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Lack of Knowledge implies Knowledge

Socrates was once opined to have said that all he knows is that he doesn’t know anything, or I know that I don’t know.

There is a formal system known as epistemic logic. It deals with an epistemic operator, K. One of the epistemic logic is known as negative knowledge, in some sense.

Negative Knowledge: ~Kp –> K~Kp or CNKpKNKp

If I don’t know p then I know that I don’t know P. Not knowing p implies knowing that don’t know p.

If I don’t know what it looks like down at the center of the Earth (or Sun), then I know that I don’t know what it looks like down at the center of the Earth (or Sun).

Furthermore, from this Axiom, we may easily show that not knowing something implies knowing something.

All we need is our axiom of negative knowledge, CNKpKNKp, and the law of contraposition. This law, basically, states that we switch the antecedent (i.e. NKp) with the consequent (i.e. KNKp), and we negate both of those propositions when we switch their places.

By the law of contraposition and negative knowledge, we obtain CNKNKpNNKp.
Now we use the law of double negation to the consequent (i.e. NNKp), and we obtain CNKNKpKp.

We obtain that if we don’t know that we don’t something then we know something.