Humean Humility

Humean Humility? Stephan Leuenberger∗ April 21, 2004

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Introduction

Are there truths that we cannot represent? On one sense of the notion, what distinguishes realism from anti-realism is its commitment to the intelligibility of the claim that there are.1 In his posthumous “Ramseyan Humility”, David Lewis not only offers a frame-work to make sense of the claim, but also an argument that it is true. Following Rae Langton in [8], Lewis calls the thesis that there are unrepresentable truths “Humility”. In his argument, he distinguishes between what a property is and what its role is, and grants only that we can gain knowledge of what roles are occupied: [T]o the extent that we know of the properties of things only as role-occupants, we have not yet identified those properties. No amount of knowledge about what roles are occupied will tell us which properties occupy which roles ([13, p.2]). ∗

Many thanks for comments to Karen Bennett, Marco Lopez, Jim Pryor, Daniel Rothschild, Jonathan Schaffer, Brett Sherman, Adam Elga and the participants in his seminar, where some of this was presented; and to Stephanie Lewis for permission to refer to “Ramseyan Humility” before its official publication. 1 This way of drawing the contrast has been explored in many fascinating variations by Michael Dummett, among others; see the papers collected in [2] and [3], passim.

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To establish this, he argues that there are worlds where those roles are swapped with respect to the actual world, or occupied by properties not existing in our world; and that somehow the difference between this world and ours is cognitively inaccessible to us. Some passages suggest that Lewis draws a skeptical conclusion from this: “There is indeed a true contingent proposition about which of the possible realizations is actual, but we can never gain evidence for this proposition, and so can never know it” ([13, p.5]). But towards the end of the paper it becomes clear that the point is not that we are capable of believing, but not of knowing this proposition. Rather, we cannot even believe it, because it cannot be entertained or represented at all: “We cannot answer the question: which property occupies that role? But worse: not only can we not answer that question, we can’t even ask it” ([13, p.13]). Lewis elaborates this point in terms borrowed from the logic of questions, and from two-dimensional semantics. But this is inessential; given suitable qualifications, Humility can be captured as the thesis that some propositions are not entertainable. Our powers of representation do not suffice to capture all the differences there are between worlds. Lewis offers two arguments for Humility, which he calls the “permutation argument” and the “replacement argument”. Their main structure is the same, and the differences between them do not matter in my discussion. My reconstruction can be fine-tuned to yield either, and my criticism applies to both. Sections 2-4 show that Lewis’s arguments crucially rely on an unacknowledged premise. Assuming that mass and positive charge are fundamental properties, that premise (which I call Structuralism) implies for example the following: a world that has the same distribution of fundamental properties as the actual world, except that mass and positive charge are swapped throughout, does not differ from our world with respect to properties we talk about in ordinary language. Structuralism seems less plausible to me than the conclusion it is used to support, and because of this I think that

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Lewis’s argument is unsuccessful. Moreover, if it were successful, it would show too much. To echo what Lewis said in [12, “Putnam’s Paradox”], it is a bomb with the very intelligibility of Contingent Materialism in the target area. I show that its premises entail that Materialism, understood in Lewis’s way, is among the many propositions that we cannot represent (section 5). I further argue that Structuralism is the premise that should be given up. For if we retain Structuralism and modify the others such that Materialism is entertainable, we are forced into ad hoc denials of what seem to be possibilities (section 6). Finally, I try to undermine the deceptive appeal of Structuralism (section 7). I should also say what my paper is not trying to do. First, it does not argue that Humility is false; it might well be true despite the failure of Lewis’s argument for it. Secondly, while it gives a reconstruction of the main arguments in that paper, it does not aim at offering an interpretation of “Ramseyan Humility”. I do not explain the philosophical concerns that are in the background, and what motivated Lewis’s project of establishing Humility. To do that, one would probably have to pay careful attention to issues which I ignore here: for example, subtleties of the debate about phenomenal and epiphenomenal qualia prompted by Jackson’s knowledgeargument. In Lewis’s paper, these issues surface only in the last section, but arguably they motivate Lewis’s premise that differences of a certain sort cannot be represented.2

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Entertainable Propositions

It lies in the nature of the claim that truth may outrun representability that we cannot verify it by example. There is a paradox of knowability ([5]) and a paradox of believability ([4]), which arguably give us examples of something 2

In my view, the discussion of epiphenomenalism in Denis Robinson’s [16] helps make clear why Lewis made the moves he did in “Ramseyan Humility”; though I cannot elaborate that here.

