Lewis’s starting point are the Leibnizian biconditionals: p is necessary iff it is true at every possible world, possible iff it is true at some possible world, and actually true iff it is true at the actual world. If these biconditionals are true, then to provide a reduction of the modal to the non-modal, one need only provide non-modal analyses of the notion of a possible world, actuality, and what it is to be true at a world. Lewis makes two assumptions in his definition of a possible world. The first is unrestricted mereological composition: for any collection of objects, the Xs, there is an object which is the sum of the Xs. The second is that there is a dyadic equivalence relation x is spatiotemporally, or analogously, related to y. That is, we can take all the things that there are and divide them into groups such that every thing is in exactly one group, each member of a group bears some spatiotemporal relation (like is 4 meters from, or is 2 years from) to each other member of that group, or at least bears some relation analogous to a spatiotemporal relation, and no two members of distinct groups stand in any spatiotemporal relation.
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