fundamental properties. In contrast, Existential Splitting directly contradicts this doctrine. Suppose Existential Splitting is true and that \(w\) and \(w^{\prime }\) are modal loci that split. Then the fundamental properties \(P_{i}\) and \(P^{\prime }_{i}\) that are respectively instantiated in their respective maximal initial segments \(s\) and \(s^{\prime }\) must be such that for each \(i\) the identity \(P_{i}=P^{\prime }_{i}\) holds. This is trivial in the case where splitting is by identity. But it is true too in the case where splitting is by duplication. For were the fundamental properties \(P^{\prime }_{i}\) that \(s^{\prime }\) instantiates not identical to the fundamental properties \(P_{i}\) that \(s\) instantiates, \(s^{\prime }\) would not be a duplicate of \(s\). However, while a combinatorialist who holds that there are concrete modal loci that include universes other than our own might invoke a counterpart theoretic treatment of fundamental properties in this way so as to reject both Concrete Plenitude and Existential Splitting, following (Schaffer (2005), Sect. II.D)), (Lewis (2009), Sect. 4) and even Black (2000), I think so doing would be misguided. It is in the nature of a fundamental property to be instantiated both here and there. What is the problem if, so to speak, “there” is an object that is not suitably externally related to us? The notion of a possible duplicate e.g. of Barak Obama is not sensitive to whether the duplicate is actual or non-actual (or existing or not existing, or metaphysical or not metaphysical): just as an actual duplicate of Barak Obama has exactly the intrinsic properties that Barak Obama has actually, so too does a duplicate of him that is merely possible. Moreover, our assumption that Concrete Plenitude is true effectively rules out a counterpart theoretic treatment of the modality of fundamental properties. The thought that there are concrete objects of every possible variety is the thought that if it is possible for there to be an object with fundamental properties \(P_{i}\) then there is an object with fundamental properties \(P_{i}\). So Concrete Plenitude as I have defined it is more or less committed to denying that fundamental properties are universe-bound and so gives the thought that they are locus-bound no purchase.
I limit this consideration to species of Concrete Plenitude that, deem the modal facts reducible to the non-modal facts. Advocates of species of Concrete Plenitude that, in contrast, resist any such reduction—e.g. so as to effect a reduction of the non-modal facts about modal loci to the modal facts, as suggested by Fine (2005: chs. 4 and 6)—tend not to think of Concrete Plenitude as a theory; consequently, one would not expect them to be impressed by considerations of economy. Put the point this way: if the concrete objects that are not suitably externally related to us are unreal who cares how many of them there are?
In effect, Lewis (1986, pp. 206–210) gives this argument. He takes one moral of (P1) to be that advocates of Our Tree would find (combinatorialist) Universal Branching attractive, i.e. because it entails something they believe. Wilson (2011, p. 382) interprets Lewis differently, however. He takes Lewis to argue more narrowly that the supposition that our universe is a temporal tree of “worlds” (i.e. modal loci) makes nonsense of our thought and talk about the future; he then suggests that Lewis’s argument can be applied equally well to the view that our universe is a tree of mere histories. This is a misinterpretation. Lewis (1986, p. 209) also explicitly applies his argument to this latter case (which he calls “branching within worlds”).
In effect, when Lewis’s argument is properly interpreted, Wilson (2011) endorses it. Consequently, Wilson’s sympathy for the view that our universe is a temporal tree of modal loci is qualified (see footnotes 22 and 32 above). In my terminology, Wilson is sympathetic to the idea that the myriad “worlds” splitting off from one another that are postulated by some no-collapse interpretations of quantum mechanics are modal loci. Consequently he is sympathetic to the idea that our universe is a temporal tree of modal loci if these “worlds” split off by branching. But he takes Lewis’s argument to show that these “worlds” are better thought of as always splitting off by diverging. For reasons similar to those advanced in Sect. 1 above, I think that in the absence of Concrete Plenitude Wilson is wrong to favour the view that the many “worlds” of the no-collapse theories in question are modal loci. (I am grateful to an anonymous referee who informs me that within the philosophical community he or she has encountered widespread sympathy with Lewis’s argument.)
