Introductory Logic 
Final Exam

Feb. 2, 2001

R. Hammack


Name: ________________________ 
Score: _________

(1) Identify the following passages as arguments or nonarguments. If a passage is an argument, underline its conclusion.
(a) Automobiles powered by gasoline engines are very inefficient
machines. Even under ideal conditions, less than 15% of the available energy
in the fuel is used to power the vehicle. The situation is much worse under
stopandgo driving in the city.
ARGUMENT (condlusion underlined)
(b) In debates about the environment the most important way of regarding
living things collectively has been to regard them as species. Thus, when environmentalists
worry about the future of the blue whale, they usually are thinking of the blue
whale as a species, rather than of individual blue whales.
NONARGUMENT (example)
(c) It is now generally agreed by scientists and philosophers of science
that theories may have considerable evidence supporting them, but no theory
can be said to be true. This is because many of the concepts and ideas in theories
refer to entities that are not directly observable, which means they are not
directly verifiable.
NONARGUMENT (explanation)
(d) Freud argued that anxiety stemmed not just from external threats but
also from internal ones, in the form of id impulses attempting to break through
into consciousness. It is this latter type of anxiety that psychodynamic theory
sees as the root of neurosis.
NONARGUMENT (expository passage)
(e) First, we know that brain processes cause mental phenomena. Mental states
are caused by and realized in the structure of the brain. From this it follows
that any system that produced mental states would have to have powers equivalent
to those of the brain.
ARGUMENT (condlusion underlined)
(2) Decide if the following arguments are deductive or inductive. Say
if each argument is valid, invalid, weak, strong, and (if possible) sound, unsound,
cogent, or uncogent.
(a) If Thanksgiving is in October, then Valentines Day is in February.
If Valentines Day is in February, then February is the shortest month. Therefore,
if Thanksgiving is in October, then February is the shortest month.
DEDUCTIVE, VALID, UNSOUND
(b) After Louis Armstrong discovered the theory of relativity, he donated
his papers to Harvard University. Therefore, those papers are likely still at
Harvard.
INDUCTIVE, STRONG, UNCOGENT
(c) For the past 10 years, RandolphMacon has had under 2000 fulltime students.
Therefore, next year RandolphMacon will have fewer than 2000 fulltime students.
INDUCTIVE, STRONG, COGENT
(d) If the Universe lacks sufficient mass, then it will continue to expand.
The Universe does not lack sufficient mass. Therefore it will not continue to
expand.
DEDUCTIVE, INVALID, UNSOUND
(3) Decide if the following pairs of statements are logically equivalent, contradictory, consistent, or inconsistent.
(a) They are consistent: 

(b) 
(

A

∨

B

)

⊃

C

(

B

•

~

C

)

∨

(

A

•

~

C

)  
T

T

T

T

T

T

F

F

T

F

T

F

F

T


T

T

T

F

F

T

T

T

F

T

T

T

T

F


T

T

F

T

T

F

F

F

T

F

T

F

F

T


T

T

F

F

F

F

F

T

F

T

T

T

T

F


F

T

T

T

T

T

F

F

T

F

F

F

F

T


F

T

T

F

F

T

T

T

F

T

F

F

T

F


F

F

F

T

T

F

F

F

T

F

F

F

F

T


F

F

F

T

F

F

F

T

F

F

F

F

T

F

They are contradictory and inconsistent
(4) Decide if the following statements are consistent or inconsistent.
(a) 
~

( 
A

⊃

D  ) 
/

(

D

∨

C  )  ⊃  E 
/

B

•

~

E 
/

~

C  ∨  K 
T

T

F

F

F

F

F

T

F

T

T

T

F

T

F

T

T 
CONSISTENT: (it is possible for them all to be true at the same time.)
(b)  A 
⊃

