Socratic Paradox

Here are two paradoxes which have been attributed to Socrates:

Paradox 1
"I know nothing at all."

In which case, how does Socrates know that he knows nothing?

Resolution
If he knows that he knows nothing, then it is false that he knows nothing, because there's (at least) one fact he has knowledge of: his lack of knowledge.

Hence "I know nothing at all" must always be false, whoever utters it.

Paradox 2
"This sentence is false."

If that sentence is false, then it must be true. But then if it's true, it must be false.

Resolution
The sentence is meaningless.

Related paradoxes abound, from Russell's Paradox through to the ultra-sophisticated Gödel's Incompleteness Theorems.

For a popular interpretation of this paradox and its relations, see Hofstadter's.