Rules for deriving conclution from two given premises

In our previous post we discussed about the concept of proposition and logical deduction techniques. Here we discuss about the important rules for deriving conclusion from two given premises.

RULES FOR DERIVING CONCLUSION FROM TWO GIVEN PREMISES:

1. The conclusion does not contain the middle term.

Example

Statements:

1. All men are girls.

2. Some girls are students.

Conclusions:

1. All girls are men.

2. Some girls are not students.

Since both the conclusions 1 and 2 contain the middle term 'girls', so neither of them can follow

2. No term can be distributed in the conclusion unless it is distributed in the premises.

Example

Statements:

1. Some dogs are goats.

2. All goats are cows.

Conclusions:

1. All cows are goats.

2. Some dogs are cows.

Statement 1 is an I-type proposition which distributes neither the subject nor the predicate.

Statement 2 is an A type proposition which distributes the subject i.e. 'goats' only.

Conclusion 1 is an A-type proposition which distributes the subject 'cow' only since the term 'cows' is distributed in conclusion 1 without being distributed in the premises, so conclusion 1 cannot follow.

3. The middle term (M) should he distribute at least once in the premises. Otherwise, the conclusion cannot follow.

For the middle term to be distributed in a premise

(i) M must be the subject if premise is an A proposition.

(ii) M must be subject or predicate if premise is an E proposition.

(iii) M must be predicate if premise is an O proposition.

Note that in an I proposition, which distributes neither the subject nor the predicate, the middle term cannot be distributed.

Example

Statements:

1. All fans are watches.

2. Some watches are black.

Conclusions:

1. All watches are fans.

2. Some fans are black.

In the premises, the middle term is 'watches'. Clearly, it is not distributed in the first premise which is an A proposition as it does not form its subject. Also, it is not distributed in the second premise which is an I proposition. Since the middle term is not distributed even once in the premises, so no conclusion follows.

4. No conclusion follows

(a) if both the premises are particular

Example

Statements:

1. Some books are pens.

2. Some pens are erasers.

Conclusions:

1. All books are erasers.

2. Some erasers are books.

Since both the premises are particular, so no definite conclusion follows.

(b) If both the premises are negative.

Example

Statements:

1. No flower is mango.

2. No mango is cherry.

Conclusions:

1. No flower is cherry.

2. Some cherries are mangoes. Since both the premises are negative, neither conclusion follows.

(c) If the major premise is particular and the minor premise is negative.

Example

Statements:

1. Some dogs are bulls.

2. No tigers are dogs.

Conclusions:

1. No dogs are tigers.

2. Some bulls are tigers.

Here, the first premise containing the middle term 'dogs' as the subject is the major premise and the second premise containing the middle term 'dogs' as the predicate is the minor premise. Since the major premise is particular and the minor premise is negative, so no conclusion follows.

5. If the middle term is distributed twice, the conclusion cannot be universal.

Example

Statements:

1. All fans are chairs.

2. No tables are fans.

Conclusions:

1. No tables are chairs.

2. Some tables are chairs.

Here, the first premise is an A proposition and so, the middle term 'fans' forming the subject is distributed. The second premise is an E proposition and so, the middle term 'fans' forming the predicate is distributed. Since the middle term is distributed twice, so the conclusion cannot be universal.

6. If one premise is negative, the conclusion must be negative.

Example

Statements:

1. All grasses are trees.

2. No tree is shrub.

Conclusions:

1. No grasses are shrubs.

2. Some shrubs are grasses.

Since one premise is negative, the conclusion must be negative. So, conclusion 2 cannot follow.

7. If one premise is particular, the conclusion must be particular.

Example

Statements:

1. Some boys are thieves.

2. All thieves are dacoits.

Conclusions:

1. Some boys are dacoits.

2. All dacoits are boys.

Since one premise is particular, the conclusion must be particular. So, conclusion 2 cannot follow.

8. If both the premises are affirmative, the conclusion must be affirmative.

Example

Statements:

1. All women are mothers.

2. All mothers are sisters.

Conclusions:

1. All women are sisters.

2. Some women are not sisters.

Since both the premises are affirmative, the conclusion must be affirmative. So, conclusion 2 cannot follow.

9. If both the premises are universal, the conclusion must be universal.

Complementary pair:

A pair of contradictory statements i.e. a pair of statements such that if one is true, the other is false and when no definite conclusion can be drawn, either of them is bound to follow, is called a complementary pair. E and I-type propositions together form a complementary pair and usually either of them follows, in a case where we cannot arrive at a definite conclusion, using the rules of syllogism.