JDEP183H
Homework
Assignment 4a
Propositional
and Predicate Logic
Assigned: October 26, 2006
Due: 12:30 p.m. November 16,
2006
(Homework 5
minutes late will not be accepted)
1. (30 points) Using only the following logical equivalences: double negation, commutative laws, associative laws, distributive laws, idempotent laws, identity laws, negation laws, De Morgan laws, implication, and equivalence, and the following logical implications: addition, and simplification to logically prove the following. You can also use “conjunction”. You must show and explain all steps clearly.
(a) (5 points) exportation law: 
(b) (5 points) reductio
ad absurdum: 
(c) (4
points) absurdity: 
(d) (4
points) modus ponens:
(e) (4
points) modus tollens: 
(f) (4
points) disjunctive syllogism: 
(g) (4
points)
2. (50 points) Give a formal proof for each of the following using only the rules given in the tables of Logical Equivalences and Implications; you can also use “conjunction”.
(a) (10
points) If 
, 
, 
, then 
.
(b) (10
points) If 
, 
, 
, then 
.
(c) (10
points) If 
and 
, then 
.
(d) (10 points) If 
, 
and 
, then 
.
(e) (10
points) If , , and 
, then 
.
3. (20 points) (Based on Ross and Wright 1988). Consider the following hypotheses:
If I take the
bus or subway, then I will be late for my appointment. If I take a cab, then I will not be late for
my appointment and I will be broke. I
will be on time for my appointment.
Which of the following conclusions must follow, i.e., can be inferred from the above hypotheses? Justify your answers. (Use a logical proof to prove that a conclusion follows; and may use a line of a truth table to prove that a conclusion does not follow.)
(a) I will take a cab.
(b) I will be broke
(c) I will not take the subway
(d) If I become broke, then I took a cab.
(e) If I take the bus, then I won’t be broke.
4.
(20 points) Use only the rules given in the tables
of Logical Equivalences and Implications in your handout 4. Given the following: If 
, 
, 
, and 
, then 
.
(a) (10 points) Give a formal proof for the above without using the contradiction approach
(b) (10 points) Give a formal proof for the following with the contradiction approach
5. (20 points) For each of the following English
statements, first translate it into symbolic notations using quantifiers and
predicates, then negate it (and bring the negation inside the quantifiers),
and then translate it back to English statements.
(a) (3 points) “Every computer scientist knows how to write
a program.”
(b) (3 points) “Not all computer scientists can count.”
(c) (4 points) “There are some computer scientists who have
been given the ACM Fellow Award.”
(d) (4 points) “Every computer scientist graduates from a
university.”
(e) (6 points) “If a computer scientist is very good, he/she will be given the ACM Fellow Award.”
6. (10 points) Show that the premises “A car in this
garage has an engine problem,” and “Every car in this garage has been sold”
imply the conclusion “A car which has been sold has an engine problem.” Let 
be “x is in
this garage,” 
be “x has an
engine problem,” and 
be “x has been
sold.” The premises are 
and 
. The conclusion is 
. Fill in the
following blanks to complete the proof:
Step Explanation
1. _____________________
2. __________ Existential instantiation from (1)
3. 
_____________________
4. _____________________
5. 
_____________________
6. 
_______________________
7. __________ Simplification from (2)
8. __________ _______________________
9. __________ Existential generalization from (8)
QED