80-110: The Nature of Mathematical Reasoning

Exercises: In class, we did 3 proofs of Leamma 1a: If P is even, then P²is even. The first proof was deductively valid, the second a good inductive argument, and the third completely irrevelvant to the claim. All the steps in all three proofs were true, and this was to illustrate that truth and proof are not the same idea. The homework is to find three arguments for Lemma 1, one deductively valid, one inductive, and one irrelevant, in which all the steps in the arguments are true.

Lemma 1: If P² is an even whole number, then P is an even whole number.

Reading Assignments: None

Due Tuesday, January 27

Exercises: Go to the library or your own bookshelves, and find two definitions, one from a non-mathematical theory, e.g., political science, sociology, drama, etc., and one from a mathematical theory, e.g., calculus, topology, etc. For each definition, write it out and identify the undefined terms just like you did with Euclid's definitions.

Reading Assignments: Glymour chapter, pp. 15-25

Due Tuesday, Feb. 3rd

Exercises: Study Questions 1 and 3, on page 31 of Glymour, chapter 1.

Reading Assignments: Glymour - 1st half of chapter 2.

Due Tuesday, Feb. 10th

Exercises: Consider the following problem. Suppose you are on a game show in which there are 3 doors, one of which will contain a terrific prize. The door that contains the prize will be decided randomly before the show, and each door has an equal chance of containing the prize. You are asked to pick a door, but you are not shown what is behind your door. To be concrete, suppose you choose door 1. Your host then shows you one of the doors you did not pick, the only restriction being that the door you are shown is empty. Say you are shown that door 2 does not have the prize. Assuming you want to maximize the chances for getting a great prize: the question is: Do you want to stay with your original pick (door 1) or switch (to door 3)? Pick one of the following answers, and justifiy your pick. Type up your answer and turn it in: A) Stay with original pick (door 1 has a better chance of having the prize than door 3). B) Switch (door 3 has a better chance of having the prize than door 1). C) It doesn't matter (door 1 and door 3 have the same chance of having the prize).

Reading Assignments: Glymour - second half of chapter 2.

Due Thursday, Feb. 12th

Exercise: Get onto the Web, and visit Lets Make a Deal :

Play the game a few times to get the hang of it. Now do the following 2 experiments, record the results in a table like the following, and turn in the table to Andrew Banas.

	Shown 2 Empty	Shown 3 Empty	Overall
Switched	1a)	1b)	1c)
Stayed	2a)	2b)	2c)

1) Play 20 times. Always choose Door #1 as your first choice. When you are shown an empty door (it has a Donkey), switch for your second choice.

In cell 1a in the table- record how often you won when you were shown door 2 empty, i.e., record two numbers: #wins when shown 2 empty / # times shown 2 empty.
In cell 1b in the table- record how often you won when you were shown door 3 empty, i.e., record 2 numbers: #wins when shown 3 empty / # times shown 3 empty.
record how often you won overall when you switched - the applet does this for you - put the answer in cell 1c.

2) Play 20 times. Always choose Door #1 as your first choice. When you are shown an empty door (it has a Donkey), stay with door 1 as your second choice.

In cell 2a in the table- record how often you won when you were shown door 2 empty, i.e., record 2 numbers: #wins when shown 2 empty / # times shown 2 empty.
In cell 2b in the table- record how often you won when you were shown door 3 empty, i.e., record 2 numbers: #wins when shown 3 empty / # times shown 3 empty.
record how often you won overall when you switched - the applet does this for you - put the answer in cell 2c.

Reading Assignments: The Language of First Order Logic, pp. 1-15.

Due Tuesday, Feb. 17th

Exercises: Problems 2 and 3, pp. 13,14 in FOL.

Reading Assignments: The Language of First Order Logic, Appendix A.

Due Thursday, Feb. 19th

Exercises : Fill in the <subject 1>, <subject 2> and predicates A and B to provide an example of an argument of the following form which is invalid:

All <subject 1> are A

All <subject 2> are B

--------------------- therefore

All <subject 1> are B

Readings: FOL, through the end of chapter 2 .

Due Thursday, March 5th

Exercises : Problem 5, p. 43

Readings: FOL, pp. 35-44 .

Due Tuesday, March 10th

Exercises : Problem 6, p. 44, and problems 12 & 13, p. 49

Readings: FOL, pp. 45-66 .

Due Thursday, March 12th

Readings : FOL, pp. 91-104

Exercises : Complete a truth table for the following sentence:

~((p & q) -> s)

On Smullyan's Island, there are only Knights and Knaves, which are identical in appearance. Knights, however, always tell the truth, and Knaves always lie. Suppose two fellows, call them A and B, come up to you, and A says: "At least one of us is a Knave." What are A and B?

Due Tuesday, March 17th

Readings : FOL, pp. 24-30, 58-82

Exercises : Ex. 27, p. 60, Ex. 39, p. 71

Due Thursday, March 19th

Readings : FOL, pp. 99-112

Exercises : Ex. 17, p. 103

Due Thursday, March 30th

Readings : FOL, pp. 115-125

Exercises : Ex. 6, p. 124, Ex. 10, p. 125

Due Thursday, April 9th

Readings : FOL, pp. 125-141

Exercises : Create a Tarski's World that is simultaneously a counterexample to all of the following arguments.

Argument1)

Premise1 Ex Large(x)

Premise 2 Ex Tet(x)

Conclusion Ex[Tet(x) & Large(x)]

Argument 2)

Premise1 Ax[Cube(x) -> Small(x)]

Conclusion Ex[Cube(x)]

Argument 3)

Premise 1 Ex[Dodec(x) & Large(x)]

Conclusion Ax[Dodec(x) & Large(x)

Due Thursday, April 16th

Readings : FOL, pp. 207-226

Exercises : None

Due Tuesday, April 21st

Readings : None

Exercises : Prob. 4, p. 211, Prob. 19,20, p. 220

Due Thursday, April 23rd

Readings : None

Exercises : Prove the the Integers have the same cardinality as the Naturals

Go to Home Page for 80-110

Maintained by: Richard Scheines