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If – then

A. If – then

“If – then” is one of the most important Logical Links. Its usage includes the affirmative as well as the negative statement. Following are some of the structured applications of this Logical Link:

  • If A happens then B happens
    Given – A has happened.
    Conclusion – B will happen.
    This is a valid conclusion.

See this through the following diagram:
 

Example 1

 

Q 1. If the terrorists’ demands are met, then lawlessness will prevail.
 

Given – The terrorist’s demands have been met.


Conclusion – Lawlessness will prevail.

  • If A happens then B happens
    Given – A has not happened.
    Conclusion – B will not happen.
    This is an invalid conclusion.
    Using the diagram given above, despite A has not happened, B might happen owing to the occurrence of either reason 2 or reason 3 etc.

Example 2

Q 2. If the terrorists’ demands are met, then lawlessness will prevail.
 

Given – The Terrorist’s demands have not been met.


Conclusion – Lawlessness will not prevail.


This is an invalid conclusion.

Example 3

Q 3. If you fall down from the roof, then your legs will get fractured.
 

Given – You have not fallen down the roof.


Conclusion – Your legs will not get fractured.


This is an invalid conclusion. What we should understand here is the fact that falling down the roof is just one of the ways by which the legs can get fractured and there can be other ways also of the legs getting fractured, e.g., meeting with an accident or getting beaten up, etc. And just because you have not fallen down the roof, we cannot be sure that the legs have not got fractured.

  • If A happens then B happens
    Given – B has happened.
    Conclusion – A would also have happened.
    This is an invalid conclusion.

Once again go through the same diagram:

 

 
 

Obviously B can happen due to reason 2 or reason 3 also and is not dependent on ‘reason 1 only’.

Example 4

Q 4. If you fall down from the roof, then your legs will get fractured.


Given – Your legs have got fractured.


Conclusion – You have fallen down the roof.


This is an invalid conclusion.

Again, going through the explanation given above, we know that falling down the roof is not the only reason for the legs getting fractured. So just because the legs have got fractured, we cannot conclude that a person has fallen down the roof.

Example 5

 

Q 5. If it rains, then farmers become happy.
 

Given – Farmers are happy.

Conclusion – It must have rained.


This is an invalid conclusion.

We do understand that farmers can have more than one reason for being happy, and having rain is just one of them. So, if farmers are happy we cannot definitely conclude that it must have rained.

  • If A happens then B happens
    Given – B has not happened.
    Conclusion – A would also not have happened.
    This is a valid conclusion.

Example 6

Q 6. If you fall down from the roof, then your legs will get fractured.
 

Given – Your legs have not got fractured.


Conclusion – You have not fallen down from the roof.


This is a valid conclusion.

Understand this—You give a contract to a goon to make Mr A fall down from the roof tonight and as a result Mr A’s legs will automatically get fractured. But the next morning you see Mr A walking and looking perfectly all right. He can walk like this because his legs are not fractured, so the direct response will be—The goon has not done his job of making Mr A fall down from the roof. With this a lot many accidents might not have happened with Mr A, but he must not have fallen down the roof too.

  • If A happens only then B happens
    Given – B has happened.
    Conclusion – A would also have happened.
    This is a valid conclusion.

See the diagram for this:
 

 

Example 7

Q 7. If you die, only then you go to heaven.

 

Given – You have gone to heaven.


Conclusion – You have died.


This is a valid conclusion.

Here, the only way in which a person can go to heaven is to die. So, if a person has gone to heaven, he must have died too.

  • If A happens only then B happens
    Given – A has happened.
    Conclusion – B will also happen.
    This is an invalid conclusion.

See the diagram for this:
 

 

Example 8

Q 8. If you die only then you go to heaven.
 

Given – You have not died.


Conclusion – You will not go to heaven.


This is a valid conclusion.

Example 9

Q 9. If you die only then you go to heaven.
 

Given – You have died.


Conclusion – You will go to heaven.


This is an invalid conclusion. However, the conclusion “You may go to heaven” is valid.

Sometimes we can even see that two compound state- ments have been joined with multiple Logical Links. One such statement can be:


If A happens, then B happens. If A does not happen then C happens. So what will be the impact on C if A has happened/A has not happened?

Let us understand this situation with an example:

Example 10

Q 10. If the terrorists’ demands are met, then lawlessness will prevail. And if the terrorists’ demands are not met, then the hostages will be murdered.


