100 Points Total - Closed book - you have 1 hour
and 20 minutes
1) Deductive Validity (20 points)
a) (5 points) Define what it is for an argument to
be deductively valid.
b) (5 points) Suppose I give you an argument with
two premises and a conclusion, and tell you that the premises
and conclusion are true. Check one of the following, and explain
your answer with a sentence or two.
The argument is valid ___________
The argument is invalid___________
I can't tell___________
c) (5 points) Suppose I give you an argument with
two premises and a conclusion. Suppose I tell you nothing about
the premises, but I tell you that the conclusion is false. Check
one of the following, and explain your answer with a sentence
or two.
The argument is valid ___________
The argument is invalid___________
I can't tell___________
d) (5 points) Provide an example of a valid argument
in which at least one of the premises is false and the conclusion
is true.
2) Definitions (15 points)
a) (5 points) Giving a reasonable definition of what
it means for one infinite collection to be larger in number than
another puzzled mathematicians for centuies. Evaluate (i.e.,
point out its pros and cons) the following candidate: An infinite
collection X is strictly larger in number than an infinite
collection Y if Y is properly contained in X.
b) (5 points) Suppose you meet a rookie mathematics
major at a party, and he defines the length of a line segment
as the number of points that lie upon it. After commenting on
what a great party this is, you ask him: ìHow long is one
point?î He says: ìA point is not even as long as
a quark - its 10-50 meters long.î Using two of
the oldest theorems in mathematics, both from Pythagoras, convince
the rookie that he should think again.
c) (5 points) What is the problem with the following system of definitions:
1) A whole number P is fragile if there is another number Q such that the product of P times Q is breakable.
2) A number X is breakable if X is not prime and there is another number Y such that X divided by Y is non-rigid.
3) A number K is non-rigid if its cube is
fragile.
3) (10 points) Explain why it is desirable to present
a theory axiomatically.
4) (10 points) Explain the difference between a concept
that is essentially subjective, one that is theoretically and
practically objective, and one that is theoretically but not practically
objective. Provide examples of each distinct from those we used
in class.
5) Fallacies (25 points)
a) (5 points) Which step in the following argument is fallacious, and why:
1) Suppose that P and R are even whole numbers.
2) If P is an even whole number, then there is another whole number Q such that P = 2Q
3) If R is an even whole number, then there is another whole number Q such that R = 2Q.
4) Since P = 2Q, and R = 2Q, then
5) P = R.
b) (10 points) St. Anselm, St. Thomas Aquinas, and
Spinoza all constructed "deductive" arguments for the
existence of God. Choose one of these, briefly describe the argument,
and explain why it is fallacious.
c) (10 points) Lets Make a Deal
Suppose you face a modified Lets Make Deal problem.
There are 4 doors that have an equal random chance of containing
the prize. You get to pick one, but you donít get to see
whats behind it. Monte, who knows where the prize is, shows you
one empty door among the three you did not choose. Suppose
a friend claims that the probability of winning if you stay with
your original choice is no better than the probability of winning
if you switch to either door that you did not originally pick
and that Monte has left closed. He presents the following argument:
The probability that the prize is in a door is the
same for all doors. After the host does his thing you have three
doors to choose among. Since the chance is no better for the prize
to be in one than the other, it makes no difference what you do.
Is this argument sound? If so why, and if not, why
not?
6) (20 Points) In a paragraph, explain a) (10 points)
why we want a theory of deductive mathematical reasoning, and
b) (10 points) why we might want a formal theory.