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Possible Worlds Notes

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Modality and possible worlds

Two kinds of modality de re = specifies the modal status of an object's exemplifying a particular trait. de dicto = ascribes necessary, possible or contingent truth or falsehood to a proposition as a whole. Suppose I am thinking of the number two. de re: the thing I am thinking about is necessarily an even number de dicto: necessarily the thing I am thinking about is an even number Here, de re modality is true, but de dicto is false, since here it tells us that a certain proposition 'the thing I am thinking about is an even number' has the property of being necessarily true, when clearly it is not so. So ascription of de re modality must be distinguished from talk about propositional necessity and possibility ('possibly P', 'necessarily P', etc.), but possible worlds theorists still wish to assert that both styles of modality can be illuminated by their theory - this requires an understanding of objects existing or failing to exist, just as propositions are true or false. Two ways to approach possible worlds: Lewis' reductive nominalism and Platinga's actualism Possible worlds In the late fifties and sixties, logicians found they could give a clear sense of modal notions through the idea of 'possible worlds'. To say a proposition is true is simply to say it is true in the possible world that is our actual world, and to say that a proposition is possibly true is simply to say it is true in at least one possible world. The idea of possible worlds solved the empiricist difficulty of explaining necessity and possibility merely by consulting the contents of everyday experience. Modal claims of these sort do not talk about the way the world is, but about the totality of possibilities. The notion of possible worlds has roots in pre-philosophical beliefs that we all share. For example: we all believe that things could have been otherwise to the way they have turned out. This is to say, we believe that how things did turn out is just one of a number of different ways things could be. We take these different ways that things could have been as the truth-makers for these rudimentary modal claims: if we believe such-and-such is necessary, we believe that whatever way things would have been, that such-andsuch would have been the case no matter what. Possible worlds theory, then, is just a regimentation of this way of conceiving of possibility that we all share, and is just putting a technical name to something we all believe in - complete or total ways that things might have been. Nominalism Nominalism arose in reaction to the problem of universals. Specifically, accounting for the fact that some things are of the same type. For example, Fluffy and Kitzler are both cats, or, the fact that certain properties are repeatable, such as: the grass, the shirt, and Kermit the Frog are green. One wants to know in virtue of what are Fluffy and Kitzler both cats, and what makes the grass, the shirt, and Kermit green.

The realist answer is that all the green things are green in virtue of the existence of a universal; a single abstract thing, in this case, that is a part of all the green things. With respect to the colour of the grass, the shirt and Kermit, one of their parts is identical. In this respect, the three parts are literally one. Greenness is repeatable because there is one thing that manifests itself wherever there are green things. Nominalism denies the existence of universals, and possible worlds allow the nominalist to formulate an account of propositions and properties which employs set theory, and avoids ontological commitment to abstract entities. Philosophers who are nominalist about possible worlds want to claim that the framework of possible worlds does more than clarify ascriptions of de re and de dicto modality - they want to claim that it enables the metaphysician to genuinely nominalistic accounts of notions such a 'property' and 'proposition'. 'Possible world nominalists' assert that possible worlds agree with the actual world in incorporating only concrete particulars, and that once the full range of possible worlds is examined through set theory, we can provide an account of what a property is: F-ness is simply the set across all worlds whose members are all and only the concrete particulars that are F. Difficulty for the nominalist view: properties that we know to be different come out as identical - all humans are featherless bipeds. All featherless bipeds are humans. So the property of being human and the property of being a featherless biped are identical. The introduction of possible world theory to this nominalist set theory may allow them to overcome this problem. Just as there is a set of objects that are featherless bipeds in the actual world and a set of things that are human beings in the actual world, there are analogous sets for all other possible worlds. So for each possible world W there is a set whose members are featherless bipeds in W and there is a set whose members are human beings in W. However, some possible worlds will be like the actual world in that these relevant sets are identical to the actual world, but many will not - there will be worlds in which featherless bipeds are not human beings, or in which human beings have feathers, so across all worlds the set of things which are featherless bipeds and the set of things which are humans will diverge. Subsequently, if we bring together each divergent branch under its own set theoretical structure, and identify each structure with the property of being a human, or being a featherless biped, then we have what the nominalist is looking for - an account of properties that shows them to be different, but, nonetheless, nothing more than a set theoretical construction of particulars. To generalise this: properties are functions from possible worlds to sets of objects. A property, Fness, is a set structured in such a way that it correlates a set of objects with each possible world - those objects that are F in that world. So properties are set theoretical objects that run through all the possible worlds and assign sets of objects to worlds. Nominalism about propositions Possible world nominalists wish to go further, however, and show that a proposition is nothing more than the set of possible worlds where the proposition is true. This is in danger of being circular, however - 'a proposition p is the set of possible worlds where p is true'. We can clarify this concept by asking what it is for a proposition p to be true in a particular possible world W. What sort of world must W be for it to be the case that p in W? Most simply, it must be a p-ish world, and possible world nominalists take ideas like that of a p-ish world to be basic. If we accept that certain facts about a possible world are ontologically basic, then the claim that propositions are sets of possible worlds is no longer circular, since this is just an extension of the nominalist account of properties - a property is a set theoretical entity whose members are things that are F, or F-ish. Thus we are invited to treat propositions like global properties that partition worlds, not their inhabitants, according to whether they pass or fail certain criteria of description.

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