Socratic perfection is impossible

Socrates thought it was important that if you didn't know something, you knew you didn't know it. And he thought that it was important to know what followed from what. Say that an agent is Socratically perfect provided that (a) for every proposition p that she doesn't know, she knows that she doesn't know p, and (b) her knowledge is closed under entailment.

Suppose Sally is Socratically perfect and consider:

1. Sally doesn’t know the proposition expressed by (1).

If Sally knows the proposition expressed by (1), then (1) is true, and so Sally doesn’t know the proposition expressed by (1). Contradiction!

If Sally doesn’t know the proposition expressed by (1), then she knows that she doesn’t know it. But that she doesn’t know the proposition expressed by (1) just is the proposition expressed by (1). So Sally doesn’t know the proposition expressed by (1). So Sally knows the proposition expressed by (1). Contradiction!

So it seems it is impossible to have a Socratically perfect agent.

(Technical note: A careful reader will notice that I never used closure of Sally’s knowledge. That’s because (1) involves dubious self-reference, and to handle that rigorously, one needs to use Goedel’s diagonal lemma, and once one does that, the modified argument will use closure.)

But what about God? After all, God is Socratically perfect, since he knows all truths. Well, in the case of God, knowledge is equivalent to truth, so (1)-type sentences just are liar sentences, and so the problem above just is the liar paradox. Alternately, maybe the above argument works for discursive knowledge, while God’s knowledge is non-discursive.