In this chapter we are going to discuss the following topics with examples.

Introduction

Types of statements

Complementary pair


Introduction

The questions in this section contain two or more statements and followed by two or more conclusions. You have to find out which of the conclusions logically follow from the given statements. The statements have to be taken true even if they seem to be at variance from the commonly known facts.

For such questions, you can take the help of Venn Diagrams. On the basis of the given statements, you should draw venn diagrams, and then derive the solution from these diagrams.


Types of Statements

The four basic statements in syllogism are,

1.All As are B   (Eg.  All Cats are Dogs)

2.Some As are B   (Eg.  Some dogs are birds)

3.No A is B   (Eg. No bird is a pig)

4.Some As are not B   (Eg. Some pigs are not birds)

These statements can be classified into two categories as shown in below table.

Alternate words

A picture is worth a thousand words

Basic Diagrams

For (i) statement, i.e. All As are B, Circle A should be inside B or A and B can be equal. But circle A should not exceed B.

For (ii) statement, i.e. Some As are B, Circle A and B should be connected always. It should not separate. 

For (iii) statement, i.e. No A is B, We should not connect circle A and circle B.

For (iv) statement, i.e. Some As are not B, We can connect circle A and circle B.


Complementary pair

In the Complementary pair, subject and predicate should be same in both the conclusions.If one conclusion is true,definitely the other conclusion will be false and vise versa. There are two complementary pairs in syllogism. 

Pair I :   All As are B & Some As are not B

If “All As are B” is true, definitely  “Some As are not B” is false. If “Some As are not B” is true, definitely “All As are B” is false.

Pair II: No A is B  & Some As are B

If “No A is B” is true, definitely  “Some As are B” is false. If “Some As are B” is true, definitely “No A is B” is false.


Procedure

Step 1 : Draw the basic diagram for the given statements. 

Step 2a : If all are positive conclusions, 

Check those conclusion in basic diagram and decide which one is true or false.

Don’t draw any other diagram if all are positive in conclusion.

Step 2b: If there is negative conclusion and it is true in basic diagram,

Try to make it false by drawing its complementary pair.

While drawing alternate diagram, it should not violate any other given statements.

If you are able to draw alternate diagram, without violating any statement. Then the negative statement is false.


Examples(Positive Conclusions)

Q.1.

Statements

  1. All grapes are apples
  2. All apples are mangoes

Conclusions:

  1. All grapes are mangoes
  2. All mangoes are grapes
  3. Some grapes are mangoes

Answer : (1) and (3) are true

 

Q.2 

Statements

  1. Some doctors are lawyers
  2. Some lawyers are circle

Conclusion

  1. Some doctors are circle
  2. All doctors are circle

Answer : Both (1) and (2) are false

Q.3.

Statements

  1. Some mobiles are rows
  2. No row is circular

Conclusion

  1. All circular are mobiles
  2. Some circular are mobiles

Answer : Both (1) and (2) are false

Explanation : 

Basic diagrams for above questions. Verify the conclusion only  in below basic diagram.(Because all are positive conclusion)


Examples (Negative Conclusion)

Q.1.

Statements

  1. All months are weeks
  2. Some week are days

Conclusions:

  1. No month is day
  2. Some weeks are months

Answer : (2) is true

 

Q.2 

Statements

  1. All right are left
  2. No left is top

Conclusion

  1. Some tops are right
  2. No top is right

Answer : only (2) is true

Q.3.

Statements

  1. All goats are good
  2. Some good are watch

Conclusion

  1. Some watch are goat
  2. No watch is goat

Answer : Either (1) and (2) are true

Explanation: 

Q.1

“No month is a day” is negative conclusion and it is true in basic diagram. So you have to make it false by drawing its complementary pair. Refer Alternate diagram. While drawing alternate diagram, it is not violating any statements. So the negative conclusion “No month is a day” is false in alternate diagram. Hence the conclusion is false. 

Q.2 

“No top is a right” is negative conclusion and it is true in basic diagram. So you have to make it false by drawing its complementary pair.  While drawing alternate diagram, it is violating the given statements. So you can’t draw alternate diagram. So the negative conclusion “No top is  a right” will always true.

Q.3

“No watch is a goat” is negative conclusion and it is true in basic diagram. So you have to make it false by drawing its complementary pair. Refer Alternate diagram. While drawing alternate diagram, it is not violating any statements. So the negative conclusion “No month is a day” is false in alternate diagram and its complementary is true. So you have to mark either (1) or (2). 

 


Thus in this post we learned about the concepts of syllogism, types of statements, complementary pair and procedure to solve different types of problems. 

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