PhD: The Metaphysics of Modality

UNIVERZITA KOMENSKÉHO V BRATISLAVE FILOZOFICKÁ FAKULTA The Metaphysics of Modality (Dizertačná práca) Študijný program Systematická filozofia Študijný odbor 2.1.2 Systematická filozofia Školiace pracovisko Filozofický Ústav SAV Školiteľ Bratislava 2016 prof. Mgr. Marián Zouhar, PhD. Mgr. Martin Vacek Abstrakt V súčasnej analytickej metafyzike je všeobecne prijímaný názor, že modálny realizmus nie je možné obohatiť o nemožné svety. Cieľom predkladanej práce je preukázať opak. V prvej kapitole predstavím problematiku modalít, predstavím modálny realizmus a motivujem diskurz nemožných svetov. V druhej kapitole sformulujem argument, podľa ktorého je modálny realizmus epistemicky prijateľná pozícia. Tretia kapitola ponúkne analýzu modálneho realizmu obohateného o nemožné svety. Nasledujúce tri kapitoly formulujú alternatívy k tzv. rozšírenému modálnemu realizmu. V štvrtej kapitole formulujem modálny dimenzionalizmus, piata kapitola sa venuje modálnemu štrukturalizmu a šiesta kapitola modálnemu fikcionalizmu. Abstract It is a widely accepted opinion among metaphysicians that Modal Realism (MR) is unable to accommodate impossible worlds. This thesis argues for the opposite. Chapter I introduces the problem of modality and one metaphysical interpretation I prefer. In particular, I sketch the basic postulates of (MR) and motivate an impossible worlds discourse. Chapter II develops an argument in support of the epistemic adequacy of (MR) as well as its extended version, Extended Modal Realism (EMR). Chapter III presents recent arguments establishing that any version (MR) fails to analyse extraordinary modal claims. I claim that (EMR) as a version of (MR) can provide such an analysis, although I agree that (EMR) is metaphysically unacceptable. The next three chapters therefore propose three different ontological frameworks which are alternatives to the ontology of (EMR). Chapter IV discusses Extended Modal Dimenionalism (EMD), Chapter V proposes Extended Modal Structuralism (EMS) and Chapter VI develops Extended Modal Fictionalism (EMF). Acknowledgement First of all, my thanks go to my supervisor, Marián Zouhar, for his optimistic, liberal and openhearted support from both professional and personal points of view. His attitude taught me to treat philosophical matters with respect and was (and is) the ideal model of how to attempt them in curious and charitable way. Apparently, this thesis has become actual due to an enormous support from the Slovak Academy of Sciences. The administration of the Institute of Philosophy (Tibor Pichler, Karol Kollár and Margita Kráľová) gave me both an excellent environment for my research and the opportunity to discuss it within and outside Slovakia. In the former, I was pleased to organise several conferences on the topic of this thesis entitled Issues on the (Im)Possible. The conference brought together researchers from all around the world. Among them were Brian Ball, Jonathan Livingstone-Banks, Francesco Berto, Alexandre Billon, Emily and Craig Bourne, Johannes Bulhof, Darragh Byrne, Sam Cowling, Ryan Christensen, Anthony Dardis, Michael De, Louis deRosset, John Divers, Vladan Djordjevic, Daniel Dohrn, Nikk Effingham, Benoit Gaultier, Karen Green, Myroslav Hryshko, Mark Jago, Alex Kaisermanm, Amy Karofsky, David Liggins, Ceth Lightfield, Theodore Locke, Toby Lovat, Luke Malik, Dan Marshall, Peter Marton, Cristina Nencha, Daniel Nolan, Vasil Penchev, Dusko Prelevic, Jiří Raclavský, Manuel Rebuschi, Janine Reinert, Maciej Sendlak, Marco Simionato, Lukas Skiba, Stephen Steward, Tuomas Tahko, Naomi Thompson, Alessandro Torza, Plato Tse, Adam Tuboly, Andriy Vasylchenko, Daniel von Wachter, Nathan Wildman, Takashi Yagisawa and Andy Yu. In the latter, I had an opportunity to discuss my research at various universities and research centres abroad. I am sure that without this opportunity to visit different parts of the world my philosophical history would have taken a very different direction. To mention just one for all, I extremely benefited from my two stays at the Australasian National University, one of the best places for doing philosophy. )iv I would also like to thank the department of analytic philosophy of the Institute. Our regular meetings, workshops, reading groups and, importantly, informal discussions in as well as outside the Academy have taught me to take great care in drawing quick conclusions. In particular, I am indebted to Lukáš Bielik, Ján Dubnička, Julian Fink, Dušan Gálik, Silvia Gáliková, Fredrik Haraldsen, Igor Sedlár, Frank Zenker and Zsófia Zvolenszky. However, the biggest thank goes to my family and friends whose love, support and occasional sarcasm turned out to be the most important impetus for my work. February 29, 2016 M.V. )v Contents ; Contents Acknowledgement iv Introduction 1 1. The Possible and the Impossible 3 1.1 Introduction 3 1.2 The Problem of Modality 4 1.3 …and the Ontology 5 1.3.1 (MR) 6 1.4 (EMR) 7 1.4.1 Impossible Worlds: What Are They? 8 1.4.2 Why (Still) Bother with Impossible Worlds? 9 1.5 …and the Ontology Again 11 1.5.1 (EMR): Problems 11 1.6 (EMR) Ultimately Defeated? 15 1.7 A Note on Methodology 17 1.8 Conclusion 18 2. On the Indispensability of (Im)Possibilia 19 2.1 Introduction 19 2.2 (MR) Again 20 2.3 An Epistemological Worry 21 2.4 … And a Reply 21 )v Contents ; 2.4.1 Stage I: Setting Things Up 22 2.4.2 Stage II: The Indispensability Argument 25 2.4.3 Stage III: Premise 3 28 2.4.4 Stage IV: Premise 5 29 2.5 (EMR) and the Problem of ‘How We Know?’ 30 2.6 Conclusion 32 3. (EMR) and Extraordinary Modalizing Problems 35 3.1 Introduction 35 3.2 Ordinary and Extraordinary Modalizing 35 3.3 Variants of (MR) and Their Problems 37 3.3.1 Variant Analyses 37 3.4 (EMR): The Analysis 44 3.4.1 (EMR) Characterized Again 44 3.4.2 (EMR) and Advanced Modalizing Problems 47 3.5 Conclusion 50 4. (Extended) Modal Dimensionalism 52 4.1 Introduction 52 4.2 (EMD) vs. (MR) 53 4.3 Some Problems for (EMD) 58 4.3.1 Problems of Possible Worlds and their Diagnosis 58 4.3.2 The Necessity Horn 61 4.3.3 Amodalism 62 4.3.4 The Contingency Horn 65 4.4 Impossible Worlds 67 4.4.1 Diagnoses 69 4.4.2 Modal Tensing Again 70 4.5 Conclusion 73 )vi Contents ; 5. Extended Modal Structuralism 75 5.1 Introduction 75 5.2 Introducing the Ontology 76 5.3 Incredulous Stares 78 5.4 Metaphysical Structures and Representation 82 5.4.1 (EMS) and Magic 84 5.5 (EMS): Still Inconsistent? 89 5.6 Conclusion 91 6. Extended Modal Fictionalism 93 6.1 Introduction 93 6.2. Modal Fictionalism 94 6.3 Some Problems with (MF) 96 6.3.1 The ‘According to the (MR)-Story’ Operator 96 6.3.2 The Brock-Rosen Objection 97 6.3.3 Hale’s Dillema 98 6.4 Some Alternatives 100 6.5 (EMF) 103 6.5.1 (EMF) and Five Objections 106 6.6 Counting the Costs 111 6.7 Conclusion 112 7. Afterword 113 References 117 )vii Introduction Introduction Possible-worlds semantics proved itself as a strong tool in analysing the statements of actuality, possibility, contingency and necessity. But impossible phenomena go beyond the expressive power of the apparatus. The proponents of possible-worlds apparatus thus owe us at least three stories. The first one is the story about ontological nature of possible worlds, the second one is the story about the theoretical role such entities play and the third one is the story about the impossible. Modal Realism (MR) provides us with a positive story regarding the first and the second, but denies impossible worlds. Extended Modal Realism (EMR) adds a positive story about the third point too. This thesis is an attempt to paraphrase extended modal realism in different metaphysical frameworks. In Chapter I, I outline the theory of modal realism, its systematic account of modality and its metaphysics. I also motivate an impossible-worlds discourse as well as a systematic appeal of extending the picture beyond the possible. I then propose several definitions of the concept of ‘impossible world’. Finally, I discuss a particular metaphysics behind the concept - extended modal realism. Chapter II considers the epistemological worry associated with (MR) and (EMR) and concludes that although the worry is justified, there can be epistemological justification of the theory. Next, I outline the so-called indispensability argument for the legitimacy of mathematical Platonism. Finally, I argue that the argument, if accepted, can be applied to metaphysics in general, to the existence of concrete possibilia (and impossibilia) in particular. Chapter III focuses on the analysis of (EMR). To be more precise, I present a socalled advanced modalising problem which seems to be infecting every genuinely realistic theory of modality. In this chapter, I propose a way of treating extraordinary modal claims by means of plurality of logical spaces. The next three chapters provide different ways of understanding (EMR). Namely, Chapter IV develops and defends Extended Modal Dimensionalism (EMD). (EMD) is realism about spaces, times and worlds—metaphysical indices that make objects spatial, )1 Introduction temporal and modal, respectively. Metaphysical indices play the role of alethic relativizers, i.e. items to which matters of truth are relativized. The chapter examines several arguments against modal dimensionalisn and shows that it offers a feasible way to understand (EMR). Chapter V offers a structural approach to possible and impossible worlds: Extended Modal Structuralism (EMS). In particular, I consider whether it makes sense to think of logical models in isolation from the concrete world but without their being divorced from all spatiotemporal totalities. The metaphysics of structure developed in this chapter assumes that structural properties of possible and impossible worlds are primitive and objective. However, I provide some characterisations of their logical and metaphysical behavior, as well as guidelines for talking about them. Finally, Chapter VI proposes yet another metaphysical framework of hybrid modal realism. I present theories of (MR), (EMR) and modal fictionalism respectively, their advantages and drawbacks. Finally, I propose a so-called hybrid view. Roughly, the view is that one might be a modal realist when it comes to possibilia, but turn into fictionalism regarding impossibilia. The approach is dubbed Extended Modal Fictionalism (EMF). )2 Chapter I The Possible and the Impossible CHAPTER I Start by doing what's necessary; then do what's possible; and suddenly you are doing the impossible. Francis of Assisi 1. The Possible and the Impossible 1.1 Introduction This thesis presents alternatives to traditionally formulated extended modal realism (hereafter EMR)1, a thesis according to which there are real possible worlds and equally real impossible worlds. In order to understand a motivation to do so, it is important to set the defence in a relevant framework. In this chapter I start with formulating a problem of modality (1.2). I then present a particular and up to now the most discussed possible-worlds solution to the problem, modal realism (hereafter MR) 2 (1.3). I also present cases that complicate the issue a bit and motivate a need to extend the solution by impossible worlds (1.4). I focus on the crucial objections against the extension (1.5), sketch criteria for a theory to be a version of (MR) rather than an ersatzist’s 3 version and introduce ways of making (EMR) a feasible position in modal metaphysics (1.6). Finally, I say a bit about my methodology (1.7) and thus pave the way for the rest of the thesis. 1 Unless stated otherwise, (EMR) refers to Yagisawa (1988) as the orthodox version of extended modal realism. 2 Unless stated otherwise, (MR) refers to Lewis (1986a) as the orthodox version of modal realism. 3 Roughly, ersatzism is a view according to which possible worlds are representations of one sort or another. )3 Chapter I The Possible and the Impossible 1.2 The Problem of Modality Actual truths abound. Propositions such as ‘Bratislava is the capital of Slovakia’, ‘I am writing this thesis’, ‘Pluto is not a planet’, and virtually infinitely many propositions are true because the world they describe is as it is. Possible truths abound too, for there is nothing controversial about claiming that ‘Bratislava could be the capital of Australia’, ‘I could be playing football’ or ‘Were the definition of ‘planet’ different, Pluto would be a planet’. It is a simple fact that many situations, although actually false, are possible. Such a plurality of possibilities calls for explanation. It is unintuitive to say that the actual world, the way things are, satisfies the conditions for infinitely many possibilities. The actual world reveals what there is, but it is far from clear that it also reveals what there might be. Philosophers have of course been aware of this limitation and, in seeking a sufficient analysis of modality, have introduced the notion of a possible world. The concept of a ‘world’ plays at least two theoretical roles in conceptual and metaphysical analysis. First, possible worlds are truth-relativizers. The proposition ‘Bratislava is the capital of Australia’ is actually false, but were the circumstances different, Bratislava could be the capital of Australia. Put differently, ‘Bratislava is the capital of Australia’ is true in a possible world that is different from the actual one. It means that the interpreted sentences are true or false relative to worlds and the very idea of there being possible worlds as truth-relativizers is the core of possible-worlds semantics. On the other hand, philosophical theories use possible worlds to provide an explanation4 of modality in which possible worlds play the roles of possibility, contingency, necessity and impossibility-localizers (impossibility being localised in no world whatsoever) 5. According to the possibility-localizer, necessity-localizer and contingencylocalizer role, it is worlds at which possible individuals (including worlds themselves 6) exist 4 Although, as Lewis has it, an explanation is more an account of etiology (Lewis 1986a, 73), I use the notion as a blanket term covering any systematization of phenomena endorsed on the grounds of its combination of conservativeness and overwhelming economy. (See Divers 2013). 5 See, for instance, (Yagisawa 1992). Here, I refer to a particular feature of modal realism that many philosophers find confusing. Namely, that worlds are also individuals and, more importantly, that de dicto modalities are just a special kind on modalities de re. 6 )4 Chapter I The Possible and the Impossible and represent actual individuals (including the actual world) being otherwise. The proposition ‘Bratislava is the capital of Australia’ is false according to the actual world, yet it is true according to worlds with different geopolitical configurations. Such an analysis, generalized in a form (P) It is possible that P if and only if there is a possible world, w, such that at w, P and supplemented by conditionals treating necessity, contingency and impossibility, respectively: (N) It is necessary that P if and only if every possible world, w, is such that at w, P (C) It is contingent that P if and only if there is a possible world, w, such that at w, P and there is a possible world, w*, such that it is not the case, that P7 (I) It is impossible that P if and only if there is no possible world world, w, is such that at w, P is widely accepted. Concepts like ‘the actual world’ and ‘a possible world’ are functional concepts that play a certain role in philosophical analysis. The aim of any metaphysical analysis is then, first, to specify theoretical roles that are associated with functional concepts and, second, to fill these roles with entities picked out by names that express those concepts8. 1.3 …and the Ontology However, the sphere of application of the concept is one thing, its metaphysical interpretation is another. For, doing semantics without doing metaphysics is just a halfway 7 A more restricted definition of contingency may be the following: it is contingent that P if and only if the actual world, @, is such that at @, P and there is a possible world, w, such that it is not the case, that P. 8 Cf. Fischer (forthcoming). )5 Chapter I The Possible and the Impossible business because it leaves the most interesting philosophical questions unanswered. Unless we are told what the concept of ‘possible world’ represents in reality, the analysis cannot even get off the ground. (MR) provides one such metaphysics. 9 1.3.1 (MR) (MR) is one of many realist theories of possible worlds. Among its basic features is a rather unpopular belief that possible worlds are concrete individuals which share a metaphysical nature with the very world of which we are a part. The ontological components of (MR) can be outlined as follows: 10 (a) Reality consists in a plurality of universes or ‘worlds’. These worlds are concrete, spatio-temporal systems. (b) One of these is what we ordinarily call the universe: the largest connected spatiotemporal system of which we are a part. It is the actual world. (c) The others are things of roughly the same kind: systems of objects, many of them concrete, connected by a network of external relations like the spatiotemporal distances that connect objects in our universe (Lewis 1986a, 74–76). (d) Each universe is isolated from the others; that is, particulars in distinct universes are not spatiotemporally related. (the other universes do not overlap; no particular inhabits two universes) (Lewis 1986a, 78). (e) The totality of universes is closed under a principle of recombination. Roughly: for any collection of objects from any number of universes, there is a single universe containing any number of duplicates of each, provided there is a spacetime large enough to hold them (Lewis 1986a, 87–90). (f) There are no arbitrary limits to the plenitude of universes (Lewis 1986a, 103). In what follows, I will not discuss alternative actualist's conceptions. See Lewis (1986a, Chapter III), Divers (2002), Menzel (2013) for a detailed introduction of the alternatives. Roughly though, actualism is the philosophical thesis that everything there is is actual. 9 10 For the sake of brevity and clarity, I borrow (with a few modifications) the exposition from Rosen (1990, 333). )6 Chapter I The Possible and the Impossible (g) Our universe is not special. That is, there is nothing remarkable about it from the point of view of the system of universes. In a nutshell, (MR) consists of the combination of the quadruple (P), (N), (C) and (I) and postulates (a)–(g). Speaking of role-fillers, the role of a ‘possible world’ is filled by maximal spatiotemporal, causally isolated systems. Besides the actual world, there exist infinitely many merely possible worlds that are ontologically on a par with the actual world. Possible worlds contain world-bound individuals; and no ordinary individual exists in more than one world. There are no gaps in the logical space, no vacancies where a world might have been, but is not. The space of worlds is plenitudinous in the sense that anything can coexist with anything else, and anything can fail to coexist with anything else. 1.4 (EMR) (EMR) goes even further. It says that beside possible worlds, there are also impossible worlds. The extended version of modal realism is obtained from the theory by replacing reference to possible worlds and their inhabitants with reference to possible and impossible worlds and their inhabitants (contra (I)). For, assuming we require a sufficiently fine-grained account of modal discourse, we might want to account for a plurality of possibilities as well as a plurality of impossibilities. (EMR) is therefore the thesis that there are possible worlds in Lewis’ sense and also impossible worlds in an equally realistic sense. Beside (MR)’s ontological postulates (a)-(g), (EMR) accepts the following additional ones: (h) There exist impossible worlds. (i) Impossible worlds inhabit different logical spaces. (j) There exists a plurality of logical spaces. (k) All worlds are possible as well as impossible in some sense, i.e. K-possible for some K (where K stands for a particular kind of possibility, be it physical possibility, )7 Chapter I The Possible and the Impossible metaphysical possibility, epistemic possibility, doxastic possibility, legal possibility, etc.) (l) For any K, some worlds are K-impossible. (m)For any K, there is another kind of possibility, K*, such that some worlds are Kimpossible but K*-possible. The central idea behind the extension is that if (MR) posits concrete possible worlds to account for possibility, it ought to posit ‘impossible worlds’ to account for impossibility. 1.4.1 Impossible Worlds: What Are They? When philosophers talk about impossible worlds, they mean one of the following things: impossible worlds are ways the worlds might not be. This definition proceeds from Lewis’s ‘argument from ways’ for the existence of possibilia and extends its application into the impossible too. Another definition concerns metaphysics: impossible worlds are worlds where different metaphysical theories are realised, be it platonistic heaven, Leibniz’s monadology or Lewis’s pluriverse. As Nolan puts it: [m]any metaphysical views seem to be such that if they are true at all, they are necessarily true, and if false, necessarily so: yet rivals understand each other, and we metaphysicians flatter ourselves that we are engaging in real debates, where argument and invocation of considerations are important: we are not babbling mere nonsense, even when some of our number (or many of our number) fall into necessary falsehood (Nolan 1997, 539). A more specific definition of impossible worlds concerns logic and says that an impossible world is a world where the laws of logic fail; more specifically, an impossible world is a world at which classical logic fails and even more specifically impossible world is a world at which the Law of Non-Contradiction fails. )8 Chapter I The Possible and the Impossible 1.4.2 Why (Still) Bother with Impossible Worlds? Finally, impossibilities abound. A reason to believe so comes from an intuition that various impossible properties, propositions, belief states, fictions and some counterfactuals are, or are about, impossible things. In order to differentiate between them, we introduce impossible worlds to play analogous theoretical roles as possible worlds do. One argument for impossible worlds relies on a ‘parity of reasoning’. It starts from an assumption that if the paraphrase argument justifies belief in worlds as ways things could have been then, the same argument justifies belief in worlds as ways things could not have been. The second argument relies on applicability of impossible worlds. To start with modality, impossible worlds are about to help us in conceptual and metaphysical analysis of modal locutions. The idea is to find a place for impossibilitylocalisers in our ontology. Such a move finds its justification in a more perspicuous ontology of truthmakers for impossibility claims. Namely, as possible worlds in (P) It is possible that P if and only if there is a possible world, w, such that at w, P play a role of possibility-localizer, talk of impossible worlds in (I*) It is impossible that P if and only if there is an impossible world, i, such that at i, P seems to be nothing but an extension of the range of modality reduction 11 12. Even stronger reasons for accepting impossible worlds come from counterpossible 11 As Divers puts it, the temptation to accept (I*) rather than (I) may consist in thinking that the former presents stronger truthmakers for impossibility claims that the latter. (Cf. Divers 2002, 70). 12 A so-called plenitude principle is widely accepted among proponents of impossible worlds. For instance, Nolan has it that ‘the most plausible comprehension principle for impossible worlds is that for every proposition which cannot be true, there is an impossible world where that proposition is true’ (Nolan 1997, 542). Jago agrees: ‘if it is impossible that p, then there exists an impossible world which represents that p’ (Jago 2014, 94, notation adjusted). For a problem with the principle, see Sedlár (manuscript). That (I*) should be formulated as a biconditional of the form ‘It is impossible that P if and only if there is an impossible world, i, such that at i, P’ has been challenged by Divers (2002, 69–73). )9 Chapter I The Possible and the Impossible reasoning. Consider the following pair of conditionals: (c*) If Martin were to square the circle, we would be surprised. (c**) If Martin were to square the circle, we would not be surprised. We seem to distinguish between the truth and the falsity of the above conditionals because we assume something to be the case and wonder what would and would not follow from that. However, the lack of (impossible) worlds at which the antecedents obtains causes (c*) and (c**) to be vacuously true. Next, for (MR), possible individuals (and worlds) serve as tools for the ontological reduction of properties and propositions, respectively. For example, the propositions <9 is a prime number> and <it is raining and it is not raining> are not (intuitively) one and the same propositions. By the same manner properties like ‘being triangular and not trilateral’, ‘being blue and green all over’ or ‘being a married bachelor’ are not one and the same property13. If a theory has worlds where impossible situations happen, it allows us to identify them with different sets of worlds. The same motivation applies to accounts of hyperintesional belief contexts that involve impossible propositions. One might believe the proposition <9 is a prime number>, yet fail to believe the proposition <it is raining and it is not raining>. Since (MR) cannot differentiate between these beliefs by means of its own theoretical toolbox, (EMR), at least in these respects, does better in the ratio of theoretical benefits to theoretical costs. Finally, one way how to make sense of metaphysical disputes is to accept impossible worlds. Nolan again: Stating and resolving metaphysical disagreements is often done using counterfactual conditionals (if that were true, then…): and counterfactuals about what things would have been like had metaphysical matters been different are often counterpossibles. So if we have an adequate analysis of counter possible conditionals in terms of impossible worlds, that will help us restate some of those Note, that other (hyperintensional) approaches to propositions might deal with the examples without impossible worlds. My intension, however, is to stay within (MR)’s framework. 13 )10 Chapter I The Possible and the Impossible disagreements. (Nolan 2013, 367). 1.5 …and the Ontology Again Speaking about the impossible so far, ‘impossible world’ figured merely as a functional concept. The crucial thing is that according to (EMR) its role-fillers are concrete impossibilia having the same ontological nature as (MR)’s universes. That is, possible and impossible worlds are concrete, spatiotemporal universes. And this feature brings several undesired consequences. 1.5.1 (EMR): Problems Namely, the addition of concrete impossibilia into (MR)’s framework causes conceptual, logical and metaphysical difficulties. First, (EMR) faces problems with logical impossibilities. If, for example, there is a contradictory world at which (P and not P) is true, then there really is such a world. But that, in turn, drags us into plain contradiction and an inconsistent hypothesis. Moreover, although (EMR) appears to be committed to a weaker claim than those of dialethists, it is still the case that there are (unrestrictedly) things we can talk about only by contradicting ourselves. In other words, the alleged truth about a thing’s contradictoriness is itself contradictory. Moreover, as Jago (2013, 2014) shows, biting the bullet and modify one’s consequence relation is not a feasible option. For the arguments hold for every single proposition we take to be false. Suppose Church’s falsity, ⊥, which entails everything. The argument then goes as follows: I) Is is impossible that ⊥. II) Some impossible world, i, is such that ⊥ III) Something is such that ⊥ (simpliciter). IV) P (for an arbitrary false proposition) )11 Chapter I The Possible and the Impossible Since, the argument concludes, any statement that entails an apparent falsity is unacceptable, (EMR) cannot be acceptable either. The second objection aims to show that (EMR) is modal in spirit. For, unlike (MR), (EMR) goes beyond the actual and merely possible. Due to its commitment to the impossible it has to offer a distinction between possible and impossible worlds. (MR) does not face this problem because its analysis prevents it from admitting impossibilia.14 But as long as we accept both possible and impossible worlds the worry persists. The analysis becomes conceptually circular by relying on modal concepts. Lycan objects that at least two concepts in (MR)’s setup are (at least implicitly) modal. He points out that unless by ‘individual’ (MR) really means possible individual, the right-to-left implication of (P) fails. Second, unless by ‘spatiotemporal relation’ (MR) refers to a possible spatiotemporal relation, (P) fails again. Since (MR)’s metaphysical setup generates impossible worlds via impossibly related sums of individuals, ‘any object including any given round square cupola is spatiotemporally related to the (actual) Sydney Harbour Bridge – albeit by some logically incoherent relation’ (Lycan 1994, 89). Although Lycan arguments against (MR) have already been answered on behalf of (MR), they are still powerful against (EMR). This is due to the fact that while the conceptual and ontological apparatus of (MR) has no room for impossible individuals, that of (EMR) explicitly presupposes it. Third, it has been shown that (EMR) suffers from extensional inaccuracy. 15 The argument from the extensional inaccuracy of (EMR) takes two assumptions for granted. Namely, given a randomly chosen impossible world in which 1) some contradictions are true and 2) the distribution and introduction of conjunctions are valid logical rules in the world, the analysis of impossibility is inaccurate. For, if (P) It is possible that P if and only if there is a possible world, w, such that at w, P is to be accepted, something along the lines of (I*) 14 For a discussion, see Cameron (2012). 