I. Symbolize each of the sentences using
the code indicated. (8 points)
P:
The class is hard.
Q:
The tests are long.
R:
The students are ready.
S:
The teacher is happy.
Fx:
x is ambitious
Gx:
x works hard
Hx:
x is successful
a:
Bob
b:
Carol
1. If the tests are long then the class
is hard.
QPC
2. The tests are long but the students
are ready.
QRA
3. Unless the students are ready the teacher
will not be happy.
RSNO
4. The students will not be ready only
if the class is hard and the tests are long.
RNPQAC
5. Anyone who is ambitious will work hard.
FGD
6. Everyone works hard and everyone is
successful.
GVHVA
7. If Bob is ambitious and Carol works
hard then both will be successful.
FaGbAHaHbAC
8. Some will be not successful only if
not everyone works hard.
HNWGVNC
II. Some of the following are not valid argument forms. In those cases supply the values for each variable that would produce a bad line. For the rest provide a direct or an indirect derivation. (8 points)
1. PQROC, QN |- PN
PQR || PQROC | QN | R
FFF FFFFT
FT F
2. PQRAC, RN |- PN
1. PQRAC Ass
2. RN
Ass
3. QRANPNC 1 Contra
4. QNRNO
2 O int
5. QRAN
4 DeM
6. PN
3,5 C elim
3. PQROC, PQNA |- R
1. PQROC Ass
2. PQNA Ass
3. P
2 A elim
4. QRO
1,3 C elim
5. QN
2 A elim
6. R
4,5 O elim
4. PQRAC, PQNA |- RN
1. PQRAC Ass
2. PQNA Ass
3. R
Hyp
4. P
2 A elim
5. QRA
1,4 C elim
6. Q
5 A elim
7. QN
2 A elim
8. QQNA
6,7 A int
9. RN
2-8 hyp elim
III. Both of the following are valid argument forms. Provide a direct or an indirect derivation. (4 points)
1. FV, FGD, GHD |- Ha
1. FV Ass
2. FGD Ass
3. GHD Ass
4. FaGaC 2 D elim
5. Fa
1 V elim
6. Ga
4,5 C elim
7. GaHaC 3 C elim
8. Ha
6,7 C elim
2. FWGVC, Fb |- Gc
1. FWGVC Ass
2. Fb
Ass
3. FW
2 W int
4. GV
1,3 C elim
5. Gc
4 V elim