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that is unknowable or unbelievable. They are generated by explicitly entertaining and discussing the putatively unknowable or unbelivable proposition. The same strategy manifestly does not work to generate a paradox of representability: we cannot represent or entertain something without representing or entertaining it. How can we argue for the claim otherwise? Roughly, by offering a theory of what kind of things bear truth or falsity, and laying down constraints on what it takes for us to represent something. Lewis conducts his discussion in the formal mode, talking about theories, Ramsey-sentences, realizations of theories etc. I reconstruct it in the material mode, talking about worlds, properties and functions, because I find it more transparent that way. At crucial steps I will defend the faithfulness of my reconstruction. The things that are true or false are propositions. I assume that each proposition is associated with exactly one set of worlds, and that every set of worlds has at least one proposition associated with it. For ease of exposition I simply equate a proposition with some set of worlds.3 If propositions are more fine-grained, Humility will be more plausible; but Lewis’s particular arguments will not be stronger. A proposition X is true in world v iff v ∈ X. It is true (simpliciter) if it is true in @ ∈ W, the actual world. A proposition X implies Y iff X ⊆ Y. X is impossible iff X = ∅. Propositions are also the objects of our attitudes.4 I use the semi-technical term “entertainable” for a proposition that we can somehow represent, that is available for beliefs, desires and other attitudes. “Entertainable” is a primitive in my exposition of the argument. What is entertainable can differ 3

For Lewis, propositions are classes of worlds. I am assuming throughout that the classes of worlds and of properties are sets. I conjecture that the arguments given here still go through if there are proper class many worlds, as argued in [15]. 4 This is another harmless simplification. In [9, “Attitudes De Dicto and De Se”, p.147], puzzles about indexicals lead Lewis to claim that classes of possibilia or equivalently centered worlds are the objects of the attitudes. In [13], he allows that the language of the final theory is not indexical-free (p.3). Obviously, though, each set of centered worlds is uniquely associated with a set of worlds, which is all the argument needs, as pointed out above.

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in different worlds, but I will only talk about what is entertainable in our world. The thesis that Lewis advocates is that some true propositions are not entertainable. In particular, we do not manage to represent truths about which properties play which roles. True propositions might fail to be entertainable for different reasons. For example, we might not be able to entertain infinite disjunctions and conjunctions.5 But this is not what really calls for Humility, in Lewis’s view. He argues that we have a more dramatic limitation. Even if we abstract from problems due to complexity, there are still propositions that we cannot represent. Thus I will idealize away from these limitations too, and assume that the set of entertainable propositions is closed under infinite conjunction and even under implication.6 Hence in my idealized sense, any proposition counts as entertainable iff it is implied by a set of propositions each of which we can represent. I assume that the impossible proposition is not entertainable; otherwise every proposition trivially comes out as entertainable, given closure under classical implication. I define worlds v and w to be separable iff there is an entertainable proposition that is true in v but not in w, or true in w but not in v; that proposition is then said to separate v and w. “Separable” is a technical term, defined in terms of the primitive “entertainable” (and not used as in topology). Intuitively, worlds are separable if we are in principle capable of representing differences between them. Inseparability is an equivalence relation. It partitions modal space into cells, as shown in the figure, where every dot stands for a world. Worlds within a single cell are not separable from each other. A proposition is not 5

Given fairly plausible assumptions, David Kaplan’s celebrated cardinality-argument can be used to show that not every proposition is entertainable in some sense (see [17, “A Problem in Possible-World Semantics”, pp.41-52] and [10, pp.104-8]). However, it cannot be used to establish Humility in the sense explained below. 6 Closure under implication implies closure under infinite disjunction. I do not assume closure under negation.

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entertainable if it contains some but not all worlds of some cell. The circle represents a truth that is not entertainable. q q q q q q q q q q q q q q q q q q

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If there are non-entertainable truths at all, then {@} is among them. Let Humility be the proposition expressed as follows: Humility There are truths that are not entertainable.7 A proposition can be entertainable in some worlds but not in others, as much as it can be true in some worlds but not in others. Thus if Humility is true, it might well be a contingent truth. Lewis tries to say more than just that {@} is not entertainable. He also tries to characterise worlds that are inseparable from @; in what ways they do not differ from it, which accounts for their inseparability; and in what way they do differ, which justifies positing them as different worlds in the first place. His arguments for Humility have the merit of proceeding from fairly minimal assumptions about how worlds can differ from each other. As I will make more precise below, he only assumes that worlds are distinct if they do not have the same space-time-structure, or if they have properties 7

Humility is true iff {@} is not entertainable.

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arranged differently in space-time. Humility would be much easier to argue for if we posited all kinds of unimaginable ways for worlds to differ. Lewis argues that propositions are not entertainable if they contain some but not all worlds that differ from the actual world only by the swapping or replacement of certain properties. In other words, he thinks that we cannot represent which of these actually play what role.

