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Syllogism

In Syllogism, we study the given statements in order to substantiate the derived conclusions.
 
The evidence provided to substantiate the con-clusions are known as premises and that which is drawn on the basis of the premises is a conclusion. Thus Syllogism can be understood to be a piece of reasoning providing ‘relational arrangement’ between premises and conclusions.
 
Let us first understand the various terms involved:
 
Statement 1: Some of the Indians are men.
 
Statement 2: All men have the potential to be good.
 
Conclusion: Some Indians have the potential to be good.
 
In the above given statements, statement 1 and statement 2 are premises and conclusion is to be verified on the basis of the premises given. It is also to be noted that:
  • Propositions and statements are not the same thing.
  • While finding out the conclusion, we should not be concerned with the vocabulary of the terms involved in the statements. Rather we should treat them in isolation, without taking their literal meaning into account. To understand it better, if it is given that ‘All boys are good’, we cannot derive any relationship between boys and bad (opposite of good), unless some association between bad and good is given in the original question.

Types of Statements and the Ways to Represent them

Famous European philosopher Aristotle and other classical logicians divided the categorical statements into four types:
  • Universally Affirmative Statement
     
    Whenever we say that “All girls are good”, we are simply accepting the relationship between the two entities ‘girls’ and ‘good’ in such a way that anything or anybody who is a girl, has to be good. We may assume that there is such a thing called girls. But the meaning of the above premise does not depend upon the assumption that girls exist. These kinds of statements are known as Universally Affirmative Statements because they give the impression of the statement being universally true. Like all the girls present anywhere can be anything else, but simultaneously have to be good also. And there cannot be any exception to this rule.
     
    Examples:
     
    All goats are animals.
     
    All actresses are beautiful.
     
    All prime numbers are natural numbers.
     
    Universally affirmative statements can be also rephrased using the word ‘only’.
     
    ‘All goats are animals’ can be written as – only animals are goats.
     
    Or, ‘All actresses are beautiful’ can be written as – only the beautiful are actresses.
     
    Or, ‘All prime numbers are natural numbers’ can be written as – only natural numbers are prime numbers.
  • Universally Negative Statement
     
    Whenever we say that, “No man is perfect”, we are simply accepting the relationship between two entities ‘man’ and ‘perfect’ in such a way that anything or anybody who is a man cannot be perfect. We may assume that there is such a thing called a man. But the meaning of the above premise does not depend upon the assumption that man exists. These kinds of statements are known as Universally Negative Statements because they give the impression of the statement being universally true. Like all the men present anywhere might not be anything else too, but simultaneously they cannot be perfect also. And there cannot be any exception to this rule.
     
    Examples:
     
    No Indian is a coward.
     
    No prime number is a fraction.
     
    No dream is unachievable.
  • Particular Affirmative Statement
     
    Whenever we say that, “Some movies are boring”, we are simply accepting the relationship between two entities ‘movies’ and ‘boring’ in such a way that some of the movies have to be boring. It also means that all the movies cannot be not-boring. We may assume that there is such a thing called ‘movies’. But the meaning of the above premise does not depend upon the assumption that movies exist. These kinds of statements are known as Particular Affirmative Statements because they give the impression of the statement being true in some particular cases and not in all the cases. Hence, it falls short of being a universal fact or a universally affirmative statement.
     
    Example:
     
    Some dogs are rich.
     
    Some people are happy.
     
    Some bosses are stupid.
  • Particular Negative Statement
     
    Whenever we say that, “Some numbers are not integers”, we are simply accepting the relationship between two entities ‘numbers’ and ‘integers’ in such a way that some of the numbers have to be non-integers. We may assume that there is such a thing called ‘numbers’. But the meaning of the above premise does not depend upon the assumption that numbers exist. These kinds of statements are known as Particular Negative Statements because they give the impression of the statement being true in some particular cases and not in all the cases.
     
    Example:
     
    Some scorpions are not honest.
     
    Some managers are not effective.
     
    Some relationships are not manageable.

