SyntheseDOI 10.1007/s11229-013-0271-7

Branching of possible worlds

Philip Percival

Received: 28 September 2012 / Accepted: 11 March 2013© Springer Science+Business Media Dordrecht 2013

Abstract The question as to whether some objects are possible worlds that have aninitial segment in common, i.e. so that their fusion is a temporal tree whose branchesare possible worlds, arises both for those who hold that our universe has the structureof a temporal tree and for those who hold that what there is includes concrete universesof every possible variety. The notion of “possible world” employed in the question isseen to be the notion of an object of a kind such that objects of that kind play a certaintheoretical role. Lewis’s discussion of the question is thereby clarified but is never-theless inadequate; his negative answer is correct but even from his combinatorialistviewpoint the rationale he provides for this answer is misguided. I explain why thecombinatorialist advocate of concrete plenitude should hold that no object is a treeof possible worlds. Then I explain that for a different reason the nomic essentialistadvocate of concrete plenitude should hold this much too.

Keywords Possible worlds · Branching · Combinatorialism · Possibilism ·Nomic essentialism · Open future · Everettian quantum mechanics · Plenitude ·Other universes · Metaphysical possibilities · Ersatzism · Modal realism ·Intrinsic properties · Laws · Meinongianism

The question as to whether some objects are possible worlds that have an initialsegment in common, i.e. so that their fusion is a temporal tree whose branches arepossible worlds, has attracted interest from two sources.1 Firstly, interest in it has asource in Our Tree—the doctrine that our universe, i.e. the object that is the fusion

1 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 metaphysicalmodality except where explicit indication is given to the contrary.

P. Percival (B)Department of Philosophy, University of Nottingham, Nottingham, UKe-mail: philip.percival@nottingham.ac.uk

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of ourselves and everything to which we are spatiotemporally or otherwise suitablyexternally related, is a temporal tree; for advocates of this doctrine must ask whetherthe branches of our universe are possible worlds or mere histories. Secondly, inter-est in it has a source in Concrete Plenitude—roughly, the doctrine that the domainof unrestricted first order quantification includes concrete objects of every possiblevariety, and in particular all possible worlds; for advocates of this doctrine mustask whether any of the possible worlds overlap so as to have an initial segment incommon.

I think it fair to say that the discussion of the question to which these sources havegiven rise has been confusing, if not confused. Amongst those who address the ques-tion from the viewpoint of the first source, MacFarlane professes to be unsure of thedistinction between a temporal tree the branches of which are possible worlds anda possible world the branches of which are mere histories;2 and amongst those whoaddress the question from the viewpoint of the second source, Lewis defines “possibleworld” in such a way as to render it analytically true that there are no temporal treesof possible worlds before treating as substantive the question as to whether there aresuch trees.3

My first concern in this paper is to dispel this confusion by clarifying the ques-tion as to whether any object is a temporal tree of possible worlds. In so doing Iargue that the notion “possible world” is ambiguous, that the question is best dis-ambiguated to the question as to whether an object is a temporal tree of what I call“modal loci,” and that Our Tree is unable to provide independent support for a pos-itive answer: pace MacFarlane, some object is a temporal tree of modal loci only ifConcrete Plenitude is true (Sect. 1).4 I then address the question from the respectiveviewpoints of two species of Concrete Plenitude. Firstly, I argue that pace Yablo,combinatorialist Concrete Plenitude should deny that there are temporal trees of pos-sible worlds, but that pace Lewis the reason for this has nothing to do with the statusof Our Tree (Sects. 2, 3).5 Secondly, I argue that nomic essentialist Concrete Plen-itude should deny that there are temporal trees of modal loci for a quite differentreason; admitting such trees would require the unsavoury possibility of fundamen-tal properties whose essences are so weak that the laws by which these propertiesare governed fail to determine even the most basic structure of the modal loci atwhich they are instantiated (Sect. 4). I end the paper by re-applying points madeearlier so as to conclude not only that advocates of Concrete Plenitude should denythat there are temporal trees of modal loci but that on any view philosophical argu-ments to the conclusion that our universe is a temporal tree of historical possibili-ties should be rejected: Our Tree cannot be true unless physics requires it to be so(Sect. 5).

2 MacFarlane (2008, footnote 1). MacFarlane uses Lewis’s (1986, pp. 206–209) terminology, which Iexplain in footnote 16 below.3 See Lewis (1986, pp. 71–78) and Lewis (1986, pp. 206–209) respectively. Lewis does not address thequestion as to whether there are temporal trees of possible worlds directly. I discuss his argument in Sect. 2.4 MacFarlane (2008).5 Yablo (1999) and Lewis (1986, pp. 206–209).

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1 The question and its sources clarified

1.1 The question clarified

1.1.1 Temporal tree

A tree is the fusion of certain objects, “branches,” any two of which properly decom-pose into a maximal common initial segment s and end segments e and e′ such thatthe first branch is the fusion of s and e and the second branch is the fusion of s ande′, s being a proper part of both branches and e and e′ having no common part that is asegment of both branches. An object is a temporal tree iff it is a tree all of the branchesof which are spatiotemporally extended and spatiotemporally unified objects any twoof which are such that their maximal common initial segment s bears the temporalrelation earlier than, or some suitable analogue of this relation, to their respectiveend segments e and e′. A temporal tree is a tree of possible worlds iff it is a tree thebranches of which are possible worlds. In contrast, a temporal tree is a tree of merehistories iff it is a universe the branches of which are not possible worlds.

For my purposes it is convenient to dissociate the notions “temporal tree of possibleworlds” and “temporal tree of mere histories” from the presumption that maximal com-mon initial segments cut the branches to which they are common across hyperplanesof absolute simultaneity. If this presumption were to hold, then, trivially, a temporaltree of relativistic possible worlds would be ruled out (and so too would be a temporaltree of mere histories that are relativistic). It does not hold generally, however; it onlyholds good in the non-relativistic case. A temporal tree of relativistic possible worlds(respectively, mere histories) must be conceived differently. One alternative is for atemporal tree of this kind to involve maximal common initial segments that cut thebranches to which they are common across hyperplanes of non-absolute simultaneity(i.e. so that the notion temporally “initial” is itself relativised).6 But since relativisticbranching is naturally conceived in terms of the light cone of a point event anotheralternative is for such a tree to involve less than this; either a maximal common initialsegment is the past light cone of a point event, as suggested by Lewis, or else the endsegments that distinguish two different branches are alternative future light cones ofa point event, as suggested by Belnap.7

6 Cf. Belnap (2003, pp. 34–35).7 For the latter case see Belnap and Müller (2010, Sect. 3) and Belnap (1992, 2003, Sect. 2); for the formercase see Lewis (1986, p. 206). Lewis (2004, footnote 13) seems to change his mind, since he appearsto say that in the absence of “worldwide branching,” by which he presumably means branching over asimultaneity hyperplane, it is inappropriate to speak of “branching.” I am grateful to an anonymous refereefor 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 anadequate representation of branching in the relativistic case. In fact, for two reasons it is a bit misleadingin even the non-relativistic case. Firstly, three dimensions are needed to represent the branching of evena one-dimensional space (one dimension representing the one-dimensional space; a second dimensionrepresenting time; and the third dimension representing the “logical space” into which the two dimensionalbranches spread out). Secondly, an object that occupies three dimensions so as to represent branching acrossa one-dimensional hyperplane of absolute simultaneity doesn’t look much like a tree: for example, sincethe initial maximal segment being represented is only two-dimensional, ideally the “trunk” of the object

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1.1.2 Possible world

The term “possible world” is structurally ambiguous but our focus will be on thepredicative parsing of it as “world and possible.”8 This structural disambiguationdoes not resolve the ambiguity of the term entirely, however, since the predicates“possible” and “world” are themselves ambiguous. The ambiguity of “possible” isrelatively unimportant and is readily resolved: I employ it to mean “actual or possiblyactual.” In contrast, the ambiguity of the term “world” is of fundamental importance.In its ordinary sense, “world” expresses the concept “universe”; in the technical senseit expresses a theoretical role concept I call “modal locus.”

The use of “world” to mean “universe” is very familiar.9 It is the technical sensein which “world” means “modal locus” that is most in need of clarification. In thissense “world” expresses the concept of an object that plays a certain theoretical role.Roughly, it is the concept of an object of a kind such that the operators “possibly”and “necessarily” are, in a loose sense, “defined” with respect to objects of that kind.Slightly less roughly, it is the concept of an object of a kind such that objects of thatkind form a set W such that, illuminatingly, modal facts are extensionally equivalentto non-modal facts about the objects in W . A bit more precisely, it is the concept ofan object of a kind such that the set W of objects of that kind is such that for someequivalence schema along the lines of “possibly ϕ iff for some w in W, at w ϕ” everyinstance of the schema is true and the resulting truths are maximally illuminating withrespect to modality.

The theoretical role concept “modal locus” that “world” in the technical senseexpresses is broad and vague, and deliberately so. It is these features that ensurethat the literature pertaining to the metaphysics and semantics of modality can beproperly represented as containing a (deep) dispute about what possible worlds are.Once “(possible) world” is disambiguated to “modal locus,” even those who hold thatthe possible worlds are sets of sentences may be properly represented as disagreeingwith those who hold that possible worlds are concrete objects akin to our universe.10

Footnote 7 continuedwould have to be as thin as possible—a sheet of paper would be getting there but e.g. a pencil would behighly misleading.8 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.”9 Outside of philosophy we often speak of “the/our world” and “the/our universe” interchangeably. Nodoubt it is because “world” has the ordinary sense “universe” that Lewis feels free to begin On the Pluralityof 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.” Hemakes these claims without giving any explanation of what “world” means and without suggesting that heis employing the term in any technical sense. It isn’t plausible to suppose that in so doing he is addressingonly those few philosophers who are familiar with the technical sense in which “(possible) world” has cometo be used.10 This view is not universally shared. For example, Bricker (2001, p. 28) writes “given the range ofdisagreement over what sort of thing “possible world” refers to, those who assert that possible worlds existcannot be taken to share a single view.” But the point that such people can, and should, be taken to share asingle 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

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Unsurprisingly, different views regarding what there is encourage very differentnarrowings and precisifications of the concept “modal locus” that “(possible) world”in its technical sense expresses. Advocates of Concrete Plenitude have sought to con-strue “at w” as a domain restricting modifier akin to the modifier “in Australia”; butopponents of Concrete Plenitude hold that there are modal loci only in so far as “atw” is construed differently. Moreover, even when the interpretation of “at w” is set-tled there is still considerable scope for fine-tuning. Opponents of Concrete Plenitudewho agree on what “at w” means have many disagreements about what the objectsin the set W are; and while I have used a classical quantifier to state the equivalenceschema, some have argued that advocates of Concrete Plenitude should use a pluralquantifier.11 Such narrowings and precisifications are all well and good but the breadthand vagueness of the ur-concept “modal locus” that captures the technical sense of“(possible) world” should not be lost sight of.