See (Belnap et al. (2001), pp. 170–176, 205–209), Belnap and Müller (2010), MacFarlane (2008), and Saunders and Wallace (2008). They make two objections. Firstly, they object that it is wrong to think that Our Tree would make nonsense of our commonsensical thought and talk about the future; they claim that a semantics can be provided that reconciles Our Tree with our ordinary thought and talk about the future. Secondly, they object that in any case common sense is hardly sacrosanct; they claim that Our Tree brings benefits that would outweigh the attractions of common sense were the two to conflict. Wilson’s (2011) sympathy for the argument is the exception.
The falsity of (P1) is so obvious that it is puzzling that anyone should have thought it true. Since Lewis believes that our universe is straight, perhaps it is because combinatorialist Universal Branching entails Universal Tree-Boundedness of Straight Modal Loci that he came to believe (P1) is true. If one starts out believing that our universe is an object that is a straight modal locus one might be led via Universal Tree-Boundedness of Straight Modal Loci to conclude that this object is a branch of a temporal tree of modal loci and, hence, pace the belief from which one started out, that our universe is a temporal tree of modal loci. But of course although one could end up changing one’s beliefs in this way logic does not require that one do so. Logic only requires that one relinquish the belief that our universe is a straight modal locus.
The thesis that our universe is not straight is very weak. In particular, it is consistent with the view that our universe is very nearly straight, as indeed is the view that our universe is a temporal tree. For this reason both combinatorialist Universal Branching and, pace (P2), Our Tree, are consistent with common sense. Indeed, to all intents and purposes they are consistent with the view that we have one future! To see this, notice that combinatorialist Concrete Plenitude holds that the universes include non-straight universes the histories of which are straight until branching occurs just once immediately before as many Big Crunches as there are branches after that moment. The supposition that our universe is identical to one of these non-straight universes does not offend against common sense; when common sense leads us to speak of “the” future we do not exclude the possibility that an otherwise linear route towards a Big Crunch is interrupted by branching nanoseconds beforehand. This consideration alone does not negate the dialectical power of Lewis’s appeal to common sense, however; in practice advocates of Our Tree maintain that our universe is constantly branching.
It might be thought that this viewpoint is undermined by Kripkean insights to the effect that some contingent propositions are a priori. Given these insights, why shouldn’t the contingent claim that our universe is not straight be a priori too? And if this claim about our universe is such that there is no obstacle to its being a priori, how can it be a defect for a theory of modality to entail it? This line of thought is powerless to rescue combinatorialist Universal Branching, however. Kripke’s examples of the contingent a priori are quite disanalogous to the proposition that our universe is not straight. What gives these examples purchase is that it appears not only that the truth values of the propositions in question are accessible in the absence of any empirical investigation, but that one has no idea what an empirical investigation into their truth values would be: one couldn’t empirically investigate e.g. whether Julius is the inventor of the zip or whether one is here now. In contrast, we have a reasonably clear idea of what empirical investigation into the spatiotemporal structure of our universe amounts to.
See Earman (2008).
The “many worlds” variant of the no-collapse interpretation of quantum mechanics stems from Everett (1957), Albert and Loewer (1988) and Lewis (2004) together comprise a good critical introduction to it. It may be subdivided according to (i) whether the “worlds” invoked are modal loci or mere histories; (ii) whether the “worlds” branch, and so constitute a temporal tree, or diverge; and (iii) whether the plurality of “worlds” pertains to the fundamental level or to an emergent macro level. The “first wave” many worlds interpretation is committed to branching and takes it to be fundamental: at the most fundamental level, spacetime itself branches. The “second wave” many worlds interpretation denies that branching occurs at the fundamental level: the division into many worlds is confined to the macro-level, and opinion is divided as to whether the worlds at this level branch or diverge. The rationale for the second wave is that first wave doctrine appears self-defeating: it aspires to interpret the deterministic core dynamical equations of quantum mechanics as they stand, without appealing to any “collapse” postulates; but its attempt to do so invokes a fundamental process—the branching of spacetime—about which, at best, quantum mechanics has nothing to say (and, at worst, with which quantum mechanics is inconsistent). For this and other difficulties with the first wave many worlds view see Albert and Loewer (1988, pp. 198–203). For discussion of whether the second wave many worlds interpretation should hold that the “worlds” branch or diverge see Wilson (2011).
