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Notes toPossible Worlds

1. For important applications of possible worlds, see the SEP entries onsupervenience,rigid designators,two-dimensional semantics,conditionals,the logic of belief revision,common knowledge, andbelief. A particularly illustrative possible worlds analysis of the concept of prudence is found in Bricker 1980.

2. A logic is often (quite legitimately) defined more generally to be a language together witheither a semantic theory or some sortof deductive apparatus, as in the definition found in theEncyclopedia entry onclassical logic. However, as the distinction between extensional and intensional logics drawn here is a purely semantic one, it is convenient for our purposes to use the more limited, semantically-oriented definition.

3. A word about the corner quotes, or “quasi-quotes”, aroundthe 2-element expressions ‘◇φ’ and ‘□φ’ here.Poss andNec aremetalinguistic statements, statements of a language — in this case, English enhanced with a bit of logical apparatus — madeabout the statements in (typically) some other language — in this case, the statements of some (unspecified) modal language ℒ. We need some type of quotation marks around ‘◇φ’ and ‘□φ’ inPoss andNec because ‘◇’ is not itself a part of our English metalanguage; rather, we are referring to certain statements of ℒ that contain it, viz., those that begin with it. Ordinary single quotation marks will not do, as they create names for the very expressions they enclose and our intent here is to refer, not to the strings ‘□φ’ and ‘◇φ’ themselves but, rather, in each case, to sentences of ℒ that have theforms those expressions indicate. This is the purpose of the quasi-quotes. Thus,Nec can be read as follows: For any sentence φ of ℒ, the sentence that results from prefixing the symbol ‘□’to φ is true if and only if φ is true in every possible world. Similarly forPoss. (Readers familiar with the programming language Lisp will recognize the similarity between quasi-quotes and the Lisp backquote operator.) For more on this topic, see Washington 1998 as well as the entryQuotation.

4. Tarski's own early work on the semantics of predicate logic was much more informal than the account here, which for the most part reflects the development found in modern expositions that came to full flower in, notably, Kemeny 1956a and 1956b. The account here differs from both Tarski's approach and its typical modern incarnations in that interpretations assign values to variables directly, much as if they were names. On the more typical approach, interpretations do not assign values to variables; rather, variablescan be assigned different values within one and the same interpretation.(See§4 of the entry onclassical logic for details.) The chief advantage of the approach adopted here (which does not alter the critical notions of validity and logical consequence in any significant way) is that it enables one to define the notion of truth under an interpretation directly (and rather more simply). For Tarski's own approach, see§2 of the entry onTarski and§2.1 of the entry onTarski's truth definitions. See also the seminal work of Etchemendy (1990) for a more philosophical examination of Tarski's work.

5. Bayart 1958 and 1959 are surprisingly unfamiliar to contemporary philosophers and logicians. In these papers, Bayart — working largely independently, it appears — developed a possible world semantics for first-order modal languages and proved the soundness and completeness of first-order S5. The articles are translated and given a very informative introduction by Cresswell (2015), whom the author thanks for introducing him to Bayart's work. See the entryModern Origins of Modal Logic as well as Copeland 2002 and Goldblatt 2003 for a comprehensive overview of the historical development of possible world semantics inthe 20th century.

6. We follow Kripke (1963) in taking names to be “rigid”, i.e., to have a fixed denotation that is independent of worlds. A more general (but, for purposes here, needlessly more complicated) approach, following Carnap 1947, is to assign intensions, not only to predicates, but to names as well. See,e.g., ch. 13 of Garson 2006, esp. §13.2, as well as§3 of the entry onintensional logic, especially §§3.3–3.5. See§3.6 of the entry for a discussion of problems arising from the assumptionthat names are rigid.

7. We again follow Kripke (1963) here and adopt avariable domain semantics for quantified modal logic to capture the intuition that, “under different circumstances, fewer, more, or other things might have existed”. A common alternative is simply to let the quantifiers range over the singlefixed domainD of all “possible objects” and, instead, introduce a primitive existence predicate ‘E!’ whose extensionvaries from world to world. Aside from the fact that Kripke's treatment of the quantifiers arguably yields an intuitively more correct semantics of quantification in modal contexts, formally, unlike a fixed domain semantics, it prevents certain controversial modal principles — most notably, the so-called Barcan Formula^⌈◇∃νφ → ∃ν◇φ^⌉ (Barcan 1946) — from being logical truths. See the articles onthe possibilism-actualism debate (especially§3.2 and§4.1) andmodal logic (especially§13) for detailed discussion.

