Syllogism (Logical Deduction)

Hello Readers,
Prakash Guru again provided us one of his magnificent post on "Syllogism (Logical Deduction)". So, here we are providing you the same. Don't forget to thank him!!!

I faced lots of problem in solving these problems until I went through the logic behind it.

Trivia: It was introduced by Aristotle.

One way to solve syllogism is the Venn-Diagram method. But the problem is that it is tedious, lengthy and complicated too. Hope everybody agrees with it. 

Hope the following note helps one and all!!

“Syllogism” is a deductive argument in which conclusion has to be drawn from two propositions referred to as premises.

1. Proposition/Premise = Quantifier+ Subject+ Copula+ Predicate.

• Quantifier: All, No, Some, Atleast, Atmost etc.
• All/No: Universal Quantifiers(as it refers to every object in a certain set)
• Some/Atleast/Atmost: Particular Quantifiers(as it refers to atleast one existing object in a certain set).

2. Subject: The subject is that about which something is said.

3. Predicate: The predicate is that part which is being affirmed /denied.

4. Copula: it denotes the relation between the subject and the predicate.

Example -

• All/Some etc. - Quantifier
• Men - Subject
• Are - Copula
• Animals - Predicate

Classification of Propositions:

• Universal affirmative proposition or Type- A: ex.  All Snakes are reptiles.
• Universal Negative Proposition or Type-E: ex. No boy is Intelligent
• Particular Affirmative or Type-I:  ex. some men are foolish.
• Particular Negative or Type-O : ex. some animals are not wild

Table 1: Summarizing the above propositions (Memorize this)

Statement form	Quantity	Quality	Distribution
All S is P	Universal	Affirmative	S only
No S is P	Universal	Negative	Both S and P
Some S is P	Particular	Affirmative	Neither S nor P
Some S is not P	Particular	Negative	Only P
S: Subject; P= Predicate

Table 2: Rule of Conversion: (Memorize this)

Statement form	Valid Conversion/ Conclusion	Examples
All S is P	Some P is S	Statement: All men are fools Valid Conclusion:  some fools are men
No S is P	No P is S	Statement: No men are fools Valid Conclusion:  No fools are men
Some S is P	Some P is S	Statement: Some men are fools Valid Conclusion: Some fools are men
Some S is not P	No valid Conversion	Statement: Some men are not fools Valid Conclusion:----
S: Subject; P= Predicate

Syllogism is concerned with 3 terms:

• Major Term: It is the predicate of the conclusion and is denoted by “P”.
• Minor term: It is the subject of the conclusion and is denoted by “S”.
• Middle term: It is the term common to both the premises and is denoted by “M”.

An Example to make you clear about the terms:

Premises:

1. All dogs are animals

2. All Tigers are dogs.

Conclusion:

All tigers are animals

Here,

Animals: is the predicate of the conclusion and hence the major term.

Tigers: is the subject of the Conclusion and hence the minor term.

Dogs: is common to both the premises and hence termed as the middle term

Note: Of the two premises, the major premise is that in which the middle term is the subject and the minor premise is that in which the middle term is the predicate. 

SO GUESS YOURSELF IN THE ABOVE PREMISES WHICH IS THE MAJOR AND WHICH IS THE MINOR ?? 

P.S: If you find the answer you have studied this well, if not read once again!!

Rules for deriving the conclusion from 2 premises:

(MAJOR PART-READ CAREFULLY AND MEMORIZE ALL THE CONDITIONS)

a. The conclusion should not contain the middle term.
Example - 
Premises
1. All mens are girls
2. Some girls are students.

Conclusion: Some girls are men ; all girls are men.
Explanation: In the above example neither of the conclusion is valid, as the middle term i.e. girls is present in both the conclusions.

b. No term can be distributed in the conclusion unless it is distributed in the premises.
Example -
Premises:
1. Some dogs are goats
2. All goats are cows.

Conclusions: All cows are dogs; some dogs are cows.
Explanation - Here, in conclusion 1 the term “Cow “is distributed but it is not distributed in the premise i.e premise-2 since it is a A-type proposition and the term “Cow” to be distributed must be the “subject” (Refer table-1) of the proposition.

c. The middle term (M) should be distributed at least once in the premises, otherwise no conclusion follows.