15 In fact, any theory of impossible worlds that accepts (I*) suffers from some version of the objection. )12 Chapter I The Possible and the Impossible (I*) It is impossible that P if and only if an impossible world, i, such that at i, P should be accepted too. But then I) It is impossible that (P and ~P) if and only if there is an impossible world, i, such that at i, (P and ~P) II. It is impossible that (P and ~P) if and only if there is an impossible world, i, such that at i, P and at i, ~P. III. It is impossible that (P and ~P) if and only if there is an impossible world, i, such that at i (P and ~P) and at i, P IV. P is possible Therefore It is not the case that there is an impossible world, i, such that at i, P if and only if P is impossible (Contra (I*)). Consequently, from the assumption that there is an impossible world at which some contradiction is true (premise 2) and the assumption that the distribution (premise 3) and introduction (premise 4) of conjunctions are valid inferential principles, we infer that particular conjuncts are true as well. Since it is not the case that the conjuncts themselves are impossible, the analysis is extensionally inaccurate. The inaccuracy rests on the fact that – contrary to the hypothesis – it is not even sufficient16 for P to be impossible that P be true at some impossible world. Divers continues and admits that (EMR) can make an additional step and enrich its analysis with an additional clause, to wit (I**) It is impossible that P if and only if ((there is an impossible world, i, such that at i, P) and (it is not the case that there is a possible world, w, such that at w, P)). Thus, as it seems, the existence of genuine impossibilia is neither a necessary nor sufficient condition for the truth and falsity of claims about the impossible. It is not a necessary condition, because Lewis himself can, via definitional (I), provide the analysis without resorting to genuine impossibilia. And it is not sufficient because it is inaccurate. 16 )13 Chapter I The Possible and the Impossible The problem with adding (I**) is that it does not improve Lewis’s initial proposal, because the proposal has to mention an absence of possibilia when talking about their impossible mates. But, Divers concludes, ‘the price of so attaining extensional accuracy is to preserve whatever disadvantages of form, or lack of ontological perspicuity, that were supposed to attach to the possibilist principle’ (Divers 2002, 71). Finally, (EMR) faces representational challenges. They involve metaphysical theories in general, (MR)’s theoretical impossibilities in particular. It is a commonly accepted thought that metaphysical theories, if true, are necessarily so. Consequently, (EMR) should have resources to provide for negations of their rivals’ postulates. But to see, or at least imagine, how such representation looks like especially when the representation goes via concrete impossibilia seem unfeasible. In Vander Laan’s words: Unless concrete worlds represent in a manner completely unlike the approaches discussed by Lewis, the above argument against concrete impossible worlds is successful. Barring a successful and hitherto unheard of theory of representation, concrete worlds must represent in a way that is not compatible with the theory that all worlds, possible and impossible, are concrete. The Achilles’ heel of a concretist theory of impossible worlds is the fact that there are certain things which concrete worlds cannot represent inaccurately: the concreteness of worlds, for example, and other facts, such as those regarding what occurs at other worlds, or certain truths about whatever transworld objects there would be. (Vander Laan 1997, 607). In contrast, …if worlds are thought to be abstract, there is nothing to prevent inaccurate representation on any topic whatsoever. It might be true in a world W that it is concrete (the proposition W is concrete might belong to its book BW), despite the fact that W is abstract and not concrete (Vander Laan 1997, 607). )14 Chapter I The Possible and the Impossible 1.6 (EMR) Ultimately Defeated? Given the above, one might think that any attempt to rescue (EMR) is doomed to failure from the very beginning. For, there are ersatz, abstractionist or actualists proposals which take possible and impossible worlds to be sets, complex properties and propositions, states of affairs, universals or situations (King 2017, Jago 2014). Even more, there are proposals that justify (MR)’s postulates for analysis of possibility, yet turn into ersatz representations to account for distinct impossibilities (Mares 1997, Berto 2010). But to go concrete all the way down is pre-theoretically as well as theoretically the most objectionable option. Yes and no. On the positive side, (MR) still belongs among the most unified, uniform, systematic and most discussed theories on the market. When it comes to impossibility though it looses against other alternatives. Its extension beyond possible worlds thus appears as the most straightforward improvement of the theory. It takes worlds, whether possible or impossible, to be worlds and not their mere representations: ‘[w]orlds are what they are, and not some other things’ (Lewis 1073, 97). Negatively speaking, the consequences of such additional step are considered as too much to swallow, and unless proponents of (EMR) offer a modified version of their systematic analyses, the theory cannot compete with its rivals. Instead of pressing the point and trying to square (EMR) with (MR) right away, I try to offer three paraphrases of (EMR). To be more precise, I try to formulate three versions of (EMR) which are a) genuine rather than ersatz, b) closer to (MR) than any actualist’s alternative and c) aim at a non-modal explanation of modal phenomena. A spectator might find such criteria misleading and very hard to conceptualise properly. I agree. However, if someone tries to formulate them, she will likely resort to one of the following ways: The Way of Parity Possible and impossible worlds exist in the way tables, chairs and continents do. )15 Chapter I The Way of Reductiveness The Possible and the Impossible Possible and impossible worlds provide a non-modal analysis of modality. The Way of the Concreteness Possible and impossible worlds represent possibility and impossibility by having a direct ontological relation to concrete mereological sums of individuals. The Way of Representation Possible and impossible worlds represent ways the world might and might not be genuinely.17 18 The four ways have something in common: any modification which counts as a modification of (EMR) must fulfil at least one of the ways and any modification of (EMR) which fulfils at least one of the ways presupposes the ontology of (MR). This is, in my view, a sufficient condition for a theory to be a version of realism, to be closer to (MR) than any actualist's alternative, and to run its analyses non-modally, respectively. One proposed alternative suggests analysing modal discourse within the framework of metaphysical structures. Such metaphysical framework posits (MR)’s worlds as well as metaphysical structures these worlds instantiate. The idea then is that the representation goes via logical structures which, however, ontologically depend on concrete universes. The second approach takes the role-fillers of possible and impossible worlds to be metaphysical indices, dimensions of concrete individuals and concrete worlds. Finally, impossible worlds according to the third approach are concrete entities which exist according to (EMR) story. For now I flesh out the notion of ‘genuine’ negatively: ‘To be sure, you might not have to be a genuine modal realist like me. You might prefer an analysis on which the modal operators are quantifiers over some sort of abstract ersatz worlds linguistic descriptions, maybe’ (Lewis 1986a, 19). 17 It is not a coincidence that my methodology evokes the methodology of Lewis in characterisation the difference between concreteness and abstractness. (cf. Lewis 1986a, 81-86). 18 )16 Chapter I The Possible and the Impossible 1.7 A Note on Methodology The reader should read the rest of this thesis as aiming at three goals. Firstly, the author is a modal realist, accepts much the theory, and thus tries to contribute to its defence. Thus, Chapter II provides an argument for epistemic adequacy of (MR). Secondly, as Chapter III shows, the author agrees with a modification of (MR) so as to include impossible worlds in order to provide for impossible world discourse. Nonetheless, he claims that impossible worlds’s roles are better played by entities different from concrete impossibilia (contra (EMR)). Chapters IV-VI offer such entities. Chapter IV develops and defends modal dimensionalism. Modal dimensionalism is realism about spaces, times and worlds—metaphysical indices that make objects spatial, temporal and modal, respectively. Metaphysical indices play the role of alethic relativizers, i.e. items to which matters of truth are relativized. The chapter examines several arguments against modal dimensionalisn and shows that it offers a feasible way to understand extended modal realism. Chapter V offers a structural approach to possible and impossible worlds. In particular, I consider whether it makes sense to think of logical models in isolation from the concrete world but without their being divorced from all spatiotemporal totalities. The metaphysics of structure developed in this chapter assumes that structural properties of possible and impossible worlds are primitive and objective. However, I provide some characterisations of their logical and metaphysical behavior, as well as guidelines for talking about them. Finally, Chapter VI proposes yet another metaphysical framework of hybrid modal realism. I present theories of (extended) modal realism and modal fictionalism respectively, their advantages and drawbacks. Finally, I propose a so-called hybrid view. Roughly, the view is that one might be a modal realist when it comes to possibilia, but turn into fictionalism regarding impossibilia. I think the most demonstrative way of sketching the rest of the thesis is to draw a simple table. Specifically, it shows how modal realism, extended modal realism and other )17 Chapter I The Possible and the Impossible alternatives play in the cost/benebit analysis. In the table, ‘✔’ stands for the ability of a theory to fulfil a criterion, while ‘✗’ expresses its theoretical limitations: Modal Realism Extended Modal Realism Extended Modal Dimensionalism Extended Modal Extended Modal Structuralism Fictionalism The Way of Parity ✔ ✔ ✔ ✔ ✗ The Way of Reductiveness ✔ ✔ ✗ ✗ ✔ The Way of the Concreteness ✔ ✔ ✔ ✔ ✔✗ The Way of Representation ✔ ✔ ✗ ✗ ✔✗ Impossible Worlds ✗ ✔ ✔ ✔ ✔ Meaningfulnes (Consistency) ✔ ✗ ✔ ✔ ✔ 1.8 Conclusion I conclude that traditionally formulated (EMR) fails. But three things have to be differentiated: (EMR) relies on the analysis based on the notion of possible and impossible worlds. I agree. For (EMR) concepts of ‘possible world’ and ‘impossible world’ are functional concepts that play certain theoretical roles in philosophical analysis. Again, I agree. For (EMR) its role-fillers are concrete possible and concrete impossible worlds. At this point, I disagree. The roles should be played respectively by metaphysical structures, metaphysical indices and concrete possible worlds plus stories about concrete impossible worlds. And only after a closer look at these proposals can we see how these versions can fare on the scale of strength. )18 Chapter II On the Indispensability of (Im)Possibilia CHAPTER II What another would have done as well as you, do not do it. What another would have said as well as you, do not say it; what another would have written as well, do not write it. Be faithful to that which exists nowhere but in yourself-and thus make yourself indispensable. Andre Gide 2. On the Indispensability of (Im)Possibilia 2.1 Introduction In this chapter19, I develop an argument in support of the epistemic adequacy of (MR). First, I briefly review (MR)’s motivation to go beyond the actual (2.2). Second, I present the ‘how do we know?’ problem (2.3) and propose a possible solution (2.4). Section 2.4.1 outlines the rough idea of my strategy, and (2.4.2) presents the so-called indispensability argument in the philosophy of mathematics and applies it to modal metaphysics. In (2.4.3) and (2.4.4) I discuss the crucial premises of the argument. Finally, (2.5) generalizes the argument and applies it to both possibilia and impossibilia. 19 With some modifications, this chapter is based on Vacek (2013b). )19 Chapter II On the Indispensability of (Im)Possibilia 2.2 (MR) Again (MR) is the thesis that the world we live in is very inclusive. It consists of us and all our surroundings, however remote in time and space. Every chair, person and city that is spatially and temporarily related to us belongs to our world. However, things might have been different in infinitely many ways. In fact, any way the world could have been is a way some real world is. We call these ways possible worlds. The argument runs as follows: I believe that there are possible worlds other than the one we happen to inhabit. If an argument is wanted, it is this. It is uncontroversially true that things might be otherwise than they are. I believe, and so do you, that things could have been different in countless ways. But what does this mean? Ordinary language permits the paraphrase: there are many ways things could have been besides the way they actually are. On the face of it, this sentence is an existential quantification. It says that there exist many entities of a certain description, to wit ‘ways things could have been’. I believe that things could have been different in countless ways; I believe permissible paraphrases of what I believe; taking the paraphrase at its face value, I therefore believe in the existence of entities that might be called ‘ways things could have been’. I prefer to call them possible worlds. (Lewis 1973, 84) Many claim that even if the argument is sound, it in fact says nothing about the nature of the entities at issue. According to this objection, the phrase ‘ways the world could have been’ can be read at face value and does not really commit us to the existence of a plethora of concrete possible worlds. As such, we should reject (MR). A second argument for (MR) – the so-called argument from utility – says that the idea of concrete possible worlds is not only a natural existential quantification entrenched in our everyday description of reality but also practically useful. Since concrete possibilia bring undeniable theoretical benefits, and since cost-benefit analysis plays an important (if not the most important) role in metaphysical methodology, their existence is worth considering. Put in more comprehensive terms, we should prefer any theory that a) contributes to the unified systematization of our pre-theoretical opinions, b) is economical )20 Chapter II On the Indispensability of (Im)Possibilia with regard to primitive (and thus unexplained) notions, c) is conservative with respect to deeply entrenched pre-theoretical opinions, and d) does well in comparison with its rivals. As Lewis (1986a) shows, (MR) fulfils many of these criteria, and we should thus prefer it to (a version of) modal ersatzism. 2.3 An Epistemological Worry Nonetheless, it has been argued that these two arguments, however persuasive they may seem, do not sufficiently support (MR). The challenge is the following: even if we have some pragmatic reasons to believe in the existence of concrete possibilia, we are completely empty handed when it comes to knowledge of them.20 Let us look at this objection more closely. Notoriously, most accepted accounts of epistemic justification include a causal component. To know something, according to these accounts, is to be causally connected to the ‘truthmaker for the known truth bearer’ (Bueno and Shalkowski 2000). But, ex hypothesi, there is no causal connection between actual and merely possible individuals in (MR)’s conception. Since (MR)’s worlds are maximal mereological sums of spatiotemporally interrelated individuals, the objection concludes, there is basically nothing beyond purely pragmatic reasons that could justify our positing the existence of concrete possibilia. In short, (MR) precludes modal knowledge. 2.4 … And a Reply Fair enough. Fortunately, however, (MR) is not the only view on the philosophical scene claiming to know something about entities that are spatio-temporally isolated from us. Famously, some philosophers21 of mathematics have considered the realm of (abstract) entities that bear no relevant relation to us. They treat numbers, classes, sets, and functions as objects (of one kind or another), subjecting them to rational examination without being in 20 For example, see Richards (1975) and Skyrms (1976). 21 To be more precise, I have mathematical realists in mind. I do not deny that other options are available. )21 Chapter II On the Indispensability of (Im)Possibilia any way casually acquainted with them. They believe in the existence of a realm of mathemata suited to the needs of the branches of mathematics (cf. Lewis 1986a, 3). To the extent that this is so, Lewis points out, the ontological commitment to possibilia is, methodologically speaking, not (fundamentally) different from the ontological commitment to the space of numbers, sets, etc. We only have to believe in the existence of possibilia, and ‘there we find what we need to advance our endeavors’ (Lewis 1986a, 4). Yes, possibilia are causally isolated and thus in some sense ‘untouchable’. But so are numbers, functions, and sets. In Lewis’s words: Set theory offers the mathematician great economy of primitives and premises, in return for accepting rather a lot of entities unknown to Homo javanensis. It offers an improvement in what Quine calls ideology, paid for in the coin of ontology. It’s an offer you can’t refuse. The price is right; the benefits in theoretical unity and economy are well worth the entities. Philosophers might like to see the subject reconstructed or reconstrued; but working mathematicians insist on pursuing their subject in paradise, and will not be driven out. Their thesis of plurality of sets is fruitful; that gives them good reason to believe that it is true. (Lewis 1986a, 4) Thus, according to Lewis, mathematicians and metaphysicians have something roughly in common. I say ‘roughly’ because the situation is much more complicated. In what follows, a closer examination of the different stages of the argument for (MR) will help to motivate, elucidate, and justify this strategy. 2.4.1 Stage I: Setting Things Up We can distinguish between platitudinous, uncontroversial claims about mathematics and controversial, philosophical claims about it. The former include a great deal of mathematical knowledge, axioms of number theory, proofs, equations, solutions, and the like: in short, the material in which mathematicians are educated and with which they engage. We do not doubt that mathematicians know what they are talking about; they )22 Chapter II On the Indispensability of (Im)Possibilia understand their subject matter, and they certainly know more about the subject than laymen. Analogically, we can distinguish between uncontroversial, platitudinous claims about possibility, necessity, and contingency and their rather controversial metaphysical interpretations. In terms of our pre-theoretical opinions, we all believe that there are donkeys, that grass is green, and that I am writing this thesis. We can also all agree that there could have been talking donkeys, that Bratislava could have been the capital of Australia, and that I could have been a poached egg. These are simply our pre-theoretical opinions, and any philosophical analysis of modality should account for (not violate) them. Philosophers of mathematics have formulated particular theories about what mathemata are. According to some, they are Platonic entities inhabiting the ‘third realm’. According to others, they are physical objects, symbols written on a piece of paper, concepts, or immanent universals. All of these mainstream views both maintain that we have good reason to think that numbers with a particular nature really exist and claim to provide the best systematization of our mathematical knowledge. The same holds for modal metaphysics. There are philosophers who take modality seriously. According to some, modality is best analyzed by means of possible worlds, considered as real, isolated, physical entities. Others have hypothesized actual surrogates for possibilia. For example, they say that possible worlds are Platonic ideas, essences, universals, set-theoretic constructions, fictions or states of affairs. Of course, there is disagreement about what the entities in fact are. What matters, however, is that philosophers agree on the content of pre-theoretical opinions, just as philosophers of mathematics agree on the content of mathematical platitudes. Given the distinction between pre-theoretical opinions and metaphysical interpretations of those opinions on the one hand, and mathematical platitudes and their metaphysical interpretations on the other, we arrive at a lattice of the following form: (i) Mathematical Platitudes (ii) Pre-theoretical Modal Opinions (iii) Philosophy of Mathematics (iv) Modal Metaphysics )23 Chapter II On the Indispensability of (Im)Possibilia Since there are few disputes about mathematical platitudes (i) and about our pre-theoretical opinions (ii), mathematical and metaphysical practices are neutral with respect to many different controversial accounts of their subject matters (cf. Bueno and Shalkovski 2000, 10).22 Bearing this in mind, in the modal case there is no dispute about what is possible (ii). Any modal realist would be willing to accept the claim that possible worlds – whatever their metaphysical nature – exist. This is because theories of both concrete and ersatz possible worlds are typically consistent with our pre-theoretical opinions about what is possible, impossible, contingent and necessary. What really varies are our philosophical interpretations of possible worlds discourse (iv). We all agree that there are donkeys, but not all of us would subscribe to the thesis that mind-independent physical objects exist. By the same token, we all agree that there could have been talking donkeys, but only a minority of people think that full-blooded talking donkeys exist in a concrete possible world. Few would argue against the notion that I could have been a poached egg, but not everybody would accept that this poached egg would be a counterpart of me rather than me. And the same seems to hold for mathematics. Various philosophical accounts of mathematics (iii) that conflict with mathematical platitudes fail as accounts of mathematics. On the other side, those philosophical accounts of mathematics that tend to be consistent with the platitudes – e.g. Platonic and nominalist theories – provide competitive accounts of the nature of mathematical entities. We can thus conclude the following. It seems that (MR) relies on the similarity between mathematics and metaphysics to construct an analogy between (i) and (iv), based on the supposition that they rely on the same reasoning. And this seems wrong. Being causally related to possible worlds is not necessary for knowledge of what is possible, just as being causally related to mathemata is not necessary for knowledge of the axioms of number theory, proofs, equations, solutions, etc. Thus only the analogy between (I) and (II) (and not between (i) and (iv)) is secure. What we really need, however, is the analogy between (iii) and (iv). Put briefly, the argument runs as follows: (a) The modal realist argues for the existence of concrete possibilia in the same way that the mathematician argues for the existence of mathematical entities. 22 In the lattice, these are (iii) and (iv). )24 Chapter II On the Indispensability of (Im)Possibilia (b) We all agree that mathematicians have some knowledge. (c) Uncontroversial mathematical knowledge is platitudinous. (d) If we take the analogy at face value, it secures only uncontroversial modal knowledge. (e) The existence of concrete possibilia is controversial modal knowledge. (f) The desired analogy is secured if and only if controversial modal knowledge is an analogue of controversial mathematical knowledge. 23 It should now be clear that the analogy that (MR) needs to establish is more controversial than first appeared. What we need is not an analogy between mathematics (i) and modal metaphysics (iv). What we in fact need in order for the analogy to work is a premise that commits us to the existence of controversial mathematical claims (iii) – a so-called mathematical realism (or Platonism). I do not think, however, that this discredits the analogy. On the contrary, given that there are (not merely) pragmatic reasons to believe in the existence of mathemata, there are (not merely) pragmatic reasons to believe in the existence of possibilia (assuming the analogy holds). I will discuss these reasons in what follows. 2.4.2 Stage II: The Indispensability Argument Famously, mathematics penetrates almost every part of human reasoning. Mathematics applies to virtually all parts of empirical and theoretical science. It also provides elegant and economical statements to many theories. It is therefore not a surprise that, given the practice and success of science, the existence of mathemata is indispensable to our theories. An argument that builds on this notion proceeds as follows: 1. We ought to be ontologically committed to all and only those entities that are indispensable to our best scientific theories. 2. Mathemata are indispensable to our best scientific theories. 23 Again, my argument has only limited power as it relies on a particular, and not exhaustive, account of the ontology of mathematical entities. )25 Chapter II On the Indispensability of (Im)Possibilia Therefore, C1. We ought to be ontologically committed to mathemata. 24 The argument, as it stands, presupposes at least two things, expressed by the following slogans: ‘To be is merely to be the value of a bound variable’ and ‘No entity without identity’. It goes without saying that there has been much debate over the success of the argument. As Quine points out, the great medieval controversy over universals has flared up anew in modern philosophy of mathematics. Formulated this way, however, the argument seems to be valid. It is an indisputable fact that, say, physics would not work without mathematics, since the results of mathematics partly constitute our knowledge of the field. Put differently, (1) serves as a general and normative premise about the considerations that govern our ontological commitments. Furthermore, given the indispensability of mathematics to physics, it is very hard to deny the existence of mathemata once one accepts that the theories of physics are true. As Shapiro suggests, many of those unmoved by indispensability arguments do not believe in the truth – in some heavy sense – of our scientific theories in the first place. But when it comes to those who do, it would seem that realism about mathematics is in some sense entailed by scientific realism. Given this assumption, mathematical entities do exist. Shapiro (2000) formulates the argument more precisely: (1a) Real analysis refers to, and has variables that range over, abstract objects called ‘real numbers’. Moreover, one who accepts the truth of the axioms of real analysis is committed to the existence of these abstract entities. (2a) Real analysis is indispensable to physics. That is, modern physics can be neither formulated nor practiced without statements of real analysis. (3a) If real analysis is indispensable to physics, then one who accepts physics as a true description of material reality is thereby committed to the truth of real analysis. (4a) Physics is true, or nearly true. Therefore, 24 This form of the argument is presented in Colyvan (2011). )26 Chapter II (5a) On the Indispensability of (Im)Possibilia Abstract entities called ‘real numbers’ exist. Shapiro suggests that if we accept the truth of physics, we are automatically ontologically committed to the existence of real numbers. If the truth of the scientific theory is accepted, explaining the further ontological commitment is a straightforward matter (Newstead and Franklin 2012). Mathematical Platonism is one metaphysical interpretation of mathematical discourse among many. Generally, it claims that mathematical theories relate to systems of abstract objects that exist independently of us, and that the statements of those theories are determinately true or false, independently of our knowledge. Put differently, mathematical Platonism is a realistic account of mathematical discourse that accounts for how mathematical statements get their truth values. Although still controversial, the issue is clearer now than it was, because we now have a more explicit standard at hand by which to decide what ontology a given theory is committed to (cf. Quine 1951). But if this is so, then we are brought back to (MR)’s analogy. To be sure, by pointing out uncontroversial mathematical platitudes on the one hand and our pre-theoretical opinions on the other – (i) and (ii) – we gain nothing by the analogy. However, by pointing out the success of a controversial mathematical theory, namely the epistemological justification of mathematical Platonism (iii), and by applying that very methodology (which is not merely based on pragmatic reasons) to (MR), the strategy might succeed. Modal realists can thus argue along the following lines: 1. We ought to be ontologically committed to all and only those entities that are indispensable to our best scientific theories. 2*. According to a strong metaphysics of mathematics, Mathematical platonism, Platonic mathemata are indispensable to our best scientific theories. C1. We ought to be ontologically committed to Platonic mathemata. 25 3. If the indispensability argument is valid in the case of mathematics, it should be applied to metaphysics as well. 25 For a summary of mathematical Platonism, see Colyvan (2011). )27 Chapter II 4. On the Indispensability of (Im)Possibilia We ought to be ontologically committed to all and only those entities that are indispensable to our best metaphysical theory. 