8. Some possible world semantic theories (e.g., Menzel 1991 and (in its own way) Jager 1982) impose the additional condition thatI_π(w) contain only (n-tuples of) things that exist inw. This condition reflects the thesis ofserious actualism that exemplification entails existence, i.e., that it is not possible to exemplify a property at a possible world without existing in that world. For more on serious actualism, see the entry on thepossibilism-actualism debate, especially§4.1.1.

9. In a more comprehensive exposition of possible world semantics, the definition of a possible world interpretation would include a binaryaccessibility relationR on the setW of worlds and the modal clause would say that^⌈◻ψ^⌉ is true atw if and only if ψ is true at all worldsu that areaccessible fromw i.e., all worldsu such thatRwu. (See§§7–8 of the entry onmodal logic.) Our semantics here is essentially the special case of this approach where all worlds accessible to each other.

10. For examples of the use of intensional entities in more sophisticated developments of possible world semantics, particularly with regard to the semantics of natural language, see, e.g., Lewis 1970, Montague 1974, Kaplan 1979, and Cresswell 1973, 1985a/b, 1988, 1990, 1994, 1996 as well as theEncyclopedia entry onintensional logic.

11. Thede re/de dicto distinction traces back to Aristotle(see Nortmann 2002) and was a matter of robust discussion in the medieval period (see theEncyclopedia entry onmedieval theories of modality).

12. More exactly, φ exhibits modalityde re if there is a subformula of φ of the form^⌈◻ψ^⌉ or^⌈◇ψ^⌉ containing either an occurrence of a name or a free occurrence of a variable. Thus, for example, ‘◇∃xFx → ∃x◇Fx’ — an instance of the controversialBarcan Formula — isde re because ‘x’ occurs free in the subformula ‘◇Fx’. This definition corresponds to Fine's notion of modalityde re in thestrict sense (1978, 143). Sentences arede re in theloose sense if they simply contain a free variable in a modal context. As he notes, the strict sense is appropriate if names are understood semantically (as they are in this exposition) to berigid designators. (Fine defines strictness and looseness for modalityde dicto, but it amounts to the same thing.)

13. This is of course not to deny that there are robust alternatives to these three. Two notable recent examples are found in McDaniel 2006 and Yagisawa 2010. Both develop detailed theories of worlds that bear important similarities to concretism but depart from it sharply on several counts, in particular in their answers to QW.

14. Note that‘part’ here does not mean ‘proper part’, nordoes “any two parts” mean “any twodistinctparts”. Even if there are (as Lewis seemed to hold) simpleobjects like spatiotemporal points that have no proper parts, they arestill parts of themselves. Moreover, such objects bear spatio-temporalrelations to themselves —coexisting at some time, forexample — and hence are connected by the given definition.

15. It is worth commenting on the fact that, the rubric “concretism” notwithstanding, the notion of concreteness is not actually part of the definition of a world. Lewis (1986, §1.7) himself acknowledges that his worlds are concrete according to several ways of understanding the notion, but is skeptical about whether the abstract/concrete distinction could be clearly made. Concrete objects, for example, are often defined to be objects whose parts all bear spatiotemporal relations to other parts. But there are conceptions of certain abstract objects (notably, sets of spatiotemporally located objects (Maddy 1980) and physical universals (Armstrong 1986b)) that arguably satisfy this description as well. Nonetheless, we are sticking with the “concretism” label for Lewis's view — albeit “advisedly” (Bricker 2008) — because it still has a good bit of traction in the philosophical literature (due in large measure to the influential van Inwagen 1986).

16. LetRxy mean thatx andy are spatiotemporally related and supposew₁ andw₂ overlap, i.e., that they share a common partc. Leta be a part ofw₁ andb a part ofw₂. Then, asw₁ andw₂ are both connected, we haveRca andRcb. Hence, on the reasonable assumption thatR is Euclidean,Rab. Hence, by maximality,b is a part ofw₁. A parallel argument shows thata must be a part ofw₂. Hence, our two worlds have the same parts and, hence, by basic principles of mereology, they are identical.