In “reject(ing) genuine branching in favour of divergence,” Lewis (1986, p. 206) defends Universal Divergence; for by “divergence” he means Universal Divergence. He says (p. 209) “[b]ranching, and the limited overlap it requires, are to be rejected as making nonsense of the way we take ourselves to be related to our futures; and divergence without overlap is to be preferred.” What Lewis means by “branching” (and by “genuine branching” in the previous quote) is Universal Branching, not Existential Branching: irrespective of whether Universal Branching entails Our Tree it is clear that Existential Branching does not entail Our Tree. Lewis (1986, p. 209) makes evident that it is Universal Branching, not Existential Branching, with which he is concerned when he writes that the problem facing those non-actual beings who are parts of non-actual worlds that are instances of branching within worlds is “not ours, as it would be if the worlds generally branched rather than diverging” (my italics).
If, as in Lewis (1986), the modal loci are exactly the universes and “at \(w\)” works by restricting the domains of quantifiers within its scope to \(w\), then it is impossible for there to be more than one universe; every universe (and so every modal locus) \(w\) is such that at \(w\), there is exactly one universe. Lewis (1986, pp. 71–73) recognizes this consequence but he bites the bullet on the grounds that the claim that it is possible for there to be island universes is not central to our customary modal views and that for this reason theory is free to reject it. His strategy is heavily undercooked, however because it is incompatible with possibility introduction, \(\varphi \, {\vert }\)– \(\diamond \varphi \); it is a central tenet of the species of Concrete Plenitude he advocates that there is more than one universe.
Upward inclusionism solves these problems as follows. Firstly, the problem of the possibility of multiple universes is solved because at a modal locus that is the fusion of at least two universes, there is more than one universe. Secondly, the problem more generally that the rule of possibility introduction is violated is solved as a result of taking the fusion of all universes to be a modal locus: the sentences that breach the rule of possibility introduction are sentences such that (i) the advocate of Concrete Plenitude (and in particular Lewis) asserts the propositions these sentences express when articulating his or her species of Concrete Plenitude and (ii) there is no universe at which any of these sentences are true. But these sentences are all true at the modal locus that is the fusion of all universes, i.e. because restricting the domains of quantifiers to parts of this modal locus is no restriction at all.
Alternative solutions to these problems have been proposed, however. In particular, although Bricker (2001) argues that the advocate of Concrete Plenitude should reject Lewis’s species of it so as to accommodate the possibility of multiple universes and Divers (1999) argues that he or she should reject it so as to accommodate the rule of possibility introduction, neither Bricker nor Divers advocates upward inclusionism. Bricker proposes to accommodate the possibility that there is more than one universe by relaxing the equivalence schema by which the role of modal loci is defined; he suggests that the right hand side of this equivalence should employ a plural quantifier i.e. so as to be of the form “for some worlds, at those worlds \(\varphi \).” In contrast, Divers suggests that the propositions of the form “Necessarily ...” be divided into “ordinary” and “advanced” propositions, i.e. with the proposition that there are multiple island universes (“worlds” in his terms) being advanced. He then proposes that the advanced propositions be treated not in the standard way but in accordance with the rule that “necessarily \(\varphi \)” is true iff \(\varphi \) is true. Bricker’s (p. 45) rationale for preferring his own proposal to upward inclusionism is a desire to preserve what he claims to be the common presumption that even in its philosophical sense the notion of a “world” is the notion of an object that is unified. (In my terminology this presumption is simply the thesis that the modal loci are all unified.)
This is the species of inclusionism that Clark (2010) focuses upon.
In effect Sider (2003, p. 196) suggests that universalist inclusionism is supported by the fact that it would accommodate the possibility of multiple universes. But we have just seen that upward inclusionism would perform this task adequately.
