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Approaching a Question

Till now we have seen two forms of representing the statements—(i) Subject-predicate form and (ii) Venn Diagram form. Each of these two methods have their own pros and cons. However, in my viewpoint, each and every question of syllogism is nothing but the juggling of words; and the only skill required to excel in syllogisms is having a clear understanding of the variety of statements used.
 
From here, we will pick up the pros of both the above mentioned forms and generate a three step hybrid method of solving the questions using the above given table.
 
Step 1: Find if the statement is affirmative or negative.
 
Step 2: Eliminate the option with a negative connotation if the statements given are affirmative and vice versa.
 
Step 3: Try to eliminate the remaining options by using the given methods. Start with the negative approach—try to prove that the given conclusion is not true and show in at least one way that the particular conclusion can be wrong. The conclusion which cannot be shown wrong in any way is definitely true.
 
Example 1:
 
Question-1
Statements:
  1. Some cars are houses.
  2. All houses are guys.
  3. All guys are cows.
Conclusions:
  1. Some cows are cars.
  2. Some cows are houses.
  3. Some cows are guys.
  4. Some guys are not cars.
Which of the above given conclusions is/are definitely true?
Solution
Let us verify the conclusions one by one.
 
Conclusion (a) Some cows are cars.
 
Using any two statements only, we cannot have any relationship between cows and cars. To find out the relationship between cows and cars, we need to use any two statements first and then using the conclusion derived from here, we will find out the relationship between cows and cars.
 
Using statement (ii) and statement (iii), All houses are guys → All guys are cows → All houses are cows. ......(iv)
 
Middle term used here is guys. So, we will use the statements where ‘guys’ is present. After combining both the statements, the middle term will get eliminated and the terms ‘guys’ and ‘cows’ will be left.
 
We can see this through a venn diagram also:
 

 
Using statement (i)
 
Some cars are houses → Some houses are cars.
 
Combining statements (i) and (iv), since all the houses are cars, so definitely anything which is a house has to be cows. Now, some houses are cars–and anything which is a house is a cow–so some cars are cows.
 
This can be seen through a Venn diagram also:
 
 
 
Conclusion (b) Some cows are houses.
 
To have a relationship between ‘cows’ and ‘houses’, we need to use all the three statements.
Using statement (ii) and statement (iii), All houses are guys → All guys are cows → All houses are cows. ......(iv)
 
Using (iv) and (i), Since all the houses are cows and some houses are cars, so obviously some cows are cars. And this fact that ‘some cows are cars’ cannot be refuted in anyway. It can be observed through the above given venn diagram also.
 
So, conclusion (b) is also definitely true.
 
Conclusion (c) Some cows are guys.
 
Using statement (c), ‘some cows are guys’ is a definitely true conclusion.
 
Conclusion (d) Some guys are not cars.
 
Since the given statements are affirmative, and the conclusion given is negative, so it cannot be a valid conclusion.
 
Hence, conclusion (a), (b) and (c) are definitely true.
 
 
Example 2 :
 
Question-2
Statements:
  1. Some panthers are cats.
  2. All cats are animals.
  3. Some animals are not panthers.
Conclusions:
  1. Some animals are panthers.
  2. All panthers are animals.
  3. All cats are panthers.
  4. All animals are panthers.
Which of the above given conclusions is/are definitely true?
 
Solution
As we have seen that statements like “Some A are not B” cannot contribute anything, so whatever conclusions are coming have to come from the first two statements.
 
Now understand the first two statements:
 
All cats are animals—Anything and everything which is a cat has to be an animal also. ‘Cat’ can be simultaneously ‘dog’ or ‘god’ or anything else too, but it has to be an animal also.
 
Some panthers are cats → Some cats are panthers
 
As we have discussed above, anything which is a cat has to be an animal also and some cats are panthers, hence, we can conclude that some animals are panthers.
 
Now look at the conclusions:
 
Conclusion (a)—It is obviously true.
 
Conclusion (b)—Using statement (b) and statement (c) together, we can have the conclusion only in ‘Some’. Since conclusion (b) has a statement in ‘ALL’, it is not a valid conclusion.
 
On the same grounds, conclusion (c) and conclusion (d) are invalid conclusions.
 

Example 3 :
 
Question-3
Statements:
  1. Some panthers are cats.
  2. No panther is a river.
  3. All rivers are roads.
Conclusions:
  1. No cat is a river.
  2. Some roads are rivers.
  3. Some cats are not a river.
  4. Some rivers are not a cat.
Which of the above given conclusions is/are definitely true?
 
Solution
We will evaluate each of the conclusions one by one:
 
Conclusion (a)—The relationship between ‘cat’ and ‘river’ can be established only through statement I and statement II. Looking upon the nature of statements (a) and (b), we can say that the conclusion can only be in “Some + Not”, and not in “No”. Hence, it is an invalid conclusion.
 
Conclusion (b)—The relationship between ‘roads’ and ‘rivers’ can be established through statement (iii). Since ‘all rivers are roads’, so it can be concluded that ‘some roads are rivers’.
 
Conclusion (c)—The relationship between ‘river’ and ‘cats’ can be established using statement i and statement ii. The conclusion has to be of the type “Some + Not”.
 
Let us understand the statements:
Statement (i) Some panthers are cats - At least one panther has to be there which is a cat. We cannot comment about the remaining panthers if they are cats or not.
 
Statement (ii) No panther is a river – Anything cannot be a panther and a river simultaneously.
 
Now use both the statements: Statement (a) tells us that at least one panther is a cat or vice versa, and statement ii tells us that If anything is a panther, then that cannot be a river and vice versa. Hence, we can conclude that at least one cat is there which is not a river. So, it can be validly concluded that ‘Some cats are not a river.’
 
Conclusion (d)—Prima facie, it appears that conclusion (d) is true. And if we see this conclusion in the absence of conclusion (c), it appears definitely true, however, it is only probably true.
 
To evaluate this conclusion, we should ideally take the recourse of a venn-diagram:
 

 
Obviously, all the rivers are cat in the above given diagram. So, conclusion (d) cannot be definitely true.
 
Hence, conclusion (b) and conclusion (c) are definitely true.
 




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