Syllogisms

Questions on syllogisms contains only the following 4 types of statement:

1. The universal affirmative : Eg: All X∗ are Y’s
2. The universal negative  :  Eg: No X∗ is Y∗
3. The particular affirmative  Eg:  Some X are Y’s
4. The particular negative Eg: Some X are not Y∗’s

Here two statements are universal (1 and 2), and two statements are particular (3 and 4).
Two statements are positive (1 and 3) and two statements are negative (2 and 4).

Here Star marks indicates "Distribution".  If a term is distributed means It covers each and every element of it. All X are Ys means X ∈ Y, But Y need not be a subset of X. So Y does not have star mark.

You should commit to memory, how to put star marks and to distinguish positive and negative statements, universal and particular statements.

I.  The Universal Affirmative:   All Xs are Ys

It states that every member of the first class is also a member of the second class.  this proposition takes the form All Xs are Ys.  The possible diagrams for this proposition are:

Take a statement "All Tamilians are Indians".  It does not necessarily follows All Indians are Tamilians.  So Indians is not distributed on Tamilians.  But some times X may equal to Y.  For example, All Tamil speaking people are Tamilians.  Here barring a few exceptions, Both sets are equal.

The general diagram for Universal  Affirmative ‘All Xs are Ys’ is

Immediate inference: Some Ys are Xs, Some Y's are X's

II.  The universal Negative: No X is Y
It states that no member of the first class is a member of the second class.  This proposition takes the form - No X is Y.  The Eular’s circle diagram for this proposition would be Two mutually exclusive circles thus:

The general diagram for Universal Negative ‘No X is Y’ is

Immediate Inference: No Y is X, Some X's are not Y's, Some Y's are not X's

III.  The Particular Affirmative:  Some Xs are Ys

It states that at least one member, but never all, of the term designated by the class ‘X’ is also a member of the class designated by the term ‘Y’. This proposition takes the form Some Xs are Ys.  This possible diagrams as shown by the Euler’s circles for this proposition are:

Immediate Inferences: Some Ys are Xs

IV.  The particular Negative: Some Xs are not Ys.
It states that at least one member of the class designated by the term ‘X’ is excluded from the whole of the class designated by the term ‘Y’.  This proposition takes the form Some Xs are not Ys.  The Euler’s circle diagrams for this proposition are as follows.

The shaded portion in each is that part of X that is not Y.

Unlike the Particular Affirmative proposition, the exclusion of a part of X from Y does not give us any information about the inclusion or exclusion of the rest of the Xs. So we cannot say anything for certain about the rest of the Xs. In each of the above cases, the diagrams fulfill the condition ‘Some Xs are not Ys’. In diagram 1, even the rest of the Xs are not Ys. Equally possible are diagram 2, where the rest of the Xs are Ys, and diagram 3, where all Ys are Xs.
Thus, there cannot be any immediate inferences, because none of these cases are certainties.

Immediate Inferences: None

How to answer Syllogims:

There are two methods to answer syllogisms.  1. Euler venn diagram method 2. Aristotle's rules Method

Euler venn diagram method is difficult to follow if there are more particular statements (starting with "some") as we have to draw more diagrams to check in each case the conclusion is true. 

But Aristotle's method initially seems to be a bit difficult to understand, as one practices good number of questions, one can easily crack these questions.  

Aristotle's Rules to solve syllogisms: 

1.   A syllogism must contain 3 terms only
2.   If both the statements are particular, no conclusion possible
 (Explanation: Statements starting with "Some" are particular)
3.   If both the statements are negative, no conclusion possible
4.   If both the statements are positive, conclusion must be positive
5.   If one statement is particular, conclusion must be particular
6.   If one statement is negative, conclusion must be negative
7.   Middle term must be distributed in atleast one of the premises
(Explanation: Middle term is the common term between two given premises, and A terms is distributed means it must have the "star = *" mark above it)
8.   If a term is distributed in the conclusion, the term must be distributed in atleast one of the premises.
(Explanation: If any term is having star mark in the conclusion, it term must have star mark in the given premises)

Solved Example 1:

P1: All MBAs are Graduates
P2: All graduates are Students
I1: All MBAs are Students
I2: Some students are MBAs

Explanation:

P1: All MBA∗s are Graduates
P2: All Graduates∗ are Students

I1: All MBAs∗ are Students
I2: Some students are MBAs

Now Let us apply rules:
1.  It contained 3 terms only (MBAs, Graduates, Students)
2. Both statements are positive, conclusion must be positive
3. Common term is Graduate and it has star mark in the second statement

Conclusion 1: MBA in the conclusion has got a star mark so it must have star mark in atleast one of the premises.  MBA in P1 has got star mark.  It satisfied all the rules. It is valid conclusion
Conclusion 2: No term in the conclusion has got a star mark so no need to check anything.  It followed all the rules.  This statement is a valid conclusion.

Solved Example 2: 

P1: All Cats are Dogs
P2: No Dog is Fish
I1: No Cat is Fish
I2: Some Cats are Fish

Explanation:

P1: All Cats∗ are Dogs
P2: No Dog∗ is Fish∗
I1: No Cats∗ is Fish∗
I2: Some Cats are Fish

Now Let us apply rules:
1. It contained 3 terms only (Cats, Dogs, Fish)
2. P2 is negative, so conclusion must be negative. So I2 is ruled out, as rule says that one statement is negative conclusion must be negative
3. Common terms is Dog and it has star mark in both the premises

Conclusion 1: In the conclusion, both the terms Cat, Fish have star marks and These two terms have star marks in at least one of the premises.  So Conclusion 1 is valid
Conclusion 2: As one of the premises is negative, conclusion must be negative.  So this conclusion is not valid