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From Quidditism to Humility

What is assumed about worlds by Lewis’s argument? Whatever else they might be, they are arrays of properties and relations arranged in space-time. If there are non-spatio-temporal worlds, let us ignore them for present purposes. For if there are any of worlds of the sorts we ignore, it can only lead to more rather than fewer non-entertainable propositions. If v and w have the same space-time-structure, I say that they are spatio-temporally isomorphic; the argument does not depend on how that notion is spelled out. For ease of exposition I pretend that spatio-temporally isomorphic worlds share their points, rather than having counterpart points. Space-time regions are loci of instantiation of properties and relations. It is worth quoting in full what Lewis says about categories of properties: I speak of ‘fundamental properties’ for short, but they fall into several categories. There are all-or-nothing monadic properties. There are all-or-nothing n-adic relations, at least for smallish n. There are properties that admit of degree, that is, magnitudes; more generally, there are scalar-valued, vector-valued, tensorvalued, ... magnitudes. There are relational magnitudes. Maybe my list is too long; maybe the magnitudes could somehow be reduced to all-or-nothing properties and relations, but that is a question I shall not take up here. (p.3)

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Not only a fundamental property, but any property falls into some category or other. Like Lewis, I simplify things and pretend that we are dealing only with properties and relations; I assume that some analogous treatment could be given for magnitudes and relational magnitudes. I will often say “property” instead of “property or relation”; the definitions extend straightforwardly to the non-monadic case. A set of properties supervenes on another set iff all worlds that differ with respect to the former also differ with respect to the latter. More formally, set B supervenes on set A iff for any worlds v and w, if there is a property P ∈ B and a region r such that P is instantiated at r in v but not in w, or vice versa, then there is Q ∈ A and a region r’ such that Q is instantiated at r’ in v but not in w, or vice versa.8 Supervenience-claims as defined are non-contingent, true in all worlds or none. Further, a set of properties is all-determining iff any set of properties supervenes on it. The set of all properties is trivially all-determining. Finally, a set of properties is minimally all-determining iff it is all-determining but none of its proper subsets is. Lewis assumes that there is a minimally all-determining set of properties. As he is well aware, the existence of such a set does not imply its uniqueness.9 But Lewis is ready to further assume that a minimally all-determining set F is privileged (in the sense of obeying a principle of recombination, to be explained below). Following him, let us call its elements the fundamental properties. It is up to science to discover them; Lewis conjectures that even a final and complete theory would count mass and charge as fundamental. Let two worlds be F-isomorphic if, intuitively, one results from swapping or replacing some fundamental properties in the other. More precisely, v and w are F-isomorphic iff they have the same space-time structure and there is a 8

“Vice versa” can be dropped if it is assumed that A and B are closed under complementation; but I do not assume any Boolean closure principles for properties. 9 A simple countermodel: There are just two point-sized worlds v and w in modal space (there is no empty world), and P is the only property in v, while Q is the only property in w. Then {P} and {Q} are both minimally exhaustive. More realistic models of modal space with that feature could be given.

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one-one mapping f from Fv (the set of fundamental properties existing in v) onto Fw (the set of fundamental properties existing in w) such that for any P ∈ Fv and any region r, P is instantiated at r in v iff f(P) is instantiated at r in w. Such a function f is then called an F-isomorphism between v and w. For any set of properties S, say that v and w are S-indiscernible iff they are spatio-temporally isomorphic and for all P ∈ S and regions r, P is instantiated at r in v iff P is instantiated at r in w. In other words, S-indiscernible worlds have the same spatio-temporal layout of S’s elements. S-indiscernibility is not compatible with properties being swapped or replaced, unlike F-isomorphism; properties in corresponding places must be numerically identical. As a special case, say that v and w are indiscernible iff they are Sindiscernible where S is the set of all properties. Indiscernability is a technical term, not having any direct epistemic implications. I assume that discernible worlds are non-identical. I do not need to assume the converse, that nonidentical worlds are discernible in my sense; thus if you are a heacceitist, you are free to assume that such indiscernible worlds display haecceitistic differences.10 Important in the following discussion is O, the set of properties that are named in what Lewis calls the “old language”. The old language is our language; it has only names for properties that exist in the actual world. Crucially, no fundamental properties are in O. At best, fundamental properties are named by advanced science, which does not use old language. Suppose TOE is the final and complete theory of our world @, entailing all true sentences that can be formulated in the old language. If TOE is also the final and complete theory of another world w, then we cannot represent differences between w and our world; that is entailed by the hy10

The set of all properties for which F is a minimal supervenience basis is best not taken to be the set of all abundant properties (which are sets of possibilia for Lewis (see [12, p.12])), but as a set whose elements are sparse to some degree. For if possibilia are world-bound, any two worlds differ trivially in the distribution of abundant properties, and the existence of an interesting all-determining set implies the identity of indiscernible worlds, something Lewis wishes to remain agnostic about.

