Introductory Logic
Test #2
Jan 19, 2001
 
R. Hammack
Name: ________________________  
Score: _________

1. Translate the following sentences into symbolic form. Use capital letters to represent affirmative English statements.

(a) Ford introduces a new model and either Chrysler raises prices or General Motors changes colors.

F • (C ∨ G)

(b) It is not the case that either General motors changes colors or Audi lays off workers.

~(G ∨ A)

(c) Ford's introducing a new model is a necessary and sufficient condition for not both Audi laying off workers and Chrysler raising prices.

F ≡ ~(A • C)

(d) If Saab increases salaries, then if Toyota opens a new plant, then Honda initiates an ad campaign.

S ⊃ (T ⊃ H)

(e) Both Russia and China do not export steel if England revalues its currency.

E ⊃ (~R • ~C)

2. Write out the truth tables for the following propositions. For each proposition, say if it is tautologous, self-contradictory, or contingent.

(a)

(A
⊃
~
 B)
⊃
(B
•
~
 A)
T
F
F
 T
T
T
F
F
 T
T
T
T
 F
F
F
F
F
 T
F
T
F
 T
T
T
T
T
 F
F
T
T
 F
F
F
F
T
 F

The statement is CONTINGENT.

(b)

~
(A
∨
 B)
≡
(~
 A
⊃
 B)
F
T
T
 T
F
F
 T
T
 T
F
T
T
 F
F
F
 T
T
 F
F
F
T
 T
F
T
 F
T
 T
T
F
F
 F
F
T
 F
F
 F

The statement is SELF-CONTRADICTORY

 

3. Determine if the following pairs of statements are logically equivalent, contradictory, consistent, or inconsistent.

(a)

B
⊃
~
 A  
A
∨
 B
T
F
F
 T  
T
T
 T
T
T
T
 F  
F
T
 T
F
T
F
 T  
T
T
 F
F
T
T
 F  
F
F
 F

The statements are CONSISTENT

(b)

A
⊃
(B
⊃
 C)  
~
(A
•
 B)
∨
 C
T
T
T
T
 T  
F
T
T
 T
T
 T
T
F
T
F
 F  
F
T
T
 T
F
 F
T
T
F
T
 T  
T
T
F
 F
T
 T
T
T
F
T
 F  
T
T
F
 F
T
 F
F
T
T
T
 T  
T
F
F
 T
T
 T
F
T
T
F
 F  
T
F
F
 T
T
 F
F
T
F
T
 T  
T
F
F
 F
T
 T
F
T
F
T
 F  
T
F
F
 F
T
 F

The two statements are LOGICALLY EQUIVALENT and CONSISTENT


4. Use any technique from Chapter 6 to decide if the following arguments are valid or invalid.

(a) If Elvis was born in Austria, then Elvis spoke German. But Elvis did not speak German. Consequently, Elvis was not born in Austria.

A ⊃ G
~G
~A

Let's write out a truth table for this one.

A
⊃
 G
/
 ~ G
//
~
 A
T
T
 T   F T  
F
 T
T
F
 F   T F  
F
 T
F
T
 T   F T  
T
 F
F
T
 F   T F  
T
 F

Notice that whenever the premises are all true (which happens only on the third line), the conclusion is also true. Therefore the argument is VALID.

 

 

(b) If a voucher system is adopted, then schools will compete for students. If schools will compete for students, then the quality of education will improve. It follows that if a voucher system is not adopted, then the quality of education will not improve.

V ⊃ S
S ⊃ Q
~V ⊃ ~Q

We will use an indirect truth table. Begin by assuming the premises are true and the conclusion is false.

V
⊃
 S
/
 S
⊃
 Q
//
 ~V
⊃
 ~Q
T
      
T
      
F
  

Now, the only way the conclusion can be false is if ~V=T and ~Q=F, so fill in that information.

V
⊃
 S
/
 S
⊃
 Q
//
 ~V
⊃
 ~Q
T
      
T
     T
F
 F

This means V=F and Q=T, so fill that in.

V
⊃
 S
/
 S
⊃
 Q
//
 ~V
⊃
 ~Q
F
T
      
T
 T   TF
F
 FT

At this point, the only thing that remains to be filled in is S, and you may fill it in as either T or F without a contradiction.

