84.
Determine whether the following symbolized argument is valid or invalid. If invalid, provide
a counterexample; if valid, construct a deduction.
~P → (Q & R)
P → ~S
Q v R/∴ ~S
Invalid.
P = F
Q = T
R = T
S = T
85.
Determine whether the following symbolized argument is valid or invalid. If invalid, provide
a counterexample; if valid, construct a deduction.
(T v Q) → R
~R/∴ ~Q
Valid. The deduction could look like this:
2. ~R
4. ~T & ~Q 3, DEM
5. ~Q 4, SIM
86.
Determine whether the following symbolized argument is valid or invalid. If invalid, provide
a counterexample; if valid, construct a deduction.
(P → T) & S
S → (T → R)
~R/∴ ~T
Valid. The deduction could look like this:
1. (P
→ T) & S
2. S
→ (T
→ R)
3. ~R/
∴
~T
4. (S & T)
→ R 2, EXPORT
5. ~(S & T) 3,4, MT
6. ~S v ~T 5, DEM
7. S 1, SIM
8. ~T 6,7, DA
87.
Determine whether the following symbolized argument is valid or invalid. If invalid, provide
a counterexample; if valid, construct a deduction.
(Q & R) → P
(R → P) → S
Q/∴ S
Valid. The deduction could look like this:
1. (Q & R)
→ P
2. (R
→ P)
→ S
3. Q/
∴
S
4. Q
→ (R
→ P) 1, EXPORT
5. R
→ P 3,4 MP
6. S 2,5 MP
88.
Determine whether the following symbolized argument is valid or invalid. If invalid, provide
a counterexample; if valid, construct a deduction.
P v ~Q
R → Q
~P
~R → ~W/∴ W
Invalid.
P = F
Q = F
R = F
W = F
89.
Determine whether the following symbolized argument is valid or invalid. If invalid, provide
a counterexample; if valid, construct a deduction.
P → Q
Q → R
R → S
(P → S) → (Q → ~P)
~P/∴ ~Q
Invalid.
P = F
Q = T
R = T
S = T
90.
Determine whether the following symbolized argument is valid or invalid. If invalid, provide
a counterexample; if valid, construct a deduction.
(P v Q) → (C & D)
~C/∴ ~Q
Valid. The deduction could look like this:
1. (P v Q)
→
(C & D)
2. ~C/
∴
~Q
3. ~C v ~D 2, ADD
4. ~(C & D) 3, DEM
5. ~(P v Q) 1,4, MT
6. ~P & ~Q 5, DEM
7. ~Q 6, SIM
91.
Determine whether the following symbolized argument is valid or invalid. If invalid, provide
a counterexample; if valid, construct a deduction.
P v (Q & K)
(P v Q) → [(L → ~X) & (~X → L)]
(L → ~X) → (X & ~Y)
(Y → Z) & (Z → M)
(Q → K) → Z/∴ Z
Invalid.
P = T
Q = T
K = F
L = F
X = T
Y = F
Z = F
92.
Symbolize the following argument, and test it for validity. If valid, construct a deduction;
if invalid, assign truth values that show that the premises can be true while the conclusion
is false. Use these letters: D = The drought will continue; S = We get an early storm;
M = Managers of the ski areas will be happy; F = There will be great fire danger next year.
The drought will continue if we don’t get a storm. If we do get a storm, the managers of
the ski areas will be happy. Since we’ll either get a storm or we won’t, it follows that either
the drought will continue or the ski area managers will be happy.
Valid.
1. ~S → D (Premise)
2. S → M (Premise)
3. S v ~S (Premise)/∴ D v M
4. D v M 1,2,3 CD
93.
Symbolize the following argument, and test it for validity. If valid, construct a deduction;
if invalid, assign truth values that show that the premises can be true while the conclusion
is false. Use these letters: D = The drought will continue.; S = We get an early storm.;
M = Managers of the ski areas will be happy.; F = There will be great fire danger next
year.
Unless an early storm moves in, the drought will continue, and there will be great danger
of fire next year. But the drought is not going to continue. Therefore, there will not be a
great danger of fire next year.
Invalid.
S v (D & F) OR ~S → (D & F)
~D/∴ ~F
S = T; D = F; F = T
94.