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pothesis that TOE is final and complete. In my terminology, such a w and @ are inseparable. Lewis then makes the crucial assumption that fundamental properties are not named in TOE other than as bearers of roles. Now consider the Ramsey-sentences of two final and complete theories, understood in the familiar way.11 It follows from Lewis’s assumption that if these Ramseysentences do not differ, then the theories which they are derived from do not differ either. Hence w’s theory having the same Ramsey-sentence as TOE is a sufficient condition for w being inseparable from @. Given how Ramsey-sentences are constructed, being F-isomorphic and O-indiscernible is a sufficient condition for two worlds having theories with the same Ramseysentence. Thus we can take Lewis as laying down the following principle about entertainability: Inseparability If worlds v and w are F-isomorphic and O-indiscernible, they are inseparable. Roughly, an argument for Inseparability would go like this: either referring directly to properties or describing different structures enables us to distinguish worlds; nothing else could enable us. Lewis’s paper does not spend much time elaborating this principle, and neither will I on this occasion; in the rest of the paper I will take it for granted. To complete the argument for Humility, Lewis needs another premise: Quidditism There is a world that is discernible from the actual world but F-isomorphic to and O-indiscernible from it.12 11

Let t1 ...tn be the theoretical terms (here assumed to stand for fundamental properties) of TOE, and let T(t1 ...tn ) be the postulate of TOE, a sentence entailing all theorems of TOE. Substitute variables for t1 ...tn to get T(x1 ...xn ). Binding all the free variables in T(x1 ...xn ) by prefixing existential quantifiers yields the Ramsey sentence of TOE. 12 A remark about terminology: In “Ramseyan Humility”, Lewis takes “quidditism” to be the claim that if two worlds differ just by a permutation of properties, they are non-identical. Quidditism in that sense already follows from my assumption, above, that

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Humility follows from Quidditism and Inseparability. By Quidditism, there is a world w 6= @ that is F-isomorphic to and O-indiscernible from @. By Inseparability, @ and w are inseparable. Hence {@} is a non-entertainable truth.13 The next section looks at how Lewis defends Quidditism, the crucial premise. Its justification will be found wanting.

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From Combinatorialism to Quidditism

Properties fall into two main sets: the set of fundamental properties F, and the set of non-fundamental or supervening S. I pointed out above that Lewis assumes that there is a minimally all-determining set of properties that is privileged in the sense that its elements obey some principle of recombination.14 A supervenience claim states a restriction of independent recombination of properties: the supervening properties cannot recombine independently from the properties in their supervenience base. Since for Lewis the fundamental properties are the supervenience base, whatever recombination discernible worlds are non-identical; Quidditism, defined here, does not. My terminological choice is motivated by Lewis’s analogy to haecceitism: “Quidditism is to properties as haecceitism is to individuals” (p. 7). Quidditism in my sense is to properties as haecceitism in Lewis’s former sense [10, p. 221] is to individuals. In John Hawthorne’s terminology, Quidditism would be the claim that there is a world quidditistically different from our world. “Anti-Structuralism is the doctrine that there are at least some cases of quidditistic difference between worlds.” [6, p.375] My use of “Structuralism” is different from his. 13 The same kind of reasoning would allow us to infer Humility from the following two claims: Duplication There is world that is indiscernible yet distinct from the actual world. Inseparability* Indiscernible worlds are inseparable. However, if you think, as Lewis arguably did, that there are no relevant differences between worlds that are indiscernible in the stipulated sense, and that there is no strong reason to posit indiscernible worlds, you are unlikely to be too much impressed by that argument. 14 There cannot be two disjoint all-determining sets such that both recombine independently from each other. In that case one set would not supervene on the other, and hence the other one would not be all-determining, contrary to the hypothesis.

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principle Lewis stipulates will only apply to them. Stating recombination principles in full generality involves some set-theoretic machinery.15 For the present argument it suffices to appeal to consequences of a general recombination principle. Lewis uses the notion of a category to state restrictions on recombination principles (see the quote above on p.7); the monadic properties are one category, the dyadic relations another one, etc. The categories form a partition of F. Now we can state the principle that is supposed to do the work in the argument. For any category C and world w, let f be any 1-1 function from C∩Fw onto some G ⊆C. Combinatorialism For any world w and category C, there is a world v such that C∩Fv = G and (i) for any region r and P ∈ C∩Fw , P is instantiated at r in w iff f(P) is instantiated at r in v, and (ii) for any region r and Q ∈ Fw \C, Q is instantiated at r in w iff Q is instantiated at r in v. Clause (i) guarantees that fundamental properties within one category can be permuted or replaced. Clause (ii) ensures that doing that does not affect those in other categories; recombinations in different categories are independent from each other. Clearly, w is F-isomorphic to any worlds v whose existence is thus guaranteed. On what is arguably the most natural reading of “Ramseyan Humility”, Lewis claims that Combinatorialism entails Quidditism. There are two relevant passages. In them, Lewis uses the notion of an n-tuple of properties realizing a final and complete theory T. We can understand this as follows: An n-tuple is a possible realization of T iff there is a world w such that the 15

It involves an infinite Cartesian product, and the Axiom of Choice to guarantee its existence. Here I am only talking about recombination principles for properties within a given space-time, not about principles guaranteeing the existence of space-times of various sizes and shapes.

