## Premises - Various Types

Broadly, premises can be classified into 4 types:

1. All A’s are B’s: It can also be stated as “No A is not B” which means that there exists no such element of A which does not belong to B which implies All A’s are B’s.

2. Some A’s are B’s

3. No A is B:  It also means that “No B is A”.

4. Some A’s are not B’s: It can also be stated as “All A’s are not B’s” which means that there are elements of A which do not belong to B.

Now let’s see how to represent all the 4 types using Venn Diagrams.

### Type 1: All A's are B's

There are two possible Venn Diagrams for this case:

In this case, there are some elements of B which do not belong to A. So, the circle representing A lies inside the circle representing B.

In this case, All the elements of B are also the elements of A, that is, All B’s are A’s as well. So, the circles representing A and B coincide.

We can conclude the following from this premise:

(a) Some B’s are A’s: This is a definite conclusion which is true for both the cases.

(b) Some B’s are not A’s: This is a probable conclusion which is valid only when the conclusion “All B’s are A’s” is not true.

Let’s practice what we have learnt. In the question below three statements are given and these statements are followed by two conclusions numbered I, II and III. You have to take the given statements to be true even if they seem to be at variance from commonly known facts. Read the conclusions and then decide which of the given conclusions logically follows from the given statements, disregarding commonly known facts.

Statements:

All burgers are noodles.

All noodles are pizzas.

All pizzas are sandwiches.

Conclusions:

I. All noodles are sandwiches.

II. All pizzas are burgers.

III.  All burgers are sandwiches.

Hint: First draw the Venn diagram representation of the statements and then try to identify the correct conclusions.

The first step to solve this problem is to draw the Venn diagram representation of the statements given which is as shown:

Now let’s see which of the conclusions are correct:

I. All noodles are sandwiches.

Since the circle representing noodles lies inside the circle representing sandwiches, this conclusion follows.

II. All pizzas are burgers.

This is not correct. The correct conclusion would be “Some pizzas are burgers”.

III. All burgers are sandwiches.

Since the circle representing burgers lies inside the circle representing sandwiches, this conclusion follows.

Therefore, both I and III are the correct conclusions.

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