CS 245 Winter 2009
Lecture 19 Suggested Exercises
Solution Set
Exercise 1
1. Every person enjoys a good time. There is a student. Therefore, some student enjoys
a good time.
We formalize this is, adding the hidden premise that all students are people, as:
∀
x
•
person
(
x
)
⇒
good
time
(
x
)
∃
x
•
student
(
x
)
Hidden Premise:
∀
x
•
student
(
x
)
⇒
person
(
x
)
Therefore,
∃
x
•
student
(
x
)
∧
good
time
(
x
)
where
•
person
(
x
) – means that
x
is a person,
•
good
time
(
x
) – means that
x
enjoys a good time, and
•
student
(
x
) – means that
x
is a student.
A proof of the argument is as follows:
1
∀
x
•
person
(
x
)
⇒
good
time
(
x
)
premise
2
∃
x
•
student
(
x
)
premise
3
∀
x
•
student
(
x
)
⇒
person
(
x
)
Hidden premise
4
x
u
student
(
x
u
)
assumption
5
student
(
x
u
)
⇒
person
(
x
u
)
∀
E 3
6
person
(
x
u
)
⇒
E 4
,
5
7
person
(
x
u
)
⇒
good
time
(
x
u
)
∀
E 1
8
good
time
(
x
u
)
⇒
E 6
,
7
9
student
(
x
u
)
∧
good
time
(
x
u
)
∧
I 4
,
8
10
∃
x
•
student
(
x
)
∧
good
time
(
x
)
∃
I 9
11
∃
x
•
student
(
x
)
∧
good
time
(
x
)
∃
E 2
,
4
−
10
1
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2. There is someone in the first row who is in front of everyone else in the first row.
Therefore, there is at most one person in the first row who is in front of everyone else
in the first row.
We formalize this is as:
∃
x
•
R
(
x
)
∧ ∀
y
•
R
(
y
)
∧ ¬
(
x
=
y
)
⇒
F
(
x,y
)
Therefore,
∀
x
• ∀
y
•
R
(
x
)
∧
R
(
y
)
∧
(
∀
z
1
•
R
(
z
1
)
∧ ¬
(
y
=
z
1
)
⇒
F
(
y,z
1
))
∧
(
∀
z
2
•
R
(
z
2
)
∧ ¬
(
z
2
=
x
)
⇒
F
(
x,z
2
))
⇒
(
y
=
x
)
and add the hidden premise that no one can be both in front of someone and also
be behind that same person:
Hidden Premise:
∀
x
• ∀
y
• ¬
(
F
(
x,y
)
∧
F
(
y,x
))
where
•
R
(
x
) – means that
x
is in the front row and
•
F
(
x,y
) – means that
y
is behind
x
. That is,
x
is in front of
y
.
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 Spring '08
 A
 Meaning of life, Xu, Konrad Zuse, Axiom, Z1, The Hidden

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