(

B 
•

C  ) 
/

A

•

D 
/

B 
⊃

(

D 
•

E  ) 
/

C 
⊃

(

E 
•

~  D)  
T 
T

T 
T

T 
T

T

T  T 
T

T 
T

T  T 
T

T 
T

F  T 
INCONSISTENT: (Contradiction highlighted.)
(5) Use only the rules of implication and replacement (and UI, EI, UG,
EG, CQ if necessary) to derive the conclusions of the following arguments.
(a)  1. A ⊃ B  
2. B ⊃ ~B  / ~A  
3. ~B ∨ ~B  2, Impl  
4. ~B  3, Taut  
5, ~A  1, 4, MT  
(b)  1. (Q ∨ ~R) ∨ S  
2. ~Q ∨ (R • ~Q)  / R ⊃ S  
3. Q ∨ (~R ∨ S)  1, Assoc  
4. Q ∨ (R ⊃ S)  3, Impl  
5. (~Q ∨ R) • (~Q ∨ ~Q)  2, Dist  
6. ~Q ∨ ~Q  5, Comm, Simp  
7. ~Q  6, Taut  
8. R ⊃ S  4, 7, DS  
(c) If the victim had money in his pockets, then robbery wasn't the motive for the crime. Either robbery or vengeance was the motive for the crime. The victim had money in his pockets. Therefore, vengeance was the motive for the crime. (M, R, V)
1. M ⊃ ~R  
2. R ∨ V  
3. M  / V  
4. ~R  1, 3, MP  
5. V  2, 4, DS  
(d) Teachers are either enthusiastic or unsuccessful. Teachers are not all unsuccessful. Therefore, there are some enthusiastic teachers. (T, E, U)
1. (x)(Tx ⊃ (Ex ∨ Ux))  
2. (∃ x)(Tx • ~Ux)  / (∃ x)(Tx • Ex)  
3. Ta • ~Ua  2, EI  
4. Ta ⊃ (Ea ∨ Ua)  1, UI  
5. Ta  3, Simp  
6, Ea ∨ Ua  4, 5, MP  
7, Ua ∨ Ea  6, Comm  
8, ~Ua  3, Comm, Simp  
9. Ea  7, 8 DS  
10. Ta • Ea  5, 9, Conj  
11. (∃ x)(Tx • Ex)  10, EG 
(6) Decide if the following arguments are valid or invalid. You may use any appropriate method.
(a) 
(

A

•

B  ) 
⊃

(

D

∨

C  ) 
/

E 
⊃

(

A

•

~  D  ) 
/

G

⊃

~  C 
//

(

E

•

G  ) 
⊃

B 
T

F

F 
T

F

F

F  T 
T

T

T

T  F 
T

T

T  F 
T

T

T 
F

F 
The argument is invalid, for the above assignments show that it's possible for the premises to be true while the conclusion is false.
(b) 
(