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proposition expressed by the Ramsey-sentence of T is true in w, and such that the elements of the n-tuple are the fundamental properties in w. The first relevant passage presents the permutation argument: Suppose we have the actual realization of T. Maybe some members of the n-tuple that realizes T are not fundamental properties, or maybe some belong to single-membered categories. Hold those ones fixed. Permute the rest within their categories to obtain a new n-tuple. It too would realize T. (p.5) Very similar in his replacement argument. We start with the unique actual realization of T; all fundamental properties except idlers and aliens are members of it. If we replace those properties by others, we get a possible realization by combinatorialism. (p.10) But in both passages, the very last transition is a non sequitur. It is true that Combinatorialism guarantees the existence of a world that is F-isomorphic to @, with the fundamental properties permuted or replaced. But clearly Combinatorialism does not guarantee that it will be O-indiscernible from @.16 If it is not, the permuted or replaced properties do not realize the final and complete theory of our world, given that the interpretation of O-terms is kept fixed. If we ignore Lewis’s formal mode, it is even easier to see that Combinatorialism does not imply Quidditism. Consider again what the latter says: Quidditism There is a world that is discernible from the actual world but F-isomorphic to and O-indiscernible from it. 16

The passages right after the permutation argument are further evidence that Lewis wants to argue from Combinatorialism to Quidditism. He reiterates that the combinatorial principle holds, but without giving due emphasis on its crucial restriction to distinct existences. He does not take up the worry that supervening properties are not distinct from fundamental properties in the sense relevant for the combinatorial principle.

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Why is this not a consequence of Combinatorialism? The latter says nothing at all about the distribution of non-fundamental properties and thus nothing about O-indiscernibility of worlds. Hence it cannot imply that swapping and replacing fundamental properties makes no difference to non-fundamental properties. In the light of the quotes above, the premise tacitly assumed in the transition from Combinatorialism to Quidditism must be: Structuralism Any world that is F-isomorphic to @ is O-indiscernible from it.17 Combinatorialism and Structuralism together immediately imply Quidditism; for the first ensures that an F-isomorphic world exists, and the latter that it is also O-indiscernible from @; and such a world is just what it takes for Quidditism to be true.18 I have tried to show exactly what premises Lewis’s argument for Humility needs. So far, nothing has been said about whether they should be accepted. In the next two sections, I show that Structuralism wreaks havoc within Lewis’s system. The concluding section raises some doubts about Structuralism on more general grounds. 17 It is true that a premise weaker than Structuralism suffices to get Humility from Combinatorialism: If there are worlds that are F-isomorphic to @, at least one of them is O-indiscernible from it. But this premise is much too specific to be a plausible candidate for a premise tacitly assumed. Moreover, assuming this without assuming Structuralism seems extremely ad hoc. All this is not to say that this premise is false; I want to emphasize again that in this paper I am not asking whether Humility and Quidditism are true, but only examining Lewis’s arguments. 18 The text suggests that Lewis was treating the O-properties as just another category in F@ \C for which clause (ii) of Combinatorialism applies. But to repeat, such a recombination principle is inconsistent with holding that O-properties supervene on fundamental properties and are not themselves fundamental.

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5

Materialism is not Entertainable

Why is the combination of premises Lewis uses in that particular argument in tension with other parts of his system? In earlier work, he had given us a definition of materialism on which it is compatible with modal space including outlandish, spirit-ridden possible worlds (see [12, New Work for a Theory of Universals, pp.33-9], [12, What Experience Teaches, pp.274-5], and [12, Reduction of Mind, pp.291-3]).19 For that purpose, he carved out a so-called “inner sphere of possibility” around each world. A world is then said to be materialistic iff among the worlds in the inner sphere around it, no two differ without differing in the distribution of physical properties. This definition takes “being physical” as a primitive second-order property. What worlds belong to the inner sphere around a given world w? Those in which no fundamental properties alien to w exist.20 If all fundamental properties in a world are physical, then it is materialistic. For, any two worlds in the inner sphere will likewise exclusively have physical fundamental properties, and hence not differ without differing physically. Conversely, if a world is materialistic, then all its fundamental properties are physical.21 So we can 19

Lewis calls this a definition of a “Minimal Materialism” in [12, New Work for a Theory of Universals, p.33]. It is thus supposed to give at least a necessary condition for a view to count as materialist. It allows for materialists to add further conditions to make stronger claims. (The argument in [7] shows that if we, unlike Lewis, do not assume pervasive recombination, then Minimal Materialism as defined is arguably too weak to capture even an undemanding variety of materialism. Given that entertainability is closed under implication, my argument below will automatically apply to any proposition that is logically stronger than Minimal Materialism as defined. 20 Maybe not even all of those, which could make the problem only worse for Lewis; see [12, Humean Supervenience Debugged, p.227]. 21 This converse implication can be proved given the following recombinatorial principle, which should be uncontroversial for Lewis (it would follow from a general principle of recombination which I have not stated here): For any world w and any fundamental property P, there is a world that differs from w in the distribution of P but not in the distribution of other fundamental properties. Now the proof goes through: Suppose a world w has a non-physical fundamental property P. By the combinatorial principle just stated, there is a world that differs from w in the

