On Oct 12, 8:07 am, Marshall <marshall.spi...@gmail.com> wrote: |I guess what I'm fixating on is what kind of values the |logic is manipulating.
It depends on what you mean.
Propositional or predicate calculus is always manipulating propositions or predicates, whether you are thinking of it as classical or intuitionistic. You get away with supposing in the classical case that "ultimately" the propositions all have "truth value" either "true" or "false" because you have assumed (pretty much by fiat) that they do, but it's not clear what the really gets you. Constructivists sometimes think differently about what propositions are, but it's somewhat difficult to convey. The kind of verificationist philosophy described in Dummett's Elements of Intuitionism might give you an idea.
For some purposes you can think of classical logic as having "values" in Boolean algebras and intuitionistic logic as having "values" in Heyting algebras. (That's essentially just a restatement of certain logical laws in algebraic form, though.)
Kripke models provide a way to think about intuitionistic logic. Instead of thinking about what is true now, think about what might or might not be discovered at future stages in the game. Then without taking off your classical thinking cap you can get a prosaic way of thinking about intuitionistic logic.