17. For this reason concretism is often referred to asmodal realism. That rubric is avoided here for two reasons. First, it is arguably tendentious, insofar as it suggests that abstractionist (§2.2 below) and (some) combinatorial (§2.3 below) alternatives to concretism are somehow less than robust forms of realism about possible worlds. (Indeed, Lewis (1986, 136) labels suchviews “ersatz” modal realism; see notes23 and34 for more on ersatzism.) Second, the rubric is arguably misleading. Some uses of ‘realism’ are intended to indicate that the fundamental locutions in a relevant body of discourse are semantically primitive and, hence, cannot be reduced to more fundamental notions. Some forms of scientific realism, for example, hold that discourse about the theoretical entities of physics is not simply shorthand for discourse about, say, meter readings and patterns on cloud chamber photographs. As discussed in §2.1.3, however, Lewis is areductionist about modal discourse— the modal operators, in particular, are not semantically primitive but, rather, are to be unpacked semantically in non-modal terms. See Plantinga 1987 for more on this point.

18. In his well known 1968 paper, Lewis axiomatizes the three predicates below (and a fourth) that express the primitives of his translation scheme explicitly, and calls the resultcounterpart theory. Hazen (1979) raises several challenges to the theory against which Hunter and Seager (1981) mount a defense that is tightened up formally by Forbes (1982). Notable more recent critiques of counterpart theory are found in Merricks 2003, Cresswell 2004, Fara and Williamson 2005, and Fara 2009.

19. It should be noted that, in Lewis's mature theory of 1986, resemblance relations are very fluid and contextually-dependent, and such relations can actually vary even within the semantics of a single sentence. See, for example, Lewis's response (ibid., 254–5) to the “there but for the grace of God” examples of Feldman (1971) and others.

20. These are of coursefirst-order intensions. Lewis does in fact allow for higher-order intensions as well, e.g., properties of properties of individuals likebeing a property possessed by every great general. Egan (2004) argues, however, that Lewis cannot generalize the definitions here and define, e.g., a property to be any set but, instead, must take intensions generally to be functions from worlds to extensions, much like the original definition above.

21. We use only the conditional rather than the biconditional here because there is no reason to think that every sentence φ* that is the translation of a modal sentence and happens to be true in Lewis's metaphysics should be the translation of an intuitive modal truth. Certain specifics of Lewis's metaphysics might well decide thetruth values of sentences that are undecided by ordinary modal intuitions.

22. Lewis (1986, 86) himself makes this point; van Inwagen (1986, 197), to whom Lewis credits the observation, makes the point as well.

23. DeRosset (2009a, 1003) argues that “pushing down” to the microphysical level to account for possibilities that do not seem to be the product of macrophysical recombination still leaves possibilities at the micro-level that seem unaccounted for byR. For more on the question of the success of Lewis's reductionism, see e.g., Shalkowski 1994 and Bricker 2008.

24. See Efird and Stoneham 2008 and Darby and Watson 2010 for a particularly interesting and sophisticated exchange.

25. I am indebted to Phillip Bricker for suggesting this line of response.

26. As noted already, Lewis (1973, 86) originally identified his concreteworlds with “ways things could have been” and realized later that the identification was vacuous. Even earlier, however, Stalnaker (1976, 68) had called him out about the use of ‘the way things are’ to indicate the concrete actual world: “If possible worlds are ways things might have been, then the actual world ought to bethe way things are rather thanI and all my surroundings.The way things are is a property or a state of the world, not the world itself.”

27. See van Inwagen 1986 for an especially clear and illuminating comparison of the two approaches and Plantinga 1987 for his own rather more tendentious comparison. See Lewis 1986 (§3.4) for anextended critique of abstractionism. Lewis (no less tendentiously) refers to abstractionism asmagical ersatzism — “ersatzism” because, in Lewis's view, concrete worlds constitute the most natural ontology for modality and abstractionistsseek to replace them with (inferior) abstractsubstitutes; “magical” because it is mysterious exactly how these abstract entities, as understood by the abstractionist, manage torepresent possible worlds at all.