Matters are rather different from the viewpoint of one who accepts that the truth conditions for modal statements are given by extensional claims about concrete modal loci but is nevertheless agnostic about whether there are concrete objects that are spatiotemporally removed from us. This viewpoint requires agnosticism about many possibility claims that are commonly believed and one who adheres to it might seek to limit how much agnosticism about such claims is required. Downward inclusionism would help in this regard: if the modal loci are exactly the universes then agnosticism would be required e.g. with respect to “possibly, there are no swans”; but if e.g. Antarctica is a modal locus then the possibility of there being no swans could be acknowledged. See Divers (2004) for the kind of agnosticism at issue here and Parsons (2007, pp. 167–168) for the point that downward inclusionism would help out in this way.
See Yablo (1999, pp. 47, 57). I have followed Yablo’s numbering but in accordance with more customary usage I have used “\(\le \)” instead of his “\(<\)” for the “part of” relation. (I am grateful to the referee who pointed out the desirability of so doing.) I have also appended “necessarily for all \(x\) and \(y\)” to the beginning of the right hand side of (3), since so doing appears to be required. Yablo argues that (3*) and (3) capture just one notion of “intrinsic”; he identifies two others for which he also gives similarly mereological definitions. These other notions raise no new issues relevant to our concerns.
Yablo (1999, p. 37) does state that he presupposes that “some worlds contain others as proper parts” but he does not state explicitly which objects he reckons (the advocate of Concrete Plenitude should hold) to be “worlds.” In effect, Parsons (2007) interprets him as advancing universalist inclusionism, however.
Parsons (2007) misses this point. Perhaps the point is clearer if one remembers that Concrete Plenitude takes the modifier “in w” to be the quantifier domain restricting modifier “at \(w\)” i.e. so that its effect is insensitive to whether the objects over which \(w\) ranges are (properly) called “modal loci.” For example, if \(w^{\prime }\) is the actual universe but \(w\) is Australia and Hilary Clinton is in the US but Bill Clinton is in Australia, then “at \(w^{\prime }\), there is someone to whom Bill Clinton is married” is true but “at \(w\), there is someone to whom Bill Clinton is married” is false, and this is so irrespective of whether Australia is reckoned to be a modal locus.
De re necessitarianism about the fundamental laws is advocated by dispositionalist essentialists about fundamental properties such as Bird (2005), while Shoemaker (1998) argues that the causal laws are necessary. To say that it is true is not to say that it is necessarily true; one might hold that the fundamental properties are such that necessarily, they are governed by such and such laws, and yet hold that there could have been fundamental properties governed by laws such that they could have been governed by different laws. But since I know of no one who has exploited this distinction and see no grounds for so doing I shall assume that if de re necessitarianism about the fundamental laws is true it is necessarily true.
Consider a deterministic world \(w\) and an initial segment \(s\) of \(w\) such that the fundamental properties distributed throughout \(s\) are the properties \(P_{i}\) and the laws \(L\) at \(w\) concerning the \(P_{i}\) are deterministic. Let world \(w^{\prime }\) be a world that splits from \(w\) after \(s\). By the definition of splitting, \(w^{\prime }\) has an initial segment \(s^{\prime }\) that is either identical to or a duplicate of \(s^{\prime }\). In either case, at \(w^{\prime }\) the laws concerning the properties \(P_{i}\) cannot be the laws \(L\): since the laws \(L\) are deterministic they would combine with \(s^{\prime }\) to require that the end segment of \(w^{\prime }\) duplicates the end segment of \(w\). So at \(w^{\prime }\) the laws concerning the \(P_{i}\) must be different laws \(L^{\prime }\). But if this conclusion is true de re necessitarianism about the fundamental laws is not necessarily true. So if de re necessitarianism about the fundamental laws is necessarily true there is no deterministic world from which some world splits.
For the many worlds no-collapse interpretation, and the first wave development of it, see footnote 39 above and the references therein.
I follow Placek and Belnap (2012, introduction), who distinguish “Laplacean” determinism, which is a matter of the strength of the laws (with indeterministic laws being too weak to determine what will happen next, given what has already happened), from “Aristotelian” determinism, which is a matter of temporal structure (with indeterministic structures being tree-like and the branches above a point being alternative historically possible futures). The notion “initial segment*” extends the notion “initial segment” to “horizontal” cuts across the tree at any point. Such a cut yields an initial segment* that is itself a temporal tree iff the cut is made after branching has occurred.
