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Introduction to Logic Third Exam (docx) Lehman College, Fall 2021 I

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Introduction to Logic Third Exam (docx)

Lehman College, Fall 2021

I. Translate the following sentences into Sentence Logic (SL) using the letters A, B, C, H, and L to stand for the following sentences: (2 pts. each)

A = Alice jogs regularly

B = Bob jogs regularly

C = Carol jogs regularly

H = Alice is healthy

L = Bob is lazy.

 

1. Alice and Bob jog regularly.

 

 

2. If Bob does not jog regularly, then Bob is lazy.

 

 

3. If Alice and Bob jog regularly, then Carol does too.

 

 

4. If Alice jogs regularly, then either Bob or Carol do too.

 

 

5. Alice jogs regularly if Bob jogs regularly.

 

 

6. Alice jogs regularly if and only if Bob jogs regularly.

 

 

7. Neither Alice nor Bob jog regularly.

 

 

II. If A, B, and C are true statements and X, Y, and Z are false statements, say whether the following symbolized compound statements are true or false. (2 pts. each)

 

 

1. ___  ~ A v C                2. ___ ~ (A v C)              3. ___  ~ A · C

 

 

 

4. ___  A É B                  5. ___  A É ~ B               6. ___  Y É ~ A

 

 

 

7. ___  ~ B É ( Y v A )            8. ___  ~ B v (~ Y É ~ A )            9. ___  ~ ( X v Y ) É (~ A · Z )

 

III. MAKE A TRUTH TABLE for each of the following arguments, and use it to prove that the argument is valid or invalid. BE SURE TO SAY whether the argument is valid or invalid. (4 pts. each)

 

1.   P ÉQ

      Q É P

\  P v Q

 

P	Q

 

 

 

 

 

 

 

 

2.  (M v N) É (M · N)

     M · N

\ M v N

 

M	N

 

 

 

 

 

 

 

 

3.   Q É R

      P É Q

 \ P É R

 

P	Q	R

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IV. Make a truth table for each of the following symbolized compound statements, and use the truth table to tell if it is a tautology, a self-contradiction, or a contingent statement. (3 pts. each)

 

1. ____________________     ( Q É Q ) É Q

 

Q

 

 

 

 

2. ____________________     ( R É S ) · ( R · ~ S )

R	S

 

 

 

 

 

 

 

3. ____________________     [ ( R É S ) É S ] É R

R	S

 

 

 

 

 

 

 

4. ____________________     A É ( ~ A É B )

A	B

 

 

 

 

 

 

 

 

V. Assume that the following statement is true: If the car starts, then there is gas in it. Given that assumption, say whether the statements below are true or false. (2 pts. each)

 

1. ___  The car starting is a sufficient condition for there to be gas in it.

 

 

2. ___  The car starting is a necessary condition for there to be gas in it.

 

 

3. ___  There being gas in the car is a necessary condition for the car to start.

 

 

4. ___  There being gas in the tank is a sufficient condition for the car to start.

VI. Translate the following arguments into Sentence Logic (SL), and then use truth tables to determine if they are valid or invalid. Use the underlined letters to symbolize the simple sentences in which they occur. (4 pts. each)

 

1. Either Argentina does not have a huge foreign debt or Brazil has a huge foreign debt. And Argentina does not have a huge foreign debt. So Brazil has a huge foreign debt.

 

 

 

 

 

 

A	B

 

 

 

 

 

 

 

 

2. If Japan continues to increase exports, then either Korea or Laos will suffer an economic decline. But Korea will not suffer an economic decline. Thus, if Japan continues to increase exports, Laos will suffer an economic decline.

 

 

 

 

 

 

 

 

 

 

J	K	L