26 5. The existence of (MR)’s possibilia is indispensable to our best metaphysical theory of the nature of possible worlds. Therefore, C2. We ought to be ontologically committed to concrete possibilia. Again, we should bear in mind that the indispensability argument for the existence of concrete possibilia can be considered the same as its mathematical counterpart. After all, we showed that a reliance on the analogy between modal and mathematical epistemology that remains neutral with regard to the nominalism/Platonism dispute only grounds the (irrelevant) justification of uncontroversial modal claims like ‘there could have been a talking donkey’, ‘I could have been doing something other than writing this chapter’, etc. This is likely neither to persuade us to believe in the existence of a counterpart of me who is not writing this chapter nor to give us any reason to believe in the existence of full-blooded talking donkeys as parts of different concrete worlds. It is the extra step, the route to mathematical Platonism, that any advocate of the analogy between mathematics and modal metaphysics should pursue. 2.4.3 Stage III: Premise 3 Premise 3 claims that if the indispensability argument is valid in the case of mathematics, it should be applied to metaphysics as well. This means that if the existence of Platonic mathemata is indispensable to our best scientific theories, the existence of, say, concrete possibilia is also indispensable (assuming that (MR) is the best metaphysical theory of what there is). And it raises a methodological worry: if we do not commit to mathematical entities in our scientific enquiries, we lose the explanatory power and predictive value provided by those theories. But what is at stake when we do not commit to possibilia? 26 Here I assume that if a metaphysical theory is true, it is necessarily so. )28 Chapter II On the Indispensability of (Im)Possibilia For (MR), the goal of philosophy is to provide an overall systematization of our pretheoretical opinions. It is pointless to build a theory, however systematized, that it would be unreasonable to believe in, and it is not even unity and systematicity alone that matter. A worthwhile theory must be credible, and it will not gain credibility if it conflicts with common sense. It is common sense – unsystematic folk theory – that we believe in anyway, and that no theory should violate. Metaphysics thus faces the following methodological imperative: never put forward a philosophical theory that you cannot believe in your least philosophical and most commonsensical moments (Lewis 1986a, 135). Moreover, other methods of philosophy govern metaphysical theorizing. For example, metaphysics concerns linguistic and conceptual analysis, applies scientific findings, and pursues the theoretical virtues of simplicity, explanatory power, systematicity, and beauty at the level of theory selection. Philosophical theories, especially metaphysical ones, simply must fulfil certain requirements in order to be accepted. If successful, (MR) combines the best balance of conservativeness (with regard to our pre-theoretical opinions) and economy (at the level of metaphysical postulates), and, when compared with different theories, its positive results outweigh those of its competitors. This points to an important conception of what the best metaphysical theory should do. For one, it is definitely not its business to undermine pre-philosophical opinions. Rather, it ought to systematize them, and it ought to do so in such a way that the postulation of metaphysical entities promotes the values of conservativeness, simplicity, explanatory power and economy. And if the advantages of a theory that meets these requirements outweighs the advantages of its rivals, we have strong, even overriding reason to accept it – together with its ontological commitments, of course. 2.4.4 Stage IV: Premise 5 I admit that deciding which theory is best when it comes to the above criteria is far from straightforward. I also admit that the existence of (MR)’s possible worlds prompts a lot of incredulous stares. Yet (MR) offers all sorts of explanations, and these explanations are for the most part successful. For example, an accurate and appropriately non-modal analysis of modality has so far not been beaten. Moreover, it can even be shown that the )29 Chapter II On the Indispensability of (Im)Possibilia applications of (MR) are greater in number than those of its actualist's counterparts, and that its ontological costs are not clearly greater than those of actualism (of one sort or another) (cf. Divers 2002). Unfortunately, the provision of a full defense of (MR) would take us far beyond the scope of this chapter. Let me therefore mention the main sources for such a defense. The most comprehensive source is Lewis’s magnum opus on (MR) (1986a). Also important is Divers (2002) which, for example, defends (MR) against objections concerning quantification over non-actuals, meets epistemological worries about the theory, and illustrates that no objection shows counterpart theory in any worse light than any other possible worlds account of de re modal content. The objection of circularity with regard to (MR)’s analyses is overcome in Divers (2002), Kiourti (2010), and Cameron (2012), among others. The above notwithstanding, I can still insist on the weaker reading of the premise. Even if we are not persuaded by arguments on behalf of (MR), my argument can be conditional. That is, no argument for the existence of concrete possible individuals is needed; instead, the existence of concrete possible individuals can be assumed in the sense that if there are concrete possible individuals, there are such and such problems and such and such potential solutions. Briefly, I pursue the following strategy: if the assumptions I am hypothetically endorsing are true, then such and such will be the case. 2.5 (EMR) and the Problem of ‘How We Know?’ What about (EMR)? As I mentioned in Chapter I, Yagisawa (1988, 1992, 2010) argues that (MR), if fully comprehensive, should also include impossible individuals in its ontology. By pointing out certain deficiencies of (MR)’s analyses, Yagisawa finds (MR) incomplete as a theory. Granted, there are ways that go beyond the way the world actually is or will be. These ways are (MR)’s possible worlds. But in addition to these ways, he adds, there are ways the world could not have been. And thus we have the argument from ways. Moreover, as (1.4.2) shows, the arguments from utility are applicable too, since impossible worlds play a theoretical role that is analogous to that played by possible worlds. )30 Chapter II On the Indispensability of (Im)Possibilia What about the indispensability argument? Can we extend the argument to demonstrate the indispensability of impossible entities? If we accept the need for impossible worlds and impossible individuals in the best theory of modal phenomena, parity of reasoning simply supports the extension of the possibilist ontology to concrete impossibilia. Moreover, if Priest is right to claim that any of the main theories about the nature of possible worlds can be applied equally to impossible worlds (cf. Priest 1997, 580–581), an indispensability argument from concrete impossibilia would run as follows: 1. We ought to be ontologically committed to all and only those entities that are indispensable to our best scientific theories. 2*. According to a strong metaphysics of mathematics, Platonic mathemata are indispensable to our best scientific theories. C1. We ought to be ontologically committed to Platonic mathemata. 4. If the indispensability argument is valid in the case of mathematics, it should be applied to metaphysics as well. 5. We ought to be ontologically committed to all and only those entities that are indispensable to our best metaphysical theory. 6. The existence of (MR)’s possibilia is indispensable to our best metaphysical theory of the nature of possible worlds. C2. We ought to be ontologically committed to (MR)’s possibilia. 7. If (MR)’s argument is valid in the case of concrete possible worlds, then it can be applied, mutatis mutandis, to the case of impossible worlds as well. Therefore, C3. We ought to be ontologically committed to concrete impossibilia. )31 Chapter II On the Indispensability of (Im)Possibilia 2.6 Conclusion Let me summarize the argument in the following table: A B C (Stage I) Mathematical platitudes (Stage I*) Pre-theoretical opinions (about the possible) (Stage I**) Pre-theoretical opinions (about the impossible) (Stage II) The indispensability of mathemata to the best scientific theories (Stage II*) The indispensability of entities postulated by the best modal metaphysical theory (Stage II**) The indispensability of entities postulated by the best modal metaphysical theory (Stage III) Philosophical disputes about the nature of mathemata (Stage III*) Philosophical disputes about the nature of possible worlds (Stage III**) Philosophical disputes about the nature of impossible worlds (Stage IV) Mathematical Platonism (Stage IV*) (MR) (Stage IV**) (EMR) Stage I represents basic mathematical truths of the type 2+2=4. Now as Quine’s and Shapiro’s arguments suggest, these truths are about something, i.e. mathemata, which in order to play any role in the truths of science must exist. Recall that one who does not accept the truth of scientific conclusions will not accept the move from Stage I to Stage II. Stage III represents philosophical disputes about the nature of mathemata. Finally, Stage IV is one particular theory of the nature of numbers, namely mathematical Platonism. The move from Stage II to Stage III is controversial. We must somehow determine the best philosophical systematization of mathematical knowledge, but first we must settle on the criteria of success for any philosophical theory, and this is highly disputed. Recall that what (MR) is looking for is a theory that a) achieves the best balance of conservativeness (with regard to pre-theoretical opinions) and economy (when it comes to )32 Chapter II On the Indispensability of (Im)Possibilia metaphysical postulates), and b) is such that its positive results outweigh the results of its competitors. With column A complete, let us move on to column B. Stage I* represents our pretheoretical opinions about what possibility is and what possibilities there are. 27 Again, determining what our pre-theoretical opinions about the possible are is tricky. Since hard cases make for bad theories, the best way to outline these opinions is the following: pretheoretical opinions are those claims that we believe to be true and that any theory (of modality) should accommodate. Premises 3 and 4 compare the practices of scientists and metaphysicians and are the most controversial assumptions of the whole argument.28 Although I will not treat the question in detail here, it is of the utmost importance to provide an account of metaphysical methodology that can both sustain the argument and describe the practice of metaphysics correctly. 29 There are surely criteria that theories must satisfy in order to avoid being dismissed from the start. What these criteria are is open to question. The move from Stage II* to Stage III* simply mirrors the move from Stage II to Stage III and is based on the indispensable role played by the existence of entities in the most successful philosophical analysis (of modality). What entities these are – and whether they are concrete possible individuals – is again decided by the success of the best theory systematizing modal phenomena. 30 In terms of methodology, the whole argument can plausibly be read as having a conditional form. It relies on highly controversial assumptions about the indispensability argument in the philosophy of mathematics, the feasibility of mathematical Platonism, the methodology of metaphysics, its similarity to scientific practices, validity, and, last but not least, the success of (MR) at the level of philosophical analysis. Any assumption certainly deserves an extensive account of its own. One can thus read every stage of the argument as either a modus ponens or a modus tollens. And I am happy for my reader to choose. 27 On the distinctness of these questions, see Cameron (2012). In response to comments by an anonymous referee of Humana.Mente Journal of Philosophical Studies, I admit that in order to be as precise as possible I should say that if the indispensability arguments in the philosophy of mathematics are ontological, their counterparts in the philosophy of modality are ontological too. 28 For an interesting contribution to the debate on the relationship between the methodology of science and the methodology of metaphysics, see French and McKenzie (2012). 29 30 I leave it to the reader to finish the exposition of the table in the case of column C. )33 Chapter II On the Indispensability of (Im)Possibilia To conclude, if (im)possible worlds are understood as other ‘remote planets’, no causal acquaintance with them is permissible. However, as I have tried to show, this limitation does not prevent (extended) modal realists from defending their view. The analogy between modal metaphysics and mathematics with regards to the existence of their subject matters must certainly be approximated carefully, since ambiguities abound. As controversial as it seems, though, the basic idea behind the indispensability argument in mathematics does not differ fundamentally from the idea behind the indispensability argument in metaphysics, and both should be taken seriously. )34 Chapter III (EMR) and Advanced Modalizing Problems CHAPTER III Ordinary people believe only in the possible. Extraordinary people visualize not what is possible or probable, but rather what is impossible. And by visualizing the impossible, they begin to see it as possible. Cherie Carter-Scott 3. (EMR) and Extraordinary Modalizing Problems 3.1 Introduction In this chapter, I present the orthodox problem with (MR) regarding extraordinary modal claims. First, I distinguish between ordinary and extraordinary modal claims (3.2). Second, I outline alternative versions of (MR) that aim to avoid the problem of extraordinary claims (3.3). I then present recent arguments establishing that the alternatives fail to rescue (MR) (3.4). Finally, I motivate an analysis (not an ontology) of (EMR) that, unlike the others, can withstand the objections (3.5). 3.2 Ordinary and Extraordinary Modalizing (MR) analyses possibility as a going-on in some possible world. In other words, possible worlds are possibility-localizers because possible situations obtain in possible worlds. Ordinary modal claims are claims that are easily analysable with the use of (P). For example, )35 Chapter III (1) (EMR) and Advanced Modalizing Problems It is possible that there are talking donkeys is a perfectly ordinary modal claim, as it fulfils both the left-hand and the right-hand side of (P): (2) It is possible that there are talking donkeys if and only if there is a possible world, w, such that there are talking donkeys at w. Consider, by contrast, (3): (3) There are possible worlds. Postulates (a)–(g) from Chapter I would seem to state explicitly that (3) is true in (MR)’s framework. (3) states something about the sum of possible worlds rather than world-bound individuals. It is an extraordinary claim, since it is not restricted to a single world. However, it reveals a problem internal to (MR). It is uncontroversial that if something is the case, it is also possible that it is the case. A so-called possibility introduction expresses a widely accepted intuition that actual things are possible, although the converse is not the case. If this is so, from (3) we get: (4) It is possible that there are possible worlds. (MR)’s systematic account of possibility paraphrases (4) as (5): (5) It is possible that there are possible worlds if and only if there is a possible world, w, such that there are possible worlds at w. It seems that (5) cannot be an appropriate analysis of (4) since it violates either (MR)’s ontological base, its account of possibility, or possibility introduction 31. Since all three Namely, (MR)’s ontological base does not permit worlds with another worlds as their parts; (MR) analyses possibility as a ‘going-on’ within a single world; and (MR)’s analysis takes situations that actually happen to be possible as well, respectively. 31 )36 Chapter III (EMR) and Advanced Modalizing Problems represent essential components of (MR), the argument concludes, (MR) is self-defeating in cases of extraordinary modalizing. 3.3 Variants of (MR) and Their Problems There are several variants of (MR). In this section, I discuss five: the advanced modalizing approach (AM), the disjunctive analysis (DA), the world-free analysis (WF), the many worlds analysis (MW), and the plurality of worlds analysis (PL). I also present objections to each and motivate a new and thus far underdeveloped alternative. 3.3.1 Variant Analyses Let us begin with advanced modalizing. Divers (1999a, 2002) presents a solution to the problem of extraordinary modal claims by distinguishing between two readings of the ‘it is possible that…’ prefix. As he puts it, ‘the semantic function of a possibility operator on a non-modal quantificational sentence is always that of quantifying in, by way of a variable that is already reserved for worlds’ (Divers 1999a, 229). Thus in some cases the phrase ‘it is possible that…’ expresses the content of ‘at w’, which means ‘there is a world, w, such that at w…’. The cases at issue are ordinary modal claims whose content is explained by confining the quantification to single worlds. Extraordinary modal claims are different because they have unrestricted content. There is no occurrence of the phrase ‘at w’ in their analysis, and prefixing that content with ‘it is possible that…’ has no effect. This analysis enables proponents of (MR) to avoid the initial problem. It is still a key claim of (MR) that ‘there are possible worlds’. But instead of a restricted analysis, (4) receives the unrestricted interpretation (AM): (AM) It is possible that there are possible worlds if and only if there are possible worlds. When it comes to extraordinary modal claims, (MR) uses a strategy of advanced modalizing. In it, ‘it is possible that…’ is semantically redundant, and as (AM) shows, it )37 Chapter III (EMR) and Advanced Modalizing Problems might, or rather should, be dropped. Moreover, such a strategy is far from being ad hoc. As Divers stresses, the distinction between ordinary and advanced modalizing finds its justification in (MR)’s setup and reflects the function and semantic effect of the ‘it is possible that…’ prefix. Jago32 admits that the advanced modalizing strategy avoids problems concerning the analysis of (4). However, he presents an example that, if correct, demonstrates the strategy’s inadequacy. The example runs as follows. Consider (i) the actual world, (ii) Anna and Bill, who are parts of this world, and (iii) the fact that Anna is taller than Bill. This gives us: (6) Anna is taller than Bill is true in the actual world. Suppose now that Anna and Bill are not worldmates, and consider the situation again (namely, the situation in which Anna is taller than Bill). By (MR)’s lights, the proposition (7) Anna is taller than Bill should be true, but only contingently so. Note that what we are dealing with now is a transworld proposition: a proposition concerning individuals from different possible worlds. A contingent truth of (7) thus cannot be analysed by means of restricting ourselves to the domain of a single world. Rather, we use the advanced modalizing strategy and get: (8) Anna is taller than Bill (unrestrictedly speaking). (8) is an extraordinary claim and isn’t sensitive to modal predication. We thus easily arrive at: (9) Necessarily, Anna is taller than Bill as a true proposition, contra the intuition that (7) is a merely contingent truth. 32 Unless stated otherwise, in this Chapter I refer to Jago (forthcoming). )38 Chapter III (EMR) and Advanced Modalizing Problems Another variant presented and defeated by Jago is the disjunctive analysis. Roughly, it is motivated by the advanced modalizing strategy, although the systematic account of possibility has a disjunctive form with ordinary and extraordinary readings as disjuncts: (DA) It is possible that P if and only if either there is a possible world, w, such that at w, P or (taking ‘it is possible that…’ to be redundant) P. (DA) faces the following problem. Consider a world-bound individual, my pug. Let us call him Charlie (c*). Charlie is not a world, but he could be since there is nothing in (MR)’s ontology that would forbid it. 33 That means that Charlie could exist without anything else, including me, as his part. Thus: (10) (c*) is not a world, but it could be a world lacking Martin (m) as a part. According to (DA) we get: (11) ¬Wc* & (∃w∃x∃y[Ww & Pxw & Cxc* & Pyw & Cym & Wx & ¬Pyx] ∨ [Wc* & ¬Pmc*]), where ‘W’ stands for ‘is a world’, ‘P’ stands for ‘is a part’ and ‘C’ stands for ‘is a counterpart of’. Apparently, ¬Wc* from the conjunction of the first disjunct contradicts the second disjunct, and (11), as it stands, thus implies: (12) ¬Wc* & (∃w∃x∃y[Ww & Pxw & Cxc* & Pyw & Cym & Wx & ¬Pyx]). This, however, is incompatible with (MR)’s metaphysics because from (13) 33 (Ww & Pxw & Cxc* & Pyw & Cym & Wx & ¬Pyx) See (f) from (MR)’s exposition in Chapter I. )39 Chapter III (EMR) and Advanced Modalizing Problems we get w = x which, in fact, leads to Pyx & ¬Pyx (contradiction). 34 Jago therefore concludes that the disjunctive analysis has problems of its own. Let us now discuss a third alternative mentioned by Jago—the world-free analysis. Roughly, such an analysis drops the quantification over worlds altogether, although the phrase ‘it is possible that…’ still has the semantic effect of quantifying over counterparts. The systematic account of possibility runs as follows: (WF) It is possible that A(c1…, cn) if and only if there are counterparts c’1..., c’n of c1…, cn, respectively, such that A(c’1…, c’n). The trick is that only counterparts (and not whole worlds) are needed in the analysis. In practice, the world-free analysis works well in understanding (4) by assigning it the unrestricted content à la Divers (1999a). For, according to counterpart theory (Lewis 1968), every individual is a counterpart of itself, and the inference from ‘A(c1…, cn)’ to ‘it is possible that A(c1…, cn)’ is thus unproblematic. All we need are counterparts c1…, cn to express the content of (4). So far, so good. Jago, however, views as problematic the apparent consequence that the world-free analysis forces us to treat every de dicto modal claim as a de re modal claim. This is because of (WF)’s analysis of (14): (14) There could have been no philosopher. The world-free analysis treats (14) as: (15) There are no philosophers (unrestrictedly speaking), which is false. What a proponent of the world-free analysis might do is to interpret (15) as (16): This is because both w and x are worlds, and x is a part of w. The consequence comes from the postulates of Lewis’s system (1968). 34 )40 Chapter III (16) (EMR) and Advanced Modalizing Problems The actual world could have contained no philosophers, meaning that the actual world has a counterpart, w, such that w does not contain any philosophers. Although such a reading is not problematic, its generalization, (17) There exists a counterpart w of the actual world such that, at w, something is not a part of w, causes serious problems. For we might admit that ‘at w’ does not restrict all quantifiers within its scope to w. Unless we restrict at least some quantifiers within its scope, however, the ‘it is possible that…’ prefix cannot play the role of modal operator at all (p.7). Therefore, the ‘something’ in (17) must be restricted to w, and this makes (17) untenable. The next two alternative analyses that I want to mention here are the many worlds analysis (MW) and the plurality of worlds analysis (PL). The former accounts for (4) by allowing the counterparts to be parts of many worlds rather than a single one. This analysis has the following form: (MW) There are worlds w1, . . . , wn and counterparts a’1,…, a’n of a1,…, an, respectively, such that each ai is a part of wi and, at the plurality of worlds w1, . . . , wn: A(a’1 ,…, a’n). It is clear that the many worlds analysis accommodates all ordinary modal claims, has resources to interpret (4) and, contra (WF), preserves the standard way of analysing de dicto modalities. The problem with (MW) arises from something else. Recall Anna and Bill from (7). Although Anna and Bill are not worldmates, the principle of unrestricted summation35 dictates that there exists a mereological sum that comprises Anna and Bill exclusively. Let us call it AnnaBill. Consequently, 35 According to (MR)’s version of unrestricted summation, sums of individuals from different worlds count as individuals, but they do not count as possibilia. )41 Chapter III (18) (EMR) and Advanced Modalizing Problems Anna and Bill are not worldmates, and there exists AnnaBill (~Wmab & AnnaBill=Anna⊔Bill), where ‘Wm’ stands for ‘is a worldmate of’. Given possibility introduction, from (18) we get: (19) It is possible that (~Wmab & AnnaBill=Anna⊔Bill), which, according to Jago, leads to the following many worlds translation: (20) ∃w1∃w2∃w3∃x1∃x2∃x2(Ww1 & Px1w1 & Cx1a & Ww2 & Px2w2 & Cx2b & Ww3 & Px3w3 & Cx3AnnaBill & at w1, w2, w3: ~Wmx1x2 & x3 = mereological sum of x1 and x2). Since we are dealing with extraordinary modalizing, the phrase ‘at w1, w2, w3’ contains no quantifiers and is redundant (see the advanced modalizing strategy). Note, however, that ‘Px3w3’ and ‘x3 = mereological sum of x1 and x2’ implies ‘Px1x2w3’ (given the transitivity of the parthood relation). If worlds are closed under the worldmate relation, then Wmx1x2 . And this contradicts (20). The last alternative analysis I want to mention is the plurality of worlds analysis. It might be seen as an improvement on the many worlds analysis because it avoids the previous problem. The shape of the analysis is as follows: (PL) There are worlds w1,…, wn and counterparts a’1,…, a’n of a1,…, an, respectively, such that each ai is a part of the plurality w1,…, wn and, at the plurality w1,…, wn: A(a’1 ,…, a’n). Jago admits that (PL) avoids the problems mentioned thus far regarding the intuitive truths of extraordinary modal claims. It accounts for (4), and it also avoids the other problems: since we can now say that ‘for some plurality of worlds, a1,…, an have counterparts somewhere in that plurality’ (p. 8, emphasis in original), none of the abovementioned problems arise. Jago points out another source of problems, however: truth simpliciter. )42 Chapter III (EMR) and Advanced Modalizing Problems The problem of truth simpliciter comes with an intuition that some propositions, just like (4), are ‘simply’ true. Such an intuition takes as its starting point that propositions uttered in the world of the utterance are true simpliciter. It is probably best to quote Jago in full on this point: Suppose we take the conservative option and continue saying that, by definition, an utterance of ‘A’ is true simpliciter iff it is true relative to the world of utterance. Then it is analytic that, for restricted contents, truth simpliciter requires truthrelative to some world. So it is also analytic (given how Lewisian metaphysics defines ‘world’) that truth simpliciter requires truth relative to some spatiotemporally connected entity. But, given the plurality-of-worlds analysis, some possible truths are not like this. ‘There are exactly two penguins, and they are not worldmates’ is false but possibly true, on the plurality of worlds analysis. The problem is that, on the present approach, it is analytic that it is false simpliciter, and an analytically false statement cannot possibly be true. So we must reject this first option. We avoid the problem if we allow that ‘A’ is true simpliciter iff it is true relative to a plurality of worlds, including the world of utterance. But there are many such pluralities. If we require an utterance ‘A’ to be true relative to all such pluralities in order for it to be true simpliciter, then very little will be true simpliciter. We won’t capture the intuitive truth (simpliciter) of ‘there are no unicorns’ (under its restricted reading), because there are pluralities of worlds which include both ours and a world of unicorns. If on the other hand we require an utterance ‘A’ to be true relative only to some such pluralities, in order for it to be true simpliciter, then we arrive at a contradiction: ‘there are no unicorns’ will come out both true (simpliciter) and not true (simpliciter). So we must reject this option, too. Note that the problem applies equally to the world-free and many-worlds analyses. (Jago, forthcoming, 9) )43 Chapter III (EMR) and Advanced Modalizing Problems When it comes to truth simpliciter, Jago concludes, the plurality of worlds analysis is among those that fall short as overall analyses of possibility. To sum up, there are various alternative versions of (MR), all of which aim to provide content for extraordinary modal claims. They do solve several problems concerning extraordinary claims; each variant, however, has problems of its own. Of course, Jago’s arguments should not be seen as conclusive, and proponents of the alternatives might (and in fact do) offer replies to the abovementioned objections. For the sake of brevity, I will not go into the details. Rather, in the rest of this chapter I present yet another theory of extraordinary modal claims—one which Jago ignores, but one which can meet his counterexamples. I call it extended modal realism (hereafter EMR). 36 3.4 (EMR): The Analysis In this section, I consider a different variant of (MR): (EMR). First, I characterize the key features of (EMR) and the crucial differences between it and (MR). Second, I provide possible responses to Jago’s objections from (EMR)’s perspective. 3.4.1 (EMR) Characterized Again (EMR) is a thesis according to which possible and impossible worlds exist. This alternative shares with (MR) the theoretical conviction that possible worlds play an important theoretical role and that in order to do so they should be understood in a full-blooded realistic sense. Its systematic account of possibility mirrors (MR)’s account of possibility: (P) It is possible that P if and only if there is a possible world, w, such that at w, P. It switches, however, to an impossibilist account of non-possibility, or impossibility via (I*): Jago might have ignored (EMR) for at least two possible reasons. He might have taken (EMR) to be incompatible with (MR), such that it cannot serve as a modification of the latter. Alternatively, he might have viewed the theory as too ‘incredible’ to be taken seriously, and thus as undeserving deeper attention. 