I know no one in the literature who explicitly denies the possibility of straight \(L\)-indeterministic laws. In particular, I know no advocate of either \(A\)-indeterminism or de re necessitarianism about the fundamental laws who denies it.
Nomic essentialism is even compatible with the species of Concrete Plenitude that reduces the modal facts to non-modal facts about the modal loci. From a viewpoint that combines nomic essentialism with reductionism about modality the explanation of the (supposed) de re metaphysical necessity of the fundamental laws proceeds in two stages. Firstly, metaphysical necessity is explained as truth at all modal loci. Secondly, nomic essences of fundamental properties are taken to explain the absence of modal loci with which the laws that govern fundamental properties are inconsistent; such objects require a pattern of fundamental property instantiation that the essences preclude. Classically, however, nomic essentialism is allied with the rejection of reductionism about modality. Its explanation of the (supposed) de re metaphysical necessity of the fundamental laws proceeds differently. Firstly, it is explained that that which is metaphysically necessary is that which is grounded in essences. Secondly, it is explained that the fundamental laws are grounded in the essences of fundamental properties.
Strictly speaking, it follows that the laws at the branches are \(L\)-indeterministic only if the discussion is restricted to temporal trees that are heterogeneous i.e. in that at least two of their branches are not duplicates. That the laws are not \(A\)-indeterministic follows even in the case of homogeneous trees, however.
In the case of \(L\)-deterministic straight laws a dispositionalist nomic essentialist can explain why no two modal loci split by supposing that, at least in effect, every straight modal locus divides into an initial segment \(s\) and end segment \(e^{\prime }\) such that the initial segment \(s\) divides into a disposition \(s_{1}\) and a stimulus \(s_{2}\), with the end segment \(e^{\prime }\) being the manifestation (cf. Bird 2005). It is hard to see how this metaphysical picture might sensibly be extended to the case of laws that are \(L\)-indeterministic. In the \(L\)-indeterministic case, as far as the states \(\sigma \) and \(\sigma ^{\prime }\) are concerned the most that can be metaphysically necessary is a real number that is properly between 0 and 1 that measures the primitive “strength” of a primitive tendency of \(\sigma \) to yield \(\sigma ^{\prime }\) (or of some ‘manifestation’ relation between \(\sigma _{1},\, \sigma _{2}\) and \(\sigma ^{\prime }\)). So the state \(\sigma ^{\prime }\) is not essential to the state \(\sigma \). But then what is essential to it? Only the chance \(\sigma \) affords \(\sigma ^{\prime }\) is left as a candidate. But what is that? Is it a fundamental property that itself has an essence? Or is it a relation (between \(\sigma \) and \(\sigma ^{\prime }\)) that likewise has an essence? I see no way to give satisfactory answers to these questions. Moreover, however they are answered the explanatory power the nomic essentialist envisages is lost. If the essence of \(\sigma \) consists in no more than the chance it affords \(\sigma ^{\prime }\) its explanatory power is diminished beyond recognition: chances explain other chances (by entailing them); chances do not explain non-chances. (See Percival (2006, Sect. 1).)
Existential Branching is the doctrine that there is a heterogeneous temporal tree of modal loci. Mutatis mutandis, the same argument shows that the nomic essentialist should also reject homogeneous temporal trees of modal loci, i.e. where a tree is homogeneous iff its branches are duplicates. The problem does not depend on the branches of the tree not being duplicates. If there is a homogeneous temporal tree of modal loci then from the viewpoint of de re necessitarianism about the fundamental laws, the fundamental properties instantiated in the maximal initial segments of the tree’s various branches are governed by laws that are \(L\)-indeterministic but neither straight nor anti-straight.
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Acknowledgments
I would like to thank my colleagues Jonathan Tallant and Harold Noonan for helpful comments on an earlier draft of this paper. I am especially grateful to an anonymous referee of Synthese for his or her conscientious, insightful, and constructive critical comments on the original submission.
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Percival, P. Branching of possible worlds. Synthese 190, 4261–4291 (2013). https://doi.org/10.1007/s11229-013-0271-7
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DOI: https://doi.org/10.1007/s11229-013-0271-7