Math 112 Beginning Logic, Spring 1997
Instructor: Julia Knight
Office: 233 CCMB
Hours: MWF 3-4 (tentatively)
MWF 1:55-2:45 First day, 119 Haggar, after that, 282 Galvin
Text: E. J. Lemmon, Beginning Logic (with supplementary material later)
The book sounds as if it was written for British schoolboys around the
turn of the century, to sit beside leather-bound books on rhetoric and
Latin grammar. The tone may be off-putting, but the book has the right
level. It does not assume any mathematical background.
Coursework and Grading:
In-class exams, three such, 15% each
Final, 30% (according to university rules)
Homework, 15% (done in groups, more about that in a moment)
Paper, talk 10% (on material beyond text, more about that later)
Dates
Exam I: Feb. 7
Exam II: March 5 (Wed. before Spring break)
Papers: due March 26 (Wed. before Easter)
Exam III: April 11
Talks: April 21-25
Final: May 5, 8:00-10:00 (first day of finals)
I'll assign homework regularly. Some of the homework will be
challenging, constructing proofs. I want you to work in groups. I
think that you will find that working together, talking about the
material helps you understand it better. Moreover, you will almost
certainly be working in groups when you take a regular job.
Assign groups of 3-4.
The first assignment will be to read through p. 8 in the text, and to
talk with the members of your group about why you signed up for the
course and what you hope to get out of it. I would like each group to
turn in a short (< 1 page) statement, jointly written, summarizing this
discussion. (Your group may designate a scribe, or each write a
paragraph, but all should read and approve the whole before you turn it
in on Friday. If you use a computer, it will be easy to fold separate
paragraphs together, or make changes.)
In the remaining time, I'd like to say a little about what logic is.
What do we do in logic ?
I. Analyze arguments
We say what is a "sound" argument.
Argument starts with some premises (assumptions) and arrives at a
conclusion. These statements may involve mathematical objects such as
numbers, or they may involve family relationships or historical facts.
In logic, we look at the form of the argument. We try to say when the
form guarantees that anyone who accepts the premises has to accept the
conclusion.
Example 1 (famous)
Premises:
All men are mortal.
Socrates is a man.
Conclusion:
Socrates is mortal.
Example 2
Premises:
For all natural numbers n, 1+2+3+...+n = n(n+1)/2
7 is a natural number
Conclusion:
1+2+3+4+5+6+7 = 7.8/2 = 28
Examples 1 and 2 have the same form (syllogism). I think you believe
both to be sound.
The next example is more complicated. I won't tell you whether it is sound.
Example 3
Premises:
If the river floods, then our entire wheat crop will be destroyed. The
river will flood if there is an early thaw. In any case, there will be
heavy rains later in the summer.
Conclusion:
If there is an early thaw, our entire community will be bankrupt unless
there are heavy rains later in the summer. Part of the difficulty in
analyzing this argument is the length of the sentences. It would be
easier to analyze something short, of the same form. Another
difficulty is that some words may be hard to interpret. This leads to
something else we do in logic.
II. Develop formal languages
Formal languages let us examine the form of an argument, or of a single
statement. Translating an argument from English into an apropriate
formal language makes the form transparent. We remove ambiguities. I
and II are the parts of logic that we shall concentrate on for most of
the course. We shall discuss some formal languages, and formal system
of proof. Homework will involve translating from English into an
appropriate formal language, and analyzing and constructing proofs in
formal system.
Students who took similar course in the past, arts and letters
students, felt that this kind of exercise should improve their ability
to analyze complex material, and their ability write to make a point.
There are other kinds of problems, other areas of logic.
III. Describe what is (or is not) computable
What can be done by a machine ? If there is an algorithm (method which
could be turned into a computer program) for doing something, you could
show this by describing the algorithm. For example, there is an
algorithm for deciding whether a natural number n is a prime. Divide n
by 2, 3, 4, ..., n-1. If you don't come to a number which divides n,
then n is prime. There is an algorithm for deciding whether a
polynomial equation in many variables has a solution in real numbers.
No such algorithm for solution in integers.
Example of polynomial equation: 7x^2y^3z^5 - 13xyz^2 + 5x^{16}z - 2xz^4 = 0
American logician, Julia Robinson, reduced problem to something else.
Finished by Russian logician, Yuri Matijasevich.
IV. Develop theory of sets
Set theory serves as foundation for mathematics. Discuss the sizes of
sets, both finite and infinite. What makes one infinite set larger
than another ?
V. Analyze expressibility, definability
Consider the relation between formal languages and real situations, or
mathematical structures. What can you say, in a particular setting,
using a particular language ? Using a natural language for talking
about numbers, in real numbers cannot define or say anything very
complicated. In rationals, situation is quite different.
How old is logic ? How did it develop ? Certainly, there were proofs,
rules for such, tied to development of geometry in Egypt. Fact that
Nile flooded and removed markers between plots of land led to squabbles
that could only be settled by sound reasoning from universally accepted
geometric premises and observations on distant landmarks.
Formal languages are more recent. Some of the motivation for study of
formal languages came from paradoxes. These also motivated more
systematic study of sets.
Examples of Paradoxes
1. Liar paradox (ancient Greece)
A man says "I am lying".
If the man is lying, then he is telling the truth, while if he is
telling the truth, then he is lying.
2. Russell's paradox (around 1900)
Intuitively, we think of a set as any collection of objects (set of
people in South Bend, set of integers).
Let A be the set of all sets X such that X does not belong to itself.
If A belongs to A, then A does not belong to A, and if A does not
belong to A, then A belongs to A.
3. Richard's Paradox
Some English phrases denote real numbers. For example, "the ratio
between the circumference and the diameter of a circle" denotes the
number pi. We can order the phrases, shortest first, those of same
length lexicographically (as in dictionary). Let r_1,r_2,r_3,... be
the numbers denoted by 1st, 2nd, 3rd phrase, in order. Now let r be
the real number with 1 in the nth decimal place if r_n does not have 1
there and 2 otherwise. r may be .12211122122211.... Then r does not
equal r_n for all n, but r is described by an English phrase.
Logicians found ways around the paradoxes by developing formal
languages, and formalizing set theory. The ideas behind the paradoxes
were used in sound arguments.
1. The idea of Russell's paradox was used to show that the set of
subsets of a given set is always larger than the set itself.
2. The idea of the Liar paradox was used to show that there is no
algorithm for deciding which sentences are true of the natural numbers
with + and ..
3. The idea of Richard's paradox was used to show that the set of
points on a line is larger than the set of integers. The paper that
you write will be on material from logic, not from the text, something
you decide to learn about and explain, possibly something on
non-computable functions, or set size. I'll say more about this later
on.