1.1.3 Temporal tree of worlds

Once “possible world” is parsed predicatively and the assumption is made that everyworld is possibly actual the possible worlds are exactly the worlds.12 In that case thequestion as to whether there are temporal trees of possible worlds collapses into thequestion as to whether there are temporal trees of worlds. In view of the ambiguity of“world” this question is ambiguous between two questions:

(1) Is there a temporal tree the branches of which are universes?(2) Is there a temporal tree the branches of which are modal loci?

1.1.3.1 Question (1) Question (1) is easily answered; it is analytically true that there areno temporal trees the branches of which are universes. This is evident upon clarificationof the concept “universe.” The concept “universe” is the concept of an object that isunified and maximal with respect to the relation that x bears to y iff either x bearsto y or y bears to x at least one suitable external relation.13 The notions “suitable”and “unified” and “maximally unified” are a matter of some dispute but for illustrativepurposes it suffices to take “suitable” as primitive and to focus on the view that “unified”

Footnote 10 continuedthose 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. Notetoo that van Inwagen does not recognize the ambiguity of “possible world” that I have identified. For theersatzist view that possible worlds are sets of sentences, see Roy (1995).11 Bricker (2001) argues against Lewis (1986) that a plural quantifier should be used in the equivalenceschema. In effect, Yablo (1999) argues against Lewis (1986) that the set W should be taken to include notjust universes but (at least some of) their proper parts too. The arguments of Bricker and Yablo are discussedin Sect. 3.2 below.12 Bricker (2001, footnote 9) and Clark (2010, footnote 1) say explicitly that they use “world” and “possibleworld” interchangeably.13 I follow Lewis (1986, p. 62) in taking a relation to be “external” iff it does not supervene on duplicatesof its relata but does supervene on duplicates of the respective fusions of its relata. I am grateful to theanonymous referee who pointed out that some hold that the notion “external relation” is prior to that of thenotion “duplicate” and is therefore not defined by this equivalence.

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signifies “globally unified.”14 An object is globally unified with respect to a relationR iff every part of it bears R to every other part of it; and it is maximally globallyunified with respect to R iff it is globally unified and everything to which some partof it bears R is a part of it.

Let E be the relation x bears to y iff x bears to y or y bears to x at least one suitableexternal relation; and let an object be a universe iff it is maximally globally unifiedwith respect to E . Suppose for reductio that O is a temporal tree of universes thathas branches b and b′ having a maximal common initial segment s and respective endsegments e and e′. Since b and b′ are universes they are maximally globally unifiedwith respect to E . Hence every part of b bears E to every other part of s and every partof s bears E to every other part of e′. So b is not maximally globally unified; someparts of it—namely, the parts of s—bear E to objects—namely the parts of e′—thatare not parts of it. So b is not a universe and O is not a temporal tree of universes. Asimilar proof can be constructed if “unified” is taken to signify the more liberal concept“locally unified.”15 There are no other alternatives. So either way it is analytically truethat there are no temporal trees the branches of which are universes.

1.1.3.2 Question (2) Question (2) disambiguates “world” as “modal locus.”16 A pos-itive answer to it amounts to the doctrine that there is a temporal tree of modal loci.If all modal loci are universes then this doctrine is false: we have just seen that it isanalytically true that there are no temporal trees of universes. However, although theclassical development of Concrete Plenitude holds that the modal loci are exactly theuniverses—in particular, in the final analysis Lewis defines “world” as I have defined“universe” and takes the worlds to play the role I have defined the modal loci as

14 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 isat most one universe. There are two ways of substantiating “suitable” that are worthy of note. One is totake spatiotemporal relatedness as a paradigm; an external relation can then be deemed “suitable” iff it isat least analogous to spatiotemporal relatedness. Another is to invoke the notion “natural” so as to holdthat a relation is suitable iff it is natural (i.e. the presumption being that the relation of non-identity is notnatural). Either way it is uncontroversial that spatiotemporal relations are suitable. For further discussionsee 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.15 For the more liberal notion of “local” unifiedness, and for the proof that locally unified “worlds” cannotoverlap (as two “worlds” that are branches of a temporal tree do), see respectively Sect. 2 and footnote10 of Bricker (1996). Lewis (1986, p. 72) is inclined to require that “worlds” be globally unified buthe also considers a liberalisation of this requirement; the liberalisation he considers falls short of localunifiedness, however. Bricker (1993, 1996, 2001) argues that no more than local unifiedness is requiredbecause spacetime is only locally unified if General Relativity is true.16 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 whichare modal loci exemplifies what he calls “branching of worlds,” while a modal locus that is a temporal treeof 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 “followingLewis, [they] obliterate the crucial distinction between histories and worlds.” Whatever the merits of thiscomplaint against Saunders and Wallace it is wrongly directed at Lewis: Lewis does not obliterate thisdistinction; on the contrary, in distinguishing “branching within worlds” from “branching of worlds” heenjoins us to attend to it.

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playing17—question (2) is not so easily answered. The concept “universe” expressedby “world” in the ordinary sense is completely different from the concept “modallocus” expressed by “world” in the technical sense. So even if every logically con-sistent instance of the form “there is a universe the intrinsic properties of which areexactly the properties ϕ” is true there is no a priori reason, so to speak, to expect thetwo concepts to be materially equivalent.

It is no surprise, therefore, that opposition has begun to emerge to the view thatConcrete Plenitude should hold that the modal loci are exactly the universes. In par-ticular, it has been argued that whereas the universes are indeed modal loci the modalloci are not confined to the universes.18 Accordingly, from this latter viewpoint thenegative answer we have given to question (1) leaves question (2) open: the fact that noobject is a temporal tree the branches of which are universes does not settle the matteras to whether any object is a temporal tree the branches of which are modal loci.

1.2 The sources clarified

It is question (2)—the question as to whether any object is a temporal tree of modalloci—that is the focus of the remainder of this paper. Interest in it has two sources,namely the doctrines I have respectively called “Our Tree” and “Concrete Pleni-tude.” In sketching these sources I shall argue that Our Tree provides no independent

17 As I read him, Lewis (1986) is guided to a hypothesis as to which objects in the concrete plenitudeare “worlds” in the technical sense of “modal loci” by his awareness of which objects in the concreteplenitude are “worlds” in the ordinary sense of “universes”; and since the resulting hypothesis that themodal loci are exactly the universes is seen by him to pass many tests he sticks to it. He suspends thishypothesis 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.18 Unfortunately, this opposition is hampered by the fact that “world” means “universe” in the ordinarysense and no other term for modal locus has become common usage. In the absence of an alternative termfor the technical sense, to deny that the universes are the modal loci on the grounds that some modal lociare not universes is to hold that some worlds are not worlds (in the ordinary sense). Philosophers who aremade nervous by this sort of thing wouldn’t have batted a eyelid had terms such as “universe” and “modallocus” 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 madenervous by it. In effect, he argues that Lewis’s hypothesis that the “worlds” (i.e. the modal loci) are exactlythe 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 thisrevision on the terminological ground that it would conflict with established “philosophical usage” (at leastin 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 “modallocus” and “universe” been in play all along, however, Bricker would have felt no pressure whatsoever onthe modification of Lewis’s theory he rejects: in effect, his view is that Lewis holds that the modal lociare exactly the universes but a better theory holds that the modal loci are the universes and the fusions ofuniverses. (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 practiceof withholding the term “universe” from the maximally unified objects that he deems “metaphysical” isunjustified and misleading; since Bricker (1996, p. 231) does not conceive the notion “particle” in sucha 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.)

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motivation for a positive answer to the question: on the contrary, it is clear that theanswer must be negative unless Concrete Plenitude is true.

1.2.1 Our tree

Our Tree is the doctrine that our universe is a temporal tree. In recent literature, OurTree has been grounded in attempts to reconcile a plurality of objective future-oriented“historical” possibilities with a B-theory of time, which takes tensed facts to be unreal,and in attempts to develop a no-collapse interpretation of quantum mechanics.19 Sincethe branches of a temporal tree are spatiotemporally extended and unified, Our Treemust not be confused with mere advocacy of the doctrine that the models of a satisfac-tory semantic theory of tenses and historical modalities invoke branching structures,however; for such advocacy may take an ersatzist form.20

Our Tree provides an independent rationale for the view that some object is a tem-poral tree of modal loci only if there is independent reason to hold that the branchesit postulates are modal loci. One might think that such a reason is forthcoming asfollows: at each node of the tree the end segments above the node are historicallypossible futures and the branches through the node are historical possibilities; buthistorical possibilities are (metaphysical) possibilities; so the branches are modal loci.Such reasoning is spurious, however. The penultimate claim may be read two ways. Ifit is construed as the claim that what is historically possible is metaphysically possibleit is true but it is too weak to support the conclusion. But if it is construed as the claimthat the set of node/branch pairs over which the operator “it is historically possiblethat” is defined is included in the set of objects over which (metaphysical) modalityis defined it begs the question.21

In fact it is clear that there is no reason independent of Concrete Plenitude to holdthat if Our Tree is true the branches of our universe are modal loci. On the contrary,advocates of Our Tree who reject Concrete Plenitude should hold that the branches ofour universe are not modal loci. This may be seen as follows.

