The Classical
Raven Paradox:
1) All ravens are black.
2) Everything that is not black is a raven.
3) Nevermore, my pet raven, is black.
4) This green (and thus not black) thing is an apple (and thus not a raven).
Ship of Theseus (My all time favorite):
"The ship wherein Theseus and the youth of Athens returned from Crete had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their place, in so much that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same."
Omnipotence Paradox:
If a being can perform any action, then it should be able to create a task which this being is unable to perform; hence, this being cannot perform all actions. Yet, on the other hand, if this being cannot create a task that it is unable to perform, then there exists something it cannot do.
"Can [an omnipotent being] create a stone so heavy that it cannot lift it?"