Statement type	Distribution Condition
A	"M" must be Subject
E	"M" must be Subject/Predicate
O	"M" must be predicate
I	Cannot be distributed/Not distributed

Ex.:

Premises:

1. All fans are watches

2. Some watches are black

In the above premises the middle term is “watches”, since it is not distributed in both premises(refer above table for distribution conditions) no conclusion can be drawn except the conversions of the premises.

d. No conclusion follows:

If both premises are particular i.e I type.

Ex.: 

1. Some balloons are flowers

2. Some flowers are petals

(No conclusion can arise from the above two premises other than the conversions).

If both premises are negative i.e. E-type.

Ex,

1. No ship is a boat

2. No boat is a vessel

(No conclusion can be deducted from the above statements, other than their conversions).

If major premise is particular and minor is negative

Ex.

1. Some dogs are bulls

2. No tigers are dogs

(Here “dogs” is the middle term and it is present in the subject part of the first premise which makes it the major premise, thus premise 2 is the minor and it is negative, thus as per the condition no conclusion can arise from the premises other than their conversions.)

e. If the middle term is distributed twice, the conclusion cannot be Universal.

Ex.

1.All fans are chairs

2. No tables are fans.

( As the middle term “fans” is distributed twice in the above premises, the conclusion cannot be universal i.e “No chairs are tables”,  rather the conclusion will be “Some chairs are not tables” i.e particular negative-O type)

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

Ex.

1. All grasses are trees

2. No tree is a shrub

Conclusion: Here the conclusion that will arise from the above premises is “No grass is a shrub” as the middle term tree  is only distributed in the second premise and one premise is negative).

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

Ex. 

1. Some boys are thieves

2. All thieves are dacoits

(Here the conclusion must be particular i.e “some boys are dacoits”, since the middle)

h. If both premises are universal affirmative, the conclusion should be universal affirmative.

Ex,

1. All womens are mothers

2. All mothers are sisters.

(Here the conclusion must be “All womens are sisters”)

The above conditions when applied on syllogism problems, makes the problems easy. Try it yourself.

What is a complementary pair?

Sometimes there is an option like “either 1 or 2 follows”. This is because of a complementary pair in the conclusions.

The pairs that make a complementary pair are I-O type, A-O type or I-E type given that the subject and predicate remains same in both the statements.

To explain this lets consider the following examples:

• Pair I:  i.) All gardens are bulbs  ii.) Some gardens are not bulbs
• Pair II: i.) Some gardens are bulbs  ii.) Some gardens are not bulbs
• Pair III : i.) Some gardens are bulbs ii.) No gardens are bulbs.
• Pair IV: i.) All gardens are bulbs  ii.) No gardens are bulbs
• Pair V:  i.) All gardens are bulbs  ii.) Some bulbs are not gardens.

Explanation: Pairs I, II and III make a complementary pair each while Pairs IV and V don’t. In pair-I the first statement is of type A and the second is of type O and thus an A-O type pair makes a complementary pair.

Similarly pair II and Pair III makes I-O and I-E type of pair respectively thus it makes a complementary pair. 

Pair IV is an A-E type of pair and thus it do not make a complementary pair.

Pair V is and A-O type pair but still do not make a complementary pair, since the subject and predicate are jot same in both the statements.

Some Important Tips:

• Practice as many problems as possible and you will automatically remember the logics that I have shared.
• Remember always about the conversions, because maximum times the conclusions are in the form of conversions only.
• Try all possibilities, as from two conclusions or from one conclusion and another premise another conclusion can be obtained.
• Refer books like BSC and Dr. R.S Aggarwal and practice problems
• Solve Quiz given by BA on syllogism, because they are really good.
• Once you do a lot of problems I am sure you needn’t even write the problems you will just look at the premises/proposition and derive all the possible conclusions. 

Now let us decipher the quiz put by BA on 17/9/2014

1. Statements: 

All buckets are mugs.  