36 )44 Chapter III (I*) (EMR) and Advanced Modalizing Problems It is impossible that P if and only if there is an impossible world, i, such that at i, P In contrast to (MR), (EMR) identifies impossibility as a going-on in some impossible world rather than a going-on in no world whatsoever. Such an extended ontology contributes to a systematization of data by allowing for fine-grained distinctions between properties, propositions and conditionals that are unavailable in standard possible worlds semantics. 37 As Yagisawa writes, (EMR)’s framework contrasts with (MR)’s in an additional way: [(MR)’s] metaphysics contains an obvious answer to this question, namely: ‘Yes, the entire logical space is a maximal universe of discourse. No variable could range over anything beyond it, for there is absolutely nothing beyond it.’ Indeed, for Lewis, our logical space is a unique maximal universe of discourse. According to extended modal realism, on the other hand, there is no maximal universe of discourse, let alone a unique one. Our logical space, namely the logical space in which our world is located, is not a maximal universe of discourse. There are other logical spaces. This last sentence contains an existential quantifier with a variable ranging over the super logical space, in which logical spaces exist with various interrelationships. Is the super logical space not a maximal universe of discourse? No. Modal statements about the super logical space will involve quantifying over the super super logical space. The hierarchy of (super)n logical space continues indefinitely. (Yagisawa 1988, 201-202)38. Not surprisingly, the motivation to go this extra mile comes partially from the need to express extraordinary modal claims (that is, claims about trans-world individuals), and 37 For such applications see Priest (1997), Nolan (1997), Berto (2009) and Jago (2012), among others. The notion of logical space is defined as a sum of all logically possible worlds. The logical laws that govern every world in logical space are the same, and alternative logics characterize alternative logical spaces. The possible-impossible distinction is thus explained via the ‘belongs to a logical space’ relation. A world, w, is logically possible according to a world, w1, if both worlds belong to the same logical space. If worlds belong to different logical spaces, they are impossible according to each other. Contra (MR), (EMR) multiplies the plurality of worlds to a plurality of a plurality of worlds and thus constructs a complicated hierarchy of logical space. 38 )45 Chapter III (EMR) and Advanced Modalizing Problems partially from a need to account for matters that are impossible. In terms of the former, Yagisawa identifies familiar propositions as examples that demonstrate the need to extend (MR)’s ontology: i) w could have been inaccessible from w’ (where w is accessible from w’). ii) There could have been more worlds than there actually are in our logical space. iii) I could have been there instead of here (where ‘there’ refers in context to another possible world and ‘here’ refers in context to the actual world α). iv) Logical facts (e.g. the law of excluded middle) could have failed to obtain. The feature common to the examples is that in order to properly interpret their truth and falsity, we need alternative logical spaces. They all come from an intuition that the propositions, whether true or false, require a ‘possibility-localizer’ (see the Introduction). (MR)’s logical space is as it is, and it cannot be otherwise. This means that not only do the above propositions turn out to be false, but (MR) cannot even express them. (EMR), by contrast, allows for quantification over different logical spaces and allows for logical spaces in which ‘w is inaccessible from w’, ‘there are more worlds than there are in the logical space of which our world is a part’, ‘I am there instead of here’ and (our) ‘logical facts fail to obtain’, respectively. This is because (EMR) treats possibility as a going-on in a possible world, impossibility as a going-on in an impossible world, and trans-world possibilities such as (i)–(iv) as relations between logical spaces. Ordinary modalizing is to be analysed in terms of possible worlds, and advanced modalizing is to be captured by appeal to a higher level of world-like entities. There is therefore general agreement that given the theoretical framework of (EMR), we gain resources to account for the content of (i)–(iv). Moreover, a notable theoretical advantage of (EMR) over (MR) is that it treats ordinary and extraordinary modal claims systematically, by localizing the possibilities to worlds and logical spaces, respectively. 39 Of course, the analysis comes at a considerable ontological cost. See Kiourti (2010, Chapter IV) for a way of avoiding the disadvantage of (EMR) regarding pre-theoretical data about the actual, the possible and the impossible. 39 )46 Chapter III (EMR) and Advanced Modalizing Problems 3.4.2 (EMR) and Advanced Modalizing Problems In this section, I briefly return to the objections from (3.3) and provide responses on (EMR)’s behalf. As I try to show, (EMR) taken as a whole has resources to overcome these problems without sacrificing (MR)’s realistic spirit. Let us start with (4). (EMR) suggests taking ‘it is possible that…’ in (4) as an existential quantification over logical spaces. (4) thus gets an interpretation which both gives a content to (4) and results in its truth: 40 (4EMR) It is possible that there are possible worlds if and only if there is a logical space such that there are possible worlds. Importantly, the phrase ‘it is possible that…’ is not redundant in (4*). It is a quantifier over world-like entities. It does not range over single worlds but is instead a higher-order quantification over logical spaces. Logical spaces include worlds as their parts, and any way the sum of worlds might be, there is a logical space that is that way. Therefore, (4) does not present a problem for (EMR). What about the intuition that (7) is true, and only contingently so? Jago correctly points out that the advance modalizing strategy falls short when it comes to analysing the data correctly. For, according to (AM), (7) turns out to be true unrestrictedly. Given the redundancy of modal operators in such situations, (7) turns out to be possible, contingent and necessary in any situation at any time. (EMR) does not face this problem. Recall that (AM) rests on making the ‘it is possible that…’ prefix redundant. What (EMR) does, by contrast, is to make the quantifier range over logical spaces. Suppose that Anna is taller than Bill, Anna and Bill are not worldmates, and it is contingent that Anna is taller than Bill. The situation is captured by (EMR) along the lines of (7EMR): Yagisawa, describing Lewis’s strategy (1986), states: ‘[t]o say that unicorns are possible is to say that there are some possibilia which are unicorns; unicorns are possible; therefore, there are some possibilia which are unicorns’ (Yagisawa 1988, 181-182). 40 )47 Chapter III (EMR) and Advanced Modalizing Problems (7EMR) It is contingent that Anna in w1 is taller than Bill in w2 if and only if there is a logical space, L1 such that counterparts of w1 and w2, w’1 and w’2, belong to L1, Anna is a part of w’1, Bill is a part of w’2, and in L1 Anna is taller than Bill. Recall now the disjunctive analysis and the objection Jago raised against it. The objection, put simply, construes a contradiction in (DA)’s analysis because (12) implies: (13) (Ww & Pxw & Cxc* & Pyw & Cym & Wx & ¬Pyx), which ipso facto implies Pyx & ¬Pyx. (DA)’s failure to capture the truth conditions correctly lies in the consequence ‘x = w’, which derives from both the impossibility of there being and not being a Charlie-world and the location of the counterpart relation within a single logical space. (EMR) offers a different analysis: (10) (c*) is not a world, but it could be a world lacking Martin (m) as a part, according to which there is a logical space in which (c*) and (m) are worldmates, and it is possible that (c*) and (m) are not worldmates. The trick is that we do not posit a single world that ‘has and does not have (m) as its part’ (contra Jago). Rather, in order for it to be true that x1 and x2 possibly exist as nonworldmates, in some logical space x1 and x2 must have counterparts that are not worldmates. Such an analysis gives us a (c*)-world at which Charlie has no worldmates. And this is enough to account for (10). What about (14)? It says that ‘there could have been no philosophers’. Jago argues that the world-free analysis delivers bad results because it is true simpliciter that there are philosophers, contra (FW)’s analysis. What can (EMR) offer? Recall again that besides worlds, (EMR) has logical spaces at its disposal, and the analysis is not confined to (MR)’s resources. I propose the following: (15EMR)There could have been no philosophers if and only if there are no philosophers in a logical space other than that of which @ is a part. )48 Chapter III (EMR) and Advanced Modalizing Problems Note that (15EMR) is not vulnerable to the consequence Jago poses because ‘there is a logical space’ behaves as a quantifier over logical spaces. Thus ‘there are no philosophers’ is not true simpliciter on this approach. Rather, the content of (15EMR) is that there are no philosophers in an alternative logical space, not that there are no philosophers (unrestrictedly or worldlessly speaking). Let me now proceed to Jago’s counterexample to (MW). It starts off by assuming that if Anna and Bill exist in different worlds, their exclusive mereological sum, AnnaBill, exists. Given possibility introduction, it is possible that Anna exists, Bill exists, Anna and Bill are not worldmates, and AnnaBill exists, which, according to (MW), is inconsistent. (EMR) interprets (20) as treating the ‘at w1w2w3’ phrase non-redundantly, namely as a quantifier over worlds in an alternative logical space: (20EMR)It is possible that (~Wmab & AnnaBill) if and only if there is an alternative logical space, L1, in which a, b and AnnaBill have counterparts, x1, x2, x3, which exist in different worlds w1, w2 and w3, respectively. Again, ‘there is an alternative logical space’ has the form of a quantifier, and therefore, contra Jago, ‘at w1, w2 and w3’ is not redundant. And this suffices to block the step from (20) to a plain contradiction. Finally, Jago admits that (PW) is immune to the advanced modalizing problems and takes it to be the best option among (MR)’s alternatives. However, he acknowledges the problem regarding truth simpliciter. Proponents of (MR) generally look at the notion of ‘truth’ itself as defined in terms of the ‘truth in’ relation. This means that a certain sentence is true at a world if and only if it is true when we quantify over all the things in that world. By the same token, when (MR) argues for the existence of merely possible individuals, it differentiates strictly between actual truths and truths simpliciter. In particular, we get the actual truths when we quantify over less than everything there is (i.e. Lewis’s plurality of worlds), thereby implicitly or explicitly restricting ourselves to the actual world and its parts. Truths simpliciter, by contrast, are not restricted to any particular part of (MR)’s pluriverse. Omitting all restrictions put on our quantifiers, we quantify over everything there is, i.e. over the whole plurality of worlds. )49 Chapter III (EMR) and Advanced Modalizing Problems (EMR) bites the bullet here and denies that there are absolute possibilities and impossibilities—that is, possibilities and impossibilities that are true irrespective of domain restrictions. In fact, this is a straightforward consequence of there being a complicated hierarchy of logical spaces. 41 In Yagisawa’s words: For any kind of possibility K, the totality of K-possible worlds (which we may call K-space) could possibly be otherwise. The ‘possibly’ here points to a different kind of possibility, K’, giving rise to K’-possible worlds. The totality of K’-possible worlds (which we may call K’-space) could possibly be otherwise, pointing to yet another kind of possibility K’’ and giving rise to K’’-possible worlds. And so on. (Yagisawa 2010, 204) Again, (EMR), unlike other variants of (MR), views logical spaces as hierarchically embedded, and this feature goes against absolute possibility and necessity and, more importantly, against truth simpliciter. 42 The strategy of evaluating modal propositions thus divides exclusively into confinement to single worlds and logical spaces. Regarding the latter, possibilities pertaining to logical space as a whole—all of those discussed in section 3 —are to be analysed via relations between logical spaces, and the theory thus avoids the unwelcome consequences discussed by Jago. 43 3.5 Conclusion Extraordinary modal claims present problems for both (MR) and its alternatives. In this chapter, I presented some of them and discussed objections to them. I also presented a When it comes to (EMR)’s home language, the situation is different. To be sure, according to (EMR), there are possible worlds, there are impossible worlds, there are logical spaces, etc. Such claims are true irrespective of domain restrictions. But this is to be expected and should not present a serious problem for (EMR). 41 Moreover, Jago relies on analyticity when constructing his argument. But it is far from clear that in (MR) ‘it is [also] analytic … that truth simpliciter requires truth relative to some spatiotemporally connected entity’ (Jago, forthcoming, p. 9). 42 There is much more to be said about the ontological setup of (EMR) since it faces the threat of inconsistency, as well as that of expressive deficiency (cf. Lewis (1986a, 7, fn. 3), Vander Laan (1997), Berto (2009) and Jago (2014), among others). For possible strategies that avoid the inconsistency and the expressive limitations, see (Vacek 2013a) and Kiourti (2010), respectively. 43 )50 Chapter III (EMR) and Advanced Modalizing Problems different option and tried to show how it can be squared with (MR) and how it can deal with advanced modalizing problems. My discussion is far from conclusive, however, as several questions regarding (EMR)’s ontology remain open. What is it for a world to belong to one logical space rather than another? How might such an account deal with other ordinary and extraordinary claims not discussed in this chapter? And how does this account fare on a cost/benefit scale in comparison with (MR)’s alternatives and with various actualist's theories? Some of these questions point to interesting avenues to be discussed in next chapters. )51 Chapter IV (Extended) Modal Dimensionalism Chapter IV Technologies that may be realized in centuries or millennium include: warp drive, traveling faster than the speed of light, parallel universes; are there other parallel dimensions and parallel realities? Time travel that we mentioned and going to the stars. Michio Kaku 4. (Extended) Modal Dimensionalism 4.1 Introduction According to (MR), possible worlds are concrete, spatio-temporal systems. (EMR) goes even further and claims that possible and impossible worlds are concrete spatiotemporal systems. However, it is commonly held that if we are willing to accept impossible worlds, they must not be conceived of as spatio-temporal systems. If we suppose that there are impossible worlds that make certain inconsistencies true, and if we suppose that those worlds represent those inconsistencies in a genuine way, then we are committed to the reality of true inconsistencies. Modal dimensionalism (hereafter EMD)44 is realism about spaces, times and worlds—metaphysical indices that make objects spatial, temporal and modal, respectively, and that play the role of alethic relativizers, i.e. items to which matters of truth are relativized. In Worlds and Individuals, Possible and Otherwise, Takashi Yagisawa characterises (EMD) as a theory which ‘shares a certain theoretical conviction Unless stated otherwise, (EMD) refers to Yagisawa (2010) as the orthodox version of modal dimensionalism. I use (EMD) instead of mere (MD) in order to stress the presence of impossible worlds in the theory. 44 )52 Chapter IV (Extended) Modal Dimensionalism with David Lewis’s classical modal realist theory and also, superficially, with anti-Lewisian actualist theories’ (Yagisawa 2010, 1; see also Yagisawa 2002). (EMD) is a metaphysical thesis according to which spatial, temporal and modal indices make objects spatial, temporal and modal, respectively. In contrast to (MR), (EMD) allows for impossible worlds—entities that have proved their utility in various branches of philosophy. In this chapter, I argue that (EMD), despite having ersatzist features, offers a feasible option when it comes to impossible worlds. In particular, I will try to show how one can be a ‘quasi’ modal realist and still have a consistent ontology of possible and impossible worlds. Firstly, I discuss the crucial difference between (MR) and (EMD) (4.2). Secondly, I present problems of (EMD) (4.3) regarding both possible worlds (4.3) and impossible worlds (4.4) as well as propose solutions. 4.2 (EMD) vs. (MR) According to (EMD), worlds are not spatio-temporally closed universes. Nor are they abstract representations of the way the world could have been. Rather, worlds are defined as modal indices that are (but do not exist) 45 along the world’s temporal and spatial indices. What (MR) describes as the actual world, or the universe, (EMD) calls the actual-worldstage of the universe. The universe is the comprehensive subject of possibility and necessity (Yagisawa 2010, 44). Possible worlds are neither concrete nor abstract, and whether they are objects at all is an open question: ‘[I] take moments of time to be real but I am noncommittal about whether they are non-concrete objects of some kind. If they are, I will be happy to accept that worlds in my sense are also non-concrete objects of some kind’ (Yagisawa 2010, 179, fn. 7). One way or the other, there is a plurality of worlds - a plurality of different world-stages of the same universe. Modal space contains many concrete objects, all of which are modal parts of one and the same universe. Some of them may be unified by spatiotemporal relatedness, some may be unified by another relation, and others might not be unified by any relation other than that of being part of the universe and whatever that requires (Yagisawa 2010, 45). 45 For Yagisawa, reality is fundamental and monadic, whereas existence is domain relative. )53 Chapter IV (Extended) Modal Dimensionalism Pivotal claims of (EMD) are summarized in the following passage: Ordinary individuals typically exist at many metaphysical indices of each of the three kinds: time, space, and world. The airplane at Heathrow exists at many temporal points and periods, many spatial points and extended regions, and many possible worlds. Suppose that it exists at different times t2 and t2 (for example, yesterday and today), different spatial regions r1 and r2 (for example, where its fuselage is and where its wings are), and different possible worlds w1 and w2. The fuselage is not identical with the airplane, but the airplane is where the fuselage is, at r1. The airplane is also where the wings are, at r2, even though the wings are not identical with the airplane. The airplane is at r1 but not wholly at r1, and at r2 but not wholly at r2. The fuselage is the plane’s spatial part, and so are the wings. The airplane is at every spatial region where some spatial part of the airplane is. Similarly for times and worlds. (Yagisawa 2010, 53) All concrete objects are temporal objects because they exist in time. It is times that make them temporal objects. Times make concrete objects temporal by being such that those objects exist at them. To exist at a temporal index means to be a temporal object. Temporal indices exist independently of the events that occur in them. Since temporal indices make objects temporal objects, it is temporal indices that do the representing of temporality. Concrete objects are spatial objects too. They exist in space: a metaphysical index responsible for their being spatial objects. Like temporal indices, spatial indices are primitive, although the way they make the object spatial is non-trivial. All concrete objects are also modal objects since they exist at different worlds. It is modal indices that are responsible for concrete objects’ being modal. Worlds are makers of modal objects, although they themselves are not modal objects. In sum, (EMD) divides reality into concrete individuals on the one side and metaphysical indices on the other. The second crucial feature of (EMD) rests on taking the analogy between spatial, temporal and modal talk seriously. I existed yesterday, I exist today, and I will (probably) exist tomorrow. Also, I exist where my arms are, where my legs are, where my head is, etc. Analogously, then, I am a PhD student, although I could be a football player. That means )54 Chapter IV (Extended) Modal Dimensionalism that I exist in the actual circumstances as well as in merely possible ones. The truth of the above sentences depends on temporal, spatial and modal indices to which we relativize their truth. More generally, temporal, spatial and modal indices are alethic relativizers—i.e. those items to which matters of truth are relativized. Ontologically, they are on a par. Although metaphysically primitive, temporal, spatial and modal indices are further explicable in more graspable terms. In terms of modality, (EMD) says that, in addition to spatial and temporal dimensions, the universe also spreads out in a modal dimension. The actual world is one of many indices, namely the one at which the universe is the way it actually is. ‘Actually’ is to be understood in the very same manner as ‘now’ is - i.e. ostensively. Importantly though, it does not refer to a concrete mereological sum of individuals any more than ‘now’ refers to something concrete. ‘Actual’, ‘here’ and ‘now’ refer to metaphysical indices.46 Generalizing the idea, (EMD) positions the temporal tense in parallel with a spatial and a modal tense, thus introducing a correspondence between tenses on the one hand and metaphysical indices on the other. Times are distinct from events that happen in them. Also, spatial points (or extended regions) are not identical with what occupies them, and worlds are not identical with (MR)’s universes. Rather, they are points in modal space in a way that is analogous to how a temporal instant is a point in temporal space and a spatial point is a point in (at least three-dimensional) spatial space. 47 A further departure from (MR) is the analogy between trans-temporal and transworld identification. Lewis sympathized with the former (which holds that we persist through time by having distinct temporal stages at different times) but formulated several objections to the latter (Lewis 1986a, 218-219). (EMD), on the other hand, accepts such an analogy and posits the so-called ‘Closest-Continuer’ relation holding between modal parts of a single individual. The relation is defined along the following lines: A modal stage x at a possible world w1 and a modal stage y at a different possible world w2 are parts of the same modally extended object of a kind K if and only if Taking the analogy seriously, Yagisawa introduces a new word, mau, combining the temporal ‘now’ with its (m)odal counterpart (cf. Yagisawa 2002, 29). 46 47 (Yagisawa 2010, 27). )55 Chapter IV (Extended) Modal Dimensionalism there is a chain of possible worlds from w1 to w2 ordered by the overall similarity relation such that x and some modal stage, x+1, at the next world in the chain are sufficiently similar to each other in relevant respects and are each other’s closest continuer at their respective worlds, x+1 and some modal stage, x+2, at the next world in the chain are sufficiently similar to each other in relevant respects and are each other’s closest continuer at their respective worlds, ... , and x+n and some modal stage, x+n+1=y, at the next world,w1, in the chain are sufficiently similar to each other in relevant respects and are each other’s closest continuer at their respective worlds, where the sufficient similarity, relevant respects, and closeness are relative to the kind K. (Yagisawa 2009, 109) This, in a nutshell, is the Closest-Continuer relation operating on the modal stages. Furthermore, (EMD), unlike (MR), accepts impossible worlds. Again, such worlds are neither concrete nor abstract, but are as real as possible worlds. Some impossible worlds are worlds at which logical impossibilities obtain. These are logically impossible worlds. Some impossible worlds are worlds at which metaphysical impossibilities obtain. In addition, there are impossible individuals. They do not exist in the domain of possible objects. They exist in the domain of metaphysically impossible objects, yet given the ‘Closest-Continuer’ relation between world-stages, they also exist at some possible worlds (by having stages that exist at those worlds). A world is impossible relative to another world if the two worlds inhabit different logical spaces.48 One logical space comprises all and only logically possible worlds, while the other logical space comprises all and only logically impossible worlds: ‘[t]he logic that governs every world in logical space is the same. So, alternative logics characterize alternative logical spaces’ (Yagisawa 2010, 184). However, instead of full modal reductionism, (EMD) prefers soft reductionism, according to which a) temporal, spatial and modal indices are taken to be metaphysically simple and b) the at-a-worldness relation is primitive. These features of the theory place it somewhere between (MR) and actualism, and more importantly, between two modes of representation: genuine and ersatz. 48 Recall that I’ve already sketched the metaphysics of logical spaces in (3.4.1) as well as in footnote 38. )56 Chapter IV (Extended) Modal Dimensionalism Next, (EMD), as opposed to (MR), introduces modal tense. To be a modal tenser might mean several things, but two aspects are especially important for my purposes. First, modal-tensed propositions belong to the everyday terminology used by metaphysicians, and modally tensing verbs is something philosophers already engage in. Second, though, the modal tense approach is not merely a conceptual approach that would systematize our use of certain words. The modal tense approach is a metaphysical approach. It takes for granted that ‘some important modal facts are modal-tensed facts, i.e. they can be designated or quantified over adequately only in modal-tensed terms, and that no important modal facts are modaltenseless facts, i.e. none of them are such that they can be designated or quantified over adequately only in modal-tenseless terms’ (Yagisawa 2010, 73). The working hypothesis of (EMD) is the existence of the actuality tense, the meremetaphysical-possibility tense, the metaphysical-impossibility tense, and, rather controversially, a tense specifically for predications concerning modal space at large, subscripted ‘a’, ‘p’, ‘i’ and ‘m’, respectively. For example, ‘Martin is actually a philosopher’, where ‘actually’ is read non-rigidly,49 is expressed as ‘Martin isa a philosopher’. By contrast, predications concerning merely possible situations require the mere possibility tense. So, ‘Martin could have been a football player’ is expressed as ‘Martin isp a football player’. Finally, the metaphysical impossibility tense helps us to articulate goings-on at some impossible worlds. Thus ‘it is impossible that Martin is a PhD student and not a PhD student at i1’ gets a modal-tensed interpretation by using ‘i’ in the tense: Martin is a PhD student and is not a PhD student at i1. Suppose now that the above sentences are uttered at the actual world. Then, the modal tensing approach interprets ‘Martin isp a football player’ as true as it is evaluated at the actual index, the actual world, if and only if Martin is a football player at some non-actual possible index and Martin isi a PhD student and not a PhD student at some non-actual impossible world. To sum up, (EMD) presents a two-categorical ontology. It posits concrete individuals and concrete worlds in the same way that traditional (MR) does. In addition, however, it posits metaphysical indices: regions to which concreta belong. The universe and its parts have spatial, temporal and modal dimensions as they extend in time, space and worlds. The universe and its parts have temporal, spatial and modal stages and it is those 49 For the difference between rigid and non-rigid uses of the actuality tense, see Yagisawa (2010, 76-77). )57 Chapter IV (Extended) Modal Dimensionalism stages that represent temporality, spatiality, and, most importantly for our purposes, modality. One thing that (EMD) has going for it is thus that the analogous theoretical roles of temporal, spatial and modal indices pave the way for a systematic and unified metaphysics of modality. 4.3 Some Problems for (EMD) In this section, I discuss a dilemma directed against (EMD)’s account of possible worlds. The dilemma comes from Jago (2012, 2013) and aims to show that (EMD)’s metaphysics is incoherent. I outline both horns of the dilemma and argue that (EMD) as a whole, rather than the fragment used in the dilemma, is not subject to it. 4.3.1 Problems of Possible Worlds and their Diagnosis To repeat, (EMD) claims that ordinary things (including the whole universe) are trans-temporal, trans-spatial and trans-modal sums. That means that the de dicto possibility schema has the form of (P): (PMD) It is possible that P if and only if there is a universe modal stage, u1, such that P holds at u1, and de re modality is expressed as (PMD*): (PMD*) An object has a modal property, G, if and only if it has a world-stage that has G as one of its properties. Note that (PMD) and (PMD*) ‘localize’ possibilities to modal indices. World-stages are neither concrete (as (MR) takes them to be) nor abstract (as modal ersatzists would insist). They are possibility-localizers in the same way that times are temporal localizers and spaces are spatial localizers. )58 Chapter IV (Extended) Modal Dimensionalism Jago’s starting point is that (EMD)’s analysis of de re modality goes against widely accepted opinions about contingent matters. The objection as follows: any possible entity whatsoever that might be F has a world-stage that is F necessarily or contingently. Although I am actually a PhD student, I could have been a football player (¬FM ∧ ◇FM). According to (EMD), there is a possible world, w, at which my football player stage exists. Call this stage fw. Now, the question is: is fw necessarily or contingently a football player? Suppose it is the former. This entails that I am possibly necessarily a football player, ◇
