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Moderation in all things, including moderation Paradoxes of self reference Is the previous slide blank? This is a paradox of self reference This sentence is false (liar paradox) liar paradox is attributed to Eubulides of Miletus in the 4th century BC. The next sentence is true. The previous sentence is false. Rule: there is an exception to every rule.

More sophisticated paradoxes of self reference Buridan’s paradox (ca. 1300 – after 1358) Imagine the following scenario: Socrates wants to cross a river and comes to a bridge guarded by Plato Plato: Socrates, if in the first proposition which you utter, you speak the truth, I will permit you to cross. But surely, if you speak falsely, I shall throw you into the water. Socrates: You will throw me into the water.

Resolution of the dilemma Is the proposition uttered by Socrates: "You are going to throw me into the water" true, or is it false? Is Plato's promise true or is it false? How can Plato fulfill his promise?” … “ought implies can” (Kant)

Fitch's paradox of knowability The existence of an unknown truth is unknowable… If all truths were knowable, it would follow that all truths are in fact known.

Suppose p is a sentence which is an unknown truth; that is, the sentence p is true, but it is not known that p is true. In such a case, the sentence "the sentence p is an unknown truth" is true if all truths are knowable, it should be possible to know that "p is an unknown truth". But this isn't possible, because as soon as we know "p is an unknown truth", we know that p is true, rendering p no longer an unknown truth, Therefore: the statement "p is an unknown truth" becomes a falsity. The statement "p is an unknown truth" cannot be both known and true at the same time. Therefore, if all truths are knowable, the set of "all truths" must not include any of the form "something is an unknown truth” Conclusion: there must be no unknown truths, and thus all truths must be known.

The barber’s paradox In a village, the barber shaves everyone who does not shave himself, but no one else. The question that prompts the paradox is this: Who shaves the barber?

Paradoxes of the infinite

Logical paradoxes I: Zeno’s paradox of Achilles and the tortoise

That which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle) Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters. each racer starts to run at some constant speed after some finite time, Achilles will have run 100 meters, During this time, the tortoise has run 10 meters. after some finite time, Achilles will have run 10 more meters, During this time, the tortoise has run 1 more meter ETC Whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. There are an infinite number of points Achilles must reach where the tortoise has already been. He can never overtake the tortoise.

If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless ” —Aristotle, Physics VI:9, 239b

Other paradoxes of the infinite Adding an element to an infinite set results in a set with the same number of elements. There is an infinite number of infinite numbers. If a hotel with infinitely many rooms is full, it can still take more guests. Banach–Tarski paradox: Cut a ball into a finite number of pieces, re-assemble the pieces to get two balls, both of equal size to the first.

Does the set of all those sets that do not contain themselves contain itself?

Homework problem Uncle Joe and Uncle Jim are walking to the barber shop. There are three barbers who live and work in the shop—Mo, Larry, and Curly—but not all of them are always in the shop. Curly is a good barber, and Uncle Jim is keen to be shaved by him. He knows that the shop is open, so at least one of them must be in. He also knows that Mo is a very nervous man, so that he never leaves the shop without Larry going with him. Uncle Joe insists that Curly is certain to be in, and then claims that he can prove it logically. Uncle Jim demands the proof. Uncle Joe reasons as follows. Suppose that Curly is out. If Curly is out, then if Mo is also out Larry would have to be in—since someone must be in the shop for it to be open. However, we know that whenever Mo goes out he takes Larry with him, and thus we know as a general rule that if Mo is out, Larry is out. So if Curly is out then the statements "if Mo is out then Larry is in" and "if Mo is out then Larry is out" would both be true at the same time. Uncle Joe notes that this seems paradoxical; the hypotheticals seem "incompatible" with each other. So, by contradiction, Curly must logically be in. Must Curly be in? Explain.