Bill Lycan once attended a talk attempting to explain what licensed us to be paternalistic to our children in Kantian ethics. His observation was that when a theory must employ complex arguments to accomodate the obvious it’s a problem for the theory even if the defense works.
I have a related suspicion of results where extremely simple methods are used to prove something extraordinary. Fitch’s paradox is an example. There’s an appealing principle that all truths are knowable: in principle, for any true statement, someone would be able to know it. Some things that could have been known can no longer be. I think there would be literally no way to know where the atoms that make up my body were 1000 years ago even given perfect knowledge of the laws of physics and perfectly accurate measuring instruments. In other cases, knowing one thing rules out knowing something else. Suppose I cat-sit breakbeat and bossanova. Kittens move around a lot, so I might have to choose to either know the location of Breakbeat at 10:53 or Bossanova is at 10:53. Any of these individual things could have been known but the principle doesn’t imply that one person, or even everyone put together could simultaneously know every true statement.
Yet, suppose there is an unknown truth, call it p. That p is an unknown truth is unknowable: if we knew “p and nobody knows that p” then p wouldn’t be unknown. So, the existence of an unknown truth, p, implies the existence of an unknowable truth. Dilemma: either every true statement is known (by someone at some time), or there are unknowable truths. Given this choice, you should accept that there are unknowable truths. The dilemma can be formally presented in quantified modal logic without any difficulty–I don’t really know modal logic at all but can follow the proof. The Stanford Encyclopaedia of Philosophy explains in more detail.
There’s a promising line of challenge to the above argument made by Dorothy Edgington, based on distinguishing between “knowing in a situation that p” and “knowing that p in a situation.” The distinction goes as follows. Let p = “Anna is walking to the door of Johnson St,” and no one in Johnson St. knows that p. So “p and no one at Johnson St knows that p” is true. But I, sitting on the curb, know “p and no one at Johnson St. knows that p.” You can scale things up from Johnson St. to the entire world and attempt to dissolve the paradox. Roughly, in our situation, no one will ever know that p, but there are alternate situations in which someone could know that p. Call him Peter. If Peter is capable of finding out that no one in our situation ever knew P the problem is dissolved. As always, there are a lot of epicycles to be had here. In particular, the talk of Peter knowing about people in our situation is extremely problematic given the standard treatment of modality. Every so often, I try to wade into the literature surrounding this solution, but inevitably get depressed by its dreariness and the way that the central issues seem to very quickly get lost.
What I really want to say in response to the Fitch’s paradox is this: you have missed the point by the way you’re treating the knowability principle. It’s attractive to treat the principle as the bare assertion that for any truth, that truth is knowable. Certainly that is sufficient for the knowability principle to be true.
I think the real assertion of the knowability principle is that there is an entirely general conceptual scheme, capable of representing any aspect of the universe. This scheme is capable of representing the objects that make up the universe, the patterns that relate them, as well as evaluative discourse surrounding them (the disciplines of ethics, aesthetics, epistemology). Moreover, a culture blessed with this conceptual scheme would be in principle able to acquire evidence about any aspect of the universe implicated in their conceptual scheme. I’m firmly committed to the existence of such a conceptual scheme. More contentiously, it’s my opinion that the contemporary western world’s science and humanities are potential ancestors of this conceptual scheme. Our competence as representers of the world is capable of increase without bound. Performance limitations will always prevent us from knowing a great many things that we’d like to know. Since it requires quite a great deal of knowledge to even have a given conceptual scheme, performance limitations will almost certainly prevent us from even acquiring the ideal conceptual scheme. I can’t argue for these claims here, though.