There are a few ways in which one might try to answer such a question: some are clearly self-contradictory, but others are not. The latter kind would be the correct way to respond.
Contradictory or useless response
Asked 'what do you not know', it would be problematic to reply 'I don't know that p' for some p. But there are different reasons why it would be problematic:
- If you do in fact know that p is true. Then you're straightforwardly lying.
- Perhaps you know not-p. Then, it is plausible that you know (or at least are in a position to know) that you don't know p (since you can't believe something that is false).[^1] But this isn't very useful. Presumably, when somebody asks 'what don't you know', they mean 'what true things don't you know'.
- You do not know whether p or not-p. Then saying 'I don't know that p' would have the same drawbacks as (2). Since you don't know whether p is true or false, you don't know if you've answered the implied question of 'what true things don't you know'.
The previous discussion was on answering in terms of a 'knowledge that' ascription of knowledge. But another kind of ascription of knowledge, which has has quite a bit of discussion[^2] is 'knowledge wh-' ascriptions. That is, ascriptions such as 'J.K. knows who wrote The Cuckoo's Calling', 'Isaac knows why things fall downwards', 'Rosalind knows what the structure of DNA is like' and so on.
Answering by stating your epistemic situation vis-a-vis a knowledge wh- statement would be perfectly fine:
- I don't know who won the World Cup in 2010 (although perhaps I should!)
- I don't know why magnets work
- I don't know what snake tastes like
- I don't know whether it's raining in San Francisco
The last of these gets us closest to the knowledge that statement. But in this case, we're not picking one of p or not-p to assert our lack of knowledge of. Instead, the effect is something like - whichever of p or not-p is true - that's a true statement that I don't know.
[^1]: We can give a proof of this, given some assumptions about epistemic logic. Write 'Kp' for 'I know that p'. Then, we have the proof:
- Kp (assumption)
- K¬p → ¬p (factivity of knowing - it is only possible to know what is true)
- Kp → p (factivity again)
- p (from 1 and 3 by modus ponens)
- ¬K¬p (from 2 and 4 by modus tollens)
- Kp → ¬K¬p (1 and 5, modus ponens, discharging 1)
- K(Kp → ¬K¬p) (Since we've proved 6 without any undischarged assumptions, we're in a position to know it. This is called the rule of necessitation)
- KKp → K¬K¬p (Knowledge is preserved under known entailment)
- Kp → K¬K¬p (Kp and KKp are equivalent. This follows from factivity in one direction, and something called the KK principle in the other. It should be noted that the KK principle is very controversial though.)
[^2]: Here are some links: Knowing the answer, Jonathan Schaffer, Questions, answers and knowledge wh-, Meghan Masto (unfortunately paywalled), Knowledge wh- bibliography on PhilPapers, T. Parent