Pb. Assuming ‘◇
A →
A’ is a theorem of the most plausible modal logic, it follows that I am necessarily a football player. But I am actually not a football player. Put more formally: 1. Martin is not a football player, but he could have been a football player (~FM & ◇FM). 2. For some possible world w, some Martin’s w-stage is a football player. [1 by MD] 3. One of Martin’s w-stages is necessarily a football player. 4. Martin is possibly necessarily a football player (◇□FM). [2, Martin has a necessarily football player world-stage] 5. Martin is a football player (FM). [4, given the S5 theorem ◇□A → A] 6. Martin is not a football player and Martin is a football player (~FM & FM). [1 and 5] Contradiction The second horn of the dilemma takes my football player stage to be a football player only contingently. By definition, my world-stages do not have their own stages, and their modal profile is explained by counterpart relations they bear to other world-stages. Suppose now that two world-stages, m and m*, are worldmates if and only if they both exist at the same world-index. Now consider any two world-stages n and n*, for which it holds that they are not worldmates (¬Wnn*). Next, what is true is possibly true, and so it is possible that they are not worldmates: ◇¬Wnn*. Since, according to (EMD), world-stages do not have stages )59 Chapter IV (Extended) Modal Dimensionalism by means of which we analyse their modal profile, the analysis must proceed via a counterpart relation. Thus, there is a world u and there are u-stages nu and n*u such that nu is a counterpart of n, n*u is a counterpart of n* and ¬W nu n*u. But since nu and n*u are both u-stages, by definition they are worldmates: W nu n*u. Contradiction. The formal representation is as follows: 1. Martin is not a football player, but he could have been a football player (~FM & ◇FM). 2. For some possible world w, some Martin’s w-stage is a football player. [1 by MD] 3*. Martin’s w-stage is contingently a football player. 4*. For some world w, Martin’s w-stage is a football player and is a counterpart of Martin’s @-stage. 5* . Martin’s w-stage and Martin’s @-stage are not worldmates (~WorldmateMwM@). 6*. Possibly, Martin’s w-stage and Martin’s @-stage are not worldmates (◇~WorldmateMwM@). [From A→◇A] 7. For some world v, and some v-stages M1v and M2v, M1v is a counterpart of Martin’s w-stage, M2v is a counterpart of Martin’s @-stage, and M1v and M2v are not worldmates (~Worldmate M1v M2v). [6*] 8. M1v and M2v are worldmates and M1v and M2v are not worldmates. Contradiction Having outlined the structure of both arguments, I now proceed to possible ways of responding to them. In particular, I identify three aspects of (EMD)’s theoretical apparatus which, taken together, avoid the undesired consequences. The first has already been identified in Yagisawa (forthcoming) and uses the strategy of rephrasing the necessity horn of the dilemma by the use of modal tensing. The second reply develops an amodalist response and applies to both parts of the dilemma. Roughly, it takes seriously the idea of forbidding predication of any modal property of any world-stage. Finally, I outline a third, )60 Chapter IV (Extended) Modal Dimensionalism more speculative view according to which there is a complex hierarchy of modal spaces. This feature enables (EMD) to meet the contingency horn of the objection. 4.3.2 The Necessity Horn Let me start with the necessity horn of the dilemma. To see how it would work, a brief summary will be helpful. According to (EMD), Martin could have been a football player if and only if one of his world-stages is a football player. If Martin’s world-stage is a football player necessarily, we get a true proposition: ‘Martin is possibly necessarily a football player’. Provided we accept ‘◇
A →
A’ as a theorem, Martin is necessarily a football player. I believe that the modal tensing approach can help here. In particular, if we disambiguate the argument in (EMD)’s light, the contradiction disappears. The disambiguation would take the following form: 1. Martin isa not a football player, but he could have been (isp) a football player (~Fm&◇Fm). 2. For some possible world w, some Martin’s w-stage isp a football player. [from 1 by (EMD)] 3. Suppose that Martin’s w-stage is necessarily a football player. 4. Martin isa possibly necessarily a football player (◇□Fm). [from 2, Martin has a world-stage that necessarily a football player] 5. Martin isp a football player (Fm). [from 4, given ‘◇□A→A’] 6. Martin isa not a football player and Martin isp a football player. Notice, that (6) is a perfectly consistent proposition now. The reason to think so is that occurrences of ‘is’ in the argument are modally tensed, depending on an index they refer to. The situation is analogous to temporal tensing. ‘Martin was a child’ is understood by (EMD) as ‘Martin (simpliciter) has a time-stage, mt, which is a child’. It is always the case that Mt is a child since Mt is a temporal stage of Martin. Martin now, Mn, is not a child, and )61 Chapter IV (Extended) Modal Dimensionalism it is always the case that Mn is not a child. Does this fact make Martin an inconsistent object? No, since Mt and Mn are Martin’s different temporal stages. Modal tensing in fact does two things in the argument. Positively, it blocks the contradiction in the way temporal tensing does. Negatively, though, the tensed version of the necessity horn of the argument invalidates the ◇□A → A theorem. For we do not get an inference from ‘Martin isa possibly necessarily a football player’ to ‘Martin isa a football player’. We do get a modified inference from ‘Martin isa possibly necessarily a football player’ to ‘Martin isp a football player’. This, however, only underwrites a feature of (EMD) - namely, that world-stages lack a modal profile. Recall that world-stages make concrete modal objects. World-stages per se are modally unextended objects, although they make modally extended objects modal objects. It is therefore not a modal predication of modally extended objects that is at stake. Rather, (EMD) forbids predication of any modal property of world-stages.50 In short, concrete objects are modal, whereas world-stages are amodal. And this brings us to the second option for (EMD): the denial of step (3). 4.3.3 Amodalism Amodalism is a negation of modal generalism; it is the view that every proposition has a modal profile. In terms of (EMD), modal generalism would be the view that every object, whether modally extended or modally unextended, has a modal profile. That is, for any modally extended and modally unextended object, we can predicate a modal property. Amodalism with respect to world-stages denies this. Amodalism says that there are some objects—world-stages—that lack modal profiles. Moreover, it turns out that we have independent reasons to prefer amodalism to modal generalism. One of these is the problem of the possibility of the whole logical space’s being otherwise. 51 For example, consider the possibility that logical space might have included more than n-worlds. If the goal of modal reductionism is to explain ordinary and Yagisawa (2015, 6, fn. 9) thinks that to forbid predication of modal properties of world-stages is a radical alternative, although I am not entirely clear on why this ought to be viewed as radical. 50 51 See Yagisawa (1988), Cowling (2011), Divers (1999a, 2002) and Jago (2014) for discussion. )62 Chapter IV (Extended) Modal Dimensionalism extraordinary 52 modal facts in terms of possible worlds and logical space respectively, neither necessity nor contingency can be attributed to logical space. Put differently, the modal reductionist cannot claim that logical space must be such that it contains n-worlds. Nor can she claim that logical space contains n-worlds only contingently. This is because for modal reductionists ‘modal facts—facts about what must and what might be the case— are ontologically and conceptually posterior to facts about the ‘shape of logical space’. 53 If this is so, modal reductionists qua modal generalists cannot analyse modal claims about the whole of logical space. A similar line of argument applies to (MR), according to which worlds are concrete universes, for traditional analysis in terms of possible worlds takes for granted that if something exists, it is also possible that it exists (p→◇p). The fact that Martin could have been a football player is represented by his counterpart, Matrin, who exists in another possible world. Therefore, both Martin and Matrin exist, although it is not possible that they both exist, since they do not inhabit the same possible world. But we still want to be able to talk about a possibility: a mereological sum that consists of [Martin, Matrin]. Amodalism, on the other hand, appears to handle both of the above limitations. Divers (1999a) quite correctly adds that the distinction between ordinary and extraordinary theorizing depends on one’s ontological preferences. If the analysis shows that some individuals have modal profiles while others do not, it is only to be expected that the distinction will play a crucial theoretical role in the theory. The case of MD is not an exception. Modal indices make concrete objects modal objects. An object is a modal object in virtue of having world-stages. World-stages make an object a modal object: a modally unextended sum. To require them to be modal goes against explanatory requirements put on metaphysical explanation. Rather, the amodalism approach stresses the legitimacy of MD when it comes to analysing modality by means of modally unextended objects that lack modal profiles. Such a stance belongs to (EMD)’s ideology and is both theoretically justified and methodologically approved. One worry concerning the modal unextendedness of world-stages remains, though. Although amodalism might have some intuitive appeal in certain cases, it is still the case 52 For the distinction between ordinary and extraordinary modalizing, see Divers (1999a) as well as Chapter III. 53 Cf. Cowling (2011, 383-384). )63 Chapter IV (Extended) Modal Dimensionalism that there is a conflict between modal logic and amodalism. Cowling formulates the argument as follows: Consider any true proposition, Q. Given the (T)-axiom, we can infer ◇Q from Q. The (T)-axiom therefore guarantees that any true amodal proposition will have a modal profile by virtue of being possibly true. As a consequence, we seem forced to choose between amodalism and modal logic (or at least any standard modal logic). (Cowling 2011, 484) The leading idea behind the worry is that there is a correspondence between modal logic and possible worlds talk. That means that any limitations of possible worlds talk are reflected in its logical formalizations, and vice versa. Put simply, this is the worry that possible worlds theory is subservient to the limited powers of modal logic. This is not the case, however. The language of boxes and diamonds provides us with formalization of a part of our possible worlds discourse, but that does not mean that the language formalizes every single bit of it. After all, if this language proves a clumsy instrument for talking about modal matters, we do better to follow the resources of MD directly.54 According to this strategy, we restrict the theoretical power of modal logic to modally extended individuals and leave unextended ones outside the expressive resources of standard modal logic. I thus conclude that the necessity horn of the dilemma can be blocked by modal tensing and by denying that world-stages have modal profiles. In the former, we deny the step from (4) to (5). In the latter, we grant to world-stages an amodal status and thus deny premise (3).55 Moreover, such strategies are in accordance with (EMD)’s ontological assumptions and present legitimate methodological options. 54 Cf. Lewis (1986a, 12-13). The validity of logic is just one part of the problem. The second is about how to build a semantics on amodalism. For a way to meet the worry see Cowling (2011, 486-491). 55 )64 Chapter IV (Extended) Modal Dimensionalism 4.3.4 The Contingency Horn Let me now proceed to the contingency horn. In nutshell, it attacks the position according to which my world-stage is a football player only contingently. Put briefly: modal realistic analysis of contingency introduces a counterpart relation, and the relation as such is incompatible with (EMD). Yagisawa identifies the structural features of the argument in the following way: . . . a world-bound object x1 exists at a possible world w1 and x1 is contingently F; so at some possible world w2, there is a world-bound object x2 which bears R to x1 and which is not F; obviously, ~Wx1x2 (x1 and x2 are not worldmates); hence ◇~Wx1x2; thus, at some possible world, there are y1 and y2 - so Wy1y2 - such that y1 and y2 bear R to x1 and x2, respectively and ~Wy1y2; therefore, at some possible world, Wy1y2 and ~Wy1y2, which is a contradiction. (Yagisawa 2015, online first, 7) At least two responses are available to (EMD). The first response was already mentioned in the previous section: the amodalist’s approach to world-stages. Put simply, it seems illegitimate to formulate the objection from the modal profile of world-stages. World-stages make objects modal objects without being modal objects themselves. And a straightforward rejection of (3*) follows from this commitment. The second response is a bit more complex and relies on the hierarchy of modal spaces. Although Yagisawa does not provide a detailed specification of the notion of ‘logical space’, he approaches it from several angles. First, a logical space consists of all and only worlds which form an equivalence class under the largest accessibility relation. Second, for any world w, the logical space that includes w includes all and only worlds that are logically accessible from w. Third, within a logical space, any world is logically accessible from (i.e. possible relative to) any world. That means that any world that lies outside a given logical space is not accessible from (possible relative to) any world in that logical space and belongs to a different logical space (Yagisawa 1988, 182). Yagisawa (2010) adds a bit more. For example, logical space contains many concrete objects, all of which are modal parts of one and the same universe; the logic that governs every world in logical space is the same, )65 Chapter IV (Extended) Modal Dimensionalism while alternative logics characterize alternative logical spaces. Logical spaces are systematized into a system K, defined in the following way: for any K (where K stands for a particular kind of possibility): (I) K-space is the totality of all K-possible worlds. (II) K-space might have been different. (III) Possible difference is to be understood in terms of a plurality of alternatives. The system of K-spaces is hierarchical, complicated, and difficult to understand completely. This, however, does not mean that we should give up exploring it. On the contrary, a lack of understanding is an impetus for further investigation.56 The response to the contingency worry thus proceeds as follows: . . . the truth condition for ‘x1 and x2 possibly exist as nonworldmates’ is not that at some possible world x1 and x2 have counterparts which are not worldmates, but instead that in some modal space x1 and x2 have counterparts which are not worldmates. Since in this modal space x1 and x2 themselves existm - x1 existsp at w1 and x2 existsp at w2 - and x1 and x2 arem not worldmates, this truth condition is satisfied. And no contradiction comes out of it. (Yagisawa, online first, 7) Notably, the ontology in (EMD) goes hand in hand with modal tensing. Recall that the ideological commitments of modal tense proponents are the actuality tense, the mere possibility tense, the impossibility tense, and the modal tense at large. Every ontological Two remarks are in order. (EMD), even if it postulates possible and impossible individuals, is not a priori committed to primitive modality. For, although there is a difference between possibility and impossibility, the difference can be handled non-modally. One way to do this is to analyse any kind of possibility as a restricted modality, while those very restrictions (usually laws) are to be understood non-modally. So (EMR), even if hard to swallow in principle, can be squared with Lewisian reductive ambitions. Second, (EMR) does not aim to violate our everyday reasoning about things actual and possible. Impossible worlds do not actually exist. They do not exist possibly either, if ‘existing possibly’ means being restricted to a particular domain. As Yagisawa puts it,‘[i]t is certainly impossible for impossibilia to exist under any possible conditions or circumstances. But that does not mean that impossibilia do not exist under any conditions or circumstances whatever. They exist under impossible conditions or circumstances’ (Yagisawa 1988, 202-203). It is therefore not the case that (EMD) automatically fails the non-reductive test, and indeed much more should be said about its commons-sense test failure. Nonetheless, it still holds that problems regarding representation of logical, metaphysical and mathematical phenomena present strong reason to reject the project. I think, however, that although controversial, (EMR) might find some resources parallel to or parasitic upon competitive accounts. 56 )66 Chapter IV (Extended) Modal Dimensionalism postulate finds its tensed interpretation, whether we talk about actuality, possibility, impossibility or extraordinary modal phenomena. This feature makes (EMD) systematic and theoretically appealing. To summarize, both the necessity and the contingency horns of Jago’s dilemma ignore crucial features of (EMD): modal tensing, the amodal status of world-stages and the iterative hierarchy of modal spaces. To the degree that we appreciate the complexity of (EMD), these arguments are revealed as either missing their target or as directed against a different position. In the next section, I turn to objections concerning (EMD)’s account of impossible worlds. 4.4 Impossible Worlds In the introduction, I pointed out an important difference between (EMD) and (MR) with regards to the acceptance of impossible worlds. However, Cameron (2010), Jago (2013, 2014, forthcoming), Kim (2011) (following Lewis 1986a), Yagisawa (1988), and Divers (2002) have formulated arguments according to which (EMD) is an inconsistent hypothesis. The dialectic of the argument proceeds from an assumption that there are real impossible worlds as legitimate objects of quantification and thus as existing in the same manner as the actual world. (EMD) is therefore literally committed to the existence of impossible things. This section presents two such arguments and provides several suggestions as to how (EMD) might respond. To begin with, provided that we accept the impossible talk and its impossible worlds interpretation, (PMD) and (PMD*) transform into their impossibilist counterparts, (IMD) and (IMD*) respectively: (IMD) If it is impossible that P, there is an impossible world-stage, i1, such that P holds at i1, and )67 Chapter IV (Extended) Modal Dimensionalism (IMD*) If it is impossible for an object to have an impossible property, G, it has an impossible world-stage that has G. For instance, it is not possible for me both to be and not to be a philosopher at the same time, both to be and not to be a football player at the same time, or to be and not to be a talking donkey at the same time. If this is so, the modal stages strategy requires that there be stages such that Martin-is-and-is-not-a-fotball-player-at-i1, Martin-is-and-is-not-a-pianist-ati2, and Martin-is-and-is-not-a-talking-donkey-at-i99. But if impossible worlds are real, there really are the abovementioned inconsistent stages. And that’s a plain contradiction because inconsistent stages turn out actually to be true. The structure of the argument is as follows: 1) It is impossible for Martin to be a football player and not to be a football player at i1. 2) There is a Martin-is-and-is-not-a-football-player world-stage. 3) Martin is a football player and it is not the case that Martin is a football player. 57 Kim (2011) proposes a finer grained argument against (EMD). It runs along the following lines: 1. Suppose, for reductio, that some world w is a logically impossible world of the kind I. 2. There is a logical contradiction that is true at w. Let such a contradiction be schematically represented as ‘P and not-P’. 3. That is, P and not-P, at w. 4. So, P at w, and not-P at w. 5. If not-P at w, then it is not the case that P at w. 6. P at w, and it is not the case that P at w. Moreover, the argument runs regardless of whether we take ‘is real’ or ‘exists’ to be primitive. See Jago (2013). For a more detailed distinction between the two, see Yagisawa (2010, Chapter II). 57 )68 Chapter IV (Extended) Modal Dimensionalism 7. At the actual world, the following is the case: P at w, and it is not the case that P at w. 8. So the actual world is a world at which a logical contradiction is true. 9. But the actual world is not a world at which a logical contradiction is true. 10. Therefore, (1) is false. That is, no world is a logically impossible world of the kind I.58 What is special about Kim’s argument is that it resists the orthodox response to it. The orthodox response, presented, among others, by Lycan (1994) and Yagisawa (2010), blocks the inconsistency in the actual world by rejecting the step from (3) to (4), because ‘we should not expect all logically impossible worlds to behave in accordance with all laws of logic. At a logically impossible world, a conjunction might be true without both conjuncts being true’ (Yagisawa 2010, 184). Kim’s argument, although weaker, attacks (EMD) on the basis that it posits an impossible world which does not exist according to (EMD). Given the principle of plenitude, this counts against (EMD), for the world Kim has in mind follows logical principles, although it does not follow all of them. The world of the kind I is a world at which a contradiction is true because both conjuncts are true. The world violates one law of logic—the law of non-contradiction—but still accepts another principle: a conjunction is true if and only if both conjuncts are true. Given such a world, we do get (4) from (3), and Yagisawa’s response fails. 4.4.1 Diagnoses Both arguments follow Lewis’s ‘no difference between a contradiction within the scope of the modifier and a plain contradiction that has the modifier within it’ denial of impossible worlds, for impossible worlds, or world-stages, are real despite the fact that reality is the most fundamental and ultimate subject of reality. Thus Jago quite correctly points out that, according to (EMD), the possibility of there being a Martin-is-a-footballplayer stage implies that there is a Martin-is-a-football-player stage. By the same reasoning, 58 The argument is due to Kim (2011, 297-298). )69 Chapter IV (Extended) Modal Dimensionalism the impossibility of there being a Martin-is-and-is-not-a-football-player-at-i1 stage implies that there is a Martin-is-and-is-not-a-football-player-at-i1 stage. Jago discusses two independent strategies for meeting the charge. One is to draw a distinction between existence and reality, 59 for existence is still relative to an aggregate while reality is not. That, however, seems only to shift the problem somewhere else rather than to solve it. Whichever direction the distinction goes, it will still be the case that Martinis-and-is-not-a-football player-at-i2 is actually true, because reality as well as existence are sufficient for the truth of a contradiction. The other option is to bite the bullet and accept that there are true contradictions. The problem with this strategy is that (EMD) accepts the so-called plenitude of possibilia. This principle says that for every possibility there is a world that makes it happen. Qua species of (MR), (EMR) is forced to accept not only the plenitude of possibilia but also a plenitude principle for impossibilia. This means that for any arbitrary false proposition, there is a world that instantiates it. Apparently, the two possible ways of meeting the worries fail. But they do not exhaust our options. In the next section, I propose two strategies available to (EMD) that, if successful, respond to both Jago’s and Kim’s challenges. The proposed strategies rely on the features of (EMD) that are ignored in the arguments. The ignored features are (again) modal tensing and the iterative conception of logical spaces. 4.4.2 Modal Tensing Again As I have already stressed, a fair criticism should pay attention to every aspect of the criticized theory. For dialectical purposes, it is important for any criticism not to overlook crucial aspects of the criticized theory. Remember that (EMD) draws a parallel between space, time and modality. Spaces, times and worlds are metaphysical relativizers that make concrete objects spatial, temporal and modal, respectively. This is, however, only one part of the story. The second part of the picture reflects another parallel between time and modality. This is the analogous use of temporal and modal tenses. Additionally, (EMD) proposes a hierarchical embedding structure of alternative modal spaces. A world is impossible according to another possible world in case it belongs to a different modal space. 59 See my footnote 45. )70 Chapter IV (Extended) Modal Dimensionalism Existence in a modal space has a particular predication concerning modal space at large, subscripted ‘m’. Finally, (EMD) is not (MR). Worlds are not concrete mereological sums. Both possible and impossible worlds are indices, or modal regions. This feature puts (EMD) somewhere between (MR) and modal ersatzism. Namely, neither de re nor de dicto representation are genuine. Taken together, these three features present resources for (EMD) when it comes to avoiding the triviality threat. The first option relies on the modal tensing strategy. With the distinction between modal tenses in mind, the triviality argument is interpreted such that no contradiction arises. This runs as follows: 1) It isa impossible that Martin isi and is not a football player at i1. 2) There isi a Martin-is-and-is-not-a-football-player stage. 3) Martin’s i1-stage isi a football player and is not a football player. C) It isi (not isa) the case that Martin is a football player and is not a football player. Premise (1) states that it is (actually) impossible that Martin is a football player and is not a football player. This means that the impossibility is predicated of a stage of Martin’s. According to (EMD) the impossibility tense is introduced in (2). The same applies to (3), resulting in the conclusion that it is still the case that Martin is a football player and is not a football player. However, ‘is’ in (C) gets an impossibility tensing instead of its actuality counterpart. Consequently, (C) is not a contradiction. Notice that the impossibility tense belongs to the basic ideological apparatus of (EMD), and its proponents are thus fully justified in applying it. Thus arguments like Jago’s share the same deficiency. They ignore the distinction between two different tenses: the actuality tense and the impossibility tense. As long as we disambiguate the two, every impossibility merely existsi, but does not exista. What about Kim’s challenge? Recall that we are dealing with a special kind of world here: one that obeys every law of logic except the law of non-contradiction. The negative answer Yagisawa proposes is that the kind of world Kim has in mind does not play any theoretical role in any analysis, and (EMD) does better to deny it.60 Positively speaking, 60 For Yagisawa’s response see Yagisawa (2011, 310). )71 Chapter IV (Extended) Modal Dimensionalism however, such worlds do not render the tensing analysis inapplicable. The special version of the counterargument simply copes with the answer to Jago’s original one: 1. Suppose that some world w ism a logically impossible world of the kind I. 2. There isi a logical contradiction that is true at w. Let such a contradiction be schematically represented as ‘P and not-P’. 