19 Regarding the first ground, see especially Belnap and Green (1994), Belnap et al. (2001), Belnap andMüller (2010), and MacFarlane (2003, 2008). Regarding the second ground, for exposition see Albert andLoewer (1988) and Lewis (2004); for more recent developments see Saunders et al. (2010).20 Those who maintain that the semantics of tenses or historical modalities demands branching structuresmay reject Our Tree by maintaining the ersatzist view that in even the intended model the branches of thestructure are abstract objects. Although ersatz branching models are popular amongst those who advocatethe “A-theoretic” doctrine that tensed facts are fundamental, and, in particular amongst those who hold ano future variant of this doctrine i.e. according to which nothing concrete is later than the present, suchersatzism does not require A-theory. Just as the modal operators have been taken to engender modal loci thatare not concrete, so too have the operators that express historical possibilities been taken to engender loci forhistorical 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 whilerejecting Our Tree is committed to what Belnap and Green (1994) call the “thin red line”; it holds thatexactly one of the many ersatz futures veridically represents the (concrete) future (and so comprises thethin red line through the ersatz branching structure).21 Just as the modal loci are the objects with respect to which the modal operators are “defined” so too areloci of other kinds objects with respect to which operators of other kinds are “defined.”

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Consider what must be the case if there are modal loci but Concrete Plenitude isfalse. There are only modal loci in so far as there is a set W for which the followingholds: for some equivalence schema along the lines of “possibly ϕ iff for some w inW, at w ϕ” every instance of the schema is true. So, since the universe could have beendifferent intrinsically in many ways, if Concrete Plenitude is false there are modalloci only if for every such set W at least one member of W is not concrete. In thatcase to suppose that the branches of the temporal tree our universe is taken to be aremodal loci would be to advocate a “hybrid” theory whereby some of the modal lociare concrete but others are not.

A “hybrid” theory of this kind is untenable. Admittedly, advocates of the classicalersatzist view, according to which the modal loci are not concrete and Our Tree is false,commonly refer to our universe as “the actual world”; and one might think that thenatural extension of this practice upon adoption of Our Tree would be to refer to thebranches of our universe as “the actual worlds.” Classical ersatzist practice does notpresuppose a hybrid theory, however. It is invariably accompanied by the qualificationthat the term “the actual world” is ambiguous and may also be taken to refer to oneof the ersatz abstract objects the modal loci are taken uniformly to be. In my terms,classical ersatzist practice exploits the ambiguity of “world”; in one sense this termmeans “universe” (on which disambiguation this practice takes the term “the actualworld” to pick out the universe of which we are a part), while in the other sense itmeans “modal locus” (on which disambiguation this practice takes the term to pickout the abstract object that represents our universe correctly). Given ersatzism, suchpractice is well advised. It cannot be that some modal loci are concrete, others not.That the set W of modal loci is not heterogenous in this way is guaranteed by the factthat the definition of “modal locus” involves a single equivalence schema. We saw thatif the modal loci are concrete, the term “at w” in the schema can be read as a restrictingmodifier; otherwise it must be read differently. Since all instances of a schema havethe same reading it cannot be that some modal loci are concrete but others are not.

We may conclude that if Concrete Plenitude is false no object is a temporal tree ofmodal loci. In that case the following propositions are true. Firstly, the variant of theno-collapse interpretation of quantum mechanics that takes our universe to be a tem-poral tree of possible worlds is untenable.22 Secondly, the B-theoretic conception ofan open future should take the form Belnap consistently advocates (i.e. whereby thebranches of Our Tree are mere (alternative) histories that serve as loci with respect towhich only historical modality is “defined”) and not the form MacFarlane has recentlyexplored (i.e. whereby the branches of Our Tree are “possible worlds” that serve asloci with respect to which both historical modality and metaphysical modality are“defined”).23

1.2.2 Concrete Plenitude

I said at the outset that Concrete Plenitude is, roughly, the following doctrine:

22 Wilson (2011, p. 371) shows qualified sympathy for this variant. See footnote 33 below for how hissympathy is qualified.23 See Belnap (1992) and Belnap and Müller (2010), and MacFarlane (2008).

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(CP1) The domain of unrestricted first order quantification includes concrete objectsof every possible variety, and in particular all possible worlds.

As such Concrete Plenitude inherits the unclarity of “concrete.” This unclarity isresolved here by means of paradigms: speaking unrestrictedly, all lumps of butter,mountains of a certain height and mass, cities with such and such populations etc.are paradigmatically concrete, whereas sets of such objects are paradigmatically notconcrete;24so a mountain having a certain height and mass is paradigmatically concreteirrespective of whether it is non-actual, or (merely) metaphysical, or non-existent.25

The vagueness of “concrete” is left unresolved.26

More importantly, Concrete Plenitude thus characterised inherits the ambiguity of“possible world.” It is most naturally understood as employing “possible world” inthe sense of “universe.” Concrete Plenitude is then merely a thesis about what thereis that makes no claims about modal loci: it does not say that there are modal loci;nor, even, does it say that if there are modal loci then modal loci are concrete. WhenConcrete Plenitude is understood in this way, however, it does give strong support tothe view that there are concrete modal loci. In particular, (CP1) gives strong supportto the following:

(CP2) There are modal loci and the modal loci are concrete.(CP3) Claim (CP2) is to be understood with respect to a precisification of the definition

of “modal locus” whereby “at w” is a modifier that restricts the domains ofquantifiers within its scope.

An advocate of Concrete Plenitude in the Lewisian tradition can be read as inferring(CP1) from a conviction that since modality is reducible the conjunction of (CP2) with(CP3) is true. But an advocate of Concrete Plenitude who occupies a quite differentMeinongian tradition infers the conjunction of (CP2) with (CP3) on the basis of aconviction that since what there is includes objects that are merely intentional, (CP1)

24 This claim about sets is to be understood in such a way as to be compatible with a reduction of sets toparadigmatically concrete objects.25 Lewis (1986) holds that there are mountains that are non-actual; Bricker (2001) holds that there aremountains that are (merely) metaphysical; Meinongian advocates of Concrete Plenitude hold that there aremountains that are non-existent. Some hold that “is a mountain” is in effect ambiguous between “exemplifiesthe property of being a mountain” and “encodes the property of being a mountain.” I do not recognize thisambiguity; those who do should respect my express intention that my use of predicates be disambiguated interms of exemplification, not encoding. Accordingly, advocates such as Zalta (2006, Sect. 2) of the doctrinethat every satisfiable predicate is such that some object encodes the property it expresses do not therebyadvocate 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.26 (CP1) remains problematic even when its vagueness is put aside, and this is why it is only a rough firstapproximation to Concrete Plenitude. (CP1) is inconsistent: the possible variety of objects includes an objectsuch 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 andthat there are no goldfish. Accordingly, Concrete Plenitude must be understood as the thesis that resultswhen the term “possible” is restricted minimally to “suitably possible” so as to restore consistency. Theproblem of defining “suitably” is akin to the so-called “characterization problem” faced by Meinongianswho would like to hold, if so doing weren’t inconsistent, that for every property F some object is completelycharacterized by the fact that it is F .

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is true.27 In view of the affinity between (CP1) and the further theses (CP2) and (CP3)it will be convenient in what follows to employ “Concrete Plenitude” for the doctrinethat is the conjunction of (CP1)–(CP3).

Concrete Plenitude is consistent with a wide variety of views as to which concreteobjects there are. From its viewpoint, such variety arises from differences of opinionregarding which (kinds of) object are possible. I shall not try to settle such differences.Nor, even, shall I try to settle whether Concrete Plenitude is true. Rather, my concernis whether some object is a temporal tree of modal loci given the truth of ConcretePlenitude. In pursuing this concern I shall focus on the two varieties of ConcretePlenitude—namely, the “combinatorialist” and “nomic essentialist” varieties—thatarise when (CP1) is refined in the two ways that are most prominent in the literature.To avoid the need for repeated use of such phrases as “given Concrete Plenitude” Ishall write as if Concrete Plenitude is an assumption shared with the reader.

2 Combinatorialism (1): existential divergence

Let a “fundamental mosaic proposition” be a proposition of the form that the universeis the fusion of such and such simple objects standing in such and such suitable externalrelations over which such and such fundamental properties are distributed in such andsuch a way. “Combinatorialism” is the view that subject only to the demands of logic(broadly construed so as to include analyticity) and the adicities of the fundamentalproperties and external relations, every fundamental mosaic proposition is possiblytrue.28

To consider the relation between combinatorialism and the doctrine that some objectis a temporal tree of modal loci let us say that two objects w and w′ “split off” (fromone another) iff some initial segment s of w is either identical to, or a duplicate of,an initial segment s′ of w′, and no other segment (initial or otherwise) of w that isnot a proper part of s is either identical to, or a duplicate of, any segment (initial orotherwise) of w′; and let us say of two objects w and w′ that split off (a) that theysplit off by branching iff the respective initial segments s of w and s′ of w′ in virtue ofwhich they split are identical, and (b) that they split off by diverging iff these segments

27 Lewis (1986) is the pre-eminent advocate of Concrete Plenitude. Of course Lewis (1990) holds thatConcrete Plenitude is an ontological thesis, and that to suppose otherwise is unintelligible. But as vanInwagen (1986, pp. 188–189) observes, “Meinongian” advocates of Concrete Plenitude, who he suspectsare 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; onemight say that he advocates Concrete Plenitude with respect to the “metaphysical” but rejects it with respectto the “physical.”28 To say that combinatorialism is true is not to say that it is necessarily true; one might coherently maintainthat although the actual fundamental properties are subject to combinatorial reshuffling there could have beenfundamental properties that are not subject to combinatorial reshuffling. To my knowledge no one has everexploited this distinction and I see no grounds to support so doing. I shall assume that if combinatorialismis true it is necessarily true.

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are duplicates.29 Articulated in these terms, the doctrine that there is a temporal treeof modal loci is the following thesis:

Existential BranchingAt least two modal loci split off by branching.

Existential Branching is only true if the following is true:

Existential SplittingAt least two modal loci split off.

Given combinatorialism, Existential Splitting is true.30 How might ExistentialBranching be obtained from Existential Splitting? One way would be for the followingthesis to be true:

Universal BranchingAny two modal loci that split off from one another do so by branching.

Clearly, Existential Branching is entailed by the conjunction of Existential Splittingwith Universal Branching.