28. Indeed, Plantinga acknowledges that states of affairs might justbe propositions, but finds it more natural to distinguish them. See Plantinga 1974, 45.

29. Note that the qualification that a world be apossible SOA is required, as every impossible SOA satisfies the definition of totality. For supposes is impossible and lett be any SOA. Then it is not possible thats obtain and, hence, trivially, it is not possible both thats obtain andt fail to obtain. Hence,s includest and, hence, trivially,s either includes or precludest, i.e.,s is total.

30. Merrill (1978) argues quite forcefully that the strict circularity ofsuch analyses does not preclude their being semantically enlightening.

31. Thus Plantinga (1987, 207):

Clearly [Lewis] is partly right: there is much about the nature of propositions we don't pretheoretically know....But we do knowsomething about the nature of propositions, prior to theory. Conceivably they could turn out to be idealized sentences or divine thoughts; but they couldn't turn out to be justanything—donkeys, or fleas, or tables, for example. We know that no propositions are donkeys, and we know that none are fleas.

The concretist would likely agree. But Plantinga (ibid., 208) continues:

Even as we can see that a proposition can't be a donkey or a flea, so we can see that a proposition can't be the unit set of a flea, or any other set of fleas or donkeys,....or a set of concrete objects of any sort. The problem, fundamentally, is that sets, like donkeys, obviously lack the relevant intentional properties — the intentional properties propositions have.

Here the concretist would balk. The reason it seems obvious that donkeys or, for that matter, individual concrete worlds, are not propositions, or any other type of intensional entity, is that it is difficult imagine a coherent, let alone natural, theory in which concrete individuals play the sorts of philosophical and logical roles that propositions are often invoked to play — exhibiting closure under boolean operations, for example. By contrast, sets of worlds (and perhaps more elaborately structured sets built up from individuals generally) are able to play those roles. As to the chargethat propositions so construed lack the relevant intentional properties, the concretist would likely argue that the charge is question-begging, insofar as (a) the properties in question are not available within the concretist framework and (b) are not obviously doing any genuine theoretical work. (Cf. the responses of Hazen (1979, 322ff) to similar arguments of Plantinga's against Lewis's analysis ofde re modality.)

32. Note, importantly, that actualism isnot the thesis that there couldn't have been anything other than the things that are, in fact, actual. Most every actualist believes there could have been. For instance, assuming his lifelong chastity, the Pope (as of April 2013), Jorge Bergoglio, has no children. However, most actualists would agree that he could have had children, i.e., that there are possible worlds in which that is the case. What the actualist denies is thatthere is a definite (non-actual)individuala such thata could have been Bergoglio's child; that there is a possible world in which it is true thata is Bergoglio's child.

33. This is an important qualification, as there is a clear sense in which concretists are not possibilists at all. One way to formulate actualism is say that there is no more general kind of being than that enjoyed by actually existing things. On this formulation, concretists are actualists. For, according to the concretist, all individuals in non-actual possible worlds exist in precisely the sameway that individuals in the actual world do. They are simply nothere (in the broadest possible sense). van Inwagen (2008, 40–41) makes essentially this point.

34. For further details of the new actualist understanding of modal truthconditions, see§2.2 of the entry on thepossibilism-actualism debate.

35. In fact, new actualists generally tend to opt for fixed domain semantics for modal logic (as discussed briefly innote 6 above) and a modal language that contains a dedicated predicate ‘C!’ to express the property of concreteness. Thus, the intuition that there could have been something that does not actually exist is understood as the claim that there are non-concrete things that could have been concrete: ∃x(¬C!x ∧ ◇C!x).

36. More exactly, a haecceity is a type ofindividual essence, i.e., a propertyP such that (i)P is possibly exemplified, (ii) necessarily, if an individuals exemplifiesP, thens exemplifiesP essentially and, moreover, (iii) necessarily, nothing other thans could exemplifyP. (s exemplifies a propertyPessentially if it is not possible thats exist and failto exemplifyP.) That haecceities are considered individual essences is enough for them to play the role Plantinga intends for them but, intuitively, haecceities can be thought of aspurely non-qualitative individual essences.

