Ontology Studies 8, 2008 119 Is really David Lewis a realist? Sorin Costreie University of Western Ontario Department of Philosophy Talbot College University of Bucharest Department of Philosophy Resumen. ¿Es David Lewis realmente un realista? Paradoxically, concerning the structure of the world, David Lewis endorses a very nominalistic point of view, whereas he approaches possible worlds from an extreme realistic position. The aim of the present paper is exactly to analyze the relation between the ontology of actual world and the possible worlds ontology in the case of David Lewis, and to see whether or not this tension between the two irreconcilable positions is based on an inner contradiction in his philosophy. Key Words: nominalism, realism, possible worlds, actual world Abstract Paradoxically, concerning the structure of the world, David Lewis endorses a very nominalistic point of view, whereas he approaches possible worlds from an extreme realistic position. The aim of the present paper is exactly to analyze the relation between the ontology of actual world and the possible worlds ontology in the case of David Lewis, and to see whether or not this tension between the two irreconcilable positions is based on an inner contradiction in his philosophy. Key Words: nominalism, realism, possible worlds, actual world In assessing David Lewis’ ontology, one seems to be trapped in the following dilemma: on one hand, he holds a nominalistic position regarding the structure of the world, and, on the other hand, he is well-known for endorsing a realistic position concerning possible worlds. Lewis says explicitly: “That ontology, though Nominalistic, is in other respects generous. It consists of possibilia – particular, individual things, some of which comprise our actual world and others of which are unactualized - together with the iterative hierarchy of classes built up from them. Thus, I already have at my disposal a theory of properties as classes of possibilia” (Lewis 1983, p. 209). Thus, in order to understand his point of view concerning what there is, we have to see what these possibilia comprised in various possible worlds are, and, consequently, how he understands properties as classes of possibilia. But Ontology Studies 8, 2008 120 let’s postpone for a while the discussion concerning his nominalism, and let’s see now what he means by possible worlds, and, after that, by properties as classes. Possible Worlds Lewis’ account of possible worlds could be characterize as holding four main tenets: 1. Modal Realism. David Lewis’ (modal) realism concerning possible worlds is a defence of the view that possible worlds and their contents are all equally real; he takes these worlds to be primitives and insists that his realistic interpretation of possibilia is merely a formalization of our common-sense thinking about modality: “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. (...) 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 1979, p.182). 2. As a direct consequence of his extreme Realism is his well-known doctrine of the Indexical Theory of Actuality: “Our actual world is only one world among others. We call it alone actual not because it differs in kind from all the rest but because it is the world we inhabit. The inhabitants of other worlds may truly call their own worlds actual, if they mean by actual what we do. (...) Actual is indexical like ‘I’ or ‘here’ or ‘now’: it depends for its reference on the circumstances of utterance, to wit the world where the utterance is located” (Lewis 1979, p.184). 3. A third main idea of Lewis’ concerning modality is his Counterpart Theory. There is no more identity through individuals, but, as a substitute between very similar individuals, we have the counterpart relation, based of the likeness of the individuals. “In general: something has for counterparts at a given world those things existing there that resemble it closely enough in important respects of intrinsic quality and extrinsic relations, and resemble it no less closely than no other things existing there. Ordinarily something will have one counterpart or none at a world, but ties in similarity may give it multiple counterparts. (...) the counterpart relation ought to be used as a substitute for transworld identity in explaining de re modality” (Lewis 1979a, p.126). 4. As a direct consequence of that, is his Theory of Worldbound Individuals, which states that an individual could exist in only one world, and no other identical individual is to be found in that or another world. Thus, his account concerning possible worlds could be summarized as holding three main claims: the existence of a plenitude of real possible Ontology Studies 8, 2008 121 worlds, each world might be seen as actual, and an individual is bound to a singular world. Properties & Classes Something must be said now about the role of classes in metaphysics1. Due to the axiom of extension we have it that two sets are equal if and only if they have the same elements. This provides a good extensional criterion for the identity of sets. So, even though these are abstract objects, they could be, if not entirely accepted, at least not rejected by a Nominalist. Accepting classes and reducing properties to classes have as effect what is called Class Nominalism, position endorsed by David Lewis. As Armstrong observes: “One attraction of Class Nominalism is that the theory of classes is well developed, which is not the case for the theory of properties. Quine says that the identity-conditions for classes are “crystal-clear”, while the identity-conditions for properties are “obscure” (Armstrong 1978, vol. I, p. 29). That’s why Lewis is entitled to maintain that: “To have a property is to be a member of a class” (Lewis, 1983, p. 210). Therefore, in this case of reductionism, to have a property F is reduced to the fact of being member of a certain class of things, namely the class of Fs2. But, since in the world we can find not only one-place predicates like properties, but also more-place predicates, like relations, how could one accommodate in a Nominalistic theory ontological facts as relations. Technically speaking, there are no problems in this case, just because formally a relation, an ordered pair of objects, could be reduced to an unordered class of unordered classes3. Observing now that we have used both ‘set’ and ‘class’, we can ask, if there is any difference between the two notions, and what this difference is about. What he means by these two notions, and also if he makes a difference at all between them, it cannot be explicitly find, for instance, in his paper New Work for a Theory of Universals, where he tackles explicitly this point. Here, according to Lewis’ account of classes, one could observe that he did not draw the distinctions from the standard set theory, postulating, beside relations as ‘to be a member of ’ or ‘to be included in’, elements like: ‘individuals’ (which are the elements on the bottom of this logical hierarchy, having as ontological correspondents the particulars), ‘sets’ and ‘classes’. The distinction between sets and classes is important. 1 In his book “Parts of Classes” Lewis argues that set theory in turn reduces, with the aid of mereology, to the theory of singleton functions. But, as long, as numbers could be also reduced to sets, it seems that Lewis, with regard to the traditional mathematical objects like numbers and sets, is embracing a Nominalistic positions concerning the status of mathematical objects. 2 � �s are not only actual objects, but also possible objects. �he class includes all �s from all pos� sible worlds. 3 In the case of relations, we can reduce the ordered pair <a,b> to an unordered class of unordered classes: a, a,b Ontology Studies 8, 2008 122 Intuitively, one might think any predicate determines a fixed set, namely the set of all things satisfying that predicate. It turns out, however, that this is an incoherent idea, as appears for instance in Russell’s paradox. Some predicates have an extension that is just too “large” to be considered a set. Examples are things like: being a set, being a set which is not a member of itself, being an ordinal number, being a cardinal number, having exactly one element. The list can go on. However, we still like to speak of the collection of things determined by a predicate, and we call this the extension of the predicate. A class is the extension of some predicates. Most of the familiar operations used in mathematics do not lead to classes, which are too large, and so do not cause any problem. It is now the practice in mathematics to distinguish between small classes, called sets, and large classes, called proper classes. For example “the class of all classes which does not include itself as a member” is not a set, being a proper class. In fact, it is not a well-founded set, as long as it generates a paradox in the standard set theory. In conclusion, sets are classes being capable of being members. An odd consequence of Lewis’ account of properties reduced to classes, since he does not draw a distinction between sets and classes, is that one could easily obtain such strange properties like the “property of all properties that are not themselves properties”.