At least from the viewpoint of the species of combinatorialism that holds that themodal facts are reducible to non-modal facts (about modal loci), Universal Branchingwould appear to be an elegant economy. It is false iff Existential Divergence is true:

29 Cf. Lewis (1986, p. 206).30 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 viewthat there are concrete modal loci. The reason is that it is compatible with a doctrine with which ExistentialSplitting is incompatible—namely, the doctrine that all fundamental properties are “locus bound” in thesense that no fundamental property is instantiated at two different modal loci. Combinatorialism is equivalentto an infinite conjunction of propositions that are each of the form “it is possible that ...” . As such it is neutralas to how exactly modalities are to be “defined” in terms of modal loci. In particular, in principle it couldbe reconciled with the doctrine that all fundamental properties are locus-bound by invoking a counterpartrelation between fundamental properties. In contrast, Existential Splitting directly contradicts this doctrine.Suppose Existential Splitting is true and that w and w′ are modal loci that split. Then the fundamentalproperties Pi and P ′

i that are respectively instantiated in their respective maximal initial segments s ands′ must be such that for each i the identity Pi = P ′

i holds. This is trivial in the case where splitting is byidentity. But it is true too in the case where splitting is by duplication. For were the fundamental propertiesP ′

i that s′ instantiates not identical to the fundamental properties Pi that s instantiates, s′ would not be aduplicate of s. However, while a combinatorialist who holds that there are concrete modal loci that includeuniverses other than our own might invoke a counterpart theoretic treatment of fundamental properties inthis 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 thenature 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. ofBarak 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 intrinsicproperties 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 themodality of fundamental properties. The thought that there are concrete objects of every possible variety isthe thought that if it is possible for there to be an object with fundamental properties Pi then there is an objectwith fundamental properties Pi . So Concrete Plenitude as I have defined it is more or less committed todenying that fundamental properties are universe-bound and so gives the thought that they are locus-boundno purchase.

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Existential DivergenceAt least two modal loci that split off from one another do so by diverging.

Prima facie, wherever a rival theory that rejects Universal Branching postulatestwo duplicate initial segments of modal loci so as to vindicate Existential Divergence,the theory that embraces Universal Branching postulates just the one common ini-tial segment. So Universal Branching would appear to be economical, and thereforetheoretically virtuous.31

The attractions of Universal Branching notwithstanding, an argument to the con-clusion that combinatorialism should reject it has been given as follows:

(P1) Combinatorialist Universal Branching entails Our Tree (i.e. the thesis that ouruniverse is a temporal tree).

(P2) Our Tree is refuted by our customary commonsensical thought and talk con-cerning “the future.”

(C) Combinatorialist Universal Branching is false.32

This argument has attracted little sympathy in the literature.33 Perhaps because OurTree is more popular than Universal Branching, critics have focused on premise (P2).34

I shall not contest their objections, however. The argument fails even if (P2) is true.

31 I limit this consideration to species of Concrete Plenitude that, deem the modal facts reducible to thenon-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 byFine (2005: chs. 4 and 6)—tend not to think of Concrete Plenitude as a theory; consequently, one would notexpect them to be impressed by considerations of economy. Put the point this way: if the concrete objectsthat are not suitably externally related to us are unreal who cares how many of them there are?32 In effect, Lewis (1986, pp. 206–210) gives this argument. He takes one moral of (P1) to be that advocatesof Our Tree would find (combinatorialist) Universal Branching attractive, i.e. because it entails somethingthey believe. Wilson (2011, p. 382) interprets Lewis differently, however. He takes Lewis to argue morenarrowly that the supposition that our universe is a temporal tree of “worlds” (i.e. modal loci) makesnonsense of our thought and talk about the future; he then suggests that Lewis’s argument can be appliedequally 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”).33 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 footnotes22 and 32 above). In my terminology, Wilson is sympathetic to the idea that the myriad “worlds” splittingoff from one another that are postulated by some no-collapse interpretations of quantum mechanics aremodal loci. Consequently he is sympathetic to the idea that our universe is a temporal tree of modal loci ifthese “worlds” split off by branching. But he takes Lewis’s argument to show that these “worlds” are betterthought of as always splitting off by diverging. For reasons similar to those advanced in Sect. 1 above, Ithink 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 informsme that within the philosophical community he or she has encountered widespread sympathy with Lewis’sargument.)34 See Belnap et al. (2001, pp. 170–176, 205–209), Belnap and Müller (2010), MacFarlane (2008), andSaunders and Wallace (2008). They make two objections. Firstly, they object that it is wrong to think thatOur Tree would make nonsense of our commonsensical thought and talk about the future; they claim thata 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 bringsbenefits that would outweigh the attractions of common sense were the two to conflict. Wilson’s (2011)sympathy for the argument is the exception.

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The reason the argument fails even if (P2) is true is that (P1) is false. (P1) claims thatUniversal Branching entails Our Tree. But this entailment cannot be. The reason is notthat Universal Branching is a claim about all modal loci that split, whereas Our Treeis a claim about our universe: although the claim that the modal loci are exactly theuniverses is controversial what is questioned is whether all modal loci are universes;it is generally acknowledged that the universes are modal loci. Rather, the reason isthat Universal Branching can only entail a claim to the effect that our universe has acertain property by entailing that all universes have this property. But combinatorialistUniversal Branching does not entail that all universes are temporal trees.

To be sure, given combinatorialism one can derive from Universal Branching a claimthat does concern temporal trees as follows. Let an object that is spatiotemporallyextended and spatiotemporally unified be “straight” iff it does not branch (with orwithout re-convergence) in either the (temporal) forwards or backwards direction.Combinatorialism yields the following:

Universal Splitting of Straight Modal LociEvery straight modal locus is such that some straight modal locus splits from it.

In combination with Universal Branching, Universal Splitting of Straight ModalLoci entails the following:

Universal Tree-Boundedness of Straight Modal LociEvery straight modal locus is a branch of a temporal tree of straight modal loci.

Universal Tree-Boundedness of Straight Modal Loci falls well short of the claim thatall universes are temporal trees, however. Its truth is compatible with the propositionthat some universes have spatiotemporal structures such that they branch exclusivelyin the direction of the past or re-converge no sooner than they branch in the directionof the future or loop back on themselves in the manner of circular time.35

Nevertheless, although (P1) is false, the argument can be repaired so as to showthat combinatorialism must reject Universal Branching. The weakness of (P1) is notthat combinatorialist Universal Branching has no consequences for all universes, andhence for our universe; it is, rather, that it does not have the consequence for uni-verses that is implicit in (P1). To repair the argument we need to reformulate (P1) sothat it correctly identifies an untenable consequence that combinatorialist UniversalBranching has for our universe, and then to reformulate (P2) so as to explain why thisconsequence is untenable. The best way to implement this strategy yields the followingargument:

(P1′) Combinatorialist Universal Branching entails that our universe is not straight.

35 The falsity of (P1) is so obvious that it is puzzling that anyone should have thought it true. Since Lewisbelieves that our universe is straight, perhaps it is because combinatorialist Universal Branching entailsUniversal Tree-Boundedness of Straight Modal Loci that he came to believe (P1) is true. If one starts outbelieving 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 modalloci and, hence, pace the belief from which one started out, that our universe is a temporal tree of modalloci. But of course although one could end up changing one’s beliefs in this way logic does not require thatone do so. Logic only requires that one relinquish the belief that our universe is a straight modal locus.

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(P2′) No tenable account of modality has any empirical consequences, e.g. suchas the consequence that our universe is not straight.(C) Combinatorialist Universal Branching is false.

Given that all universes are modal loci, the truth of (P1′) is implicit in the fact thatUniversal Branching entails Universal Tree-Boundedness of Straight Modal Loci:since it is analytically true that no universe is a branch of a temporal tree, if everystraight modal locus is a branch of a temporal tree, then no universe is a straight modallocus; but then given that all universes are modal loci it follows that no universe isstraight and hence that our universe is not straight.36

It remains only to defend (P2′). In effect, this amounts to two claims: the first isthat a tenable theory of modality does not have empirical consequences; the second isthat the thesis that our universe is not straight is empirical. Since the first claim willbe readily accepted—a tenable theory of modality cannot be hostage to science—it isthe second claim that requires defence.

The reason the second claim is true is simply this: spacetime structure is the provinceof physics, and “spacetime structure” here refers to the structure of our universe in itsentirety, not to the mere structure of branches should our universe be a temporal tree(even if the branches have a common structure).37 This rationale sides with one of themorals of Earman’s critique of Belnap’s advocacy of Our Tree.38 In developing OurTree Belnap is content for physical theory—for the laws of physics—to be accountableto the branches of the tree; he does not require that these laws be accountable to thetree itself. In contrast, I read Earman’s preoccupation with the question of whetherthe theory of relativity, and in particular the general theory, has a model in whichspacetime has a tree structure, to amount to the protest that if our universe is not straightthen physics must model it as such. This rationale also has the authority of scientific

36 The thesis that our universe is not straight is very weak. In particular, it is consistent with the view thatour universe is very nearly straight, as indeed is the view that our universe is a temporal tree. For this reasonboth 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 thehistories of which are straight until branching occurs just once immediately before as many Big Crunchesas 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 interruptedby branching nanoseconds beforehand. This consideration alone does not negate the dialectical power ofLewis’s appeal to common sense, however; in practice advocates of Our Tree maintain that our universe isconstantly branching.37 It might be thought that this viewpoint is undermined by Kripkean insights to the effect that somecontingent propositions are a priori. Given these insights, why shouldn’t the contingent claim that ouruniverse is not straight be a priori too? And if this claim about our universe is such that there is no obstacleto its being a priori, how can it be a defect for a theory of modality to entail it? This line of thought ispowerless to rescue combinatorialist Universal Branching, however. Kripke’s examples of the contingent apriori are quite disanalogous to the proposition that our universe is not straight. What gives these examplespurchase is that it appears not only that the truth values of the propositions in question are accessible in theabsence of any empirical investigation, but that one has no idea what an empirical investigation into theirtruth values would be: one couldn’t empirically investigate e.g. whether Julius is the inventor of the zip orwhether one is here now. In contrast, we have a reasonably clear idea of what empirical investigation intothe spatiotemporal structure of our universe amounts to.38 See Earman (2008).

