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"

Let us take an example: If it rains, It will be cloudy

let us explore the above statement in various cases

X as the conditional clause

Y as the conditional clause

Hence,
Premise: If X then Y.
Immediate inference: If X happens then Y should happen.  If Y did not happen then X should not happen.
Symbolically we write as If X then Y gives two inferences X⇒YorY⇒−X

Other Structures:

Only If


Premise: Only if X then Y
Immediate inference: If Y happens then X should happen.  If X did not happen then Y should not happen.
Symbolically we write as Only If Y then X gives two inferences Y⇒Xor−X⇒−Y

When

When is same as If.

Premise: When X then Y.
Immediate inference: When X happens then Y should happen.  If Y did not happen then X should not happen.
Symbolically we write as When X then Y gives two inferences X⇒Yor−Y⇒−X




Unless
Premise: X unless Y
This statement can be re-written as a hypothetical statement as −X⇒Y
Premise: If not X then Y
Immediate Inference: When X did not happen then Y should happen.  If Y  did not happen the X should happen
Symbolically we write as  X unless Y gives two inferences −X⇒Yor−Y⇒X

Disjunctive Argument (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'.  The proposition presents a disjunction. Any immediate inference with respect to any one of 'X' or 'Y' will be subject to a condition imposed on the other.   Hence all immediate inferences will be hypothetical propositions.

Structure of a Disjunctive argument

Premise 1:   Either X or Y
Premise 2:   not X        
Conclusion: Therefore, Y
 or

Premise 1:   Either X or Y
Premise 2:   not Y        
Conclusion: Therefore, X


Here we have two occurrences where at least one of the elements has to occur.

So the seeming logic is that if not one the other event will surely occur, the drinking of Pepsi or eating of a sandwich.

Analysis of Disjunctive proposition: 

X as the conditional clause

Y as the conditional clause

Premise: Either X or Y.
Immediate inference: If not X, then Y or If not Y, then X
Symbolically we write as  Either X or Y
∼X⇒Y or∼Y⇒X

Example: 

Premise 1: I will study or I will fail ........... Disjunctive proposition
Premise 2: I will not study.         ........... Categorical proposition
Conclusion: Therefore I will fail ........... Categorical proposition

Solved Examples

1. Sam is either black or white.

A. Sam is not white  B. Sam is white

C. Sam is black. D. Sam is not black.

a. CB b. BA

c. DB d. DC

Solution:  We know that If not black then White or If not white then black. So AC or DB correct.  Correct option C.

2. Rohit is in the class when Puneet is in the lab.

A. Puneet is in the lab.

B. Rohit is in the park.

C. Puneet is not in the lab.

D. Rohit is in the class.

a. CA b. AD

c. BC d. BD

Solution: When X then Y.  So When puneet is in the lab, then Rohit is in the class or Rohit is not in the class then Puneet is not in the lab. So AD is correct.

3. You will add more value to the brand if strategic planning is done.

A. Stratigic planning was done.

B. More value was not added to the brand.

C. More value was added to the brand.

D. Stratigic planning was not done.

a. BD b. DB

c. BC d. DC

Solution: If strategic planning was done then you added more value to the brand or you did not add more value then Strategic planning was not done. So AC or BD correct. So choice A.

4. She sleeps only when her boss is away from the office.

A. The boss is away B. She did not sleep.

C. She slept. D. The boss ins in the office.

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 slept means boss is away, or Boss is not away then She did not sleep.
Option A.

5. If Berty and Oly are selected in that order, Phil and Santhi cannot be selected.

A. Phil and Santhi are selected in that order.

B. Oly and Berty are selected in that order.

C. Berty and Oly are selected in that order.

D. Phil 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