Phil1001: Introduction to Logic
Name:
Fall 2021
Group:
Problem Set 10: Sentences of FOL - Solutions
Due: N/A (This problem set is optional)
1. (3pts)
Sentences of FOL:
Mark
true
or
false
for each of the following:
a.
T / F
∃
xP
(
x
)
is a sentence of FOL. – TRUE
b.
T / F
∃
y
∀
xP
(
x
)
is a sentence of FOL. – TRUE
c.
T / F
∃
y
(
∀
xP
(
x
))
is a sentence of FOL. – FALSE (this formula has incorrect parenthe-
ses)
d.
T / F
(
P
(
x
)
∧
Q
(
x
))
is a sentence of FOL. – FALSE (this formula has free variables)
e.
T / F
(
P
(
a
)
∧
Q
(
a
))
is a sentence of FOL. – TRUE
2. (2pts)
Atomic Formulas of FOL:
Looking at the definition of atomic formulas in FOL (
§
26.2),
which of the following are atomic formulas? If it is not, explain why.
a.
G
(
a
)
— Atomic formula
b.
(
∀
xG
(
x
)
∧ ∃
yF
(
y
))
— Not an atomic formula because it uses quantifiers and
∧
c.
x
1
– Atomic formula
d.
x
=
x
1
– Atomic formula
3. (2pts)
Terms of FOL:
What is a
term
in FOL? Give an example.
In first-order logic, a term is any variable or name. So,
x
would be a term.
4. (2pts)
Free and Bound Variables:
For each of the following formulas, is the variable x
free
or
bound
? See
§
26.3 for reference.
a.
∀
yP
(
x
)
– Bound
b.
G
(
x
)
– Free
c.
∃
x
(
P
(
y
)
∧
P
(
x
))
– Bound
d.
∃
xP
(
y
)
∧
P
(
x
)
– Free