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practice: when the ‘first-wave’ Everettians postulated many worlds as branches of atemporal tree in an attempt to avoid wave function collapse the immediate complaint byphysicists that no dynamics of branching had been offered presupposes that physicaltheory is answerable to our universe in its entirety.39

3 Combinatorialism (2): universal divergence

To reject Universal Branching is to embrace Existential Divergence. But ExistentialDivergence is compatible with Existential Branching. Existential Branching is equiv-alent to the falsity of Universal Divergence:

Universal DivergenceAny two modal loci that split off do so by diverging.

Having rejected Universal Branching, Lewis embraces Universal Divergence, andso rejects Existential Branching. Viewed as an inference, this is a non sequitur; it is ofthe form “It is false that all A’s are B’s, so all A’s are not B’s.”40 So the combinatorialistwho does no more than reject Universal Branching has as yet done nothing to decidethe status of Existential Branching. Can he or she do better in this regard?

3.1 Competing theoretical virtues?

The combinatorialist might try to choose between Universal Divergence and the con-junction of Existential Divergence with Existential Branching on the basis of theirrespective theoretical virtues.

39 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 pluralityof “worlds” pertains to the fundamental level or to an emergent macro level. The “first wave” many worldsinterpretation 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 atthe fundamental level: the division into many worlds is confined to the macro-level, and opinion is dividedas to whether the worlds at this level branch or diverge. The rationale for the second wave is that first wavedoctrine appears self-defeating: it aspires to interpret the deterministic core dynamical equations of quantummechanics as they stand, without appealing to any “collapse” postulates; but its attempt to do so invokes afundamental process—the branching of spacetime—about which, at best, quantum mechanics has nothingto say (and, at worst, with which quantum mechanics is inconsistent). For this and other difficulties withthe first wave many worlds view see Albert and Loewer (1988, pp. 198–203). For discussion of whetherthe second wave many worlds interpretation should hold that the “worlds” branch or diverge see Wilson(2011).40 In “reject(ing) genuine branching in favour of divergence,” Lewis (1986, p. 206) defends UniversalDivergence; for by “divergence” he means Universal Divergence. He says (p. 209) “[b]ranching, and thelimited overlap it requires, are to be rejected as making nonsense of the way we take ourselves to be related toour 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: irrespectiveof whether Universal Branching entails Our Tree it is clear that Existential Branching does not entail OurTree. Lewis (1986, p. 209) makes evident that it is Universal Branching, not Existential Branching, withwhich he is concerned when he writes that the problem facing those non-actual beings who are parts ofnon-actual worlds that are instances of branching within worlds is “not ours, as it would be if the worldsgenerally branched rather than diverging” (my italics).

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Let W be a set of concrete objects with respect to which an equivalence schema ofthe form “possibly ϕ iff for some w in W, at w ϕ” is such that every instance of theschema is true. Since the combinatorialist holds that it is possible that the universe isstraight and governed by indeterministic laws, the combinatorialist must hold that ifW is the set of modal loci it contains straight modal loci that split. Let T be the theorythat the modal loci are exactly the members of W and that Universal Divergence holdsfor such loci. There are several ways to construct rivals to T that reject UniversalDivergence.

The first way is the way of addition. The rival theory T + concedes to T that UniversalDivergence holds with respect to the objects in W but claims that T is wrong aboutthe modal loci; it claims that the modal loci are the objects in a set W + that containsthe members of W and in addition some duplicates of straight modal loci in W thatunlike the straight modal loci in W split by branching. According to T + UniversalDivergence only holds when restricted to the subset W of the set W + of modal loci;Universal Branching holds with respect to the modal loci that belong to W + but notto W , and since some such loci split, Existential Branching holds for them too.

The species of combinatorialism that obtains Universal Branching by the way ofaddition is untenable. By hypothesis, W is adequate with respect to the relevant equiv-alence schema that defines the role modal loci play. So nothing in even the reductionof modality by modal loci (a fortiori, nothing in Concrete Plenitude) requires that themodal loci be the members of the extension W + of W . Hence T + is an ontologicalextravagance; it postulates objects in addition to those T says that there are for no goodreason. Accordingly, faced with the choice between adopting Universal Divergenceand grounding the conjunction of Existential Divergence with Existential Branchingin the way of addition, the combinatorialist should adopt Universal Divergence; itstheoretical virtues are superior.

The second way to construct a rival to T is the way of subtraction. Here the rivaltheory T − agrees that the modal loci are exactly the members of W but it rejects T ’sclaim that Universal Divergence holds with respect to W ; instead T − claims that bothExistential Divergence and Existential Branching hold with respect to W . In contrast tothe way of addition the way of subtraction does appear to posit Existential Branchingto some purpose. It eliminates some of the commitments of Universal Divergence;in some but not all cases in which Universal Divergence claims that there are twoduplicate objects—namely, duplicate initial segments of certain modal loci that splitfrom one another—it claims that there is only one object—namely, an object that istheir common initial segment.

How much economy the way of subtraction affords depends on how frequently thetheory T − takes splitting by branching to occur. The combinatorialist cannot allow italways to occur; that would result in Universal Branching. But the truth of ExistentialDivergence requires no more than that two modal loci split by diverging. Accordingly,if T − holds that all but two of the straight modal loci in W split by branching, itseconomy is maximised. The resulting competition between Universal Divergence andthe conjunction of Existential Divergence with Existential Branching now seems moreequal. Perhaps the gain T − makes in economy outweighs T ’s greater simplicity, anddoes so despite the fact that the choice as to which modal loci split by diverging appearsarbitrary.

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The case for T −, and hence for rejecting Universal Divergence in favour of ground-ing the conjunction of Existential Divergence with Existential Branching in the wayof subtraction, is irredeemably flawed, however. Consider the case in which T − holdsthat exactly two modal loci in W split by diverging and let these loci be w1 and w2.Then T − has the consequence that if our universe is straight it is identical to w1 or w2.But this consequence is empirical and we agreed that a theory of modality must nothave empirical content. By similar reasoning it can be seen that the way of subtrac-tion gives empirical content to Existential Branching even in the case at the oppositeextreme, i.e. in which T − holds that exactly two modal loci in W split by branching.Let these loci be w3 and w4. The species of T − that holds this much is the minimaltheory that accommodates Existential Branching by the way of subtraction. But eventhis species of T − still has empirical content; it entails that if our universe is straightour universe is not identical to w3 or w4.

The way of addition is an ontological extravagance. Improperly, the way of sub-traction has empirical content. So the combinatorialist advocate of Concrete Plenitudemight feel on the verge of a decision in favour of Universal Divergence. For what thirdway could there be? How can splitting by branching be obtained from the set W otherthan by means of the ways of addition or subtraction with respect to the straight modalloci that W contains?

This question is a case of failing to see the trees for the wood. To pursue a thirdway by which to construct a rival to T one should ignore the straight modal loci inW . Instead one should focus on the members of W that are temporal trees. From thecombinatorialist viewpoint W must contain such objects if it is the set of modal loci; thecombinatorialist holds that it is possible that the universe is a temporal tree of variouskinds and, hence, that many universes, and so many modal loci, are temporal trees.Moreover, the combinatorialist also holds that some of the modal loci are temporaltrees the branches of which are duplicates (but not of course indiscernible duplicates)of the straight modal loci in W . Accordingly, to obtain a rival to T it suffices to “dissect”the temporal trees in W by taking their branches to be modal loci. Whereas the originaltheory T sees in W modal loci the branches of which are not modal loci, the theory T ∗that is obtained by the way of dissection sees in W modal loci the branches of whichare modal loci.

Unlike the theory T −, T ∗ concedes that W contains straight modal loci that all splitby diverging; and like T + it holds that there are modal loci that are not members of W .Unlike T +, however, T ∗ holds that the modal loci that are not members of W are objectsT already recognises: for these modal loci are simply parts of members of W . UnlikeT +, therefore, T ∗ does not disagree with T over which objects there are. Whereas T +adds objects to the objects T says there are and T − deletes objects from the objects Tsays there are (i.e. by introducing identities where T sees non-identities), T ∗ agreeswith T over which objects there are but simply disagrees with T over which of theseobjects are modal loci; it identifies some of the objects T says there are—the branchesof modal loci that are temporal trees—and it claims that these objects are modal loci too.

The way of dissection shows that the conjunction of Existential Divergence withExistential Branching can avoid the charges of extravagance and untenable empiricalcontent respectively brought against the ways of addition and subtraction. Equally,however, in so doing it shows that the appearance that splitting by branching promises

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cost-free theoretical economy is unfulfilled; the supposition that certain straight modalloci split by branching, not by diverging, brings economy only at T −’s untenable costof empirical content.

It is perhaps surprising that the way of dissection is the only means that is notobviously flawed by which the combinatorialist can reconcile Existential Branchingwith Existential Divergence. The way of dissection is committed to the “inclusionist”doctrine that some modal loci have proper parts that are modal loci; it takes modalloci and asserts that some of their proper parts—their branches—are modal loci. Initself Existential Branching carries no such commitment, however; it only requiresmodal loci that are not universes because they overlap so as to have a common initialsegment. Consequently, the suspicion might remain that the combinatorialist mustbe able to reconcile Existential Branching with Existential Divergence without goinginclusionist and without incurring the charge of ontological extravagance or untenableempirical content.

Any such suspicion is unfounded, however. That Existential Branching forces thecombinatorialist to be inclusionist may be seen as follows. In accordance with thesupposition that Existential Branching is true let O be a temporal tree of modal locithe branches of which are exactly the modal loci b1 and b2. Suppose for reductio thatO is not a modal locus. Then the combinatorialist must hold that there is a modallocus O ′ such that its branches b1′ and b2′ are respective duplicates of b1 and b2; forcombinatorialism holds that it is possible for the universe to have exactly the intrinsicproperties O has and so requires a modal locus O ′ with respect to which this possibil-ity is “defined.” In that case the object O is a duplicate of an object O ′ that is a modallocus. But this can only be if there is a relevant difference between O and O ′; it cannotbe a primitive fact that modality is “defined” with respect to some object but not withrespect to a duplicate of this object. Since no relevant difference is forthcoming itmust be concluded that the reductio is successful and that O must be a modal locusafter all. And since O was defined to be a temporal tree having modal loci as properparts (namely, its branches), it follows that some modal locus has a proper part thatis a modal locus. So, as the way of dissection recognises, Existential Branching doesindeed oblige the combinatorialist to be inclusionist.

