Introductory Logic
Test #2
January 13, 2006
 
Name:_________________________
R. Hammack
Score: _________

1. (20 points) Translate the following sentences into symbolic form. Use capital letters to represent simple statements. For each letter you use, please indicate what statement it stands for (e.g. N = "my nose itches").

(a) A day is a test day if and only if the day is a Friday and the month is January.
D = "A day is a test day"
F = " the day is Friday"
J = "the month is January"

D ≡ (F • J)


(b) I will give you a make-up test only if you request it.

M = "I will give you a make-up test"
Y = "you request it"

M ⊃ Y

(c) If the Internet use continues to grow, then more people will become cyberaddicts, and normal human relations will deteriorate.
I = "the Internet use continues to grow"
M = "more people will become cyberaddicts"
N = "normal human relations will deteriorate"

I ⊃ (M • N)


(d) We will have a picnic unless it rains.
P = "we will have a picnic"
R = "it rains"

(P ∨ R) • ~(P • R)

Note: I gave nearly full credit for P∨R or ~R⊃P. However, neither of theses answers is completely correct. To see why, notice that the statement "We will have a picnic unless it rains." is FALSE (a lie) when P = T and R = T, but both P∨R and ~R⊃P are TRUE for this substitution. Thus neither P∨R nor ~R⊃P is an accurate translation. Reason: this is the exclusive sense of unless.


(e)

If you hold down the shift key and press the delete button, then your computer will explode, and you'll have to buy a new one and rewrite all your files.
H = "you hold down the shift key"
P = "you press the delete button"
E = "your computer will explode"
B = "you'll have to buy a new one"
R = "you'll have to rewrite all your files

(H • P) ⊃ (E • B • R)


2. (20 points) Write out the truth tables for the following propositions. For each proposition, say if it is tautologous, self-contradictory, or contingent.

(a)
S
⊃
[
(
R
•
S
)
∨
~
R
]
T
T
T
T
T
T
F
T
T
T
F
F
T
T
T
F
F
T
T
F
F
F
F
T
F
T
F
F
F
T
T
F
TAUTOLOGOUS


(b)
~
[
~
(
K
⊃
 H )
∨
(
H
≡
K ) ]
F
F
T
T
 T  
T
T
T
 T    
F
T
T
F
 F  
T
F
F
 T    
T
F
F
T
 T  
F
T
F
 F    
F
F
F
F
 F  
T
F
T
 F    
CONTINGENT


3. (20 points) Determine if the following pairs of statements are logically equivalent, contradictory, consistent, or inconsistent.

(a)
~
(
X
⊃
Y )  
Y
∨
~
 X
F
T
T
 T    
T
T
F
 T
T
T
F
 F    
F
F
F
 T
F
F
T
 T    
T
T
T
 F
F
F
T
 F    
F
T
T
 F
CONTRADICTORY
and
INCONSISTENT


(b)
A
∨
(
B
•
C )  
(
A
∨
 B )
•
(
A
∨
 C )
T
T
T
T
 T    
T
T
 T  
T
T
T
 T  
T
T
T
F
 F    
T
T
 T  
T
T
T
 F  
T
T
F
F
 T    
T
T
 F  
T
T
T
 T  
T
T
F
F
 F    
T
T
 F  
T
T
T
 F  
F
T
T
T
 T    
F
T
 T  
T
F
T
 T  
F
F
T
F
 F    
F
T
 T  
F
F
F
 F  
F
F
F
F
 T    
F
F
 F  
F
F
T
 T  
F
F
F
F
 F    
F
F
 F  
F
F
F
 F  
LOGICALLY EQUIVALENT
and
CONSISTENT



4. (20 points) Use indirect truth tables to decide if the following sets of statements are consistent or inconsistent.

(a)
K
≡
(
A
•
~
P
 )
/
A
⊃
(
P
•
~
S
 )
/
S
⊃
~
 K
/
A
•
~
 K
 
F
T
T
F
F
T
  
T
T
T
T
T
F
  
F
T
T
 F
T
T
T
 F

There is no contradiction, so the statents are CONSISTENT.

(b)
(
Q
∨
K )
⊃
 C
/
(
C
•
 P )
⊃
(
N
∨
 L )
/
C
⊃
(
P
•
~
 L )
/
Q
•
~
 N
 
T
T
 ?  
T
 T
T
T
 T  
T
F
F
 F  
T
T
T
T
T
 F  
T
T
T
 F

A contradiction is highlighted, so the statents are INCONSISTENT.
(Note that the value of K cannot be determined, but this does not matter, because, since Q=T, the statement Q∨K is true no matter what the value of K is.)


5. (20 points) Use any technique from Chapter 6 to decide if the following arguments are valid or invalid.

(a) Elvis was a space alien or he was not a hound dog. If Elvis was a space alien, then he's still alive. Thus, if Elvis was a hound dog, then he's still alive.

S = "Elvis was a space alien"
H = "Elvis was a hound dog"
A = "Elvis is still alive"

Writing the argument in symbolic form, and filliing out an indirect truth table with true premises and false conclusion, we get:

S
∨
~
 H
/
S
⊃
 A
//
H
⊃
 A
F
T
F
 T
F
T
 F
T
F
 F

A contradiction is highlighted. Thus the argument is VALID


(b)
J ⊃ (~L ⊃ ~K)
K ⊃ (~L ⊃ M)
(L ∨ M) ⊃ N
J ⊃ N

Filling out an indirect truth table, with true premises and false conclusion, we get:

J
⊃
(
~
L
⊃
~
 K )
/
K
⊃
(
~
L
⊃
 M )
/
(
L
∨
 M )
⊃
 N
//
J
⊃
 N  
T
T
T
F
T
T
 F  
F
T
T
F
F
 F  
F
F
 F  
T
 F
T
F
 F  

There is no contradiction, so the argument is INVALID.