(
x
y P(x, y))
x
(
y P(x, y))
x
y
P(x, y)Unit 9 Answers
1.
( x
y P(x, y))
x
(
y P(x, y))
x
y
P(x, y)
2.
x
(S(x)
D(x)).
Using universal instantiation and the last premise, S
(John) D
(John) follows. Applying modus ponens to this conclusion and the second premise,
D(John) follows. Using conjunction and the first premise, C
(John) 
D
(John) follows. Finally, using existential generalization, the desired conclusion,
x
(C(x)
D(x))
follows
x (C(x)
H(x))
and
x
(H(x)
B(x)).
Using existential instantiation and the first premise
C(d)
H(d)
for some person d. From this by simplification H(d) is obtained.
For that person
d, by universal instantiation and the second premise,
H(d)
B(d).
Hence by modus ponens B(d) follows. Also by simplification from
C(d)
H(d)
, C(d) is obtained.
Using the conjunction on the last two conclusions,
C(d)
B(d)
is obtained.
Then applying existential generalization to this
x
(C(x)
B(x))
follows x
(C(x)
P(x)),
x
(P(x)
I(x))
and C(John). Using universal instantiation and the first premise,
C(John)
P(John) follows. Applying modus ponens to this conclusion and the last
premise,
P(John) follows. Using universal instantiation and the second premise,
P(John)
I(John) follows. From the last two conclusions by modus ponens
I(John) follows. Hence from this conclusion and the last premise by conjunction
C(John)
I(John)
follows. x
(C(x)
P(x))
and
x
(C(x)
I(x)).
Using existential instantiation and the second premise
C(d)
I(d)
for some person d. Hence by simplification C(d).
For that person d, by universal instantiation
and the first premise C(d)
P(d).
>From this and the previous conclusion, by modus ponens P(d).
Also from C(d)
I(d)
by simplification
I(d).
Hence by conjunction P(d)
I(d).
Hence by existential generalization
x
(C(x)
I(x)).
3.