Friday, June 1, 2012

A moderately smart being that knows all necessary truths can know everything

Suppose Fred knows all necessary truths and is at least as smart as the author of this post. Fred wants to know whether a proposition p is true. So Fred says: "I stipulate that P is the singleton set {p} and that S is the subset of all the members of P that are true." But sets have their members essentially. So S is necessarily empty or necessarily non-empty. If S is necessarily empty, then Fred knows that, and if S is necessarily non-empty, then Fred knows that, too. Since Fred is at least as smart as the author of this post, if Fred knows that S is necessarily empty, he can figure out that therefore S is empty, and hence that all the propositions in P are false, and hence that p is not true. And if Fred knows that S is necessarily non-empty, then Fred can figure out that therefore S is non-empty, and hence that p is true. In either case, then, Fred can figure out whether p is true.

2 comments:

Brian said...

Another route to the same conclusion: For any truth p, actually(p) is a necessary truth. Consider an arbitrary truth q. Fred will know actually(q), since actually(q) is necessary. Now the conditional-- if actually(q), then q-- though not necessary (assuming q isn't necessary), is nonetheless a priori knowable by moderately smart folks. So Fred will be able to know this conditional, and then use modus ponens to reason to the truth of q. So Fred will be able to know q. Hence, for any truth, Fred will be in a position to know it.

Alexander R Pruss said...

Yeah, though the actually operator here is contrived and does not match our ordinary use of "actually". "If the temperatures in Alaska were 100 degrees higher, Alaska would actually be uninhabitable."

One can also do it with "the actual world". There is a multitude of ways.