Introductory Logic
Test #2
October 21, 2005
 
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) If Harriet Miers is nominated to the Supreme Court, then Democrats will object and Republicans will too.
M ="Harriet Miers is nominated to the Supreme Court"
D = "Democrats will object"
R = "Republicans will object"
M ⊃ ( D • R )

(b) If you did poorly on the first test, the grade will be dropped if you do well on the second.

P = "you did poorly on the first test"
D = " the grade will be dropped"
W = "you do well on the second test"
P ⊃ ( W ⊃ D )

(c) Cigarette manufacturers are neither honest nor socially responsible.

H = "Cigarette manufacturers honest"
S = "Cigarette manufacturers are socially responsible"
~H • ~S

(d) You get credit for a course if and only if you take it and you pass it.

C = "You get credit for a course"
T = "you take the course"
P = "you pass the course"
C ≡ (T • P)

(e)

Either the train is on time or the train is late.

O = "the train is on time"
L = "the train is late"

(O∨ L ) • ~(O• L ) (Exclusive sense of or)


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 )


( S
⊃
 R )
•
( S
•
~ R
T
T
 T
F
T
F
 F T
T
F
 F
F
T
T
 T F
F
T
 T
F
F
F
 F T
F
T
 F
F
F
F
 T F

SELF-CONTRADICTORY


 

(b) ( ~ K⊃ H ) ≡ ( H ∨ K )

 

( ~
K
⊃
H)
≡
(H
∨
K)
F
T
T
T
T
T
T
 T
F
T
T
F
T
F
T
 T
T
F
T
T
T
T
T
 F
T
F
F
F
T
F
F
 F

TAUTOLOGOUS




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

(a) ~ ( B ∨ A ) ~ B • ~ A

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

LOGICALLY EQUIVALENT





(b) ~ A ≡ X ~ ( ~ A ∨ X ) ∨ ( A • ~ X )

 

~
A
≡
X
~
(
~
A
∨
X
)
∨
(
A
•
~
X
)
F
T
F
T
F
F
T
T
T
F
T
F
F
T
F
T
T
F
T
F
T
F
F
T
T
T
T
F
T
F
F
T
F
T
F
T
T
F
F
F
F
T
T
F
T
F
F
T
F
T
F
F
F
F
T
F

CONSISTENT





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

(a) K ≡ ( R ∨ M ) K • ~ R M ⊃ ~ K

K
≡
(
R
∨
M
)
/
K
•
~
R
/
M
⊃
~
K
T
T
F
T
T
T
T
T
F
T
T
F
T

Contradiction highlighted. STATEMENTS ARE INCONSISTENT.






(b) ( N ∨ C ) ≡ E N ⊃ ~ ( C ∨ H ) H ⊃ E C ⊃ H

(
N
∨
C
)
≡
E
/
N
⊃
~
(
C
∨
H
)
/
H
⊃
E
/
C
⊃
H
F
T
T
T
T
F
T
F
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
F
T
T
T
T
F
T
T


Begin by looking at all the ways the last statement can be true. There are three lines to examine. But the very first one leads to no contradiction, so the statements are CONSISTENT.






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

(a) If high school graduates are deficient in reading, then they will not be able to compete in the modern world. If high school graduates are deficient in writing, then they will not be able to compete in the modern world. Therefore if high school graduates are deficient in reading, then they are deficient in writing.

R = "high school graduates are deficient in reading"
W = "high school graduates are deficient in writing"
C = "high school graduates will not be able to compete in the modern world"

R
⊃
C
/
W
⊃
C
//
R
⊃
W
T
T
T
F
T
T
T
F
F

No contradiction. Argument is INVALID.

 

(b)
G ⊃ H
H ⊃ I
~J ⊃ G
~I
J

 

G
⊃
H
/
H
⊃
I
/
~
J
⊃
G
/
~
I
//
J
T
T
F
F
T
F
T
F
T
T
T
F
F


A contradiction is highlighted. The argument is VALID.