Wednesday, July 7, 2010
Reducing sets to propositions
Lewis tried to reduce propositions to sets. I think that doesn't work. But maybe the other way around does. Plausibly, for any xs there is a proposition that those xs exist. One can then identify collections with those propositions that affirm the existence of one or more things. Then x is a member of a collection c if and only if c represents, perhaps inter alia, x’s existing. We can then pick out our favorite axioms of set theory, and stipulate that any collection of collections that satisfies these axioms counts as a universe of sets. We will have to make a decision when we defined a collection as a proposition that affirms the existence of one or more things whether nonexistent things are permitted. If they are, and if propositions are all necessary beings, then we will have sets of nonexistent things. If they are not, then we won’t.