37. A set Γ of sentences in the language ℒ of a logic is (semantically)consistent if it has a model, i.e., if there isan interpretation of ℒ under which every member of Γ is true. Γ ismaximally consistent if it is consistent and no proper superset of Γ is consistent. Consistency is also often defined proof theoretically — Γ is (proof theoretically) consistent if there is no proof of a contradiction from Γ — but the definition above comports with the definition of a logic used in§1.1 simply as a language together with a semantics in which truth, validity, etc are definable.

38. Lewis 1986, 142–165. See note23 above for the explanation of ‘ersatzism’. Carnap's (1947)notion of astate description can be seen as an early version of linguistic ersatzism (see esp. §41). See Roy 1995 andSider 2002 for particularly sophisticated developments of this approach. Bricker 1987 provides an illuminating account of linguisticersatzism and a cogent assessment of its prospects.

39. The use of “the complex fact” here assumes that facts with the same conjuncts are identical; nothing terribly important hangs on this principle, but it is a natural one for the combinatorialist and, hence, it will be introduced explicitly below (proposition(29).

40. See Armstrong (1978b, 69ff). Note that, on this definition, “[s]tructural properties may or may not involve certain relations among the part of the particulars having the properties.” Thus, on this definition, properties of, e.g., theformbeing exactly two Fs appear to count as structural, even for simple universals F. Armstrong gives the examplebeing exactlytwo electrons, although of course he does not claim thatbeingan electron is a simple). However, in a later incarnation, Armstrong (1997, 32) suggests that, for a propertyU to count as structural, at least one of the universals constitutingU must be a relation that at least two of the simpler parts of any instance ofU bear to one another.

41. Armstrong developed this notion of constituency in response to serious objections from Lewis (1986a, §2.3.1) in regard to the notion of a structural universal, understood “pictorially” as isomorphs of their instances in the manner laid out in §2.3.1. Lewis's strongest objection suggests that the notion is ultimately inconsistent. Consider the structural universalWater inFigure 1. As can be seen there, theWater involves the universalhydrogen two times over; likewise for thebonded relationB. Lewis then asks:

But what can it mean for something to have a part [two] times over? What are there [two] of? There are not [two] of the universalhydrogen, or the universalbonded; there is only one. The pictorial conception...has many virtues, but consistency is not one of them.

Armstrong (1997, 119–123) argues in response that the objectionpresupposes that metaphysical complexity can only be mereological and, hence, that structural universals are simply sums of their constituent universals. To the contrary, as noted, structural universals enjoy a richer “non-mereological mode of composition”. Armstrong illustrates the idea chiefly by pointing to cases of distinct states of affairs with identical constituents. For example, there is obviously more to the state of affairsa's loving b than the sum ofa,b, and the universalloving, lest it be identical to the state of affairsb's loving a. The states of affairs must therefore in general exhibit a richer sort of composition than the mere mereological composition of its constituents to explain this difference. As structural universals are simply abstractions from states of affairs, the argument presumably concludes, it is reasonable to appeal to this richer sort of composition to explain their structural complexity. The response is promising but, unfortunately, Armstrong does not develop the concept of non-mereological composition any further.

42. This sense of ‘gunk’ originates in Lewis 1991, 19–21, 133–136. For reasons discussed in§2.3.3, structural facts cannot beidentified with the sum or conjunction of the simple facts they are grounded in. Problems that arise when the assumption that the constituency relation is well-founded (and hence that structural facts cannot be infinitely decomposed) is lifted are discussed in the supplemental documentFurther Problems with Combinatorialism.

43. That is, assumingO is the thin particularo's only intrinsic property, the atom/facto is a “thick“ particular, i.e., the thin particularo “considered along with all of its intrinsic...properties” (ibid., 434).

44. The example illustrates the sort of thing Armstrong has in mind in suggesting that complex universals enjoy a “non-mereological mode of composition”. If the complexity of the structural universalWater were simple mereological complexity, there would only be just the one universal, hydrogen (H) and the onebonding relation (B) and, hence, the structure of the universalWater could not be isomorphic to that of our moleculeM, in whichH andB are exemplified twice. The “non-mereological” composition of the structural universal, by contrast, permits there to be, as it were, two “copies” of bothH andB inWater so as to reflect accurately the structure ofM.

