## 80-110 The Nature of Mathematical Reasoning

**Spring 2001**

*Dirk Schlimm*
# Homework No.10

Thursday, April 5, 2001

Due Tuesday, April 10, 2001

**Note 1:**
Since this homework is more technical than
usual, you are allowed and *encouraged* to work on it in small groups
(of two or three) and hand in *one* homework together. However, do not just split
the work into equal parts, but *discuss the problems together*, so that
everybody understands them.
**Note 2:**
Since this is also more work than usual, it will technically (for the
calculation of the final grade) count as
two homework sets, each with maximal 10 points.

**1.** (2 points)
Read FOL, pages 91-103 (Sections
4.1-4.4). Do the following problems:

- Page 95, Problem 4.

Hand in your answers on
a disk (in PC/IBM format). Name your file 4-4.sen (as it is specified
in the problem).
- Page 104, Problem 21.

An example for a similar proof is given on page 102 of FOL. Make sure
to understand this example before attempting to solve this problem.

**2.** (3 points)

Read FOL, page 112 (Section
4.7).

- Write up five questions from the readings assigned in 1. and 2.,
and give short
answers to them.

(Discuss the questions with your group partners. This helps you to reflect about the important parts of the readings.)

**3.** (3 points)

Give a *semantic* and a *syntactic* proof
for each of the following valid steps
of reasoning:

- Modus tollens: From "A implies B" and "not B" infer "not A".
- Strenghening the antecedent: From "B implies C" infer "(A and B)
implies C".
- Weakening the consequent: From "A implies B" infer "A implies (B
or C)".

**4.** (5 points)

Read carefully FOL, pages 115-127
(Sections 5.1-5.8).
Do the following problems:

- Page 119, Problem 1 (Filename: 5-1.sen).
- Page 124, Problem 6 (Filename: 5-6.wld).
- Page 125, Problem 10.
- Page 127, Problem 11 (Filename: 5-11.sen).
- Page 131, Problem 17.

**5.** (3 points)

Read FOL, pages 132-141 (Sections
5.9-5.10), and pages 152 (Section 5.13).

- Write up five questions from the readings assigned in 4. and 5.,
and give short
answers to them.

(Discuss the questions with your group partners. This helps you to reflect about the important parts of the readings.)

**6.** (4 points)

Read carefully FOL, pages 154-168 (Section 6.1-6.5)
Do the following problem:

- Page 164, Problem 21 (Filename: 6-21.sen).