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The Euler’s Circles or Venn Diagrams

There is a pictorial way of representing the propositions, formulated by Euler, an ancient mathematician. Suppose that the proposition is trying to relate the subject (S) with the predicate (P). Then, there are four ways in which the relation could be made according to the four propositions.
  1. Diagrammatic representation of ‘A’ proposition.
     
    ‘All cows are animals’ will be shown as:
     
     
    In ‘A’ proposition, the subjective term is fully included in the predicative term. But the predicative term cannot be fully included in the subjective term. It implies that the subjective term is distributed and the predicative term is not distributed.
  2. Diagrammatic representation of ‘E’ proposition.
     
    ‘No truck is a bus’ is represented as:
     
    Description: 110353.png
     
    In ‘E’ proposition, the subjective term and the predicative terms are mutually exclusive; they have nothing in common with each other. This implies that the subjective term refers to all the things for which it stands and the predicative term also refers to all the things which it denotes. The diagrammatic representation will be two independent circles.
  3. Diagrammatic representation of ‘I’ proposition.
     
    ‘Some candles are pencils’ is represented as:
     
    Description: 110388.png
     
    In ‘I’ proposition, both the terms are undistributed i.e., they do not refer to all the things for which they stand.
     
    In the above example, only the shaded part of the intersecting circles represent ‘I’ proposition. Thus, neither of the terms is distributed.
  4. Diagrammatic representation of ‘O’ proposition.
     
    ‘Some fruits are not vegetables.’
     
    Description: 110394.png
     
    The subjective term does not stand for ‘all’ but the predicative term refers to all because part of the subjective term referred to in the statement has nothing to do with the predicative term.
Based on the above analysis, we can tabulate the concepts as in the following table.
 
If the Type of the Given Proposition is
Then its Diagrammatic Representation is
‘A’
All S are P.
Always
Description: 110655.png
‘E’
No S are P.
Always
Description: 110649.png
‘I’
Some S are P.
Either Description: 110644.png Some S are P.
(Some S are not P.)
or, Description: 110980.png Some S are P.
(All S are P.)
or, Description: 110989.png Some S are P.
(All P are S.)
‘O’
Some S are not P.
Either
Description: 110615.png Some S are not P.
(Some S are P.)
or, Description: 110607.png Some S are not P.
(All P are S.)
or, Description: 110599.png Some S are not P.
(No S are P.)





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