45. For Armstrong, supervenience has significant ontological heft: “[w]hat supervenes is no addition of being” (Armstrong 1997, 12). That is, in particular, complex states of affairs — whether mere conjunctions of simple, atomic states of affairs or structural states of affairs like our water moleculeW —are nothing, ontologically, over and above the states of affairs on which they supervene (see alsoTLP 2.032; Armstrong 1989, 113;Armstrong 1997, 44ff). Lewis (1992, 216) finds the claim difficult tounderstand in terms of a standard notion of ontology and, hence, proposes that “[Armstrong's] question is not: what is there? But rather: what does it take to provide truth-makers for all the truths. That way, it makes perfect sense to say that supervenient entities add nothing to our ontology. A supervenient entity is still an entity, but it is altogether superfluous as a truth-maker.”

46. This claim is a bit ambiguous. Non-structural states of affairs do not supervene on their atomic conjuncts individually, but jointly, a claim that, it appears, must be understood via plural quantification: forany non-structural state of affairss, there are some states of affairs such that (a)s supervenes on them and (b) something is one of them if and only if it is atomic ands includes it.

47. The astute reader will have noticed that we have avoided displaying the exact form of the state of affairsW's being water. The grammatical form suggests that it has the form [Water,W], but that would not be correct, in light of the fact noted in §2.3.1 that the constituent object in states of affairs of this form are thin particulars, andW is, by assumption, thick. Armstrong himself is not terribly clear on this point but, extrapolating from Armstrong 1993 (433–434) and 1997 (34–35) (which are not obviously consistent), the most plausible formofW's being water is [Water,w], wherew is the mereological sumo+h₁+h₂ of the thin particulars underlying the oxygen and hydrogen moleculeso,h₁, andh₂. In fact, the notion of exemplification in constituent ontologies like Armstrong's —or, at least, like our simplified reconstruction here — is a rather tricky business. See, e.g., Moreland 2011 and Pickavance 2014.

48. It is natural to understand spoilers in primitive modal terms, e.g.,a is a spoiler forS just in case it is not possible that botha andS exist. But there does not seem to be any reason the combinatorialist cannot simply deny the primitiveness of the modality here and assert instead that it is just a fact about recombination thata andS do not co-exist in any combinatorial possible world, that a world in whicha exists is not one whereS does and that this non-modal fact is the truthmaker for the modal proposition above. Thus, a world in which a third hydrogen atom bonds to our the oxygen atomo in Figure 1is simply not a world in whichW exists; it is not a world in whicho,h₁, andh₂, even bonded as they are inW, jointly constitute an instance ofWater.

49. Notably, §§1.11–1.12: “The world is determined by thefacts, and by their beingall the facts. For the totality offacts determines what is the case, and also whatever is not the case.”

50. A bit more specifically, Armstrong (2004, 74) takes a totality state of affairs to be a higher-order atomic state of affairs of the form [Tot,T,being a state of affairs], whereTis a sum of states of affairs — intuitively, the state of affairs that the sum in question is a totality with regard to the property in question. Heil (2007) argues cogently that the perceived need for totality states of affairs is a confusion based upon a “shady ‘linguisticizing’ tendency to conflate features of descriptions and features of what is described“ (ibid., 233). Keller (2007; 2009, 125–126) argues, more ominously, that the idea of a totality state of affairs is paradoxical and, hence, that such states of affairs simply cannot exist. However, the argument involves the assumption that if there are totality states of affairs, then they, too, form a totality. But,as Armstrong (2007) points out in a reply to Keller, it seems quite consistent to accept the existence of totality states of affairs and deny that they jointly constitute a further totality, much in the waythat ZF set theory denies that the sets jointly constitute a further set.

51. That said, Armstrong goes a bit soft on his commitment to modal reductionism in light of, especially, apparently irreducible modal properties of certain relations. See 1997, Ch. 9, esp. 147.

52. Armstrong (1989, 47) adds a further condition on possible worlds: they must not include any (simple) bare particulars, but the requirement stems more from Armstrong's more global metaphysical views and does not seem essential to combinatorialismper se.

53. Totality states of affairs and so-called law states of affairs, amongothers, are “higher-order” for Armstrong, insofar as theyinvolve properties of, or relations among, properties. Law states of affairs in particular are causal relations between universals (Armstrong 1986, §15.2).

54. As Armstrong (1997, 169) notes, on the combinatorialist picture, animal bodies and other large, complex, highly structured physical objects can only be “loosely” identified with similar objects in other possible worlds (or, for that matter, at other times(ibid., 104–7)), as any difference in the internal structure of complex objectso ando′ entails thato ≠o′. Cf. the discussion of loose and strict identity (ibid., 14–17).

55. Armstrong himself argues from more or less these premises to the conclusion thatO must be identical to one of its proper partsbut it is not entirely clear how to reconstruct the argument. Lewis (1992, 212) questions the reductive adequacy of Armstrong's account of analytic necessities and impossibilities, as the claim of analyticity seems to rest on themodal assumption “that a whole cannot share a universal with its part.” But, Lewis asks, why not? “If something has a mass of [two] kilograms by consisting of [two] one-kilogram parts, how does that prevent it from also having a mass of one kilogram?” See also Eddon 2007 for a detailed and incisive critique of Armstrong's analysis of quantitative properties.

56. Somewhat surprisingly, Armstrong himself does not seem to avail himself of explanations of exactly the sort given in the paragraph containing the reference to this footnote. He does, however, explicitly appeal to the notion of emergence. (Indeed, he defines thenotion of an emergent property as a non-structural “anomoeomerous” property, i.e., a non-structural propertysuch that some particular that has it has a part that does not have it; see Armstrong 1978b, 69–71, 171.) In his 1997 (152–3), he uses the notion of emergence with regard to the prospect of asimple propertyQ of a complex individuala+b that arises nomically froma andb having certain specific properties other thanQ and standing in specific simple relations. (It is unclear exactly how such aQ is to be distinguished from a structural property whose constituents are those simple properties and relations, much like thepropertiesB andH of Figure 1.) And in 2004b (14–15) Armstrong again appeals to the notion with regard to complex relations that are, e.g., asymmetric, and, hence, give rise to such impossibilities as: IfR is necessarily asymmetric, there can be no state of affairs that includes both [R,a,b] and [R,b,a]. (Recall that, due to the unrestricted nature of recombination, no simple relation can be intrinsically asymmetric.) For more on emergence, see theEncyclopedia entry onemergent properties, especially§3.

57. More complex real-world examples of incompatible states of affairs can be found in molecular biology. The misfolded prions responsible for the class of TSE diseases (Novakofskiet al. 2005) constitute a particularly dramatic example of the impossibility of a given complex object simultaneously exemplifying structural properties that it can exemplify serially. Instancesp of the normal prion protein,PrP, are routine on the surface of mammalian neurons and consist of a polypeptide sequences folded into a certain structure. Thus, both the component amino acidsofs and the structural relations they bear to one another areconstitutive of that instance of the structural propertyPrP. If that structure is altered in a certain way,s ceases to be such a protein and transforms instead into a pathogenicPrPSc proteinp′ with very different causal properties. (The “Sc” suffix here derives from the TSE disease scrapie that affects sheep and goats.) The conjunction [p &p′] of the two component structural states of affairs, therefore, is impossible simply in virtue of the fact that the structure that defines either (and endows it with its particular causal properties) involves relations between the constituents ofs that are not part of the structure that defines the other.

58. As withAW3, Armstrong himself (1989, 61–5) adds a proscription against bare particulars. The continuation ofthe quote above in Armstrong 1989 (579) reflects the modifications ofAW3′ that allow for contracted worlds: “Such possible atomic states of affairs may then be combined inall ways to yield possible molecular states of affairs. If such a possible molecular situation is thought of as the totality of being, then it is apossible world.”

59. This is of course the problem ofalien universals andalien particulars (Lewis 1986), which is discussed in the context of concretism in the supplemental documentFurther Problems with Concretism. In a detailed discussion, Schneider (2001) argues that Armstrong (1997) plausibly meets the challenge of alien particulars but fails with regard to alien universals.

60. The problem with possible world fictionalism identified in Rosen 1993in particular led to the revised account in Armstrong 1997.