If the combinatorialist had some independent reason for advocating ExistentialBranching he or she might go on the defensive when confronted with the result thatExistential Branching brings commitment to inclusionism. No such reason has asyet been offered, however. Moreover, in the absence of any such reason the way ofdissection is ad hoc: what is so special about the branches of modal loci that are tem-poral trees such that just these parts of modal loci should be reckoned modal loci?The only hope for a combinatorialist advocate of Existential Branching is to offer anindependent reason for a species of inclusionism that is sufficiently broad to have theconsequence that amongst the proper parts of modal loci that are themselves modalloci are branches of at least some of the modal loci that are temporal trees.

3.2 Inclusionism

According to Lewis’s classical account the modal loci are exactly the universes. Thereare two kinds of consideration in favour of inclusionism. Considerations of the first

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kind are seen to be powerful but irrelevant. Considerations of the second kind are seento be relevant but powerless.

The first inclusionist modification of the classical account retains the view that theuniverses are modal loci but adds to it the view that the universes are proper parts ofmodal loci as follows:

Upward inclusionismThe modal loci are the universes and the fusions of universes.

Upward inclusionism is motivated by the fact that the classical account appearsunable to accommodate certain apparent possibilities: firstly, the classical accountappears incompatible with the possibility that there is more than one universe; sec-ondly, the classical account appears incompatible with the possibility of several claimsthat are integral to Concrete Plenitude and so violates the rule of possibility introduc-tion, ϕ|– �ϕ. Upward inclusionism provides a convenient solution to these problems;it yields the possibility that there is more than one universe and it restores the rule ofpossibility introduction.41

Such arguments are powerless to motivate Existential Branching, however.Although upward inclusionism has the consequence that some modal loci have properparts that are modal loci it cannot provide a reason for thinking that some modal loci

41 If, as in Lewis (1986), the modal loci are exactly the universes and “at w” works by restricting thedomains 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 possiblefor there to be island universes is not central to our customary modal views and that for this reason theoryis free to reject it. His strategy is heavily undercooked, however because it is incompatible with possibilityintroduction, ϕ |– �ϕ; it is a central tenet of the species of Concrete Plenitude he advocates that there ismore than one universe.Upward inclusionism solves these problems as follows. Firstly, the problem of the possibility of multipleuniverses is solved because at a modal locus that is the fusion of at least two universes, there is more thanone universe. Secondly, the problem more generally that the rule of possibility introduction is violated issolved as a result of taking the fusion of all universes to be a modal locus: the sentences that breach the ruleof possibility introduction are sentences such that (i) the advocate of Concrete Plenitude (and in particularLewis) asserts the propositions these sentences express when articulating his or her species of ConcretePlenitude and (ii) there is no universe at which any of these sentences are true. But these sentences are alltrue at the modal locus that is the fusion of all universes, i.e. because restricting the domains of quantifiersto 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 thepossibility of multiple universes and Divers (1999) argues that he or she should reject it so as to accommodatethe rule of possibility introduction, neither Bricker nor Divers advocates upward inclusionism. Brickerproposes to accommodate the possibility that there is more than one universe by relaxing the equivalenceschema by which the role of modal loci is defined; he suggests that the right hand side of this equivalenceshould employ a plural quantifier i.e. so as to be of the form “for some worlds, at those worlds ϕ.” Incontrast, 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” inhis terms) being advanced. He then proposes that the advanced propositions be treated not in the standardway but in accordance with the rule that “necessarily ϕ” is true iff ϕ is true. Bricker’s (p. 45) rationale forpreferring his own proposal to upward inclusionism is a desire to preserve what he claims to be the commonpresumption that even in its philosophical sense the notion of a “world” is the notion of an object that isunified. (In my terminology this presumption is simply the thesis that the modal loci are all unified.)

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are temporal trees the branches of which are modal loci. For none of the modal locithat upward inclusionism adds to the universes are temporal trees.

The second inclusionist modification of the classical account proceeds in the oppo-site direction and takes the modal loci to be the universes and their proper parts:

Downward inclusionismThe modal loci are exactly the concrete parts of universes (a universe being partof itself).

Since, given combinatorialism, some universes are temporal trees, downward inclu-sionism yields Existential Branching immediately.42

The conjunction of downward inclusionism with upward inclusionism takes usmore than half way to an especially simple account:

Univeralist inclusionismEvery concrete object is a modal locus.

Universalist inclusionism entails both upward and downward inclusionism. Some con-siderations that have been advanced in its favour are in reality merely grounds forupward inclusionism.43 But one argument that has been taken to support universalistinclusionism really would support downward inclusionism were it successful.

To approach this argument, let us consider how downward inclusionism might bemotivated directly. Since the modal loci are the objects with respect to which the modaloperators are “defined” one would expect an argument for downward inclusionism tobe similar in form to the argument we considered for upward inclusionism; one wouldexpect it to invoke possibilities that the classical account is unable to accommodate. Itis clear that there can be no de dicto possibility of this kind, however. Such a possibilitywould require there to be an x that is a proper part of a universe w and a truth “possiblyS” such that the following conditions are satisfied: (i) at x , S; and (ii) for no universe w′is it that case that at w′, S. Given combinatorialism, however, these conditions cannotbe satisfied simultaneously. Since “at x” works by restricting the domain of quantifi-cation to parts of the referent of “x” the truth of “at x , S” must turn on the intrinsicproperties of x . But according to combinatorialism every proper part of a universe issuch that some universe w′ is a duplicate of it. So if condition (i) is met condition (ii) isnot met.44

42 This is the species of inclusionism that Clark (2010) focuses upon.43 In effect Sider (2003, p. 196) suggests that universalist inclusionism is supported by the fact that it wouldaccommodate the possibility of multiple universes. But we have just seen that upward inclusionism wouldperform this task adequately.44 Matters are rather different from the viewpoint of one who accepts that the truth conditions for modalstatements are given by extensional claims about concrete modal loci but is nevertheless agnostic aboutwhether there are concrete objects that are spatiotemporally removed from us. This viewpoint requiresagnosticism about many possibility claims that are commonly believed and one who adheres to it mightseek to limit how much agnosticism about such claims is required. Downward inclusionism would help inthis regard: if the modal loci are exactly the universes then agnosticism would be required e.g. with respectto “possibly, there are no swans”; but if e.g. Antarctica is a modal locus then the possibility of there beingno 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.

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What then of de re possibility claims? One could be forgiven for thinking that it hasbeen argued that there are de re possibility claims that only universalist inclusionismcan accommodate and that once this much is recognized the notion of an intrinsicproperty can be defined in mereological terms. For it has been suggested that

(3*) G is intrinsic iff[necessarily for all x and y] necessarily: if a mass of atoms that composes xwould, had a more inclusive world obtained, have composed y, then x is G iffy would have been G;

and that an advocate of Concrete Plenitude should express this definition as

(3) G is intrinsic ifffor all x ≤ w ≤ w′, x has G in w iff every copy x ′ of x has G in w′.

In (3) the variables “w” and “w′” are “world-variables,” “≤” signifies the mereologicalrelation of part to whole, and “x ′ is a copy of x” is short for “x ′ is constituted in w′by the x-portion of w.”45

One might think that two different arguments for universal inclusionism are implicitin the definitions (3*) and (3). Both turn on the fact that in (3) the “world variables”must range over “worlds” in the technical sense of modal loci, not in the ordinarysense of universes; for whereas (3) plainly fails unless some “world” is a proper partof another it is analytic that no universe is a proper part of another. Moreover, when“world” is read as “modal locus” it can be reasonably supposed that (3) requires uni-versalist inclusionism; (3) gets stronger, and its prospects for success greater, as moreproper parts of “worlds” are themselves deemed “worlds”; so on this reading at thevery least universal inclusionism provides (3) with the best chance of success.46

The first argument for universalist inclusionism one might think to be implicit in(3*) and (3) is this: from the viewpoint of (combinatorialist) Concrete Plenitude uni-versal inclusionism is a sufficient condition of (3)’s successfully defining “intrinsic”;so this viewpoint should endorse universalist inclusionism so as to reap the rewardsof a successful definition. This argument is untenable, however. The reason is that(3) is a successful definition of “intrinsic” only if an alternative definition that rejectsuniversal inclusionism is likewise successful. This alternative is (4):

(4) G is intrinsic ifffor all x ≤ z ≤ w′, in z, x has G iff in w′, every copy x ′ of x has G.

(4) differs from (3) in two respects. Firstly, in (3) both w and w′ range over modal locibut in (4) only w′ ranges over modal loci; z ranges over parts of modal loci. Secondly,

45 See Yablo (1999, pp. 47, 57). I have followed Yablo’s numbering but in accordance with more customaryusage I have used “≤” instead of his “<” for the “part of” relation. (I am grateful to the referee who pointedout the desirability of so doing.) I have also appended “necessarily for all x and y” to the beginning of theright hand side of (3), since so doing appears to be required. Yablo argues that (3*) and (3) capture just onenotion 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.46 Yablo (1999, p. 37) does state that he presupposes that “some worlds contain others as proper parts” buthe does not state explicitly which objects he reckons (the advocate of Concrete Plenitude should hold) tobe “worlds.” In effect, Parsons (2007) interprets him as advancing universalist inclusionism, however.

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whereas (3) is allied with universal inclusionism (4) is allied with the classical doc-trine that the modal loci are exactly the universes. Clearly, given our assumption thatConcrete Plenitude is true if (4) doesn’t work then (3) won’t work either. The matterof whether to call the proper parts of universes “modal loci” or not is irrelevant in thepresent context; given the definition of “modal locus” this matter must turn entirelyon nuances pertaining to the behaviour and “definition” of modal operators that areextraneous to whether “intrinsic” is definable in mereological terms. It is easy to dispelany lingering suspicion that taking the proper parts of universes to be modal loci so asto switch from (4) to (3) strengthens the definition by so to speak giving it tacit modalforce. From the viewpoint of Concrete Plenitude the values of w include all possibleuniverses, i.e. so that the values of z in (4) include all possible parts of all possibleuniverses and the desired modal force is there already.47

The second argument for upward inclusionism that one might think to be implicit in(3*) and (3) turns not on what is required for the success of (3) but on what is requiredfor the expression of (3*). The argument is to the effect that in the absence of univer-sal inclusionism (combinatorialist) Concrete Plenitude is unable even to express thecondition that (3*) takes to be necessary and sufficient for a property to be intrinsic.

