Where modal propositions get their truth values from
A common question asked in Philosophy of Logic is as follows: In virtue of what can a proposition asserting either the possibility, impossibility, or necessity of something be said to be true or false? A theory of modality is needed to explain it. Some, like David Lewis, claim that all possible worlds exist. However, the metaphysics of this claim is outlandash. Therefore, I have a more reasonable account.
A possible world is a set of internally consistent propositions. By internally consistent I mean that there is not a contradiction, nor is it possible to prove from those propositions, a contradiction. Therefore, when one says that some proposition P is possible, they are asserting that there is either a set, T (or one can construct a set, depending upon your philosophy of mathematics) that P is a member of T, and ~P is disjoined from T. A necessary proposition Q, is such that for any set T, Q is a member of T. The concequence of this, is that every possible world is infinite, since there are infinitely many true propositions. A contradiction is any set T, which as P as a member unioned with a set T' which has as its member, ~P. This union is strictly forbidden.
Now, why would I choose to modal a theory of modality within Set Theory? For the simple reason that all of mathematics can be reduced to set theory. Numbers are given set theoretic definitions [the number 1, for example, is defined as the set of all singletons, the number 2 as the set of all doubles, and so on]. Hence, the success mathematics has with set theory can be extended to an explanation of where modal logic propositions derive their truth values from.
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions