Image taken from Kierkegaard for Beginners:
Soren Kierkegaard wants to tell us the truth about human existence. Trouble is, by his own account, everything he writes is a lie! He says as much in his books. So is what he has written true or false? If it’s true, then because it’s written, it’s false. If it’s false, then because he claims it in writing, it’s true. Either way, it’s both true and false. Such is the liar’s paradox.
Here’s another paradox. In the set below, each whole number can be paired off with a corresponding even number in an ascending pattern:
The list can go on forever; there is no point where the progression could stop. Therefore, we can conclude that the number of even numbers is exactly the same as the number of whole numbers. But this seems to be clearly false! How can an infinite set of elements subtract half of its elements and still be infinite? This question fascinated Galileo and his contemporaries, and so it can be called Galileo’s paradox.
A paradox is a logically valid argument that contains sound premises which yields a (seemingly?) false conclusion. Philosophers don’t like paradoxes, because they spell trouble for logic itself. Jettisoning the law on noncontradiction is not an option, for without it we could not communicate. Perhaps it has its limits, but it is very difficult to articulate how and where these limits apply. Most philosophers try solve the paradox by explaining how the conclusion can be true by modifying or adding a premise to the original argument, thus eliminating the contradiction. Others bite the bullet and put limits on logic and admit that some contradictions… are actually true!
HT: Tim Pickavance.