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equivalently take Materialism to be the proposition that is true in all worlds in which all fundamental properties are physical. To be a materialist is to claim that ours is one of those worlds.22 Lewis wants to hold that Materialism is true, but only contingently so; i.e. all actual fundamental properties are physical, while some aliens are non-physical. However, this spells trouble in combination with the premises of his argument for Humility: Suppose Materialism is contingently true. It seems that by Combinatorialism, there is a non-materialistic world w that is F-isomorphic to @. By Structuralism, such a world is O-indiscernible from the actual world. By Inseparability, any entertainable proposition that contains @ also contains w. Since Materialism contains @ but not w, it is not entertainable. This seems to be a reductio of the conjunction of Lewis’s premises. The argument just given has just one gap that I now need to fill. It moves from the existence of non-physical aliens to the existence of a world that is F-isomorphic to @ and has such a non-physical fundamental property. This presupposes that there is at least one non-physical alien in the same category as at least some actual fundamental property. So whoever endorses Lewis’s argument for Humility along with the claim that Materialism is true and entertainable needs to deny that presupposition. But though such a denial is compatible with the letter of the claim that Materialism is contingent, it is surely not compatible with its spirit. Remember what categories are: the monadic properties form a category, as do the dyadic relations, etc. Holding, plausibly enough, that there are actual monadic fundamental properties commits the denier to maintain that necessarily, all monadic properties are physical. But what independent reason could there be to think that there distribution of P but not in the distribution of physical fundamental properties. Hence w is not materialistic. 22 Lewis himself uses a formulation of that sort in [12, Reduction of Mind, p.292]: “[A]ll fundamental properties and relations that actually occur are physical. This is the thesis of materialism.”

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are non-physical relations but not (mondic) properties? Since the maneuver considered is extremely ad hoc, my argument shows that Materialism, if it is true, is not entertainable. There should be little tempatation to argue that since it is entertainable, it is false, in a sort of parody of Meditations III. A premise only slightly stronger than Structuralism yields the following verdict independently of the truth of Materialism: if there are non-physical properties in every category, Materialism is not entertainable.23 That stronger premise is: General Structuralism Any worlds v and w that are F-isomorphic are O-indiscernible.24 Structuralism is a special case of General Structuralism, covering only worlds that are F-isomorphic to ours. In my reconstruction of Lewis’s argument, I refrained from claiming that General Structuralism is tacitly assumed, for the weaker premise is sufficient for the argument to go through and is only moderately ad hoc. However, I think that it also defensible to reconstruct the argument as assuming the stronger premise. If Structuralism were true and General Structuralism false, then the actual world had a special distinction not shared by all worlds, namely that its true description in the old language is fully determined by the pattern of instantiation of fundamental properties. For Lewis, the default view is that our world is not metaphysically special, and thus he would have made it explicit that he was relying on one of its contingent features. 23

There is an alternative route to the claim that Materialism is not entertainable that does not need a strengthening of Structuralism. It assumes that there are possible physical properties in any category. I am less confident about this assumption than its converse, that there are possible non-physical properties in any category. 24 The argument for the claim just stated then goes as follows: Let w be a materialistic world, and P one of its properties. Then since any category has non-physical properties, there is a non-physical Q in the same category as P. By Combinatorialism, there is a world v that is F-isomorphic to w, but where P is uniformly “replaced” by Q. By General Structuralism, w and v are O-indiscernible. By Inseparability they are inseparable; i.e. any entertainable proposition that contains w also contains v. Hence Materialism is not entertainable.

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The argument in this section does not only work against Materialism, but against pretty much any contingent supervenience claim, except the Leibnizian doctrine that all relations supervene on monadic properties. At least this lesson is taught by the argument: if one makes claims about modal space from a perspective outside a particualar world, and one then goes on to make claims about which distinctions are thinkable and which are not, one should check whether some cherished first-order philosophical claims can still count as intelligible.

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Fine-tuning Categories

My reductio shows that there must be something wrong with the premises of Lewis’s argument. This section looks at a proposal to avoid the verdict that Materialism is not entertainable while retaining Structuralism, Inseparability and some fairly strong combinatorial assumptions. I argue that minor modifications of that sort are not promising. My argument only works if there there is a category with both a nonphysical and an actual property. I pointed out that given how categories are understood, a defender of the contingent truth of Materialism cannot plausibly deny that assumption. But maybe categories should be understood differently. After all, Lewis did not discuss them at length, and maybe the passage quoted is just intended to convey the general idea of what they are, without committing to a particular partition of the set of all possible fundamental properties. One might say, for example, that properties only instantiated at point-sized-regions are one category; we can call them Humean. Other types of regions, characterized in terms of size and shape, yield other categories, on that proposal.25 This makes the sort of region at which it is 25

If fewer properties and relations are in the same category, then Combinatorialism becomes a correspondingly weaker claim. I am not trying to assess here whether this should be considered a significant price to pay; my objection to the strategy considered here will not depend on this.