Conceptually, this argument is an improvement on the first; at least it recognizesthat in view of the way in which the notion “modal locus” is defined any challengeto the classical account, whereby the modal loci are taken to be the universes, mustcite modal phenomena this account cannot accommodate. Nevertheless, it too is veryweak. It is simply untrue that in the absence of universalist inclusionism (combina-torialist) Concrete Plenitude is unable to express what is expressed by “necessarilyfor all x and y, necessarily: if a mass of atoms that composes x would, had a moreinclusive world obtained, have composed y, then x is G iff y would have been G.”To see this it suffices to focus on the constituent “if a mass of atoms that composesx would, had a more inclusive world obtained, have composed y, then x is G iff ywould have been G.” The term “world” must be given its ordinary sense, i.e. so thatthe counterfactual’s antecedent amounts to “had the universe been more inclusive.”But then what is envisaged in the antecedent is just a counterfactual scenario in whichthe universe has, so to speak, an extra outer layer. From the viewpoint of ConcretePlenitude this scenario no more demands modal loci that have modal loci as properparts than does a counterfactual scenario in which Obama is never President.

4 Nomic essentialism against existential branching

My argument to the conclusion that combinatorialism should reject Existential Branch-ing was in two steps; at the first step Universal Branching was rejected in favour of

47 Parsons (2007) misses this point. Perhaps the point is clearer if one remembers that Concrete Plenitudetakes the modifier “in w” to be the quantifier domain restricting modifier “at w” i.e. so that its effect isinsensitive to whether the objects over which w ranges are (properly) called “modal loci.” For example, ifw′ is the actual universe but w is Australia and Hilary Clinton is in the US but Bill Clinton is in Australia,then “at w′, there is someone to whom Bill Clinton is married” is true but “at w, there is someone to whomBill Clinton is married” is false, and this is so irrespective of whether Australia is reckoned to be a modallocus.

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Existential Divergence; at the second step the conjunction of Existential Divergencewith Existential Branching was rejected in favour of Universal Divergence. The argu-ment for the first step relied heavily on the fact that combinatorialism entails that allstraight modal loci participate in splitting. A popular alternative to combinatorialismdoes not entail that all straight modal loci are subject to splitting, however. This alter-native is de re necessitarianism about the fundamental laws, i.e. the doctrine that thefundamental properties are such that the laws that govern them do so necessarily.48

De re necessitarianism about the fundamental laws is so far from entailing that allstraight modal loci participate in splitting that one might sensibly wonder whetherit yields so much as the doctrine that some modal loci split. Consequently, from itsviewpoint Universal Branching looks more attractive. Nevertheless it should still resisteven Existential Branching. Why this is so involves quite different considerations tothose adduced against combinatorialist Existential Branching, however.

4.1 Existential splitting

We called the doctrine that some modal loci split “Existential Splitting”:

Existential SplittingAt least two modal loci split off from one another.

From the combinatorialist viewpoint the only threat to Existential Splitting is the doc-trine that fundamental properties are locus-bound. Trans-locus identity for fundamen-tal properties is not sufficient for Existential Splitting, however, even in combinationwith the contingency of some truths concerning such properties. To obtain ExistentialSplitting the right truths concerning fundamental properties must be contingent.

To see how de re necessitarianism about the fundamental laws might lead one to theview that even Existential Splitting fails, notice that if de re necessitarianism about thefundamental laws is necessarily true, the restriction of Existential Splitting to modalloci the fundamental properties of which are governed by deterministic laws is false.49

Hence, Existential Splitting is incompatible with the following conjunction of views:de re necessitarianism about the fundamental laws is necessarily true and determinism

48 De re necessitarianism about the fundamental laws is advocated by dispositionalist essentialists aboutfundamental properties such as Bird (2005), while Shoemaker (1998) argues that the causal laws arenecessary. To say that it is true is not to say that it is necessarily true; one might hold that the fundamentalproperties are such that necessarily, they are governed by such and such laws, and yet hold that there couldhave been fundamental properties governed by laws such that they could have been governed by differentlaws. But since I know of no one who has exploited this distinction and see no grounds for so doing I shallassume that if de re necessitarianism about the fundamental laws is true it is necessarily true.49 Consider a deterministic world w and an initial segment s of w such that the fundamental propertiesdistributed throughout s are the properties Pi and the laws L at w concerning the Pi are deterministic. Letworld w′ be a world that splits from w after s. By the definition of splitting, w′ has an initial segment s′that is either identical to or a duplicate of s′. In either case, at w′ the laws concerning the properties Picannot be the laws L: since the laws L are deterministic they would combine with s′ to require that the endsegment of w′ duplicates the end segment of w. So at w′ the laws concerning the Pi must be different lawsL ′. 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 worldfrom which some world splits.

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is necessarily true (i.e. in that it is necessary that the laws are deterministic). Call thisconjunction of views the doctrine that de re necessitarian determinism is necessarilytrue (i.e. “de re necessitarian determinism” being the conjunction of determinism andde re necessitarianism about the fundamental laws).

The incompatibility of Existential Splitting with the view that de re necessitariandeterminism is necessarily true might be thought irrelevant. Admittedly, de re neces-sitarianism about the fundamental laws is quite the rage and those who embrace it donot take it to be contingent. But isn’t it obvious that de re necessitarian determinismis not necessarily true? After all, isn’t it obviously conceivable, and so possible, thatdeterminism is false? Indeed, doesn’t quantum mechanics make manifest to those wholack imagination, or who doubt the inference from conceivability to possibility, thateven determinism is false?

The correct answer to these questions is “No”: there is nothing obvious aboutsuch matters. Quantum mechanics has a “no-collapse” interpretation on which it is adeterministic theory; so it is presumptuous to maintain that the empirical successesof quantum mechanics make the possibility of indeterministic laws manifest. More-over, as the “first wave” many worlds development of this interpretation illustrates,it appears possible for there to be deterministic laws that require that the universehave the structure of a temporal tree; and a proponent of the view that determinism isnecessarily true might try to invoke this consideration so as to try to explain away asillusory a conviction that it is possible for the laws not to be deterministic—for evenlaws that determine that the universe is a temporal tree may be “indeterministic” inwhat Belnap calls the “Aristotelian” sense.50

To clarify this point let us disambiguate the notions of “determinism” and “inde-terminism” as presently used in the literature. Let “A-indeterminism” (respectively,“A-determinism”) be the doctrine that the universe is (respectively, is not) a tempo-ral tree such that the branches above any point are alternative historically possiblefutures, and let “L-determinism” (respectively, “L-indeterminism”) be the doctrinethat the laws are such that together with any initial segment* of the universe theydetermine (respectively, fail to determine) the whole universe.51 A-determinism andL-determinism are logically independent. So there are two varieties of de re neces-sitarian determinism i.e. depending on which variety of determinism is intended. Inthese terms, the idea behind the aforementioned attempt to explain away a convictionthat the falsity of determinism is possible is to observe the consistency of the claimthat necessarily, de re necessitarian L-determinism is true with the possibility of lawsthat are A-indeterministic.

50 For the many worlds no-collapse interpretation, and the first wave development of it, see footnote 39above and the references therein.51 I follow Placek and Belnap (2012, introduction), who distinguish “Laplacean” determinism, which isa matter of the strength of the laws (with indeterministic laws being too weak to determine what willhappen next, given what has already happened), from “Aristotelian” determinism, which is a matter oftemporal structure (with indeterministic structures being tree-like and the branches above a point beingalternative 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 temporaltree iff the cut is made after branching has occurred.

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One can imagine a de re necessitarian about the fundamental laws being drawntowards the doctrine that both L-determinism and A-indeterminism are necessarilytrue. He or she need only embrace two further theses that have been found inde-pendently attractive: (i) the fundamental properties instantiated in our universe—i.e.the “actual” fundamental properties—are governed by L-deterministic laws that areA-indeterministic; and (ii) it is not possible for fundamental properties other than theactual fundamental properties—i.e. “alien” fundamental properties—to be instanti-ated.

Be this as it may, I cannot believe the doctrine that necessarily, the fundamental prop-erties are such that necessarily, the laws that govern them are either L-deterministic orA-indeterministic, and I know of no one who has defended it. The attempt to explainaway the conviction that it is possible for the laws to be indeterministic as the con-viction, simply, that it is possible for the laws to be A-indeterministic is unsuccessful;for once the relevant distinctions are drawn the appearance that it is possible for thefundamental laws to be L-indeterministic and A-deterministic is just as strong. Whenpush comes to shove, then, the de re necessitarian about the fundamental laws willdraw back from the conjunction of (i) and (ii). Since it is an empirical matter whether(i) is true, suspicion will fall on (ii). De re necessitarians about the fundamental lawsdo tend to be wary of (ii) in any case, since rejecting it enables them to accommodatethe popular opinion that the fundamental laws could have been different from whatthey are actually.

Once (ii) is rejected, the possibility of the fundamental laws being L-indeterministicand “straight,” i.e. in the sense that they require that the universe be straight (and sothat A-determinism be true) may be embraced with a clear conscience irrespective ofwhat the actual fundamental laws should turn out to be. For there is no reason to limitthe possibility of non-actual fundamental properties to fundamental properties that areakin to the actual fundamental properties in respect of the kind of laws by which theyare governed.52

In the light of the possibility that the fundamental laws should have beenL-indeterministic and straight we may safely conclude that Existential Splittingis true even if de re necessitarianism about the fundamental laws is necessarilytrue.

4.2 Universal divergence

Existential Branching sits very uncomfortably with de re necessitarianism about thefundamental laws, and especially so with nomic essentialism, i.e. the species of de renecessitarianism about the fundamental laws according to which fundamental proper-ties have essences and these essences explain both the necessity of the laws to whichthe fundamental properties are subject and the fact that the fundamental properties

52 I know no one in the literature who explicitly denies the possibility of straight L-indeterministic laws. Inparticular, I know no advocate of either A-indeterminism or de re necessitarianism about the fundamentallaws who denies it.