Consider the statements given below:

  1. Lawlessness has not prevailed.
  2. The hostages have been murdered.
  3. The terrorists’ demands have been met.
  4. The terrorists’ demands have not been met.

Which of the following options does not show a proper cause-effect relationship?

(a) i-ii
(b) iv-ii

(c) iii-ii
(d) ii-iv


Let us take the options one by one:


Option (a) i-ii

Since lawlessness has not prevailed (i), it means that the terrorists’ demands have not been met. And if the terrorists’ demands have not been met, then the hostages will be murdered (ii). So, this option shows a valid relationship.


Option (b) iv-ii

Since the terrorists’ demands have not been met, the hostages will be murdered. This is a direct conclusion. So, it is a valid relationship.


Option (c) iii-ii

Since the terrorists’ demands have been met, lawlessness will prevail. Now, from here we cannot deduce that the hostages will be murdered. So, it is not a valid relationship.


Option (d) ii-iv

Since the hostages have been murdered, the terrorists’ demands must not have been met. So, this is also a valid relationship.


Hence, option (c) is not a valid cause-effect relationship.

Either – or

We use a lot many statements involving “Either – or” as a logical link in our day-to-day life too. In case of reasoning questions also, we see very similar statements with very similar interpretations. However, for a proper understanding, we should be very clear with the structure of these statements. Some of these structured statements are given here under.


There are two situations possible here:

  • Inclusive It means ‘and/or’ where at least one term must be true or they can both be true simultaneously. This is the case of set A∪B, where any member of this set
    (A union B) should be present in at least one of the sets.
  • Exclusive It means one must be true and the other must be false. Both terms cannot be true and both cannot be false.

The popular English language concept of or is often ambiguous between these two meanings, but the difference is pivotal in evaluating the Logical Link questions asked in CAT.


Understand this argument:

Either P or Q.

Not P.


Therefore, Q is valid and indifferent between both inclusive and exclusive meanings. However, only in the ‘exclusive’ meaning is the following form valid:

Either P or Q (exclusive).

P.

Therefore, not Q.


With the ‘inclusive’ meaning, we cannot draw a definitely true conclusion from the situation given above.

Remember, for the purpose of CAT we have to consider the statements in their ‘exclusive’ sense.

  • Either A will happen or B will happen
    Given – A will happen.
    Conclusion – B will not happen
    This is a valid conclusion.
  • Either A will happen or B will happen
    Given – A will not happen.
    Conclusion – B will happen
    This is a valid conclusion.
  • Either A will happen or B will happen
    Given – B will happen.
    Conclusion – A will not happen
    This is a valid conclusion.
  • Either A will happen or B will happen
    Given – B will not happen.
    Conclusion – A will happen
    This is a valid conclusion.

Understandably, in case of ‘Either A will happen or B will happen’, not happening of one part ensures the happening of the other part because one of the parts has to be true. Similarly, we should remember that happening of one part ensures ‘not happening’ of the other part.


Summarizing the above discussion,

 

Either A will happen or B will happen

Validity

Truth-metre

A has
happened

B will not happen

Valid

Definitely True

A has not happened

B will happen

Valid

Definitely True

B has
happened

A will not happen

Valid

Definitely True

B has not happened

A will happen

Valid

Definitely True


Further, it should be understood that the statement ‘Either A will happen or B will happen’ and statement ‘A will happen or B will happen’ are one and the same.


Example :-

 

Q 11. Either A is 200 m from B and B is 100 m from C or A is 300 m from C.

  1. A is 200 m from B and B is 100 m from C.
  2. A is 300 m from C.
  3. A is not 200 m from B and B is not 100 m from C.
  4. A is not 300 m from C.

Which of the following is a correct ordered relationship?

(a) ii-iii
(b) i-iv

(c) iii-ii
(d) i-iii


Solution

Go through the options.


Option (a) ii-iii

A being 300 m from C ensures that A is not 200 m from B and B is not 100 m from C. So, ii-iii is a correct relationship.


Option (b) i-iv

We definitely have a true pair here as well.


Option (c) iii-ii

Since one of the two statements given in the example has to be true; and in this case one statement is given to be wrong; so, the other statement has to be true.


Hence, if A is not 200 m from B and B is not 100 m from C then the statement A is 300 m from C has to be true.

So, this is a valid conclusion.


Option (d) i-iii

Obviously, this is not true.





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