# Logical Reasoning and Reasoning Errors

## Four Logical statements with examples

### General examples

1. Proposition: If you do X, then you may do Y.
2. Inverse: If you do NOT do X, then you may NOT do Y.
3. Converse: If you do Y then you do X.
4. Contrapositive: If you do NOT do Y then you will NOT do X.

### Specific examples

1. Proposition: If there is rain, then there are clouds.
2. Inverse: If there isn't rain, then there aren't clouds.
3. Converse: If there are clouds, then there is rain.
4. Contrapositive: If there aren't clouds, then there isn't rain.

### Statements equivalent to the proposition:

Proposition: If there is rain, then there are clouds.

• Rain implies there are clouds.
• If there is rain, then there are clouds.
• There is rain only if there are clouds.
• There are clouds if rain.
• If there are no clouds, then there is no rain.
• Rain is sufficient for clouds.
• Clouds are necessary for rain.

### Statements not equivalent to the proposition:

Proposition: If there is rain, then there are clouds.

• If there are clouds, then there is rain.
• If there isn't rain, then there aren't clouds.
• There are clouds only if there is rain.

### Statements coded:

1. Proposition: If X, then Y.
2. Inverse: If ~X, then ~Y.
3. Converse: If Y, then X.
4. Contrapositive: If ~Y, then ~X.

Equivalent statements:

X -> Y
if X, then Y
X only if Y
Y if X
~Y -> ~X
X is sufficient for Y
Y is necessary for X

### Examples - Label each of the following statements from the proposition and tell which is true if the proposition is true.

Proposition: If John has work to do, then he is in his seat.

If John doesn't have work, then he isn't in his seat.

If John is in his seat, then he has work to do.

If John isn't in his seat, then he doesn't have work to do.

### Label each of the following statements from the proposition and tell which is true if the proposition is true.

Proposition: If Mary doesn't do her work, then she will not go to recess.

If Mary does her work, then she will go to recess.

If Mary will not go to recess, then she did not do her work.

If Mary goes to recess, then she did her work.

1. Marty asked her teacher if she should do a report. The teacher said, "Marty, you can earn an A only if you do the report." Marty did the report, but she did not get an A. Did the teacher lie?

2. A teacher told Jim that if he didn't do his work he wouldn't go outside to recess. Jim didn't do his work and the teacher let him go to recess. Did the teacher lie?

### More examples

Assume that the first statement is true, tell if the following statements are true, false, or questionable.

1. If you wear nice clothes to school, then your teachers will be impressed.

2. If your teachers will be impressed, then you wear nice clothes to school.

3. If your teachers won't be impressed, then you don't wear nice clothes to school.

4. If you don't wear nice clothes to school, then your teachers won't be impressed.

For the four statements above this time assume the first statement is false, now tell if the same three statements are true, false, or questionable.

### Common errors in reasoning:

Circular reasoning:

James: "Grass is green."

Lynn: "Why do you think so?"

James: "Because it is green."

### Proof by selected instances:

John has never seen his parents fight. He thinks that all married couples never fight.

### Proof by failure to find a counter example:

Mary has never seen a purple elephant, therefore they do not exist.

### Proof by counter example:

You are never on time. What do you mean? I was yesterday.

### Avoiding the question:

Is it bad doc? It's too early to say.

Dad: "Slow down your going 60 mph and that's over the speed limit."

Son: "You drive 65 mph".

Dad: "That's different. I know what I'm doing."

### Faking a connection:

A dog is an animal. A cat is an animal. Therefore a dog is a cat.

### Tell if the following are logical or not and what kind of faulty reasoning is being used, if they are faulty.

1. It doesn't make any difference to me whether you believe me or not, because I don't care one way or another whether or not you believe me.

2. Development of solid foundations of reasoning is essential to the proper education of a person. Therefore, if someone is to be educated properly, then she must at least acquire a thorough knowledge of the basics of reasoning.

3. "Dad, can I use the car tonight?"
"What do you think?"
"Nope. You told me not till next week."

4. I know that I can not learn a foreign language, because when I took Spanish I received an F.

5. My mother can cook anything. She is always trying new recipes and they always turn out great.

6. Daughter: "Why won't you let me go downtown? You said you did when you were my age.
Mother: "That's different. The times have changed, the size of the town, and the ways of the people."

7. John must be a very good person, because I have never heard anything bad about him.

8. "Sara, will you go out with me tonight?"
"Get lost."

9. All teachers are important people. You are a teacher. Therefore, you are an important person.

10. "Mom, Why is the sky blue?"

11. "Mom, why is the sky blue?"
"because red wouldn't look so good."

12. "Mom why is the sky blue?"
"I don't know."

13. All traditional teachers favor corporal punishment. George favors corporal punishment. George is a traditional teacher.

14. You said you would let us go to recess only if we did all our work, and we did all our work, so now you have to let us go to recess.

15. Teacher: "I can not let you use the phone, because the rule says no one uses the phone during class time."
Student: "Please just for me, I won't tell anyone and I have to make this call before mom leaves for work."

16. All teachers like children. You like children. So you are a teacher.