Representing the Statements and Standard Deductions

Normally, the statements given above construct a major part of any question in a syllogism. To represent these, we can either apply the subject-predicate form or the Venn-diagram method. We will see both these methods one by one:
  • Universally Affirmative Statement
     
    Consider the example–All A are B. Subject-predicate form: These types of statements are known as ‘A-type’ statements. Venn-Diagram form: In the Venn-diagram form, we can represent A-type statements in the following ways:
     
     
    In the above form, A is included inside B. And obviously set B is bigger than set A.
     
     
    In the above form, set A and set B are of the same size.
     
    Following are the deductions which can be made from the above given statement:
    1. Some A are B.
    2. Some B are A.
The given deductions are definitely true, however, we can derive some more ‘probably true’ deductions from the above statement. For example—‘Some B are not A’ is probably a true statement. And similarly, ‘some A are not B’ is a definitely false statement.
  • Universally Negative Statement
     
    Consider the example–No A are B.
     
    Subject-predicate form: These types of statements are known as ‘E-type’ statements.
     
    Venn-Diagram form:
     
     
    Following are the deductions which can be made from the above given statement:
    1. No B are A.
    2. Some A are not B.
    3. Some B are not A.
The above given deductions are definitely true.
  • Particular Affirmative Statement
     
    Consider the example–Some A are B.
     
    Subject–Predicate form: These types of statements are known as ‘I-type’ statements.
     
    Venn-Diagram form:
     
     
    Following are the deductions which can be made from the above given statement:​
    1. Some B are A.
The above given deduction is definitely true, however we can derive some more ‘probably true’ deductions from the above statement. E.g., ‘Some B are not A’ is probably a true statement.
  • Particular Negative Statement
     
    Consider the example–Some A are not B.
     
    Subject-Predicate form: These types of statements are known as ‘O-type’ statements.
     
    Venn-Diagram form:
     
     
    The following are the deductions which can be made from the above given statement:
    1. All A are not B.
However, we cannot make any deduction of A, E, I or O format from this statement.
 
Summarizing the whole discussion till now, we can have the following conclusions drawn:
 
 
Affirmative
Negative
Universal
All (A)
No (E)
Particular
Some or Many (I)
Some not or Many not (O)


We can see the summary of all the standard deductions in a table format also:
 
Given statement
Deduction
Truth-metre
Summary
All
A are B
Some A are B
Definitely True
‘All’ can give only ‘Some’ as definitely true statement.
Some B are A
Definitely True
Some B are not A
Probably True
Some A are not B
Definitely False
Some
A are B
Some B are A
Definitely True
‘Some’ can give only ‘some’ as definitely true statement.
Some B are not A
Probably True
Some A are not B
Probably True
Some
A are not B
No ‘Definitely True’ deduction possible
Some B are not A
Probably True
 
No A are B
No B are A
Definitely True
‘No’ can give only ‘No’ or ‘Some + Not’ as definitely true statement.
Some A are not B
Definitely True
Some B are not A
Definitely True


Remember that
  • No positive statement can give rise to any negative definitely true conclusion.
  • No negative statement can give rise to any positive definitely true conclusion.
Besides the standard AEIO statements, there are a few more statements which are used in Syllogism:
 
Given statement
Deduction
Truth-meter
Summary
Only A
are B
All B
are A
Definitely True
All the conclusions related to the Some format will be true.
Some A are B
Definitely True
Some B are A
Definitely True
All A is not B
Some A are not B
Definitely True
 
Some B are not A
Probably True
 


Deductions of two or more than two statements of AEIO type together:

 

First Statement Type type
Second Statement type
Possible ‘definitely true’ conclusions
Not possible
All
Some
Some
No/Some Not
All
All/Some
No/Some not
No
Some not/No
Some All
Some not
No conclusion is possible
Some
All
Some
All/Some not/No
Some
No such conclusion possible
No
Some not
All/No/Some
Some not
No conclussion is possible
No
All
So/Some not
All/Some
Some
Some not
All/No/Some
No
No/Some not
All Some
Some not
No conclusion is possible
Some not
All
No conclusion is possible
Some
Some not
No




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