Reasoning (Concepts): Syllogism

Reasoning (Concepts): Syllogism 

The concept of 'Complimentary Pairs' is very important in syllogisms.

Consider the statement: x ≥ y

Conclusions: 1. x > y   2. x = y

Now, if you seethe first conclusiononly, it may or may not be true; hence it is independently false.
And if you see the second conclusiononly, it may or may not be true; hence it is independently false as well.

But if you conclude that both the conclusions are false at the same time, then itwould imply x<y, which negates the given statement. Hence, the conclusion should be either 1 or 2 is true i.e. both cannot be false simultaneously. So conclusions 1 and 2 make a complimentary pair.

The following are the three conditions for the statements to be complementary to each other:

• The conclusions to be included in the complimentary pairs are those, which are not definite otherwise.
• The subject and the predicate should be same i.e. some x are not y or some x are z, cannot be included in the complimentary pairs.
• This is because the subject is same but the predicate is not the same.

The following three kinds of pairs can be included in complimentary pairs.

1.  ‘Some are .… NO’ 

Example 1:

Statements:
1. Some men are intelligent.
2. Some intelligent are honest.

Conclusions:
I. Some men are honest
II. No man is honest.

Solution: The statements can be depicted using Venn diagrams as:

From the above diagrams,it is clear that nothing can be definitely concluded about the positions of men and honest. So independently,each conclusion is false. But if we consider only the first conclusion i.e. some men are honest; it implies that men and honest will either partially overlap or all men will lie inside honest or all honest will lie inside men…As some means few or all.

In the second conclusion,it says that men and honest will not intersect each other at all. Now, both the conclusions cannot be false at the same time as there is no other possible position for the two circles representing honest and men.

Hence, with respect to the position of the two circles representing honest and men, there can only be two possibilities:

1. either they partially intersect or
2. not intersect at all. Hence one of the conclusions has to be true.

So the answer is either conclusion 1 or 2.

2. ‘Some not .… All’

Example 2:

Statements:
1. Some men are intelligent.
2. Some intelligent are honest.

Solution: 
Conclusions:
1. Some men are not honest.
2. All men are honest.

As discussed earlier, each conclusion is false, when taken independently. But if you consider only the first conclusion i.e. some men are not honest, it means that men circle will beat least partially outside the honest circle.

The 2^nd conclusion implies that all men are inside the honest circle. Now both the conclusions cannot be false simultaneously as there is no third possible position for the two circles honest and men. Hence, one of the conclusions has to be true. Therefore, the answer is either conclusion 1 or 2.

3. ‘Some are .… Some not’

Example 3:

Statements:
1. Some men are intelligent.
2. Some intelligent are honest.

Conclusions:
1. Some men are not honest
2. Some men are honest 

Solution: Again, each conclusion is false, when taken independently. But if we consider only the first conclusion i.e. some men are not honest, it means that men and honest will be at least partially non-intersecting or two absolutely different circles.

Further, the 2^nd conclusion implies that either the men circle will at least partially intersect with the honest circle. Now, both the conclusions cannot be false at the same time as there no third possible position for the two circles honest and men. Hence, one of the conclusions has to be true. Therefore, the answer is either conclusion 1 or 2.