All lunch boxes are pencil boxes.  

Some pencil boxes are mugs.  

Some buckets are not drums.

Conclusions: 

I. All mugs are lunchboxes.  

II. All lunchboxes are buckets.

III. Some lunchboxes are mugs. 

IV. Some drums are not mugs.

(1) Only III follows   

(2) Only I and III follow

(3) Only II and III follow  

(4) None of these

(5) Only II follows 

Solution with explanation: Here, premise 2 and premise 3 have pencil boxes as the middle term, but it is not distributed in any of the premises since premise 2 is A-type, so the middle term must be the subject but it is in the predicate, and premise 3 is I-type, so middle term cannot be distributed, thus no conclusion can arise from the statements. In premise 1 and premise 4 the middle term is buckets and is also distributed in one premise that is premise 1 but the predicate if different, thus no conclusion follows. So, now check the conclusions for a complementary pair or a conversion of the any statements, and a careful analysis shows nothing such. Thus the answer is 4-None follows.

2. Statements: 

Some chairs are tables.  

Some desks are benches.

All benches are tables.  

Some woods are not desks.

Conclusions: 

I. Some benches are desks.  

II. Some tables are benches.

III. Some woods are not benches. 

IV. Some desks are tables.

(1) Only III follows   

(2) Only I and III follow

(3) Only I, II and IV follow   

(4) Only II and IV follow  

(5) None of these

Solution with explanation: Here we can only compare, premise 2 and premise 3. In the said premises the middle term is benches and it is distributed in premise 3 and one premise is particular so the conclusion shall be particular and without the middle term i.e “Some desks are tables” which matches the conclusion IV and there is no complementary and conclusion I and II are conversions of premise 2 and premise 3. Thus Only I,II and IV follows-that is option 3.

3. Statements: 

No bank is a market.  

Some markets are offices.

All restaurants are offices.  

Some banks are rooms.

Conclusions: 

I. Some markets are not rooms. 

II. Some offices are not banks.

III. Some restaurants are markets. 

IV. Some rooms are restaurants.

(1) Only I and II follow  

(2) Only II follows

(3) Only I, II and III follow   

(4) Only I and III follow  

(5) None of these

Solution with explanations: See premise 1 is negative and premise 2 is particular and the middle term “markets“ is distributed in premise 1 thus the conclusion shall be particular negative without the middle term i.e “some offices are not banks”, now premise 1 and premise 4 have the same conditions, middle term being “banks” so conclusion shall be particular negative without the middle term i.e “some rooms are not market” but conclusion 1 is “some markets are not rooms” which is not similar as there is no valid conversion for a particular negative statement. So only conclusion II follows- option 2.

4. Statements: 

All ones are twos.   

Some threes are fours.

All three are ones.   

All fives are fours.

Conclusions: 

I. Some fives are threes.  

II. Some ones are fives.

III. Some twos are fives.  

IV. Some twos are fours.

(1) Only I and IV follow  

(2) Only IV follows

(3) Only II and IV follow  

(4) None of these

(5) Only I follows  

Solution with explanation: From premise 2 and 3 we get “some fours are ones” now making this as a premise and comparing with premise 1 i.e “All one are twos” we get  “some fours are twos”, the conversion being “some twos are fours” which is conclusion IV, thus only Conclusion IV follows-option 2 is your answer. 

5. Statements: 

Some ice-creams are cakes.  

All cakes are biscuits.

Some biscuits are parles.  

Some parles are toffees.

Conclusions: 

I. All ice-creams are biscuits. 

II. Some ice-creams are biscuits.

III. Some toffees are biscuits. 

IV. Some cakes are biscuits.

(1) Only I and III follow  

(2) Only I and II follow

(3) Only II follows   

(4) Only II and IV follow  

(5) None of these

Solution with Explanation: Comparing Premise 1 and 2 we get the conclusion as “Some ice-creams are biscuits” with is conclusion II. Now statement 1’s conversion is “Some biscuits are cakes”, the conversion of this conversion is “Some cakes are biscuits” which is conclusion IV thus only II and IV follows.- option 4 is your answer