3. That is, P and not-P, at w. 4. So, P at w, and not-P at w. 5. If not-P at w, then it isa not the case that P at w. 6. P at w, and it isi not the case that P at w. 7. At the actual world, the following isi the case: P at w, and it is not the case that P at w. C. So the actual world isa not a world at which a logical contradiction is true. Interestingly, (1*) has a special modal tense attached to it—one that goes beyond the local metaphysical space. Although one might think that such a tense is utterly ad hoc, seen from (EMD)’s point of view it simply fits into and reflects (EMD)’s ontological picture. (EMD) introduces an iterative hierarchy of modal spaces. Every modal space contains K-possible worlds only where K stands for a certain modality (be it physical, logical or metaphysical possibility). Indeed, the system of logical spaces (K-spaces, for different Ks) is hierarchical and difficult to understand completely. Note however, that complete understanding is not a sufficient condition for accepting a theory. Nor does it has to be a necessary condition. Among the relevant theoretical virtues that play a role in choosing between modal metaphysics are explanatory power, consistency, simplicity, elegance, strength, and consistency with what we already know. Unless it is shown that (EMD) as a whole violates one of the virtues and does not bring anything in return to the overall picture, commitment to the plurality of modal spaces appears unreasonable. Put differently, insofar as we locate some feature of the theory that makes it in some respect superior to its rivals, why not take the theory seriously? One such application is an ability to block the necessity horn of Jago’s dilemma. Jagos’s argument, if valid, shows that any realistic position falls short in the case of analysis of possibilities )72 Chapter IV (Extended) Modal Dimensionalism pertaining to modal space as a whole.61 This is because such analyses localize possibilities to single worlds, and the possibility of worldmateship and non-worldmateship receive the single-world analysis. (EMD), on the other hand, does not confine such (extraordinary) possibilities to a single world. Such modalities are understood via relations between modal spaces. A possibility of non-worldmateship switches from world analysis to modal space analysis. Considering (EMD) as a whole, it is thus not an ad hoc move to have a plurality of worlds, modal spaces and modal tenses that reflect the ontology. When speaking of actuality, we use the actuality tense. Moving to possibility, we switch to the merely possible tense. Extending possibility must be followed by the impossibility tense. Finally, varieties of modal space as such must be mediated by a unique tense: the modal tense at large.62 Again, these are ontological and ideological elements of (EMD)’s toolbox and should not be attacked separately. To be sure, they can be attacked individually, but the extent to which such arguments make their point remains an open question. Additionally, there is an uncontroversial piece of pre-theoretical modal knowledge, accessible to human beings, that (EMD) accepts and does not aim to revise. There is a general agreement between (EMD) and its rivals about ordinary modal claims. We agree on what actually exists, what is merely possible and what is impossible. The disagreement comes with the interpretation of the modal discourse where incredulous stares not only play a minor role but also have no business being in the game in the first place. And extraordinary modalizing is one such case. 4.5 Conclusion This closes my defence of (EMD) as a way of making (EMR) a meaningful alternative. So far, I do not claim ultimately to have defeated objections against (EMD). Rather, I have tried to point out that any critique of it must consider the theory as a complex 61 Cf. Jago (forthcoming). There is yet another option available to (EMD). I have already pointed out that modal indices, unlike Lewis’s worlds, are not concrete. This feature puts it somewhere between (MR) and modal ersatzism. A version of the latter represents modality not genuinely but, to use Lewis’s label, by magic. But as I argue in Vacek (forthcoming) and the next chapter there are ways to meet the challenge from magic. If my responses work, I do not detect a serious reason not to apply them to (EMD) as well. 62 )73 Chapter IV (Extended) Modal Dimensionalism whole. As the arguments were meant to show, (EMD) taken as a complex thesis blocks the arguments outlined above. The theory’s essential components are not limited to spatial, temporal and modal indices and consequent spatial-stages, temporal-stages and modalstages analyses. (EMD) also includes a modal tensing approach to modality and, as the argument above indicates, this feature is ignored in both the possible and the impossible world challenges. To the extent that we are both modal indicers and modal tensers, the objections can be handled on independent grounds. )74 Chapter V Extended Modal Structuralism Chapter V It amazes me sometimes that even intelligent people will analyze a situation or make a judgement after only recognizing the standard or traditional structure of a piece. David Bowie 5. Extended Modal Structuralism 5.1 Introduction In this chapter63 I pursue a strategy according to which logical impossibility is analyzed as logical inaccessibility as well as consider whether it makes sense to think of logical models in isolation from the concrete world but without their being divorced from all spatiotemporal totalities. The metaphysics of structure developed in this chapter assumes that structural properties of possible and impossible worlds are primitive and objective. However, I provide some characterizations of their logical and metaphysical behavior, as well as guidelines for talking about them. Namely, I develop a modified version of the (EMR) according to which there are structural properties grounded in concrete worlds themselves (5.2). To justify the move, I discuss the argument from the incredulous stares (5.3), present a problem of representation (5.4). I then propose a ‘magical’ account of representation (5.4.1) in order to avoid the inconsistency worry (5.5). 63 With some modifications, this chapter is based on Vacek (forthcoming). )75 Chapter V Extended Modal Structuralism 5.2 Introducing the Ontology This section proposes a particular version of (EMR), namely Extended Modal Structuralism (hereafter EMS). This branch of (MR) is fully realistic in a sense that impossible worlds exist as full-blooded entities. However, impossible worlds are not mere merelogical sums. Rather, I introduce a two-categorical ontology according to which there exist world-cum-structure entities. On one side, I agree with (MR) that there exist maximal mereological sums of interrelated individuals. On the other side, the sums do not exhaust the modal space. In order for them to represent the actual, the possible and the impossible, they have to instantiate the so-called structures. Let me explain. According to (MR), possible worlds are maximal mereological sums of spatiotemporally interrelated individuals. Every way the world could have been – that is, every such sum – displays enormous spatiotemporal structural complexity. By way of example, think about the actual world. The world we live in is a very inclusive thing. Every stick, every dog, every chair and every stone you have ever seen is a part of it. It is therefore natural to say that different worlds differ from each other on the basis of what’s going on in them. Put differently, worlds differ structurally. However, there are mutually exclusive ways of fleshing out this notion. We might, together with Lewis, think that worlds have their own parts, which determine worlds as wholes. More precisely, the order and configuration of parts structure worlds; worlds differ from each other by being structured differently – by having different, variously ordered parts. On another conception, concrete worlds have enormous structural complexity and enormous local variability, yet they do not have genuine parts. They of course display different structures, since things happen differently in them. But their structural variety is not determined by their parts, for indeed they have none. Rather, this structural variety is derivative. In truth, both of these conceptions aim at the same target: they aim not only to systematize our common sense view about the actual world, but also to account for the ways in which reality might be, must be, and cannot be, respectively. This chapter proposes a defense of the latter conception: the notion of a world – WORLD – is a composite notion, constituted by the notion of a concrete simple and the notion of a metaphysical structure. Every concrete simple instantiates a metaphysical )76 Chapter V Extended Modal Structuralism structure. WORLDS are not fully concrete entities, but nor are they primitive (abstract) indices. WORLDS are combinations of the two: they are simple-cum-structure pairs. In effect, we should not confuse the universe that surrounds us, the entity we all inhabit, with the actual world. They are not the same entity. WORLDS are not maximal mereological sums of spatio-temporally interrelated individuals. The structural component of a WORLD is a structure according to which things in general are a certain way. A WORLD is impossible according to another WORLD if and only if they are parts of different logical spaces, meaning that their components are paired with mutually incompatible regions: metaphysical structures. Traditional modal realists suppose that, irrespective of the variation across the plurality of Lewis’s worlds, the domain of the abstract is unchangeable.64 Thus, concrete things are contingent and vary across worlds, while abstract entities exist in every possible world. Contra Lewis, I understand the relation between a concrete simple and its metaphysical structure to be factive – that is, grounded upon and posterior to it. For instance, there might be conjunctive properties of the form A&B that cannot be further broken down into their individual components, A and B. In such worlds, simplification fails to be a valid rule of inference (Kiourti 2010, 151). One might protest against this priority talk from at least two points of view. First, one might reject such talk on the basis of meaningfulness, arguing that the priority relation is confusing and explanatorily useless. However, such an objection overlooks the very motivation behind metaphysical explanation. For if the subject matter of metaphysical inquiry is the notion of that which is fundamental – where fundamental means prior – one must have a pre-theoretical grasp of this notion. Otherwise, supervenience relations, setmembership relations, and reduction relations turn out to be theoretically vacuous. Secondly, one might object that the notion of asymmetry is irrelevant to modality. However, what modal metaphysicians – genuine modal realists in this case – aim to do is to explain (away) modality in terms of non-modal terms. They aim to explain modality via ‘because’ or ‘in virtue of’ claims, which requires general asymmetric explanation. It is thus not fair to accuse accounts like that proposed here of being meaningless because it takes the Interestingly though, Lewis admits that this might not be the case. He writes: ‘As for the parts of worlds, certainly some of them are concrete, such as the other-worldly donkeys and protons and puddles and stars. But if universals or tropes are non-spatiotemporal parts of ordinary particulars that in turn are parts of worlds, then here we have abstractions that are parts of worlds’ (Lewis 1986a, 86). 64 )77 Chapter V Extended Modal Structuralism concrete simple/structure relation to be asymmetric. Given the order of explanation, the carving relation must be asymmetric, unless one accepts circular arguments. Methodological requirements based on the notion of explanation thus prevent the relation between a simple and its structure from being idle. We are forced to dispense with symmetry in the interest of ensuring that the relationship between concrete individuals and their internal structural complexity is informative and thus deals meaningfully with the question of fundamentality. Mutually exclusive answers to priority question correspond to mutually exclusive ways of carving nature at its joints. Either the structural complexity is prior to the concrete simple, or the concrete simple is prior to its structure. I assume the latter: concrete simples are basic, but ontologically posterior structures make them extremely complex. 5.3 Incredulous Stares The idea of there being a metaphysical simple, parts of which are merely derivative, is not a novelty in metaphysics. Philosophers have been asking the question ‘How many things fundamentally exist?’ for decades and have more or less provided three mutually incompatible answers: (1) there is only one (actual) thing (monism); (2) there is a plurality of (actual) things (pluralism); and (3) there are no (actual) things (nihilism). Since I defend a version of monism in this chapter, I will digress a little and discuss some objections to this view. For, the arguments from incredulous stares in metaphysics have a similar structure and usually rest on a confusion between two different data. We can differentiate two objections that underlie the ‘incredulous stare’: one concerning both existence monism and priority monism, the other concerning only the latter. According to existence monism, exactly one concrete object exists, despite the fact that we experience more than one existing thing. According to priority monism, exactly one basic concrete object exists, and many other concrete objects exist only derivatively. Although these positions might seem similar, it is important to distinguish between them. Unlike existence monism, priority monism does not deny that tables, dogs and chairs exist. )78 Chapter V Extended Modal Structuralism What it denies is that they are fundamental. Only concrete simples are fundamental, whereas particulars are merely derivative. However, existence monism appears to be inconsistent with an evident datum of experience (as does priority monism, if the argument is read as including ‘fundamentally’), for there (fundamentally) is a plurality of things: a plurality of material things. Put in the form of a simple argument: 1. It is obvious that there (fundamentally) is a plurality of concrete objects. 2. If it is obvious that there (fundamentally) is a plurality of concrete objects, then we have strong reason to believe that there is a plurality of concrete objects. 3. There is prima facie reason to believe that there is a plurality of concrete objects. Recall that according to monism, only one concrete thing (fundamentally) exists, whereas according to pluralism many concrete things exist, and according to nihilism no concrete things exist at all. As the argument shows, common sense favors pluralism over the remaining two positions since our common way of speaking about (and, more generally, conceptualizing) the world assumes that there is more than one individual. After all, a key part of our pre-theoretical grasp of the world includes the notion that the world contains chairs, tables and many other countable things. Monism supposes the contrary. It is the doctrine that there (fundamentally) is exactly one concrete simple. This means that if we want to affirm the existence of at least two chairs in front of us, we either have to deny concreteness to one of them or deem them identical. Since both options fail the common sense test, monism’s tenability depends on reinterpreting the common sense data. On further consideration, however, reinterpretation that makes space for the second option is actually relatively straightforward. What is at stake here is a reinterpretation of the data that justifies the appearance of a plurality of individuals but is consistent with there (fundamentally) being only one concrete simple. For example, consider the Moorean fact ‘this is my right hand’. A monist might say that this sentence is true when paraphrased as ‘the world is handish here’. And, although the first sentence would be false in a world with only one concrete simple, the truth of the paraphrase is enough to block the objection. )79 Chapter V Extended Modal Structuralism Moreover, there doesn’t seem to be anything wrong with saying that if truth-makers are required, the truthmaker for the Moorean truism is simply the world.65 66 It is therefore far from clear that we should deny monism on common sense grounds, for it is far from clear what those grounds are. Is it the fact that we cannot represent a plurality of things? Monism does not deny this. Is it the claim that a plurality of things does not (fundamentally) exist? Monism agrees. These are two different claims, however, and unless the objector differentiates between them, her argument misses the target. Let us therefore consider another line of argument. This argument concerns the common sense argument against priority monism only – namely, the apparent problem of the priority of the whole to its parts. It proceeds as follows: 1. Common sense holds that a part is prior to its whole. 2. If common sense holds that a part is prior to its whole, then there is reason to think that a part is prior to its whole. C. 65 There is reason to think that a part is prior to its whole. See Schaffer (2007) for more details. Horgan and Potrč (2000) pursue an analogous strategy. They argue for the common sense feasibility of existence monism by advancing the following ontological and semantic theses: 66 a) b) c) d) There really is just one concrete particular, viz. the whole universe (let us call it the ‘blobject’). The blobject has enormous spatiotemporal structural complexity and enormous local variability, although it does not have any genuine parts. Many of the postulates of common sense and science are true, despite the fact that nothing in the world answers directly to these postulates. Truth, for such statements, consists in indirect language-world correspondence. Horgan and Potrč’s strategy thus employs an indirect correspondence theory of truth, according to which Moorean truisms can count as true in lax contexts. This means that the relevant construal of truth entails a commitment not to the ultimate metaphysical existence of a plurality of common sense objects, but rather to their lightweight ontic, mind- and languageinvolving existence. This so-called ‘blobjectivism’ thus claims the following: A statement’s truth results from the interaction of two factors: the contextually operative semantic standards, and how things stand with the mind-independent world. When the semantic standards operate in such a way that a given statement can be correct semantically (i.e., true) even though the statement posits (i.e., quantifies over) certain items that are not there in reality, then truth (for discourse governed by such semantic standards) thereby becomes an indirect form of language/world correspondence. (Horgan & Potrč, 2000, 253) In effect, such a position is not relativistic in spirit. Rather, it amounts to eliminating chairs, tables, dogs and other concrete objects that concern our ontological commitments in such a way that everyday statements about them can be true. Cf. French (2014, 174). )80 Chapter V Extended Modal Structuralism Methodologically speaking, it is not at all obvious that common sense is a reliable arbiter of the priority question in the first place. Recall that ontological priority is a highly theoretical notion; metaphysical status simply cannot be determined by consulting our intuitions. Therefore, it is unlikely that there are platitudes that would prefer priority pluralism to priority monism. Let us, however, put this quick rejoinder to the side and see what else a priority monist might offer in order to block the argument. One such answer, proposed by Schaffer (2010), appeals to a distinction between mere aggregates and integrated wholes. As he argues, although common sense might appeal to the priority of parts in cases of mere aggregation, it hardly endorses the priority of integrated wholes. Take, for example, a heap of sand on the one hand and a circle and its arbitrary partitions on the other. It seems right to say that parts of the heap are prior to the heap. But it is not similarly clear that any arbitrary partition of the circle is prior to the circle. In this case, the integrated circle just is prior to any semicircle carved from an arbitrary portion of it. 67 The opponent of priority monism ignores this distinction. For him, or more generally, for anyone who subscribes to the argument, mere aggregates and integrated wholes are metaphysically on a par and deserve the same philosophical analysis. But if they are not, the objection runs into difficulties. For is the claim that common sense holds that a part is prior to its whole, whether an integrated whole or a mere aggregate? The priority monist denies this. Or is the claim that common sense holds that a part is prior to a mere aggregate? A priority monist would not disagree. Finally, is the claim that common sense holds that a part is prior to its integrated whole? Here, the disagreement arises once again. Of course, my aim here is not to fully defend monism as the best systematization of our pre-theoretical data. For now, it suffices to demonstrate the metaphysical acceptability of the position according to which a concrete simple is both fundamental and in possession of a structural complexity that (a) derives from it and (b) is ontologically dependent on it. For, as we will see in a moment, WORLDS are pictured as monistic simples that give rise to metaphysical structures. Some of them represent things that are possible, some of them things that are impossible. The question, however, is how the representation is supposed to work so as to avoid both the inconsistency and certain limitations to representing ‘abstract’ 67 Cf. O’Conaill & Tahko (2012). )81 Chapter V Extended Modal Structuralism impossibilities. It is thus of the utmost importance to represent plain inconsistencies and to preserve the theory’s consistency. Let us therefore turn to the representation problem. 5.4 Metaphysical Structures and Representation It is often considered a virtue of (MR) that it represents our possible situations in terms of genuine worlds. For modal realists, something is possible if and only if there is a world that is that way, something is necessary if and only if every world is that way, and something is impossible if and only if there is no such world. And this stands in opposition to other accounts of possible worlds according to which it makes sense to speak of what is the case according to them. (EMS) says that it is not Lewis’s worlds themselves but simple-cum-structure pairs that do the representing. This feature of the theory places it somewhere between (MR) and actualism and, more importantly, between two modes of representation: genuine and ersatz. While the former causes inconsistency of a kind mentioned earlier, the latter does not necessarily do so. WORLDS do not represent in the way that Lewis thought, even though they have concrete constituents. Concrete ‘stuff’ does not do the representing. Rather, it is the concrete simples together with metaphysical structures that do the representing. Metaphysical structures are grounded in concrete simples, and every structure is ontologically dependent on a simple. Again, it is not simples but simples-cum-structures that represent something as possible, contingent, necessary or impossible. In On the Plurality of Worlds (Chapter III), Lewis spent much time arguing that representing modal phenomena in non-genuine terms gives rise to many obscure consequences. In particular, he attacks a so-called magical ersatzism, according to which an element E represents that so-and-so (or it is the case that so-and-so according to E) if and only if, necessarily, if E is selected, then so-and-so. This is how maximal elements in particular represent. The maximal elements are the ersatz worlds (Lewis 1986a, 175). The relation of selection is supposed to connect concrete simples with metaphysical structures. For Lewis, the problem concerns whether the relation of ‘selection’ is an internal relation or an external one. Suppose that the relation is internal. Then it holds in virtue of )82 Chapter V Extended Modal Structuralism the intrinsic natures of its relata, the concrete simple and the abstract element. For instance, if part of what goes on within a WORLD is that there is a flying pig, this means that some elements will be selected and others not. Given the nature of internal relation, it is the intrinsic nature of the selected element that plays a role in the selection – for if its intrinsic nature were different, it would not be selected. In fact, Lewis attacks the internal conception of the selection relation from three different points of view. First, he voices a metaphysical worry: the elements do not have familiar sorts of intrinsic features. They are neither spatiotemporal nor set-theoretic entities (as in the case of linguistic or pictorial ersatzism). They do not seem to exhibit any internal structure at all, and it is magic that pairs the elements with ways the simples might be. Secondly, Lewis claims that the selection relation raises epistemological worries. Here, the idea is that since the elements are abstract, their causal isolation makes their individual natures inaccessible to us. So, the objection goes, we cannot know about a range of elements and their connections with concrete goings-on because they are causally isolated from us. How do we know that the relation of selection ever happens when we have no access to one of its relata – namely, the element? Finally, Lewis argues that the relation of selection is doubtful on rational grounds. Magical ersatzism is accompanied by a certain unintelligibility, and ersatzists themselves are not in a position to understand what they are saying. Sure, we know something substantial about the elements. For instance, they are not all alike, they differ from each other, and their nature must be rich enough to permit enormous variation. But when it comes to selection itself, ‘we have not the slightest idea what the ‘representational properties’ are’ (Lewis 1986a, 178). All we have is a schema saying that if there is one element that represents that a donkey talks, then one is selected if and only if a donkey talks. There is nothing that would clarify the ‘selection’ relation. With all of this noted, Lewis considers the selection relation to be external. This reading of ‘selection’ views the relation as being like a distance relation between space-time points. Such a relation does not obtain in virtue of the distinctive intrinsic natures of the selected elements, because all there is to them is their place in a relational system (Lewis 1986a, 179). So the relation now obtains between the concrete cosmoi and the element, but it is not the natures of the relata that determine it. )83 Chapter V Extended Modal Structuralism Again, Lewis argues that the relation is suspicious from both a metaphysical and an epistemic point of view. With regards to the latter, he identifies the same acquaintance problem as in the case of internal relations. That is to say, it is not clear to him how a relation, one relatum of which is abstract and causally isolated from us, the other concrete, can ever come within reach of our thought and language (Lewis 1986a, 179). With regard to the former, this selection is not any ordinary external relation; it is a modal relation. He writes: Necessarily, if a donkey talks, then the concrete world selects these elements; if a cat philosophizes, it selects those; and so on. I ask: how can these connections be necessary? It seems to be one fact that somewhere within the concrete world, a donkey talks; and an entirely independent fact that the concrete world enters into a certain external relation with this element and not with that. What stops it from going the other way? Why can't anything coexist with anything here: any pattern of goings-on within the concrete world, and any pattern of external relations of the concrete world to the abstract simples? (Lewis 1986a, 180) To sum up, Lewis quite clearly denies that magical ersatzism provides a complete and accurate analysis of modality. Either way the ersatzist articulates her theory, she faces epistemological, metaphysical and even rational worries regarding how the theory is supposed to work. 5.4.1 (EMS) and Magic Let me now go through Lewis’s objections to magical representation. Hopefully, my replies to them will shed light on the account I prefer and, to some extent at least, help to rehabilitate (EMR)’s credibility in the eyes of those who stare incredulously 68. The objections are mainly due to Lewis (1986a, section 3.4). See also Nolan (forthcoming) as a representative of a slightly different party. 68 )84 Chapter V Extended Modal Structuralism Objection Any theory that treats impossible worlds as real is incoherent in nature: if it is impossible that P, where P stands for whatever you take to be false, then P. Answer Let us consider first the well-known objection from the inconsistency of (EMR) in general and then see how it applies to the proposal at hand. One version of the argument goes like this: consider an impossible world such that if it exists, then p. If there are impossible worlds, there is this impossible world. Now, take any falsehood you like; plug p into this argument, and you will get an argument that the falsehood is true – not true at the impossible world at issue, just true simpliciter (Cameron 2010, 791). So, given the real existence of impossible worlds, any false proposition turns out to be true in the actual world. Three assumptions relied on in this argument are important here. First, impossible worlds exist. Second, they represent something as impossible by really being impossible. Finally, this argument applies exclusively to conceptions that ascribe to the first and the second assumptions. It is easily refuted by other conceptions – say, one according to which we cannot conclude from ‘there is a set, S, containing the proposition that if S exists then p’ and ’S exists’ that p is the case. Although I agree with the argument from inconsistency, given all the above assumptions, it is far from clear how it threatens my own proposal. The first assumption surely applies, and I have nothing to say against it. But the second assumption does not. I am not saying that WORLDS genuinely represent inconsistencies by being inconsistent. Again, simples do not represent. Their structures do, although what structures there are is determined by what simples there are. Nonetheless, the representation is not genuine. It’s a kind of magic. Objection Since the structures are not concrete, their causal isolation makes their individual natures inaccessible to us. )85 Chapter V Extended Modal Structuralism Answer Fair enough. Speaking in a negative way, metaphysical structures are not concrete in the sense that Lewis’s worlds are. They neither display causal powers nor enter into causal relations. However, we grasp them via the spatio-temporal system we inhabit. Since we have causal access to the world we inhabit – it is us and all our surroundings – there is at least something positive that a WORLDS theorist can say about the structural component. Namely, it suffices to show that we can grasp some abstract features of the concrete stuff we inhabit through interaction with it. In doing so, we grasp at least some objective features of the structure of the world we are part of. One way of pursuing this line is to follow Mortensen (1989). Mortensen writes: Our world has very general structural features too, for instance very general aspects of its differential topology. It is possible to present General Relativity, Quantum Mechanics, Gauge Theory, even Newtonian Dynamics in very abstract fashion. Considered in isolation from the concrete universe out of which they arise, it can be difficult to grasp their connection with our world. I suggest that things might well be that way with abstract-looking logical countermodels too. […] There is, I suggest, no reason why such very general or abstract structures should not be realized. (Mortensen 1989, 328) In other words, the fact that we describe physical reality in an abstract way and model various features of it does not give us a reason to deny the concreteness of physical reality. Physical reality is concrete and does display phenomena that physics works toward systematizing. If that is so, things might well be that way with abstract-looking logical countermodels too. Moreover, we certainly engage in logical debates, so why not admit that the debates partly concern WORLDS themselves rather than mere conventions?69 Any Lewisian about possible worlds might therefore rather look for a deeper and metaphysically more robust account of logical laws. It is simply a consequence of her metaphysical position 69 In Vacek (2011, 57-58) I discuss two ways of understanding logical laws in more details. )86 Chapter V Extended Modal Structuralism that its logical space is independent of the way we speak about it.70 It’s a metaphysical structure. Objection But a certain unintelligibility attaches to your theory because magical ersatzists themselves are not in a position to understand what they are saying. Answer I understand this objection as a follow-up to the previous one. Nonetheless, it is more general, and instead of raising a substantial epistemological challenge 71, it accuses magical ersatzism of meaninglessness rather than epistemic fallaciousness, for according to this objection, the selection relation – whether internal or external – is unintelligible and nonsensical at its core. But if that were so, Lewis would be committing himself to nontrivial counter-possible reasoning. For Lewis does describe how the ‘selection’ relation would work if magical ersatzism were true. He very clearly describes and even explains both horns of the dilemma. But if magical ersatzism does not make sense at all, how can it be so precisely criticized? Moreover, recall that my proposal has simples as well as metaphysical structures among its postulates. The ‘selection’ relation in this case is a relation determined by the intrinsic nature of the metaphysical structure, which is determined by the simple in which it is grounded. Anybody who understands the terms ‘concrete world’, ‘intrinsic property’, ‘quantification’ and the other ingredients just does understand what I am claiming. Of course, I might be wrong. But there is a difference between being wrong and being unintelligible. The second horn of the dilemma takes the ‘selection relation’ to be external, meaning that for every way the world might be there is exactly one metaphysical structure that stands in the selection relation to its simple. Is it the existence of concrete mereological sums that is unintelligible? That would make (MR) nonsensical, despite the amount of The fact that there are plenty of mutually incompatible logics on the market does not contradict the assumption. We can still consider various logics as approaching the best description of reality. But it is a matter of fact which logic does so accurately. 