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do not exhibit certain patterns of instantiation.53 Nomic essentialism is not ordinarilyallied with Concrete Plenitude. The two doctrines might be combined, however.54

Conceivably, nomic essentialism about the fundamental properties is true but onlycontingently so: it might be both that the actual fundamental properties are suchthat the de re metaphysical necessity of the fundamental laws that govern them isexplained by their essences and that it is metaphysically possible for there to have beenfundamental properties that lack nomic essences. In practice, however, the distinctionbetween the truth and the necessary truth of nomic essentialism is not exploited. This isunsurprising. Were one to think that there could have been fundamental properties thatlack nomic essences, what reason could one have for thinking that in this respect theactual fundamental properties are different? I shall assume that if nomic essentialismabout the fundamental properties is true it is necessarily true.

The difficulty Existential Branching poses for nomic essentialism may be seen asfollows. Just as laws are “straight” iff they require that the universe has a straightspatiotemporal structure let laws be “anti-straight” iff they require that the universe isnot straight. Assume that nomic essentialism is true and that Existential Branching istrue. Let w and w′ be branches of a temporal tree O of modal loci and let s be theirmaximal common initial segment. Consider the fundamental properties Pi instantiatedin s. Since the laws that govern these properties at w are the same as the laws thatgovern them at w′, these laws are not L-deterministic: w′ splits from w and vice-versa.55 Moreover, by the same token, nor are these laws anti-straight: w and w′ areboth straight. So at each of O’s branches the fundamental laws are L-indeterministicbut they are not anti-straight. One might be forgiven for inferring that since the laws arenot anti-straight they are straight. Given nomic essentialism, however, this inferenceis blocked. This can be seen by dilemma.

First horn: Suppose that the tree O is a modal locus. It follows that the laws thatgovern the fundamental properties Pi at w are not straight: these laws hold at O (by dere necessitarianism about the fundamental laws; a fortiori, by nomic essentialism) andtherefore cannot require that the universe be straight, and in particular not a temporaltree (since O is a modal locus that is a temporal tree); so they are not straight.

Second horn: Suppose that O is not a modal locus. If some duplicate O ′ of O isa modal locus then by parallel reasoning, at O ′ the laws governing the fundamental

53 Nomic essentialism is advocated by Fine (2005, chap. 7) and Bird (2005).54 Nomic essentialism is even compatible with the species of Concrete Plenitude that reduces the modalfacts to non-modal facts about the modal loci. From a viewpoint that combines nomic essentialism withreductionism about modality the explanation of the (supposed) de re metaphysical necessity of the funda-mental 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 withwhich the laws that govern fundamental properties are inconsistent; such objects require a pattern of funda-mental property instantiation that the essences preclude. Classically, however, nomic essentialism is alliedwith the rejection of reductionism about modality. Its explanation of the (supposed) de re metaphysicalnecessity of the fundamental laws proceeds differently. Firstly, it is explained that that which is metaphys-ically necessary is that which is grounded in essences. Secondly, it is explained that the fundamental lawsare grounded in the essences of fundamental properties.55 Strictly speaking, it follows that the laws at the branches are L-indeterministic only if the discussion isrestricted 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.

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properties Pi are not straight: these laws must hold at O ′ and O ′ is a temporal tree.Suppose then that no duplicate of O is a modal locus. There is severe tension betweenthis view and de re necessitarianism about the fundamental laws; for the questionarises as to why metaphysical possibility is defined over objects such as w and w′ butnot over objects such as O (or any duplicates of O). Perhaps the de re necessitarianabout the fundamental laws might try to respond by simply insisting that this much isjust brute fact. Be this as it may, from the viewpoint of nomic essentialism the tensionis unbearable. Nomic essentialism requires that if the laws that govern the fundamentalproperties Pi are straight, then the essences of the Pi ’s explain the fact that no modallocus at which the Pi are instantiated is a temporal tree. Such essences cannot do this,however, if some modal locus at which the Pi are instantiated is a part of a temporaltree. For in that case the essences cannot preclude a temporal tree structure: propertiesthat are instantiated in the branches of an object that is a temporal tree cannot haveessences that preclude their instantiation in a temporal tree.

On both horns of the dilemma just considered it follows that if Existential Branch-ing and nomic essentialism are both true there could have been fundamental propertiesgoverned by L-indeterministic laws that are neither straight nor anti-straight. Such lawsare a significant departure from even neo-classical laws. Classical laws are straight—they determine a straight structure—and they are L-deterministic (respectivelyL-indeterministic) iff they combine with what has happened to determine (respec-tively, not to determine) what happens next. In contrast, neo-classical laws include notonly classical laws but laws that are “tree” laws—i.e. anti-straight laws that deter-mine a (certain) branching structure—and can be classified as L-deterministic orL-indeterministic in terms analogous to those just given (once the notion “initial seg-ment” is extended to a notion “initial segment*” that covers “horizontal” cuts across allthe branches of a temporal tree). Given de re necessitarianism about the fundamentallaws, however, Existential Branching would require the possibility of L-indeterministiclaws that are not even neo-classical i.e. since they are neither straight nor anti-straight.Such laws fail to determine even the even the most basic spatiotemporal structure ofwhat happens next.

To my knowledge, the idea that it is possible for there to be fundamental proper-ties governed by L-indeterministic laws that are neither straight nor anti-straight hasbeen given no credence in the mainstream philosophical community. That Existen-tial Branching would require the de re necessitarian about the fundamental laws toadvocate the possibility of such properties gives him or her reason to reject Existen-tial Branching. This is especially true in the case of nomic essentialism. Consider amodal locus in states σ at t and σ ′ at some time t ′ shortly after t , and let us condensetalk of the essences of the fundamental properties instantiated by parts of σ to talkof the essence of σ itself. It is hard enough for nomic essentialists to countenancelaws that are L-indeterministic.56 But once the nomic essentialist makes the further

56 In the case of L-deterministic straight laws a dispositionalist nomic essentialist can explain why notwo modal loci split by supposing that, at least in effect, every straight modal locus divides into an initialsegment s and end segment e′ such that the initial segment s divides into a disposition s1 and a stimuluss2, with the end segment e′ being the manifestation (cf. Bird 2005). It is hard to see how this metaphysicalpicture might sensibly be extended to the case of laws that are L-indeterministic. In the L-indeterministiccase, as far as the states σ and σ ′ are concerned the most that can be metaphysically necessary is a real

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concession that the laws governing the fundamental properties instantiated by parts ofσ are L-indeterministic but neither straight nor anti-straight, however, the essences ofσ and on the one hand, and of the fundamental properties distributed within σ on theother hand, are etiolated to the point of no return. An “essence” of σ that cannot evendetermine the fundamental structure of a possible successor state σ ′—that cannot evendetermine whether σ ′ involves branching or not—is unworthy of the name.57

5 Conclusion

In this paper I have focused on the technical sense of “(metaphysically) possible world”as “modal locus” (of metaphysical modality). I have argued that Our Tree providesno support independent of Concrete Plenitude for the view that in this sense there aretemporal trees of possible worlds. I have also argued that although Lewis’s argumentagainst the view fails the view should be rejected by both the combinatorialist and thenomic essentialist species of Concrete Plenitude.

I would like to end by returning briefly to Our Tree. In effect, I have focusedon the species of Our Tree according to which our universe is a temporal tree ofmetaphysically possible worlds (in the sense of modal loci) at the expense of thespecies of Our Tree according to which our universe is a temporal tree the branchesof which are historical possibilities. One might think that MacFarlane’s flirtation withthe former species is an aberration and that my discussion does not engage the latterspecies. This is not true, however. An observation I made at the end of Sect. 2 abovedoes engage it. I observed there that physics directs its concern with spatiotemporalstructure at our universe in its entirety and that as such its theories model our entireuniverse. Consequently, unless fundamental physical theory holds that it is nomicallypossible that our universe is a temporal tree it conflicts with all species of Our Treehowever these species gloss the branches. And contemporary physical theory does nothold that it is nomically possible that our universe is a temporal tree: as “second wave”Everettians recognise, even “no collapse” quantum mechanics eschews branching atthe fundamental level. The moral is that a Belnapian B-theoretic reduction of historicalmodalities via Our Tree is in this respect less credible than a Lewisian reduction of

Footnote 56 continuednumber that is properly between 0 and 1 that measures the primitive “strength” of a primitive tendency of σ

to yield σ ′ (or of some ‘manifestation’ relation between σ1, σ2 and σ ′). So the state σ ′ is not essential to thestate σ . But then what is essential to it? Only the chance σ affords σ ′ 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 σ and σ ′) that likewisehas an essence? I see no way to give satisfactory answers to these questions. Moreover, however they areanswered the explanatory power the nomic essentialist envisages is lost. If the essence of σ consists in nomore than the chance it affords σ ′ its explanatory power is diminished beyond recognition: chances explainother chances (by entailing them); chances do not explain non-chances. (See Percival (2006, Sect. 1).)57 Existential Branching is the doctrine that there is a heterogeneous temporal tree of modal loci. Mutatismutandis, the same argument shows that the nomic essentialist should also reject homogeneous temporaltrees of modal loci, i.e. where a tree is homogeneous iff its branches are duplicates. The problem doesnot depend on the branches of the tree not being duplicates. If there is a homogeneous temporal tree ofmodal loci then from the viewpoint of de re necessitarianism about the fundamental laws, the fundamentalproperties instantiated in the maximal initial segments of the tree’s various branches are governed by lawsthat are L-indeterministic but neither straight nor anti-straight.

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metaphysical modalities via Concrete Plenitude. Empirical science has no concernwith concrete objects to which we are not suitably externally related. So it does notrule out such objects and does not conflict with a Lewisian reduction of metaphysicalmodality.

Acknowledgments I would like to thank my colleagues Jonathan Tallant and Harold Noonan for helpfulcomments on an earlier draft of this paper. I am especially grateful to an anonymous referee of Synthesefor his or her conscientious, insightful, and constructive critical comments on the original submission.

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