A

•

B  ) 
⊃

(

D

∨

C  ) 
/

E 
⊃

(

A 
•

~  D  ) 
/

H

⊃

(  G 
•

~  C  ) 
//

(

H

•

E  ) 
⊃

~  B 
T

T

T 
T

F

T

F  T 
T

T 
T

T  F 
T

T

T 
T

T  F 
T

T

T 
F

F  T 
The argument is valid, because in filling in the indirect truth table, a contradiction (highlighted) is produced.
(c) Some diplomats are clever. Some diplomats are spies. Therefore, some diplomats are clever spies.
Counterexample Method:
Some D are C Some D are S  Some D are C and S 
D = Dogs C = Male S= Female 
Some dogs are male Some dogs are female  Some dogs are male and female 
Since the argument form can be translated into an argument with true premises and false conclusion, the argument is INVAID.
(7) Use the methods of conditional proof or indirect proof to derive the conclusions of the following arguments.
(a) If there are any voters, then all politicians are astute. If there are any politicians, then whoever is astute is clever. Therefore, if there are any voters, then all politicians are clever. (V, P, A, C)
1. (∃ x)Vx ⊃ (x)(Px • Ax)  
2. (∃ x)Px ⊃ (x)(Ax ⊃ Cx)  / (∃ x)Vx ⊃ (x)(Px • Cx)  
 3. (∃ x)Vx  ACP  
 4. (x)(Px • Ax)  1, 3, MP  
 5. Py • Ay  4, UI  
 6. Py  5, Simp  
 7. (∃ x)Px  6, EG  
 8. (x)(Ax ⊃ Cx)  2, 7, MP  
 9. Ay ⊃ Cy  8, UI  
10. Ay  5, Comm, Simp  
11. Cy  9, 10, MP  
12. Py • Cy  6, 11, Conj  
13. (x)(Px • Cx)  12, UG  
14 (∃ x)Vx ⊃ (x)(Px • Cx)  313 CP 
(b)  1. (A ∨ B) ⊃ (C • D)  
2. (C ∨ P) ⊃ (D ⊃ ~A)  / ~A  
3. ~~A  AIP  
4. A  3, DN  
5. A ∨ B  4, Add  
6. C • D  1, 5, MP  
7. C  6, Simp  
8. D  6, Comm, Simp  
9. C ∨ P  7, Add  
10. D ⊃ ~A  2, 9, MP  
11. ~D  3, 10, MT  
12. ~D • D  8, 11, Conj  
13 ~A  312, IP 
(8) Use the Finite Universe Method to show that the following argument is invalid.
(x)(Jx ⊃ Hx)  
(∃x)(Jx • Hx) ⊃ (∃x)(Px • Ox)  / (x)(Jx ⊃ Ox) 
Consider a universe with just one thing, a. Working out an indirect truth table
for this situation gives:
Ja

⊃

Ha

/

(

Ja

•

Ha

) 
⊃

(

Pa

•

Oa

) 
//

Ja

⊃

Oa

T

T

T

T

T

T

T

T

F

T

F

F

No matter how you fill in the last entry for Pa, there is a contradiction,
so this argument is valid in a universe with one element.
Next, consider a universe with two elements a and b, and work out the indirect
truth table for the argument:
(

Ja

⊃

Ha

) 
•

(

Jb

⊃

Hb

) 
/

[

( 
Ja

•

Ha

) 
∨

(

Jb

•

Hb

)  ] 
⊃

[

(

Pa

•

Oa

) 
∨

(

Pb

•

Ob

)  ] 
//

(

Ja

⊃

Oa

) 
•

(

Jb

⊃

Ob

) 
T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

F

F

T

T

T

F

T

F

F

Notice that if we let Ja, Ha, Jb, Hb, Pa, Pb, and Oa all be true, but set Ob
= F, then the premises are true and the conclusion is false (without contradiction).
Thus the argument is invalid in a universe with two elements, so it's INVALID.
(9) Each of the following arguments is either valid or invalid. Decide which are valid and which are invalid. If an argument is valid, use the rules of implication and replacement to derive its conclusion. If an argument is invalid use any appropriate method to show this.
(a)  (∃x)Px  
(∃x)Qx  / (∃x)(Px • Qx) 
Let P = "is a cat"
Let Q = "is a dog"
The argument is now:
There exists a cat.
There exists a dog.
Thus there is someting that's a both a dog and a cat.
Here the premises are true and the conclusion is false, so the argument is
INVALID by the counterexample method.
(c) Some bad judges are lawyers, since some lawyers are prejudiced magistrates, and all good judges are unprejudiced magistrates.
The conclusion is underlined.
Let G="Good judge," L="Lawyer,"
and P="Prejudiced magistrate." Then the argument and deduction
are as follows.
1. (x)(Gx ⊃ ~Px)  
2. (∃ x)(Lx • Px)  / (∃ x)(~Gx • Lx)  
3. Lm • Pm  2, EI  
4. Gm ⊃ ~Pm  1, UI  
5. Lm  3, Simp  
6, Pm  3, Comm, Simp  
7, ~~Pm  6, DN  
8, ~Gm  4, 7, MT  
9. ~Gm • Lm  5, 8, Conj  
10. (∃ x)(~Gx • Lx)  9, EG  