# 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.

#1I don't have any answers for you, I only have one nit-picky comment.

Chaoslord2004 wrote:Errrrrr.... not really. Almost all of it can, but set theory can't quite deal with collections "larger" than sets (like proper classes). Unfortunately, such "larger" collections are a natural conclusion of using set theory, so set theory does a very good job of explaining other things but not entirely itself. Plus, there's category theory, which is basically abstract nonsense.

That said, set theory is a fine base for looking at modal truth values.

#2Yiab wrote:The set-class distinction is controversal.

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