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instantiated essential to a property. Humeanism is then the claim that all actual fundamental properties are Humean. Is Humeanism entertainable, given Lewis’s premises? Combinatorialism, as stated, only allows recombination within a category. Thus if properties instantiated in point-sized regions form their own category, the assumption that there are possible non-Humean monadic properties is not enough to show that Humeanism is not entertainable. If Humean properties are defined, as above, to be the category of properties instantiated at points, my argument does not apply to Humeanism, and cannot show that it is not entertainable. But this is small comfort for materialists, for two reasons. First, Humeanism cannot count any more as a promising empirical hypothesis. There are strong empirical reasons to think that it is false (see [14], for example); our best physical theories just are not local in the relevant sense. Thus a materialist should not commit herself to Humeanism, for she should avoid making claims that run counter to current physics. Secondly, if Humean properties are defined as point-sized properties, then Humeanism cannot be taken to be a version of materialism - unless every point-sized property is physical. But that assumption would also be ad hoc. (Ad hominem, it would also be contrary to Lewis’s view.26 ) Thus since Humeanism does not capture her view, the move to take categories as more-fine-grained does not help the materialist. For reasons similar to those given above, it appears that other claims that are stated in terms of a category of properties (of the form “all actual fundamental properties belong to one of the following categories: ...) are not suited to capture the view either. Hence the move does not avoid the conclusion that materialism is not entertainable. One might now be tempted to change the conception of categories fur26

As expressed in [11, Introduction, p.x]:“I take it that materialism is metaphysics built to endorse the truth and descriptive completeness of physics more or less as we know it; and it just might be that Humean supervenience is true, but our best physics is dead wrong in its inventory of the qualities.”

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ther, such that properties may belong to different categories although they are instantiated in regions of the same type. One could then define an Fisomorphism* to be an F-isomorphism that preserves membership in a category. Structuralism* could then be understood as the claim that all Fisomorphic* worlds are O-indiscernible. Now it is open to stipulate that no physical property belongs to the same category as a non-physical property, and the problem is avoided. I do not have any objection against this move. But clearly, Structuralism* is not Structuralism, hence this cannot count as a solution that preserves Structuralism.27 I will not examine further how somebody committed to Structuralism might react to the argument, for I do not see a promising route. Rather I want to end with some considerations regarding Structuralism that are independent of the argument against the entertainability of materialism.

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Doubts about Structuralism

What does Structuralism involve? Lewis thinks it likely that mass and charge are among the fundamental properties (p.3). Assume that they are, and that they belong to the same category. Now consider the example I introduced earlier: a world that has mass at each space-time point where there is positive charge in our world, and vice versa, and that has the same distribution of the other fundamental properties. Structuralism implies that such a world will be O-indiscernible from our world; i.e. as far as properties named in the old language are concerned, it is exactly the same. Of course, the laws governing the interactions of mass and charge are different in such a world, but advocates of Structuralism will happily grant that. There is another sort of law, linking distributions of fundamental properties to more familiar features of the world. Let us take the heaviness of a desk as an example. If Struc27

In addition, combinatorialism is weakened further, in a way that might be considered costly for somebody strongly committed to it, such as Lewis.

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turalism is true, a desk is not heavy because it is mass that is instantiated so many times (or in a large magnitude) in the region occupied by the desk; for if positive charge were in its place, it would still be as heavy. I do not have any strong intuitions that a desk with mass and charge swapped is not heavy, but neither do I find it obvious that it is. In general, I do not know how to solicit intuitions about de re possibilities for properties that I am not acquainted with, and according to Lewis I am not acquainted with mass or charge. I note this consequence of Structuralism to raise a question for its defenders, but without giving it probative force. I cannot do more than sketch my main objection against Structuralism here, but it is roughly this: it seems possible that there is a world that is F-isomorphic to the actual world in which there are no solid bodies. No description of a world that just gives a pattern of instantiation of fundamental properties implies that it displays familiar features on the macroscopic level, or that it does not. Think of a world of spatio-temporally located immaterial spirits, of just as many sorts as there are fundamental properties in our world. If this is a real and not just a seeming possibility, then Structuralism is false, since “solid” belongs to the old language and has its interpretation kept fixed. To be sure, a defender of Structuralism can deny that the seeming possibility of a world that is F-isomorphic to the actual world yet has no solid bodies is a real possibility. But this should be recognized for what it is - the assertion of a brute, strong necessity, in Chalmers’s terms (see [1, pp.136138]. If she is willing to make such a claim, why should she not be willing to say that non-materialistic worlds only seem possible, which dispenses from the task of formulating materialism in a way that allows for its contingency? Why is it that Structuralism can seem plausible? Because we assume more about fundamental properties than we are entitled to. Sometimes we are in the grip of a certain picture of their nature that is more often presupposed than articulated. It depicts fundamental properties as having one manifestation: they appear as matter-like. The motley of our world is due

