# Logical Consistency

Suppose your father promised you a new bike, if you get good marks in your engineering.  What if, you don't get good marks? Our analysis explore possibilities of your father buying a new bike for you, even if you don't get good marks!  This type of reasoning is classified under a head called "Logical Consistency"

Structure 1: If ..., Then ....:
Let us take an example: If it rains, It will be cloudy
Let us explore the above statement in various cases
Case 1: It rained then we say, It should be cloudy.  So If x happened then y should happen. x ⇒ y
Case 2: There are no clouds, So there is no rain. ∼ y ⇒ ∼ x
Case 3: It is not raining.  Uncertain.  As there may be clouds or may not be.
Case 4: It is cloudy. Uncertain. As it may rain may not rain.
So of the above 4 cases Case 1 and Case 2 holds good.

Structure 2: Only If
Let us take an example, Only If you work hard, you will be successful.

Write the above statement like below. whenever there is 'onlyif' make sure it is in the middle of the two given statements.
Now to become successful, there is only one condition. To work hard. So we say, If one is succeeded means he must have work hard. So x ⇒ y
Also you did not work hard means, you are not succeeded. ∼ y ⇒ ∼ x

Structure 3: When / Whenever
When / Whenever is similar to Structure 1: If.. then.  So when x then y.
So possible conclusions are
1. x ⇒ y
2. ∼ y ⇒ ∼ x

Structure 4: Unless
Unless means "If not"
Unless you work hard, you fail = If you don't work hard, then you fail.
Again x ⇒ y and ∼ y ⇒ ∼ x are true.

Structure 5: Either / or
Take the Proposition: Either I will drink Pepsi or I will eat a sandwich.  Let 'I will drink Pepsi' be 'X' and 'I will eat a sandwich' be 'Y'.
I drank Pepsi, then one cannot say whether I ate sandwich or not. But If did not drink Pepsi, then one can say that I must have eaten sandwich. So
Possible conclusions:
1. ∼ x ⇒ y
2. ∼ y ⇒ x

Solved Examples

1. Sharma is either an Engineer or a Scientist.
A. Sharma is not a scientist
B. Sharma is a scientist
C. Sharma is an engineer.
D. Sharma is not an engineer.
a. CB
b. BA
c. DB
d. DC
Solution:  We know that If not an engineer then a scientist or If not a scientist then an engineer. So AC or DB correct.  Correct option C.  Please note if a person is an Engineer he can still be a scientist.

2. Raju is in the class when Puja is in the library.
A. Puja is in the library.
B. Raju is in the park.
C. Puja is not in the library.
D. Raju is in the class.
a. CA
c. BC
d. BD
Solution: When X then Y.  So When Puja is in the library, then Raju is in the class or Raju is not in the class then Puja is not in the library. So AD is correct.

3. You will become successful if you work hard
A. You worked hard.
B. You did not become successful.
C. You became successful .
D. You did not work hard.
a. BD
b. DB
c. BC
d. DC
Solution: If you work hard then you will become successful  or you did not become successful then you did not work hard. So AC or BD correct. So choice A.

4. She chats on her mobile only when her boss is away for lunch.
A. The boss is away for lunch
B. She did not chat on her mobile.
C. She chatted on her mobile.
D. The boss in not away for lunch.
a. DB
b. AB
c. DC
d. BC
Solution: Only when X then Y means Y happen then X happens or its contra-positive X did not happen then Y did not happen. So We say, She chatted on her mobile means boss is away for lunch (or) Boss is not away for lunch then She did not chat on her mobile.
Option A.

5. If Bobby and Danny are selected in that order, Parvez and Santhi cannot be selected.
A. Parvez and Santhi are selected in that order.
B. Danny and Bobby are selected in that order.
C. Bobby and Danny are selected in that order.
D. Parvez and Santhi are not selected.
a. BC
b. CD
c. BD
d. DB
Solution: this is called compound hypothetical.  If A and B then not C and D, Then C and D then not A or not B. Option B

Level 2

6.  My house has got a number.
If it is a multiple of 3, then it is in between 50 and 59.
If it is not a multiple of 4, then it is in between 60 and 69
If it is not a multiple of 6, then it is in between 70 to 79
What is my house number?
Solution:  If the house number has to be in 50 to 59, then "If "conditions 2nd and 3rd statements should not happen. i.e., It is a multiple of 4 and 6.  Now we know that if a number is a multiple of both 4 and 6, then it is a multiple of 12.  But no 12 multiple exists between 50 to 59.  So house number should not be in between 50 to 59
If the house number has to be in 60 to 69, then "if" conditions of 1st and 3rd statements should not happen.  i.e., the number should not be a multiple of 3 but multiple of 6. All multiple of 6 should be multiples of 3.  So no number exists in between 60 to 69
So the house number should exists between 70 to 79.  Then It should not be a multiple of 3 but multiple of 4.   Between 70 to 79, 72 and 76 are multiples of 4 but only 76 is not a multiple of 3.  So my house number is 76