LOGICAL REASONING

In logical reasoning one draws inferences or conclusions from statements or premises and test the validity of such inferences. The following terms are useful in understanding the principles of logical reasoning.

Proposition :  A proposition is a statement of a relationship between two terms.
A categorical statement to be in standard form,  the sentence expressing that statement must begin with quantifier such as “all”,  “no”, “some”.  It must then present the subject term – the term designating the set of objects the statements is about  - followed by copula – either  “are” or “are not” – followed by the predicative term.

Quantifier + subject + copula + predicative term.

A – All graduates are literates.
E – No graduates are criminals.
O – Some graduates are not rich people.

Statement A  - Universal affirmative
Statement E  - Universal negative
Statement I   - Particular affirmative
Statement O  - Particular negative

Distribution of terms

A term is distributed in such a sentence if it refers to all members of the set of objects denoted by the term, otherwise it is said to be undistributed.

Ex : 1 All graduates are literates.

Literate  - undistributed.

Ex : 2 No graduates are criminals.

Subject & predicate distributed.

Neither subject nor predicate distributed.

Ex : 4 Some graduates are not rich people.

Only predicate distributed.

Following are the rules used in solving logical reasoning.

Rule 1 : If both propositions are particular or one affirmative and one negative or both affirmative, the inference will be uncertain.

The inference is uncertain.

Rule 2 : If one premise is negative, the inference must be negative.  If  the inference is universal affirmative, it is definitely false.  But, if it is particular affirmative inference will be correct. Because, that will be the implication of the particular negative proposition.

Some rice people are not graduates.
So all rich people are men.
Some rich people are men.

Rule 3 : In a categorical argument there must be a middle term.  If there is no middle term no inference can be derived.

All birds fly
No human being flies
So,  no human being is a bird.

Inference is incorrect.

My hand touches the chair
The chair touches the ground.
My hand touches the ground.

Inference is incorrect.

Rule 4 :   The middle term must be distributed at least once in the premises,  otherwise inference is uncertain.

All chairs are furniture.
All tables are furniture.
So all tables are chairs.

Middle term furniture is undistributed.

Rule 5 : If a middle term is distributed in the inference, it must be distributed in the major premise, otherwise the inference is false.

All chairs are pens.
No table is a book.
So, no table is a chair.

Rule 6 :  If  a minor term is distributed in the inference it must be distributed in the major premise, otherwise inference will be false.

No horse is dog.
All dogs are rats.
So,  no rat is a horse.

Inference is false.

Exercise :

Directions : Two statements are followed by 2 possible inferences. You have to point out which of the following answer choices applies to each of the following questions :

a)    Only inference  I is followed.
b)    Only inference II is followed.
c)    Both inference I & II are followed.
d)    Neither inference I nor II followed.

1. All horses are donkeys.  All donkeys are monkeys.
i)                     All horses are monkeys.
ii)                   All monkeys are horses

1. No rice is food, No food is bread.
ii)
1. All papers are books,  Some papers are files.
i)             Some files are books
iv)                 Some books are files
ii)

1. Some pens are books,  No pencil is a rubber.
i)             Some rubbers are not books
v)                   Some books are not rubbers.
ii)

1. No crow is a sparrow, All hens are eagles.
i)             No eagle is a sparrow
ii)            No sparrow is a eagle.

Directions :  Choose the best alternative.

1. He cannot vote because he is a minor.
a)    Some minors cannot vote in elections.
b)    Some who cannot vote are minors.
c)    No minor can vote.
d)    All minors cannot vote.

b) All good are not cheat.
c) No good person is cheat.
d) All cheat are not good.

8.  He is wise because all graduates are wise.
b)    He is not a graduate.
d)    Some graduates are not he.

9. Asoka was popular, because he was fair.
a) All popular are fair
b) All fair are popular
c) Only fair are popular
d) None.

10. He cannot read, because he is a blind.
c) All who can read are not blind
d) Some who read are not blind.

11. He wear turban, so he is a sikh.
a)  All who wear turban are Sikhs
b) All Sikhs wear turban.
c) Only Sikhs wear turban
d) Some Sikhs wear turban.

12. He wears  sweater, because he is sick.
a) All who wear sweater are sick.
b) All sick wear sweater
c) No sick wear sweater
d) All sick do not wear sweater.

13. Pick pockets are hated, because they are criminals.
a)  All who are hated are pick pockets.
b) All criminals are hated.
c) All hated are criminals.
d) Some criminals are hated.

14. He is unhappy,  because he has failed in exam.
a) All unhappy fails.
b) Some who fail are unhappy.
c) Some unhappy fails.
d) All who fails, are unhappy.

15. He lived long,  because he was healthy.
a) All who lived long are healthy.
b) All healthy live long.
c) No healthy live long.
d) Some healthy live long.

Directions :  An interrogative sentence is followed by  2 arguments.  One beginning with “yes” and the other beginning with “no”.  You are to pick one of the  following answer choices which is applicable to each of the following questions :
A. Only argument 1 is forceful.
B. Only argument 2 is forceful.
C. Both arguments are forceful.
D. Neither 1 nor 2 is forceful.

16. Has India been consistently non-aligned?
1.       Yes.  Because India has not sided with any big power.
2.       No. Because India did not condemn Russian occupation of Afganistan.

17. Should education be job oriented?
1. Yes.  Because purpose of education is to prepare us for earning.
2. No. because education should not be for the sake of education.

18. Should India remain secular?
1. Yes. Because people belonging to different religions live in India.
2. No. Because the different religious group do not see eye to eye
with one  another.

19. Should strikes be banned in essential services?
1. Yes. Because strike disturbs the normal life.
2. No.  Because it is a democratic right of the people.

20. In India,  should income tax be abolished?
1. Yes, because it is an unnecessary burden on the wage earners.
2. No, because it is a good source of revenue.

21. Barking dogs never bite.
1. Dogs always bite.
2. Some dogs bite.

22. Man is born free.
1. Freedom is the birth right of man.
2. The very birth confirms right on man.

23. I pray to god  whenever I am in difficulty.
1. Prayers can solve problems.
2. Prayers to god is the highest duty of  man.

24. Laugh, and the world will laugh with you.
1. People generally laugh.
2. Laughter symbolizes happiness.

25. Owing to stress,  you digestive system is affected.
1. There is a stress in life.
2. Digestive system has no defensive.

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