Overview: Syllogism | CSAT Preparation - UPSC PDF Download

Syllogism, derived from a Greek term meaning "inference" or "deduction," is a critical aspect of logical reasoning. It tests a candidate's analytical and logical skills to determine relationships between given statements (premises), which may not necessarily align with universal truths. The task is to derive logical conclusions based on all possible cases or alternatives presented. A syllogism represents a logical argument where conclusions are drawn from one or more specific premises. In exams, these questions assess a candidate’s ability to deduce valid inferences from a set of given propositions.

Proposition

A proposition is a statement that expresses a specific relationship between two terms. Every proposition either affirms or denies something. It consists of three components: the subject, the predicate, and the copula.

• Subject: The subject refers to what the proposition is about.
• Predicate: The predicate describes or states something about the subject.
• Copula: The copula indicates the connection or relationship between the subject and the predicate.

Here are some examples of propositions:

• All men are animals — subject, copula, predicate
• No birds are mammals — subject, copula, predicate
• Some pens are erasers — subject, copula, predicate

Types of Propositions

Propositions can be classified into two main categories: universal and particular. Each type is further divided based on whether it affirms or denies a relationship.

• Universal Propositions: These propositions either entirely include or exclude the subject. They are further classified into:
  (a) Universal Affirmative Proposition (denoted by A): These assert complete inclusion of the subject in the predicate.
  Example: “All men are boys” is a universal affirmative proposition because the subject “All men” is fully included in the predicate.
  (b) Universal Negative Proposition (denoted by E): These assert complete exclusion of the subject from the predicate.
  Example: “No criminal is innocent” is a universal negative proposition because the subject "criminal" is entirely excluded from the predicate.
• Particular Propositions: These propositions only partially include or exclude the subject. They are divided into:
  (a) Particular Affirmative Proposition (denoted by I): These indicate partial inclusion of the subject in the predicate.
  Example: “Some boys are cute” is a particular affirmative proposition because it partially includes the subject "boys" by specifying “Some boys.”
  (b) Particular Negative Proposition (denoted by O): These indicate partial exclusion of the subject from the predicate.

“Some mandarins are not secretary” is an example of particular negative propositions because the subject “Mandarins” are partly excluded from the predicate.
The above facts can be summarised as follows:

Here, quantity represents whether the proposition is universal or particular and quality represents whether the proposition is affirmative or negative.

Venn Diagram or Euler’s Circle

Venn diagram or Euler’s circle is a method of pictorial representation of the propositions. These circles represent the relationship between subject and predicate stated in the given proposition.Have a look on the table given below in which four major type of propositions discussed earlier are represented by Euler’s circle with all possibilities.
Venn Diagram or Euler’s Circles and representation of the four propositions

Types of Premises and their Trivial Conclusions

The types of premises seen in syllogisms and their trivial (immediate) conclusions are as summarised in the table below

Following types of questions are asked under syllogism:

Statement and Conclusions Based Question

In these types of questions a set of statements (two, three or four) is given followed by a set of conclusions (two, three or four) on the basis of statements given the correctness of conclusion is checked or conclusions follow from the statement or not.
Example: Consider the following statements 
1. Some locks are keys. 
2. All keys are made from metal. 
3. Some metals are hard. 
Which of the following conclusion can be drawn from the above statement? 
(a) All locks are keys 
(b) All keys are hard 
(c) Some locks are hard 
(d) None of these
Ans: (d) 
Sol: Here, we make a Venn diagram from the statement.

So, from the above diagram, it can be concluded that none of the conclusion is correct.

Correctness of the Statements

Following questions consists of four statements, of these four statements, we have to check for the two statements which cannot be true but both can be false. Study the statements carefully and identify the two that satisfy the above condition.
Example: Examine the following statements 
1. All fruits are vegetables. 
2. Some fruits are not vegetables. 
3. Fruits are not vegetables. 
4. Some fruits are vegetables. 
Which of the following two statements cannot both be true?
(a) 1 and 2 
(b) 2 and 3 
(c) 1 and 3 
(d) 3 and 4
Ans: (c) 
Sol: The statement can be understood as follows 
1. All fruits are vegetables.

2. Some fruits are not vegetables or atleast one of the fruits is a vegetable

3. Fruits are not vegetables.
4. Some fruits are vegetables.

So, from above four statements only Statements 1 and 3 cannot be true.

Choosing Logically Related Statements

In these types of questions a set of six statements is given, out of which four combinations of three statements can be made, from these six statements we have to choose a set of statements or a combination in which third statement will logically follow the first two statements.
Example: The following question contains six statements followed by four sets of combinations of three. Choose the set in which the third statement is a logical conclusion of the first two. 
1. Men are batsmen. 
2. Boys are fielders. 
3. Boys are not batsmen. 
4. Some batsmen are bowlers. 
5. Boys are not bowlers. 
6. Some men are bowlers. 
(a) 123 
(b) 143 
(c)  231 
(d) 164
Ans: (d)
Sol: Here, note that a given statement can be looked at as a premise as well as conclusion. In such questions, consider each answer option at a time and verify where the third statement given in the option logically follows from the first two. Based on the outcome, mark or eliminate the answer options. Taking each answer option at a time, observe that only in option (d) the third statement follows the first two.

All men are batsmen and some men are bowlers. Hence, the men who are bowlers are also batsmen. So, there are some batsmen who are also bowlers. The Venn diagram for option (d) is shown is figure. Hence, the correct answer is option (d).

Checking Correctness of Set of Statements

In these type of questions, few set of statements are given from which we have to choose the correct set in which the third statement can be inferred from the first two statements.
Example: In this question, there are three sets of statements given. Select the set that is most logical i.e., the third statement can be concluded from the first two statements. 
1. All watermelons are green. All greens are healthy. All watermelons are healthy. 
2. All men have a business. Mohan has a business. Mohan is a man. 
3. All roses are fragrant. All flowers are fragrant. All flowers are roses.
(a) Only 2 follows 
(b) 1 and 3 follow 
(c) 1 and 2 follow 
(d) Only 1 follows
Ans: (d) 
Sol: Here, consider the first set and verify it. Based on the outcome, eliminate answer options. Now, based on the options left, verify the remaining sets.
Consider Set 1 Since, all watermelons are green, the set of watermelons is a subset of the set of green objects. Similarly, since all greens are healthy, the set of green objects is a subset of the set of the healthy objects. Thus, the set of watermelons is a subset of the set of healthy objects.
Therefore, all watermelons are healthy. So, set 1 is logical. Hence, option (a) can be eliminated.
Consider Set 2 All men have a business. Mohan also has a business. Thus, the set of men as well as the set of Mohan are subsets of the set of people having a business. Now, there can be a set of people who are not men but have a business. Mohan can clearly belong to this set. So, the Mohan may or may be a part of the set of men.
Thus, the conclusion does not follow. Therefore, set 2 is not logical. Hence, option (c) can be eliminated.
Consider Set 3 Using the same logic as for set 2, the set of flowers and roses may or may not overlap. Thus, the conclusion does not follow. Therefore, set 3 is not logical. Hence, option (b) can be eliminated.
Hence, the correct answer is option (d).