The Knowability Paradox is the surprisingly easy argument that if there are unknown truths, there are unknowable truths. Use "Kp" to mean that p is known (say, to humans). Then, suppose p is an unknown truth and let q be the proposition (p and ~Kp). Then q is true, but cannot be known, since if q were known, its first conjunct, namely p, would also be known, and both of its conjuncts would be true (since knowledge entails truth), so that we would have both Kp and ~Kp, which is absurd. For reasons I don't quite understand, some people think this is a paradox rather than just a perfectly good, and highly intuitive ("if p is an unknown truth, then that p is an unknown truth is an unknowable truth"—isn't that obvious?) argument that the existence of unknown truths entails that of unknowable ones.
Here's something less trivial—a reductio of an interesting quintuple of premises.
- (Premise) Human epistemic states concerning natural states of the world, including knowledge of natural states of the world, supervene on natural states of the world.
- (Premise) Every true proposition reporting only natural states of the world can be known (by humans).
- (Premise) Some true proposition reporting only natural states of the world is not known.
- (Premise) There is a true proposition p reporting solely natural states of the world is such that any world in which p holds is an exact duplicate of our world in respect of natural states.
- (Premise) The conjunction of propositions reporting solely natural states of the world reports solely natural states of the world.
- Let q be a true proposition reporting only natural states of the world that is not known. (3)
- Let r be the conjunction of p and q. (4 and 6)
- r is true. (4, 6 and 7)
- r reports only natural states of the world. (4, 5, 6, 7)
- r is knowable. (2, 8 and 9)
- Let w be a world at which r is known. (10)
- At w, r is true. (11: knowledge entails truth)
- At w, p is true. (7 and 12)
- w is an exact duplicate of the actual world in respect of natural states. (4 and 13)
- At w, Kr. (11)
- Actually, Kr. (1, 14, 15)
- Therefore, Kq. (7 and 16: if a conjunction is known, so are its conjuncts)
- Therefore Kq and ~Kq. (6 and 17)
If we take 2-5 to be very plausible, this gives us a nice argument against 1. I think there are some technical difficulties with 4. To make 4 work, a description of natural states of the world has to be able to use not only first order naturalistic vocabulary ("electrons", "equids", etc.) but also "natural", so it can, after giving a complete catalog of natural states, say: "And there are no other natural states." This reading of "reporting natural states" is not the standard, I think. This broader reading of "reporting natural states" only makes 1 and 3 more plausible, though it makes 2 a bit less plausible.
The other way to get 4 working with only first order naturalistic vocabulary is to have an infinite proposition that inter alia denies the existence of all possible natural kinds other than the ones that are exemplified. This has some problems, but can be defended (how well?).