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to the complexity in the arrangement of bits of matter. For many familiar features of the world, we do have a conception of how an arrangement of matter in space-time gives rise to them. Apart from their matter-like appearance, fundamental properties, on this picture, play a role in the way earlier time-slices of the world lead to later time-slices. That role is captured by those dynamical laws postulated by the final physical theory that describe the evolution of the pattern of fundamental properties. Given the supervenience of laws on the distribution of properties, F-isomorphic worlds have exactly the same laws involving fundamental properties, modulo permutation and replacement within the laws. Thus they have the same property-roles occupied, in a sense that can be made precise. But since all of these fundamental properties have the same matter-like appearance, the worlds are the same in all of their familiar features. Thus the picture leads us straight into Structuralism.28 There might be other, more theoretical motivations for holding it, discussion of which would require a different paper. However, in a case where Structuralism is not explicitly endorsed, but only presupposed, as in Lewis’s argument, I think it is likely that the matter-in-motion-picture of the world just sketched is in the background. That picture has been popular in natural philosophy since the Early Modern period. However, I do not see in what sense it is forced on us. It is probably misleading even with respect to some actual physical fundamental properties.29 Much less is it compulsory for all possible non-physical ones. I have offered three types of considerations against Structuralism: first, if Lewis’s other premises are plausible, then my argument that materialism is not entertainable already gives us a reason to reject Structuralism; secondly, if 28

It is not clear how fundamental relations should be understood on this picture. The grip of the picture might thus explain why some philosophers are reluctant to give up otherwise under-motivated views on which only monadic properties are fundamental, in particular Humeanism. 29 I am influenced here by a talk by Jeremy Butterfield, and by conversation with him.

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conceivability-arguments have force, then the view is in trouble; and thirdly, we can explain away intuitions in its favour. Although my brief discussion has left many issues untouched, whether or not Structuralism is ultimately tenable, we should not take its truth for granted. In the following I briefly explore the consequences of its falsity. What conception of possible fundamental properties should replace the matter-in-motion-picture? No particular one; those properties come in many varieties. Some manifest themselves in the way matter does, others do not. Once we give up Structuralism, there is a need to distinguish two notions of what the “role” of a property is. In the presence of a certain pattern among its peers, a fundamental property gives rise to non-fundamental ones. One might call this its “thick role”; on the matter-in-motion-picture, the thick role of every fundamental property is to give rise to matter. Its “thin role”, in contrast, is its place in the pattern of fundamental properties only; it is summarized by the dynamical laws describing the evolution of that pattern. In a sense that can be made precise, Combinatorialism implies that thin roles are contingent, and Structuralism implies that every fundamental property in a given category has exactly the same thick role. To express differently one of the reasons why I think Structuralism should be rejected: it does not seem that every possible property has the same thick role.30 I hope that the distinction between thin and thick roles has potential to shed light on metaphysical disputes about properties, but this is not the place to examine this. Here my aim has only been to clarify what assumptions go into Lewis’s argument for Humility, and to argue that one of them should be rejected. 30

An analogous distinction can be made between thin and thick laws. Roughly, thick laws are inter-level-laws, while thin laws govern the bottom-level.

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References [1] David J. Chalmers. The Consicous Mind. In Search of a Fundamental Theory. OUP, 1996. [2] Michael Dummett. Truth and Other Enigmas. Harvard UP, 1978. [3] Michael Dummett. The Seas of Language. Clarendon Press, 1993. [4] Michael Fara. The Paradox of Believability. Personal Webpage, 2003. [5] F.B. Fitch. A logical analysis of some value concepts. Journal of Symbolic Logic, 28:135–42, 1963. [6] John Hawthorne. Causal Structuralism. Philosophical Perspectives, pages 361–378, 2001. [7] John Hawthorne. Blocking Definitions of Materialism. Philosophical Studies, pages 103–113, 2002. [8] Rae Langton. Kantian Humility. OUP, 1998. [9] David Lewis. Philosophical Papers, Vol.I. OUP, 1983. [10] David Lewis. On the Plurality of Worlds. Blackwell, 1986. [11] David Lewis. Philosophical Papers, Vol.II. OUP, 1986. [12] David Lewis. Papers in Metaphysics and Epistemology. CUP, 1998. [13] David Lewis. Ramseyan Humility. In David Braddon-Mitchell and Robert Nola, editors, The Canberra Plan. Forthcoming. [14] Barry Loewer. Humean Supervenience. Philosophical Topics, 1996. [15] Daniel Nolan. Recombination Unbound. Philosophical Studies 84.2-3, pages 239–262, 1996. 24

[16] Denis Robinson. Epiphenomenalism, Laws and Properties. Philosophical Studies, pages 1–30, 1993. [17] Walter Sinnott-Armstrong, editor. Modality, Morality, and Belief. Essays in Honour of Ruth Barcan Marcus. 1995.

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