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Lecture 1, MATH 210G.04, Spring 2011

Lecture 1, MATH 210G.04, Spring 2011. Mathematical Reasoning, Part I. Causality. Ben is a dog. all dogs are green Ben is green. . Some questionable statements. If a=5 then 2*a =2*5 All triangles have three sides If I study hard I will get a good grade

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Lecture 1, MATH 210G.04, Spring 2011

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  1. Lecture 1, MATH 210G.04, Spring 2011 Mathematical Reasoning, Part I

  2. Causality • Ben is a dog. all dogs are green Ben is green.

  3. Some questionable statements • If a=5 then 2*a =2*5 • All triangles have three sides • If I study hard I will get a good grade • If I major in math then I will get a better job than if I major in english. • If I take this pill then I will feel better. • If Sarah Palin is elected president then the world will end in 2012.

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

  5. 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 metres. • each racer starts runs at some constant speed • after some finite time, Achilles will have run 100 metres, • During this time, the tortoise has run 10 metres. • after some finite time, Achilles will have run 10 more metres, • During this time, the tortoise has run 1 more metre • 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.

  6. 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?

  7. The paradox of the liar • This sentence is false • liar paradox is attributed to Eubulides of Miletus in the 4th century BC.

  8. Logical arguments I: The syllogism • Aristotle, Prior Analytics: a syllogism is "a discourse in which, certain things having been supposed, something different from the things supposed results of necessity because these things are so.”

  9. Syllogism cont. • A categorical syllogism consists of three parts: the major premise, the minor premise and the conclusion. • Major premise: All men are mortal. • Minor premise: Socrates is a man. • Conclusion: Socrates is mortal. • Major premise: All mortals die. • Minor premise: All men are mortals. • Conclusion: All men die.

  10. Modus ponens • If P, then Q. P. Therefore, Q • If Socrates is a man then Socrates is mortal • Socrates is a man • Socrates is mortal

  11. Law of the excluded middle

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