# Shortcut to solve syllogism by rules

## Day 1 Topic: Arrangement of Sentence (FIRST & LAST RULE)

Today we will discuss the basic methods that will help us in solving syllogisms in a fast and effective manner. The real rules will start from Day 3. But without the knowledge of the concepts that we will discuss today, you cannot move on to rules directly. So understand these concepts well.

Basic Terms used in syllogism that we will follow in our post

Sample example of sentence in Syllogism: Some A are B

• Entity/Term: The terms between which relation is given is referred as entity. Here A and B are entity.
• Relation: Some, All, No, Some Not are examples of relation.
• Proposition/Sentence: The statement from which we have to derive the conclusion. We can refer to it by statement also in general.
• Conclusion: The derived solution from the proposition.

The main concept of Syllogism: FIRST AND LAST RULE

Our main aim in syllogism is to arrange the given proposition in such a way that the entities which are common in the multiple propositions always come together. Let’s understand it by an example.

Example: Consider these three statements: But this may not be the case always. Some time in question, we may get propositions which are not arranged. We have to manually arrange them and then start solving the question. So we will use the FIST AND LAST RULE as our starting step in solving syllogism.

FIRST & LAST RULE: The propositions should be arranged in such a manner that, the two entities between which the conclusion is to be found out should come at the beginning and at the last while the common term should be in middle. i.e One should be come at the beginning and the other should be at the last. Example: In the above example 1: The entity at the beginning is A and at the last is D. So using these three statements together we can find the relation between A and D.
If we consider only statements b) and c) then the first entity is B and Last entity is D. So using these two propositions we can find the relation between B and D.
If we consider only statement a) and b) then the FIRST entity is A and the last entity is B, so using these two together we can find the relation between A and B.

We must note that in Example 1, the propositions are already arranged in the desired manner. But in reality it may not be arranged. We have to arrange them by using certain methods. We will discuss two methods to achieve this.

(i) Rearrangement of Proposition – Day 1
(ii) Reversal of Proposition – Day 2

(i) Rearrangement of Proposition

First identify the common terms. And arrange them accordingly. With non-common entity at FIRST and LAST and the common entity coming together as shown in example 1 (b) highlighted entity B,B and C,C comes together. Example: Now if we want to find relation between G and C, we will use just proposition (a) and (b)
If we want to find relation between F and Z, we will use the proposition (c) and (a)
For relation between F and C we will use all (c), (a) and (b).

For concept of rearrangement to work, one of the common entity should be at the last of the proposition and the other should be starting.

Example: All A are B. Some B are C. Here the common entity is B. It comes at the end in proposition 1 while at the start in Proposition 2.
Example: All A are B. Some A are C. Here the common term is A. But in both the proposition A is at the beginning. So in such case we cannot use the concept of rearrangement. The method to solve such a case will be discussed later.

More Examples:

Exercise 1) Arrange these propositions in such a way that common entity comes together:

(a) No tiger is lion
(b) Some elephant are tiger
(c) All lion are monkey

Solution: The common terms are tiger and lion. Arrangement is (b),(a),(c) i.e
Some elephant are tiger. No tiger is lion. All lion are monkey.

Exercise 2) Arrange these propositions in such a way that common entity comes together:

(a) Some red are blue
(b) All black are pink
(c) some blue are black
(d) No pink is violet

Solution:  (a),(c),(b),(d)
so we have now
(a) Some red are blue
(c) some blue are black
(b) All black are pink
(d) No pink is violet

Ques) Which propositions are required to get a relation between blue and violet.
Answer: (c),(b),(d) [FIRST and LAST RULE] – The two entity between which relation is to be found out should come at beginning and at last.

Ques) Which propositions are required to get a relation between red and pink.
Answer: (a),(c),(b) [FIRST and LAST RULE

LIMITATION of REARRANGEMENT

Consider this example: We will discuss the solution to this problem on Day 2.

## Exercise for the Day

Q1) Arrange the given statement according to the FIRST & LAST rule to find a relation between CONE and RECTANGLE.
(a) All cones are cylinders.
(b) No square is a rectangle.
(c) All cylinders are squares.

Correct Order is: (a)(c)(b)
Explanation
: As we have to find relation between CONE and RECTANLGE, they should come at the beginning and at the end. So start with the statement that has CONE in the start. i.e (a).  Now the common term cylinder is in (a) and (c). So next proposition is (c) followed by (b).

Q2) Arrange the given statement according to the FIRST & LAST rule to find a relation between GRAPE and SWEET
(a) No orange is apple.
(b) All grapes are oranges.
(c) Some apples are sweet.

(b) (a) (c)

Q3) Arrange the given statement according to the FIRST & LAST rule to find a relation between PRETTY and BOYS

(a) All girls are pretty.
(b) Some smart are girls.
(c) Some boys are smart.

(c) (b) (a)
Explanation
: Start with the proposition which has either BOY or PRETTY in it. We have (c) and (a) . But choose (c) because it has boy in the starting. It will be easy. While (a) has PRETTY at the end. (c) follow by (b) [due to common term smart] followed by (a) [] due to common term girls

Q4) Arrange the given statement according to the FIRST & LAST rule to find a relation between BUSSES and PLANE

(a) No plane is metro
(b) All busses are rails.
(c) Some rail is plane.

(b) (c)
Explanation
: Start with (b) it has BUSSES in starting. Common term rails is in (b) (c) SO we now have BUSSES and PLANE in starting and end. So FIRST LAST RULE is satisfied.

Q5) Arrange the given statement according to the FIRST & LAST rule to find a relation between SILVER and PLATINUM

(a) No gold is silver.
(b) Some platinum is gold.
(c) Some silver are diamond.