V
⊃
 S
/
 S
⊃
 Q
//
 ~V
⊃
 ~Q
F
T
 F   F
T
 T   TF
F
 FT

It follows that the substitutions V=F, S=T, and Q=T make the premises true and the conclusion false. Therefore, the argument is INVALID.

 

 

(c)
A ≡ (B ∨ C)
(D ∨ E) ⊃ B
E • G
D ⊃ A

We will make an indirect truth table. Begin by making the premises true and the conclusion false.

A
≡
(B
∨
 C)
/
(D
∨
 E)
⊃
 B
/
E
•
 G
//
D
⊃
A
T
 
 
T
    
T
    
F
  

The last two statements give E=T. G=T. D=T, and A=F. Fill all this in.

A
≡
(B
∨
 C)
/
(D
∨
 E)
⊃
 B
/
E
•
 G
//
D
⊃
A
F
T
   
T
T
T
    
T
T
 T  
T
F
 F

Now you can see that the or operators in the first and second statements must be F and T, respectively. Fill this in.

A
≡
(B
∨
 C)
/
(D
∨
 E)
⊃
 B
/
E
•
 G
//
D
⊃
A
F
T
F
  
T
T
 T
T
    
T
T
 T  
T
F
 F

The first statement now implies B=F and C=D. Fill this in.

A
≡
(B
∨
 C)
/
(D
∨
 E)
⊃
 B
/
E
•
 G
//
D
⊃
A
F
T
F
F
 F  
T
T
 T
T
 F  
T
T
 T
T
F
 F

Now there's a contradiction in the second statement. Therefore, the argument is VALID.

 


5. Decide if the following sets of statements are consistent or inconsistent.(a) First, let's assume each statement is true.

~
 (A
•
 B)
/
~
 (B
⊃
 C)
/
~
 B
∨
 A
/
(A
•
 D)
•
 C
T
  
   
T
  
 
T
  
 
 
T
  

Now the table can be filled in as follows.

~
 (A
•
 B)
/
~
 (B
⊃
 C)
/
~
 B
∨
 A
/
(A
•
 D)
•
 C
T
  
F
    
T
  
F
    
T
  
F
    
T
  
T
 T

Now, look at the last statement and fill in T for A and D.

~
(A
•
 B)
/
~
(B
⊃
 C)
/
~
 B
∨
 A
/
(A
•
 D)
•
 C
T
T
F
    
T
F
 T  
T
  
F
 T  
T
T
 T
T
 T

Now the first statement says B has to be false.

~
(A
•
 B)
/
~
(B
⊃
 C)
/
~
 B
∨
 A
/
(A
•
 D)
•
 C
T
T
F
 F  
T
F
F
 T  
T
 F
F
 T  
T
T
 T
T
 T

There's a contradiction in the second statement, which says F⊃ T is false. Condlusion: The statements are INCONSISTENT.

 

(b) To begin, assume each statement is true.

A
⊃
 (B • C)
/
 B
⊃
 (D ∨ ~E)
/
C
⊃
 (E
∨
 ~D
/
A
•
 E
/
D
T
      
 
T
      
T
  
 
T
  
T

Next the last statement says D=T, and the next-to-the-last says A=T and E=T. Fill this in.

A
⊃
 (B • C)
/
 B
⊃
(D
 ∨ ~E)
/
C
⊃
 (E
∨
~D
/
A
•
 E
/
D
T
T
      
 
T
T
  
T
T
T
FT
T
T
 T
T

Now, from the first statement, B and C must be true. Fill this in.

A
⊃
 (B • C)
/
 B
⊃
(D
 ∨ ~E)
/
C
⊃
 (E
∨
~D
/
A
•
 E
/
D
T
T
 T   T
T
T
T
  
T
T
T
T
FT
T
T
 T
T

Finally, the remaining slots can be filled in.

A
⊃
 (B
•
 C)
/
 B
⊃
(D
∨
~E)
/
C
⊃
 (E
∨
~D
/
A
•
 E
/
D
T
T
 T
T
 T
T
T
T
T
FT
T
T
T
T
FT
T
T
 T
T

No contradictions here. This shows the statements are consistent. The following substitutions make them all true: A=T, B=T, C=T, D=T, E=T.