70 71 Cf. Vacek (2013b). )87 Chapter V Extended Modal Structuralism literature dedicated to the doctrine. Or, is it the metaphysical structure that gives rise to the nonsensical consequences? If this is so, philosophers defending some sorts of ontological dependence might be offended. Finally, is it the necessary connection that requires independent rational justification? Although the necessary co-existence problem is certainly tricky, to call non-Humeans unintelligible seems too hasty. I therefore conclude that the argument from unintelligibility fails. At base, it is actually a version of the incredulous stare, which results from how difficult it is to believe in this ‘selection’. But incredulity does not imply unintelligibility. And, taking a page from Lewis himself, unless supported by further arguments against the hypothesis, this objection is not sufficient. Objection Ersatz worlds do not seem to exhibit any internal ‘structure’ at all; it is as if by ‘magic’ that elements are paired with possible ways the world might have been. Answer This objection, as it stands, is strong enough to make its point, at least when it comes to orthodox examples of magical ersatzism. Recall, however, that my version of (MR) is a thesis according to which there are simple-cum-structure entities, rather than mere Lewisian worlds. Such entities consist of one-way ontologically dependent simple-structure pairs. The structures are grounded in simples themselves and thus mirror their derivative complexity. It is therefore not the case that the WORLDS represent qua abstract simples. The structures that do the representing are complex. Objection The proposal presented is not in line with the Humean supervenience project. Answer Metaphysical structures are not worlds, but they ontologically depend on simples. This means that there is a tight connection between a concrete simple and its structure. Even more, the connection is such that it is impossible for a concrete simple to exist but for its )88 Chapter V Extended Modal Structuralism structure not to. Also, if a concrete simple exists, its structure necessarily does too. If this is so, I am apparently forced to admit that the proposal violates the Humean picture of reality. According to this picture, reality does not contain necessary connections between entities; rather, our connecting entities in such a way is attributable to mere habit. I propose two responses. Firstly, the Humean notion of necessary connections between existing entities only concerns individuals. It would be unreasonable to require the principle to hold without restriction, since such a principle would fail intuitively valid tests. For instance: is it problematic to posit a necessary connection between a set and its members, between me and my singleton, or between ‘a fact’ and ‘the fact that it is a fact’? The principle – understood unrestrictedly – is simply too demanding. Secondly, Lewis himself concedes that Humean supervenience is at best contingently true. He writes: Two worlds might indeed differ only in unHumean ways, if one or both of them is a world where Humean supervenience fails. Perhaps there might be extra, irreducible external relations, besides the spatiotemporal ones; there might be emergent natural properties of more-than-point-sized things; there might be things that endure identically through time or space, and trace out loci that cut across all lines of qualitative continuity. It is not, alas, unintelligible that there might be suchlike rubbish. Some worlds have it. And when they do, it can make differences between worlds even if they match perfectly in their arrangements of qualities. (Lewis 1986b, x). So even Lewis admits that the Humean supervenience thesis may hold only contingently. I therefore think that none of these objections presents a lethal argument against the proposal. Incredulous stares are sure to remain. But if we have reason to coherently believe in a variety of worlds-cum-structures, why not postulate them? Moreover, changes to our theories need not imply changes with respect to how we reason about actuality, since the entirety of reality does not need to fit into a single logical picture. 5.5 (EMS): Still Inconsistent? Let me end with the very problem we began with. That is, one might still object that the representation, however magically you construe it, does not avoid the inconsistency in the first place. Briefly, the objection runs as follows: you want your concrete basis to be )89 Chapter V Extended Modal Structuralism consistent, so that your metaphysical structures inherit this consistency and can nonetheless represent (logical) inconsistencies. So how can something consistent represent plain inconsistencies? There are different answers to this question, depending on which particular kind of ersatzism one prefers. First of all, Lewis is clear that if impossible worlds were sets of sentences – that is, if impossible worlds were replaced by their stories – there would indeed be room for worlds according to which contradictions are true (Lewis 1986a, 7, fn.3). ‘According to the Bible’ and ‘Fred says that’ are not restricting modifiers, which means that they do not pass through the truth-functional connectives. Similarly, impossible worlds, conceived as abstract states of affairs, do not bring plain inconsistencies into existence. Again, this is because of the denial of the move from ‘according to w, Px’ to ‘something is such that Px’. Ersatz worlds, whether states of affairs, maximal properties, or sets of sentences, are mere representations of impossibility and do not require that anything posses impossible properties per se. Now it seems that my proposal requires that there are plain inconsistencies out there in reality, because structures representing impossibilities ontologically depend on concrete stuff. But if the concrete is consistent, how can it ground such structures? Put differently: how can something concrete ground something that represents plain inconsistencies? I am afraid that this objection, as it stands, proves too demanding. Take, as a counterexample, the hybrid (MR) proposed by Divers (2002) and further elaborated by Berto (2010)72. In it, (MR) is taken for granted in the analysis of possibility, but ersatzism is taken to account for impossibility. Thus, while concrete possible worlds are ‘localizers’ of all possible phenomena, true contradictions are represented by sets of sets of them. Here is an example: suppose that metaphysical space consists of exactly six worlds {w1, w2, w3, w4, w5, w6}. Provided that the proposition ‘it is raining’, A, is identified with the set {w1, w2, w3} and the proposition ‘it is not raining’, ~A, with the set {w4, w5, w6}, the contradictory proposition ‘it is raining and it is not raining’ - (A and ~A) - is, by the same reasoning, identified with the set of the above sets, namely {{w1, w2, w3} {w4, w5, w6}}. The resultant My proposal is one among many. Since I cannot discuss then all here I mention them at least. Beside Berto (2010), there is McDaniel (2004)’s version according to which Lewis’s worlds overlap and provide thus for various impossibilities. Another realistic option is Yagisawa (2010) which takes worlds to be as real as times and spaces. I defend it in previous chapter. 72 )90 Chapter V Extended Modal Structuralism set is an impossible world, i1, because it represents a contradiction. Now, let us also suppose that the proposition ‘the sun is shinning’, B, is identified with the set {w1, w3, w5} and its negation, ~B, with {w2, w4, w6}. Similarly, the contradictory proposition ‘the sun is shining and the sun is not shining’ – (B and ~B) – is then the set {{w1, w3, w5} {w2, w4, w6}}. Let us dub this impossible world i2. Impossible worlds i1 and i2 are undoubtedly different. Whereas i1 is identified with the set of the form {{w1, w2, w3} {w4, w5, w6}}, the form of i2 is quite different: {{w1, w3, w5} {w2, w4, w6}}.73 Apparently, we have a set of worlds with consistent members that nonetheless represent plain inconsistencies. More generally, we have sets that represent consistencies as well as sets that represent inconsistencies, even though in both cases their members are selfconsistent. Although this does constitute a kind of magic, it definitely does not result in big metaphysical controversies 74. I therefore conclude that this version of (MR) is not committed to an inconsistent basis75. It is simply unreasonable to demand that consistent entities represent only consistent phenomena. Consistent concreta can represent inconsistencies, as Berto’s proposal demonstrates. If this is so, then structures can also represent impossibilities, even when they are based on exclusively consistent matter. 5.6 Conclusion In this chapter, I argued for an extended version of (MR), according to which there are concrete simples and metaphysical structures. These structures depend on concrete simples. Also, they represent ways the worlds might (and might not) have been, although not in a genuine way. My primary aim was to deal with simple impossibilities: that is, plain 73 For a more detailed discussion, see Vacek (2013a). Does the theory have any consequences for what the correct logic of modality is? I understand modal logic as a tool to formalize our ontological commitments. I do not, however, think, that modal logic is prior to them. The language of boxes and diamonds provides us with formalization of a part of our possible worlds discourse, but that does not mean that the language formalizes every single bit of it. After all, if this language proves to be a clumsy instrument for talking about modal matters, we do better to follow the ontological postulates directly. Cf. Lewis (1986a, 12-13). 74 Indeed, one might ask why we should prefer my version of (EMR) rather than Berto’s. I confess I have not a definite answer as the comparison of my and Berto’s proposals would be too complex to be pursued here. Nonetheless, the reader might consider my proposal as yet another contribution to the debate without any ambition to be indispensable. 75 )91 Chapter V Extended Modal Structuralism contradictions. How such a project might deal with mathematical and metaphysical impossibilities remains an open question to be addressed elsewhere. )92 Chapter VI Extended Modal Fictionalism CHAPTER VI A film is - or should be more like music than like fiction. It should be a progression of moods and feelings. The theme, what's behind the emotion, the meaning, all that comes later. Stanley Kubrick 6. Extended Modal Fictionalism 6.1 Introduction Recall again, that according to (EMR), there exist possible worlds and there exist impossible worlds. As I have shown in Chapter I such a theory has inconsistent consequences. This chapter proposes yet another emendation of (EMR) I provisionally call Extended Modal Fictionalism (EMF). In (6.2) I present the basic postulate a theory of modality called basis, Modal Fictionalism (MF). In (6.3) lists some problems of (MF), namely the status of ‘according to the (MR)-story operator’ (6.3.1), The Brock-Rosen Objection (6.3.2) and a so-called Hale’s Dilemma (6.3.3). I next consider possible alternatives available to (MF) (6.4). Then I propose a view according to which there are (MR)’s possible worlds, but (MF)’s impossible worlds (6.5). In (6.5.1) I provide reasons to endorse such a view although I admit some of its drawbacks (6.6). )93 Chapter VI Extended Modal Fictionalism 6.2. Modal Fictionalism Modal Fictionalism (hereafter MF) 76 is seen as a kind of deflationism about modal truth. That is, the fictionalist is prepared to assert propositions whose truth is not to be taken as a literal truth, but is regarded as a sort of ‘truth in fiction’. In particular, (MF) takes modal discourse seriously, accepts its interpretation in a possible-worlds framework, yet avoids an ontological commitment to these worlds. Such a strategy claims to utilize the apparatus of possible worlds with all its benefits without being committed to their existence. The essential component of (MF) is thus that possible-worlds sentences are to be interpreted in the same way as discourses in which a suppressed ‘story prefix’ is invoked. Therefore, the statement (1) It is possible that there are talking donkeys is not to be understood as expressing a factual proposition. Rather, (1) is interpreted as a statement about a particular story – an (MR)-story. Inspired by and parasitizing on (MR), (MF) construes the story as a set of already mentioned postulates. Here are again: (a) Reality consists in a plurality of universes or ‘worlds’. (b) One of these is what we ordinarily call the universe: the largest connected spatiotemporal system of which we are parts. (c) The others are things of roughly the same kind: systems of objects, many of them concrete, connected by a network of external relations like the spatiotemporal distances that connect objects in our universe. (Lewis 1986a, 74–76) (d) Each universe is isolated from the others; that is, particulars in distinct universes are not spatiotemporally related. (It follows that universes do not overlap; no particular inhabits two universes.) (Lewis 1986a, 78) (e) The totality of universes is closed under a principle of recombination. Roughly: for any collection of objects from any number of universes, there is a single universe 76 Unless stated otherwise, (MF) refers to Rosen (1990) as the orthodox version of modal fictionalism. )94 Chapter VI Extended Modal Fictionalism containing any number of duplicates of each, provided there is a spacetime large enough to hold them. (Lewis 1986a, 87–90) (f) There are no arbitrary limits on the plenitude of universes. (Lewis 1986a, 103) (g) Our universe is not special. That is, there is nothing remarkable about it from the point of view of the system of universes. Apparently, (MF) takes (almost) everything (MR) claims except the very existence of the worlds.77 The denial of the existence of such worlds gives us (P), (N), (C), and (I) fictionalist, ontologically innocent, readings: (PMF) It is possible that P if and only if ‘according to the (MR)-story’ there is a possible world, w, such that at w, P. (NMF) It is necessary that P if and only if ‘according to the (MR)-story’ every world, w, is such that at w, P. (CMF) It is contingent that P if and only if ‘according to the (MR)-story’ there is a possible world, w, such that at w, P and ‘according to the MR-story’ there is a possible world, w*, such that it is not the case, that P 78. (IMF) It is impossible that P if and only if ‘according to the (MR)-story’ there is no world, w, is such that at w, P. (MF) seems more appealing than (MR) from both ontological and epistemological points of view. To begin with ontology, (MR) (and (EMR)) suffer substantially from the objection that they are simply incredible. And although strangeness cannot be a decisive knock-down argument against any metaphysical theory, (MF)’s commitment to stories does not, at least intuitively, seem so weighty as (MR)’s (and (EMR)’s) commitment to real possibilia (and impossibilia). The epistemological advantage of (MF) over (MR) (and (EMR)) rests, again at least intuitively, on the grasp of stories in comparison to real, spatiotemporal systems casually isolated from us. In particular, our imaginative skills enable us to understand (MF) has going for it that the debate between (MR) and ersatzism has been settled in (MR)’s favour. Cf. Rosen (1990, 329). 77 78 For a more restricted version, see my footnote 7 and translate it to (MF)’s terms accordingly. )95 Chapter VI Extended Modal Fictionalism stories in a straightforward way and thus provide a proper identification of mystical modal phenomena with imaginative experiments. In sum, at a first glance (MF) does not appear to suffer from the most frequent objections raised against (MR) and (and (EMR)). This, however, does not mean that (MF) is a better alternative. For now, it only presents another way of understanding and systematizing modal discourse. The next section points to some problems at the core of (MF). 6.3 Some Problems with (MF) In this section, I discuss several arguments against (MF). First, I look at a rather problematic understanding of the ‘according to the (MR)-story’ operator. I then proceed to the Brock-Rosen objection, according to which (MF) is a self-defeating position. Finally, I outline the so-called Hale’s dilemma, which challenges the expressive power of (MF). I discuss both horns of the dilemma and go through some possible responses. 6.3.1 The ‘According to the (MR)-Story’ Operator At the core of (MF)’s analysis of modality is the ‘according to the (MR)-story’ operator. The original formulation takes the fictive prefix to be ‘a potentially puzzling creature’, whose hidden complexity complicates, rather than elucidating modal matters. For, if the operator is taken as primitive, that is, as one that is not further analyzable in either modal or non-modal terms, it is very unsatisfying. Of course, every theory has its primitives. But the strategy of counting primitives leaves (MF) at a disadvantage to (MR). This is due to the fact that (MF) inherits the ideology of (MR) and even adds one more – the story operator – to avoid its ontology. (MF) might point out that the additional primitive is easy to understand. A preliminary indication of the way it works is to see it in light of alternative paraphrases. What we mean by ‘according to the (MR)-story’ means, for instance, Were the (MR)-story true, such and such would be true; alternatively assuming that the (MR)-story is true, such and such happens; or It cannot be that the (MR)-story is true and such and such is not; etc. In other words, (MF) can provide a paraphrasing of the ‘according to the (MR)-story’ )96 Chapter VI Extended Modal Fictionalism operator that can explicate it in more graspable and understandable terms. The problem is that even if the ‘according to the (MR)-story’ operator is not modal in spirit, its paraphrases are. Another option for (MF) is to bite the bullet and take the ‘according to the (MR)story’ operator to be a modal operator. In so doing, (MF) resigns from a reductive analysis of modality, yet still has the resources to provide a guide for understanding such notions. The input for such an eliminative reduction is a whole variety of modal concepts, while its output is the ‘according to the (MR)-story’ operator alone. This might be considered a substantive advantage of (MF) over its rival, especially if, as some theorists have argued, theories including primitive modal operators whose vocabulary contains boxes and diamonds only fall short when it comes to expressing some modal judgments. 79 The ideological cost of accepting (MF)’s operator is thus (at least) threefold. Either (MF) takes its leading element to be primitive without any attempt to explicate its content in more detail. Alternatively, it might approach the operator through various paraphrases, although as far as I can see any such attempt presupposes (implicitly or explicitly) a modal notion. Finally, (MF) can admit that its core is properly modal, although the modality brings a substantive explanatory benefits. Of course, this debate is not conclusive, since the literature offers various slight modifications of the above options. I do not attempt to give a precise and exhaustive account of the ‘according to the (MR)-story’ operator. Rather, I delineated a framework in the boundaries of which (MF) runs its analysis. Whether anything in this framework is worth doing is a separate and methodologically intriguing question to be settled on other grounds. And unless it is settled otherwise, (MF) is obliged to tell us something about its pivotal primitive postulate. 6.3.2 The Brock-Rosen Objection The Brock-Rosen objection concerns the necessary status of metaphysical theories. Think of a metaphysical picture that commits one to claims about the ontological nature of 79 Cf. Hazen (1976). )97 Chapter VI Extended Modal Fictionalism possible worlds. It seems that if its claims are true, they are necessarily so. Applied to (MR), the (MR)-story is true in every possible world. Given this, any fictionalist who translates modal claims into claims about possible worlds is forced to admit that the truth of the story is necessary. Since necessity implies actuality, (MR) is, even according to (MF), actually true. Put in a form of argument, the Brock-Rosen objection goes as follows80: (1) According to the (MR)-story, in all possible worlds there is a plurality of worlds. (2) Necessarily, there is a plurality of worlds if and only if according to the (MR)story, in all possible worlds there is a plurality of worlds. (3) Therefore, necessarily, there is a plurality of worlds. (4) Therefore, there is a plurality of worlds. If (MF) is a story about (MR), premise (1) must hold. This is implied by (MR)’s ontological setup plus the assumption that things could have been otherwise. For, that possible worlds exist is not a truth relativized to a particular world. Possible worlds exist, according to (MR), full stop. Premise (2) is a translation of (1). Premise (3) follows from (1) and (2) by biconditional elimination; and premise (4) follows from premise (3) because if something is necessary, it holds of the actual world too. 6.3.3 Hale’s Dillema Bob Hale (Hale 1995) poses a different objection to (MF) in the form of a simple dilemma. According to the (MR)-story, the story is not literally true. This poses a question about the modal status of the fiction: is it necessarily false, or false only contingently? Both options, Hale argues, lead (MF) into trouble. Consider first, that the (MR)-story is false contingently. Then, following the definition of ‘contingent’ as ‘possibly true’, the story might have been literally true. Put otherwise, this would mean that the (MR)-story is false (in the actual world) but possibly true. This modal intuition raises problems again when we try to restate it within the (MF)’s framework. It is probably best to quote Hale in full: 80 Cf. Dardis (2015). )98 Chapter VI Extended Modal Fictionalism If…[the (MF)-ist] opts for the view that the (MR)-story, though false, is no worse than contingently so, he must hold that the (MR)-story might be (or might have been) true. But how is this modal claim – the claim that possibly the (MR)-story is true – to be understood? If we apply he usual fictionalist recipe, what we get is: ‘According to the (MR)-story there is a possible world at which the (MR)-story is true’, which is equivalent to the conditional: ‘if the (MR)-story were true, there would be a world at which the (MR)-story is true’. Since what the antecedent hypothesizes is the (MR)-story’s truth at the actual world @, this conditional is an immediate consequence of ‘If the (MR)-story were true at @, the (MR)-story would be true at @’. But this conditional is merely an instance of the schema ‘If A were true at @, A would be true at @’, which holds whatever proposition A may be – even an impossible one. In particular, ‘If the (MR)-story were true at @, the (MR)story would be true at @’ – and hence its consequence ‘According to the (MR)story, there is a possible world at which the (MR)-story is true’ – would be true, even if the (MR)-story were impossible. Thus the official fictionalist paraphrase certainly cannot adequately capture the content of the claim that possibly the (MR)story is true. (Hale 1995, 65)81. On the other hand, suppose that the (MR)-story is false necessarily, and thus is impossible. Then, if the ‘according to the (MR)-story’ prefix is to be read as ‘were the (MR)-story true, then p would be true’, where p stands for any proposition, the theory turns out to be trivial. For conditional claims with antecedents that are necessarily false are automatically true and so any conditional of the form ‘were the fiction of possible worlds true then p’ will be true. This is, however, unintuitive since ‘were the (MR)-story true, there would be no worlds’ is not true, period. 81 Here I use ‘(MR)-story’ phrase instead of the original ‘PW’. )99 Chapter VI Extended Modal Fictionalism 6.4 Some Alternatives There are several routes (MF) might take in order to block the above arguments. When it comes to the Brock-Rosen argument, Rosen (1995), following Noonan (1994), relies on counterpart theory as presented in Lewis (1968) to make premise (1) false.82 Kim (2002) proposes another reply. Instead of a single-worlds analysis only, Kim adds tuples (pairs, triples, etc.) of worlds that play roles of possibility and necessity localizers. In so doing, (MF) might propose the following translation schemas: Possibly P iff Acc to the (MR)-story there is a one-world multiverse at which P or there is 2-world multiverse at which P or …, or P is true unrestrictedly Necessarily P iff Acc to the (MR)-story at all one-world multiverses, P and at all 2-world-multiverses, P and …, and P is true unrestrictedly83. Yet another way of dealing with the Brock-Rosen argument is presented by (Woodward 2008) and Liggins (2008). In slightly different ways, they both think that (MF) is best understood as a paraphrase strategy. In particular, (MF) should interpret modal statements in natural language as meaning the equivalent claims about the fiction. In practice, this paraphrasing strategy enables (MF) to accept the truth of the sentence ‘there is a plurality of worlds’ without countenancing the existence of the worlds. In Woodwards’s words: Firstly, we have a sentence, S1, which apparently quantifies over Fs. Secondly, we have another sentence, S2, which is the candidate paraphrase of S1 and does not apparently quantify over Fs. In the case of the modal fictionalist, if S1 were ‘there is 82 Cf. Lewis (1968) or Dardis (2015). 83 Cf. Dardis (2015). )100 Chapter VI Extended Modal Fictionalism a possible-world at which there are blue swans’, S2 would be ‘According to (MR), there is a possible-world at which there are blue swans’. Corresponding to each sentence is a proposition. <P1> is a proposition whose logical form is the facevalue interpretation of S1. <P2> is a proposition whose logical form is the facevalue interpretation of S2. Hence, whereas the truth of <P1> entails the existence of Fs, the truth of <P2> does not. Returning to the case of the modal fictionalist, if <P1> were <There is a possible-world at which there are blue swans>, <P2> would be <According to (MR), There is a possible-world at which there are blue swans>. With this situation in place, it immediately becomes apparent that ontological commitment is incurred at the level of propositions, not the level of sentences. (Woodward 2008, 276)84. The core of Woodward’s strategy is to construe (MF)’s theory as metalinguistic. In particular, (MF) specifies its interpretation of modal claims metalinguistically, meaning that (MF) is committed to the sentence ‘There is a plurality of worlds’ without being committed to the worlds themselves. Divers (1999b), stresses the inadequacy of (Lewis 1968) because counterpart theory itself lacks resources of expressing extraordinary modal claims. Divers therefore proposes a ‘redundancy interpretation’ according to which only ordinary modal claims receive worldbound interpretation. Modal operators in ordinary modal claims restrict quantification to a single world, while modal operators in extraordinary modal claims have no such effect85. Extraordinary modal claims are true or false but their truth and falsity is not relativized to single worlds. Provided the above, (MR) can represent the argument (Arg) There are many worlds It is possible that there are many worlds as the sound 84 Here, I use ‘(MR)’ instead of the original ‘GR’. 