Symbolize the following argument, and test it for validity. If valid, construct a deduction;
if invalid, assign truth values that show that the premises can be true while the conclusion
is false. Use these letters: D = The drought will continue; S = We get an early storm;
M = Managers of the ski areas will be happy; F = There will be great fire danger next year.
If there’s no early storm, the drought will continue. And if the drought continues, there will
be a great danger of fire next year. So, if there is to be no great danger of fire next year,
there must be an early storm.
Valid.
1. ~S → D (Premise)
2. D → F (Premise)/∴ ~F → S
3. ~S → F 1,2, CA
4. ~F → S 3, CONTR/DN
95.
Symbolize the following argument, and test it for validity. If valid, construct a deduction;
if invalid, assign truth values that show that the premises can be true while the conclusion
is false. Use these letters: D = The drought will continue; S = We get an early storm;
M = Managers of the ski areas will be happy; F = There will be great fire danger next year.
There will be a great danger of fire next year only if the drought continues, and it will
continue unless we get an early storm. However, if we do get an early storm, the ski area
managers will be happy. So if the ski area managers are not happy, it’ll mean that there’s
going to be a great danger of fire next year.
Invalid.
(F → D) & (D v S) OR (F → D) & (~S → D)
~S → M/∴ ~M → F
F = F; S = T; D = F; M = F
96.
Symbolize the following argument, and test it for validity. If valid, construct a deduction;
if invalid, assign truth values that show that the premises can be true while the conclusion
is false. Use these letters: D = The drought will continue.; S = We get an early storm.;
M = Managers of the ski areas will be happy.; F = There will be great fire danger next
year.
Either there will be an early storm, or the drought will continue. If there’s no continuation
of the drought, then the managers of the ski areas will be happy and there will be no great
danger of fire next year. So if we’re to avoid any great danger of fire next year and to make
the ski area managers happy, it will be necessary for there to be an early storm.
Invalid.
S v D
~D → (M & ~F)/∴ (~F & M) → S
Make either D = T; M = T; F = F; S = F
97.
Determine whether the following is valid or invalid:
If a flyer is on one of the bulletin boards, then you know it was approved by the Associated
Students, and that flyer was approved. Therefore, it’s on one of the bulletin boards
somewhere.
Invalid.
98.
Determine whether the following is valid or invalid:
If it’s going to be a cool spring, then the azaleas will need extra fertilizer. Indications are,
however, that it’s not going to be a cool spring.
Invalid. [Unstated conclusion: The azaleas won’t need extra fertilizer.]
99.
Determine whether the following is valid or invalid:
If she can play the tuba, then she can darn well play the baritone. And if she can play the
baritone, then she can play the French horn. We can conclude that as she can play the
tuba, she can play the French horn.
Invalid.
100.
Determine whether the following is valid or invalid:
If we don’t see the movie tonight, we won’t be able to talk about it in class. And if we can’t
talk about it in class, we won’t make Ms. Schmidt very happy. So let’s see the movie. Then
Ms. Schmidt will be happy.
Invalid.
101.
Determine whether the following is valid or invalid:
Look, if the check bounced, then you’re right: Two things will happen. First, we’d owe a
penalty. And second, our records would be out of whack. Now, if our records are out of
whack or if we pay a penalty—either way—then we’re in big trouble, just like you said. The
check isn’t going to bounce, though. So relax. We’re not in any trouble.
Invalid.
102.
Determine whether the following is valid or invalid:
If he doesn’t think he’ll pass the class, then either he’ll be talking to someone, or he won’t
be paying attention, or both. Well, look at him. He’s talking to someone. And he’s not
paying the least bit of attention. Clearly he doesn’t think he’ll pass the class.
Invalid.
103.
Determine whether the following is valid or invalid:
Either she ordered the eggplant, or she ordered the calamari, though possibly she might
have ordered both. Well, she ordered the eggplant. So, she didn’t order the calamari.
Invalid.
10–55
104.
Determine whether the following is valid or invalid:
She must not have ordered the eggplant, ’cause if she had ordered it, then she wouldn’t be
eating any dessert like she’s doing right now.
Valid.
105.
Determine whether the following is valid or invalid:
It’s easy enough to do logic if you think logically. Fortunately, I have no trouble doing logic,
so I guess I think logically.
Invalid.
106.
Determine whether the following is valid or invalid:
It’s time to leave when they start putting up the chairs, as they’re doing right now.
Valid. [Unstated conclusion: It’s time to leave.]