85 Cf. Chapter III. )101 Chapter VI Extended Modal Fictionalism ∃y∃z(Wy & Wz & y≠z) (Arg*) ∃y∃z(Wy & Wz & y≠z) because once we accept the distinction between the ordinary and the extraordinary readings86, the possibility operator in (Arg) is redundant. An answer to the Brock-Rosen argument would go as follows: (F) It is possible that P if and only if ‘according to the (MR)-story’, P which is ambiguous. One reading takes (F) as ordinary modal claim: (FO) It is possible P if and only if ‘according to the (MR)-story’, for some world, P while the other reading is advanced: (FA) It is possible P if and only if ‘according to the (MR)-story’, P. Now, if the (MF)-ist interprets (P) as (FA) rather than (FO), we get a truth of (FO-P): (FO-P) It is possible that there is a plurality of worlds if and only if ‘according to the (MR)-story’ there is plurality of worlds. since the right-hand side of (FO-P) is true 87. The (MF)-ist is able to claim both that it is not actually the case that that are many worlds and that it is possible that there are many worlds. Speaking of Hale’s dilemma, several ways of responding are available, aiming at both horns of the dilemma. For instance, (MF) might treat conditionals in such a way that 86 Recall the distinction from (3.2). 87 In order to avoid a confusion I am using Divers’s original notation. )102 Chapter VI Extended Modal Fictionalism conditionals with necessarily false, that is impossible, antecedents are not trivially true. Alternatively, one might endorse Rosen’s (1995) proposal to the extent that the (MR)-story lacks truth-value altogether. As I have pointed out elsewhere, such a move is far from being ad hoc because the dilemma is directed primarily against fictionalists who take their fiction to be false. It does not touch those theories that ascribe some other status to their stories and is thus not general enough to challenge every fictionalist theory on the market. Nolan (unpublished), on the other hand, challenges Hale’s statement that ‘If the (MR)-story were true’ is equivalent to ‘if the (MR)-story we true at @’. To sum up, there are various objections against (MF) and various ways of modifying it that block the objection. Note also that none of the responses are conclusive and any of them might be attacked from various angles. I will not address this further, however. Rather, in the next section I present yet another modification that fills a gap in the existing literature on the metaphysics of modality in general, in (MR) and (MF) in particular. My proposal is hybrid in spirit in the sense that it accepts (MR) as the best analysis of possibility 88, though turns to (MF) as the best analysis of impossibility. I dub the theory extended modal fictionalism (EMF). 6.5 (EMF) In what follows I make clear what I mean by (EMF). First I recapitulate crucial features of (MR) and (EMR). Second, I present the (EMR)-story, that is, the story about real impossible worlds. Third, a theory combining (MR) and (MF), (EMF), is presented. Finally, I discuss some problems with both (MR) and (MF) and show that (EMF) has means to respond them. (MR) is a thesis that there are possible worlds in Lewis’s sense. (EMR) is a thesis that there are possible worlds in Lewis’s sense and impossible worlds in an equally realistic sense. Beside this digression from (MR), (EMR) denies that there is a maximal universe of discourse. Beside (MR)’s ontological postulates (a)–(g), (EMR) accepts the following additional postulates: 88 One of the reasons is to save possible-worlds semantics. )103 Chapter VI Extended Modal Fictionalism (h) There exist impossible worlds. (i) Impossible worlds inhabit different logical spaces. (j) There exists a plurality of logical spaces. (k) All worlds are possible in some sense, i.e. K-possible for some K. (l) For any K, some worlds are K-impossible. (m)For any K, there is another kind of possibility, K*, such that some worlds are Kimpossible but K*-possible. There are various reasons for extending (MR) by postulates (h)–(m). 89 To recall, counterfactuals with an impossible antecedent have been put to heavy work as they do not always appear to be trivially true. In addition, propositional attitudes that happen to be inconsistent sometimes cannot be explained by possible-worlds talk without extending worlds by impossible worlds. Furthermore, metaphysical theories have appeared which, if one of them is true, the rest are impossible; or various mutually inconsistent epistemic and belief systems. These, but not only these, motivate us to introducing impossible - together with possible worlds. However, as my introduction indicated, the existence of concrete impossible worlds has damaging consequences for (EMR). (MF), on the other hand, does not postulate possible worlds but formulates stories about them. A hybrid view commits its proponents to concrete possible worlds but avoids any commitment to impossibilia, yet formulates a story about them. This is a combination of (MR) with the (EMR)-story and its goal is to sustain the theoretical virtues of (MR), (MF), and (EMR), while avoiding their unwelcome consequences. (EMF) is a metaphysical picture according to which possible worlds exist in a genuine way and impossible worlds exist (only) according to a particular metaphysical story. Possible worlds are maximal isolated systems as (MR) defines them. They are concrete and maximally interrelated and internally unified wholes. The (EMR)-story, on the other hand, is a set of (informal) postulates (h)–(m) that adds more ontological commitments to (MR)’s picture. The additional commitments, although false, say that our Cf., among others, Priest (1997), Nolan (1997), Berto (2009) or Jago (2014). They all agree that impossible worlds play an analogous role to possible worlds. For the opposite view, see, for instance, Perszyk (1993). 89 )104 Chapter VI Extended Modal Fictionalism logical space is not the only logical space. According to (EMR)-story, there exist a complicated hierarchy of such spaces. Apparently, the new analysis must mirror (EMF)’s ontological and ideological setup. For possibility and impossibility become fundamentally different metaphysical categories and for possible and impossible propositions ontologically depend on possible worlds and the story of impossible worlds, respectively. To start with possibility, contingency, and necessity discourse, the analysis plays out as usual: (P) It is possible that P if and only if there is a possible world, w, such that at w, P (C) It is contingent that P if and only if there is a possible world, w, such that at w, P and there is a possible world, w*, such that it is not the case, that P (N) It is necessary that P if and only if every world, w, is such that at w, P. Crucially, the analysis of impossibility is radically different. Instead of (MR)’s (I) It is impossible that P if and only if there is no world, w, such that at w, P (EMF) introduces its systematic account of impossibility along the lines of (IEMF): (IEMF) It is impossible that P if and only if ‘according (EMR)-story’ there is an impossible word, i, such that at i, P. In short, the leading motivation behind (EMF) is to sustain the advantage of (MR), sustain the advantages of (MF), provide for a finer-grained analysis of modality by using impossible worlds, and yet avoid the problems of (EMR). If it turns out that a theory might do all this, this fact is enough to take the theory seriously. Surely, such a two-sided approach to modality immediately raises an objection about why possibility and impossibility should be analysed in such an unsystematic and nonunified way. But I will not consider the objection here.90 Rather, I want to go through the 90 I pursue a way of doing this in Vacek (2013b). )105 Chapter VI Extended Modal Fictionalism above-mentioned objections raised against (MR), (EMR), and (MF) and offer answers on behalf of (EMF). For if it turns out that (EMF) can (at least partially) meet the objections, this would give its proponents an advantage over their metaphysical rivals. Again, this would not be a conclusive advantage. But it is at least a good starting point for a further elaboration. I will discuss the objections in turn. 6.5.1 (EMF) and Five Objections For the sake of simplicity, let’s take for granted that (EMR) is an inconsistent hypothesis 91. This means that a proper proponent of (EMR) would agree that (EMR) commits her to real inconsistencies that, in consequence, results in an inconsistent hypothesis. 92 (EMF) is another story since it bites the bullet somewhere else: it introduces a dual analysis based on both (MR) and (MF) to analyze possibility and impossibility, respectively. To speak about the impossible is to speak about real impossible worlds. It is to speak about worlds that exist according to the (EMR)-story. Importantly, such stories are not restricting modifiers that pass through truth-functional connectives. The objection from inconsistency can be blocked very simply using two interrelated assumptions taken from (EMF)’s theoretical background. First of all, impossible worlds do not exist, although they exist according to the (EMR)-story. This means that, second, ‘there would indeed be room for worlds according to which contradictions are true’ since ‘[t]he sad truth about the prevarications of these worlds would not itself be contradictory’ (Lewis 1986a, 7, fn.3). If so, (EMF) does not present real inconsistences a la (EMR). Let me now approach the problem of extensional inaccuracy. As is clear from the exposition, the crucial premise of the argument states that any world at which (P & ∼P) is a world at which P. In other words, possible as well as impossible worlds belong to domains of restricting modifiers ‘at w’ and ‘at i’ respectively, which, as a mater of fact, distribute across conjunctions. Recall, however, that (EMF) paraphrases any locution ‘at i, P’ as I hope to have shown this in Chapter I. However, some people might not see this as a problem. For, ‘[w]hy can you not tell the truth about an impossible thing by contradicting yourself? It seems that you have to contradict yourself to tell the truth about impossible thing. What else would we expect? Impossible things are impossible!’ (Yagisawa 1988, 203). 91 Some philosophers are willing to bite the bullet and agree that the only way to speak about the impossible is to contradict ourselves. 92 )106 Chapter VI Extended Modal Fictionalism ‘according to the (EMR)-story there is an impossible world, i, is such that at i, P’. Therefore, in light of the previous section, (EMF) is inconsistent with each of the tripartite theses that back up the objection: i) there exists a real impossible world, i ii) the distribution and introduction of conjunctions are valid logical rules in this world ‘at i’ is a restricting modifier that distributes across conjunctions. iii) Of course, (EMR) (qua modal realism) admits that ‘at w’ and ‘at i’ function as restricting modifiers. Consequently, (EMR) (qua modal realists) agrees both that there is no difference between a contradiction within the scope of the modifier and a plain contradiction that has the modifier within it. But (EMF) is not (EMR). (EMF)’s systematic account of possibility is (P) It is possible that P if and only if there is a possible world, w, such that at w, P while its account of impossibility has fictionalist features entrenched in (IEMF): (IEMF) It is impossible that P if and only if ‘according to the (EMR)-story’ there is an impossible word, i, such that at i, P. Again, ‘according to the (EMR)-story’ in (IEMF) is not a restricting modifier and therefore does not necessarily pass through the truth-functional connectives (contra (iii)). What about non-modal analysis? It is widely accepted that among the reasons for preferring (MR) to (various versions of) modal ersatzism is its non-modal analysis of modality. If we accept impossible worlds, though, the non-modal status is questioned. That is, one might challenge (EMR) for making a modal step in using the notion of a possible world. The challenge goes as follows: ‘How can you, the (EMR)-ist, avoid admitting )107 Chapter VI Extended Modal Fictionalism impossibilia? Don’t you need ‘world’ to mean ‘possible world’ in contrast to ‘impossible world’ for this to be the case?’93 Cameron (2012), on the other side, argues that the (MR), as opposed to (EMR), can formulate a non-modal account of what possibility is and also provide a story of its extent. That is, accounts of what possibility is and what possibility there is differ, although any theory should provide a package that answers both questions in a unified manner. As we already know, possibility is identified with going-on in some world, necessity is identified with a going-on in every world, and impossible things take place in no world, whatsoever. What possibility there is is given by the recombination principle by patching together parts of different possible worlds. Although this formulation of the principle is rather metaphorical, it secures at least two things. There are neither gaps in logical space nor necessary connections between distinct existences. 94 (MR)’s principles thus enable the modal realist to delineate possibility from impossibility non-modally as well as to provide the extent of possibility and impossibility in the same manner. For, as long as we admit (P) and the recombination principle (together with several background assumptions concerning the logic of our home language), there is no question of there being a world corresponding to a way the world might not be. The analysis states what it means to be possible: namely to be true in some world. The analysis also states what possibilities there are: possibilia result from the recombination principle. Whatever worlds exist, they correspond to possibilities, and no really existing world corresponds to any impossibility. Cameron, when paraphrasing what Lewis would have said to address the objection, makes this very point: I [referring to Lewis] need no ‘prior modal constraints’ on the nature of worlds to ensure this: what I mean by ‘possible’ ensures this. Similarly, there is no question of there not being enough worlds - i.e., that some possible circumstance be unrepresented by a world. Given what my analysis says possibility is, it simply follows that whatever the extent of the space of worlds happens to be, that is the extent of what is possible. Again, no prior constraints on the nature of worlds is 93 The objection in a slightly modified version is from Lycan (1994). For a response see, for example, Cameron (2012). 94 To get a full story about the plenitude principle, see Lewis (1986a, §1.8). )108 Chapter VI Extended Modal Fictionalism necessary to ensure that there is a world for every possibility: this is guaranteed by what I mean by ‘possibility’. (Cameron 2012, 7–8) If Cameron is right, (MR) but not (EMR) escapes the challenge. So far so good. (EMR) has a problem in delineating possible and impossible worlds non-modally. However, (EMF) as a kind of (EMR), can be parasitical on (MR)’s response. Recall that (EMF) is full-blooded realism about (possible) worlds, but fictionalism about impossible worlds. It is therefore still the case that possibility is (unrestricted) existence because only worlds that represent possibility exist. When impossibility comes into the game, (EMF) introduces its (EMR)-story about the hierarchy of different logical spaces. Importantly, other logical spaces do not exist and so Lewis and Cameron’s own answers still hold: whatever worlds exist, they correspond to possibilities, and no world corresponds to an impossibility. If the analysis is correct then there are worlds enough to cover possibility, full stop.95 To sum up, (EMF) has the resources to non-modally differentiate between possible and impossible worlds to the same extent as (MR). That is, possibility is identified with unrestricted existence and every world that exists is a possible world. On the other hand, impossible worlds do not exist, although there is a story – the (EMR)-story – that makes discourse about them comprehensive and meaningful. Finally, let’s return to problems with (MF), namely to the Brock-Rosen argument and Hale’s dilemma. I show that (EMF) has the resources to meet both challenges since, contra (MF), it commits us to the existence of possible worlds and only ‘pretend’ that there are real impossibilia. First, I address the Brock-Rosen argument and then discuss Hale’s dilemma. Note that (EMF) does not deny the existence of possible worlds. That is, it is still true that modal realism is true and, moreover, that it is so in every possible world. The 95 Again, the question ‘What is possibility?’ is quite different from ‘What possibility is there?’. The answer to the former in entrenched in (P), whereas the answer to the latter depends on the content of the recombination principle. A more interesting, and up to now underdeveloped worry arises when it comes to the impossible. Namely: ‘What is impossibility?’ and ‘What impossibility is there?’. To be sure, one part of the ‘What is impossibility?’ question is captured in (IEMF). However, how non-modal it is is still an open question since, as I mentioned above, ‘according to the (EMR)-story’ is thought to be primitive. Even deeper will be the answer to the second question, ‘What impossibility is there?’. Is it recombination principle that we can rely on here? Or is there another way of systematizing the hierarchy of different modal spaces? For reasons of space, I cannot address this problem here. )109 Chapter VI Extended Modal Fictionalism consequence that it is also actually true is perfectly fine, then. Given this, the Brock-Rosen argument does not get off the ground. The reason for this is the following: uncontroversially, possibility and necessity are restricted to possible worlds only. That means that the conditions for negation, conjunction, disjunction, the material conditional, and even the modal operators of necessity □ and possibility ◊ are defined in the usual way. By hypothesis, those worlds are MR-worlds and the consequence that the actual world is one of them is perfectly in accordance with (EMF) ontology. If, however, a world is an impossible world, then the truth conditions for modalizers are defined differently, as: Vw(□A) = 0 Vw(◊A) = 1.96 This suggests that the traditional understanding of necessity as truth in every possible world is in accordance with (EMF) 97. On the other hand, there is no analogical worry that there exist concrete impossible worlds. Impossible worlds exist only according to the (EMR)story. The story is necessarily false, meaning false in every possible world including the actual world. Put another way, if the Brock-Rosen argument is run to show that (MR) realism is actually true, (EMF) agrees. If, on the other hand, we want to extend the argument and show that (EMR) is actually true, this attempt will fail in principle. An analogous strategy applies to Hale’s dilemma too. For the (EMF)-ist takes the second horn of the dilemma and grants that if the (EMR)-story is false, it is necessarily so. The (EMF) is necessarily false because no (possible) world is such that it makes it hold within it. Again, the definition of necessity enables the (EMF)-ist to state that the story is necessarily false. Nevertheless, the (EMR)-story represents various impossibilities as existing and thus enriches the explanatory and expressive power of (EMF). This again happens without a straightforward commitment to impossible worlds. It appears that (EMF) does not suffer from the same problems as (MF). On one hand the theory is able to accept the existence of possible worlds and define necessity as truth in 96 Cf. Berto (2009). 97 See also my footnote 86. )110 Chapter VI Extended Modal Fictionalism every possible world. It is also willing to accept the (EMR)-story as necessarily false, meaning both not true in any possible world, but not vacuously so. For the (EMR)-story provides a story of there being different impossible worlds that represent distinct impossibilities and, in combination with (MR), benefits from both. 6.6 Counting the Costs Let me finish with a critical evaluation of (EMF) and a sketchy comparison of the proposal to (MR), (EMR), and (MF). Note that the project of (EMF) has just commenced and much more work has to be done in order to properly contrast it to its rivals. Some preliminary advantages and drawbacks have already appeared. In the above I tried to outline the advantages (EMF) brings to the analysis of modality and, to some extent, improve the traditional (EMR). This final section discusses the immediate negative consequences of the proposal. (MR) comes with a unified ontology. All the worlds are concrete. (MF) comes with a unified ontology too. For, according to the (MR)-story all and only worlds that there are are concrete. (EMR) stretches an extra mile and claims that possible as well as impossible worlds are equally real and equally concrete. Such an ontologically unified and elegant treatment, based on a single comprehensive ontology, is an attractive option. (EMF) lacks this attractiveness, as instead of one-categorical ontology it poses two fundamentally different ways of analyzing modality. I am inclined to agree. (EMF) violates both kinds of parsimony, qualitative and quantitative. In the former, the theory does not keep the number of instances of the kinds it posits down. It accepts (MR)’s pluriverse and inherits all its ontological commitments going beyond the actual. The proposal is not qualitatively parsimonious either, as it does not keep down the number of fundamentally different kinds of entities. Since (MR) and (MF) have different ontologies, their combination must accommodate them both. As a result, (EMF) does not only inherit all the advantages of (MR) and (MF), but all their disadvantages too. Again, (EMF) is in any case just a sketch of a theory and it is yet to be seen how it fares on the scale of strength. It might be appealing for those who endorse (MR)’s reductionism yet engage in impossible-world discourse. It might be appealing for those who )111 Chapter VI Extended Modal Fictionalism endorse impossible worlds discourse yet do not want to reduce the worlds to abstract constructions that represent them. Some worlds of (EMF) are of the same kind as the actual world. They are not identified with sets of sentences, states of affairs, or structural universals. Some of them exist, some of them don’t. The worlds of (EMF) are designed to play various theoretical roles. They are possibility and impossibility-localizers and truthrelativizers, although the systematic accounts of possibility and impossibility are structurally different. 6.7 Conclusion Unsurprisingly, I do not claim to have conclusively shown that (EMF) is in a better position than its rivals. I have tried to introduce another, so far underdeveloped, position. This combines two strong approaches to modality and, if successful, it benefits from both. Of course, one’s modus ponens is another’s modus tollens and, seen from a different angle, (EMF) inherits all the disadvantages of both theories. I, however, leave it to the reader which point of view to prefer. )112 Afterword . 7. Afterword This thesis approached a controversial theory of modality. The theory at issue takes modal discourse to be describing real things, be it actual, possible and even impossible. On a closer reflection though, the real aim was a bit modest. Rather than a real defence of (EMR), the thesis offered ways of making the theory meaningful in the first place. I admit that (EMR), as originally presented, is very hard to understand and, if at all, even harder to defend. A certain reinterpretation of the theory in more graspable and theoretically acceptable terms turned out to be more plausible option. I proposed three such reinterpretations: the dimensional approach (EMD), the structural approach (EMS), and the fictionalist’s approach (EMF). Notably, the interpretations are mutually inconsistent and if one prefers one of them, she is forced to deny the other two. Nonetheless, all three approaches have something in common: they all are to be a version of (EMR) (with respect to criteria formulated in Chapter I), a thesis according to which possible and impossible worlds are real entities. Also, they all are hybrid in nature because beside (MR)’s spatiotemporal systems, they posit metaphysical indices, metaphysical structures and ‘according to (EMR)-stories’, respectively. There is a legitimate worry as whether these proposals are versions of (EMR) rather than versions of other sort of theories. For, the metaphysical frameworks substantially differ from the original setup of (EMR) and, more importantly, from (MR) itself. My preferable response to such worry is that it is only to be expected that something has to go if we modify one’s theory to such an extent. (MR) gives us a rich ontology, although has no room for impossible worlds. Importantly, my proposals sustains (MR)’s ontological commitments and, at the same time, provide room for impossible worlds. Rival alternatives do the opposite: they start with an assumption that (MR)’s ontology is too much to bite and any attempt to extend it is mistaken from the very beginning. The methodology behind my proposals rejects the assumption want, in my opinion, gives us an independent reason to think of them as alternatives to (EMR). )113 Afterword . One might grant me this point, yet press the doubt that (EMR) is an ‘too incredible’ and any attempt to paraphrase it in terms on different metaphysical terms lacks a basic theoretical justification. For, (EMR) violates our firm pre-theoretical opinion about what exist. I agree. Unless philosophers find a theory in accordance with their firmly entrenched opinions, or unless philosophers understand a theory at issue, we should not be surprise about their resistance. But alternatives I am proposing do, in my modest opinion, make sense of (EMR) and avoid the very inconstancy and incredibility of the hypothesis. Moreover, we should pay an extra attention what exactly we understand by the data. (EMD), (EMS) and (EMF) try to draw such a line. A worry remains. Namely, to what extent the alternatives fulfil the criteria I introduced in Chapter I? To recall, the criteria at issue are The Way of Parity, The Way of Reductiveness, The Way of the Concreteness and The Way of Representation. Each of the ways characterises a particular features of (EMR) and if a theory fulfils one of them while another theory does not, we have a reason to believe that the former is closer to (EMR) than the latter. Given the Ways, (EMD) presents an alternative to (EMR) with respect to The Way of Parity and The Way of the Concreteness. In the former, recall that possible and impossible worlds are indices and, as such, are not reducible to something else. They exist in the very same way as times and spaces exist and, derivatively, as table chairs and continents do. Speaking about The Way of the Concreteness (EMD) fulfils the condition since, by definition, spatial, temporal and modal indices make concrete objects (including worlds) spatial temporal and modal objects, respectively. On the other hand, (EMD) does not aim at a fully reductive account of modality (contra The Way of Reductiveness). Rather, the theory argues for a so-called soft reductionism, according to which a) temporal, spatial and modal indices are taken to be metaphysically simple and b) the at-a-worldness relation is primitive. This feature localises (EMD) somewhere between (EMR) and modal ersatzism. Similarly, (EMD) does not aim at genuine representation. Although possible and impossible worlds exist, the ways they represent possibility and impossibility differ from those of (MR) and (EMR). However, as (5.4.1) shows, magical account can be squared with (MR)’s desiderata regarding representation. )114 Afterword . What about (EMS)? I have for it that this approach plays well when The Way of Parity and The Way of Concreteness are at issue. To begin with the latter, recall that the crucial feature of (EMS) is the relation of ontological dependence of structures on (MR)’s worlds. Since (MR)’s worlds are concrete, (EMS) coheres with (MR) and (EMR) regarding the concreteness. Moreover, possible and impossible worlds exist out there is reality, in accordance with The Way of Parity. As in the case of (EMD) though, (EMS) cannot provide non-modal analysis of modality. For, the notion of ontological dependence is modal in nature. I however leave on the reader to decide how bad the consequence is, given other (positive and negative) features of the proposal. Also, (EMS) refrains from genuine representation and prefers representation by magic instead. Finally (EMF). It is undeniable that it respects The Way of Concreteness since concrete possible worlds exist, and concrete impossible worlds exist according to (EMR)story. I am inclined to think that The Way of Reductiveness, The Way of Representation are compatible with (EMF) too. Namely, as (6.5.1) shows, (EMF) draws a non-modal line between possibility and impossibility in the very same way as (MR) does. Also, (EMF), at least in the case of possibility, represents modality in a genuine way since possible worlds are spatiotemporal universes. However, (EMR) violates and The Way of Parity. Indisputably, opinions about the importance, relevance, and priority of other criteria for the evaluation of metaphysical theories vary. Consistency, however, is still taken as a necessary feature of any theory and in this respect (EMF) is preferable to (EMR), even at the cost of accepting the ‘according to the (EMR)-story’ operator. (MR) and (MF) are still preferable when it comes to qualitative parsimony and systematic and unified analysis, although they fall short in a finer-grained representation of distinct impossibilities. (MF) is preferable to (EMF) in cases of qualitative and quantitative parsimony as well as systematic and unified analysis of modality. However, it is still an open question how ‘modal’ its analyses are in comparison to (EMF) and how the representation of both possible and impossible phenomena works. I do not find the debate conclusive, though. Every alternative to (EMR) I presented deserves an extra attention with regard to all its consequences, its advantages and disadvantages in comparison to its rivals as well as one’s metaphysical preferences. But I )115 Afterword . hope to have shown at least two things: first, (EMR) can find its meaningful metaphysical interpretation; second, it deserves a new start. This thesis aimed at such resurrection. )116 References . 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