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Philosophy 103: Introduction to Logic
Venn Diagrams of Standard Form Categorical Propositions

Abstract: The technique of representing statements by means of pictures is explained.

I. One way to view the "logical geography" of the standard-form categorical propositions is to use diagrams invented by John Venn, a friend of Lewis Carroll.
A. Perhaps, you have been introduced to diagrams used in set theory; the Venn Diagrams are somewhat different.
B. Most descriptions of Venn Diagrams introduce the three symbols used as follows.
1. An empty circle is used to represent a subject class or a predicate class and is generally so labeled with an S or a P. Putting the name of the actual subject or predicate class next to the circle is preferred. The area inside the circle represents members of the class in question, if there are any. The area outside the circle represents all other individuals (the complementary class) if there are any. Note that the label "things" is written outside the circle, even though "things," if there are any, would be inside the circle. things.gif (1122 bytes)
2. Shading or many parallel lines are used to indicate areas which are known to be empty. I.e., there are no individuals existing in that area. E.g., the diagram to the right represents the class of "Yeti." shading.gif (1164 bytes)
3. The third symbol used is an "X" which represents "at least one" or "some" individual exists in the area in which it is placed. The diagram to the right indicates "some thing." things1.gif (1134 bytes)
C. Perhaps the most important symbol of all is the blank area where no marks of any kind are made. If an area is not shaded or has no "X," then it is not considered empty, but the blank area represents "no information is known." In other words, a blank area represents the possibility of something existing in that area, nothing more. It is also worth noting, that if an "X" is drawn on a line, the "X" represents only the possibility of being "on either side," but where it is exactly is not known.
II. Given this interpretation, the four standard-form categorical propositions are diagrammed as follows.
1. The A form, "All S is P," is shown in the diagram to the right. Notice that all of the S's are pushed out, so to speak, into the P class. If S's exist, they must be inside the P circle since the left-hand lune of the diagram is shaded and so is empty. venn_a.gif (1566 bytes)
2. The E form, "No S is P," is shown in the diagram to the right. Notice that the lens area of the diagram is shaded and so no individual can exist in this area. The lens area is where S and P are in common; hence, "No S is P." All S, if there are any, are in the left-hand lune, and all P, if there are any, are relegated to the right-hand lune. venn_e.gif (1533 bytes)
3. The I form, "Some S is P," is much more easily seen. The "X" in the lens, as shown in the diagram to the right, indicates at least one individual in the S class is also in the P class. Note that the blank lunes indicate that we do not know whether or not there are individuals in these areas. In fact, we have no information. venn_i.gif (1452 bytes)
4. The O form, "Some S is not P," is also easily drawn. The S that is not a P is marked with an "X" in the S-lune. This area is not within the P circle and so is not a P. It is worth while to note, that from this diagram we cannot conclude that "Some S is P" because there is no "X" in the lens area. Thus, studying this diagram will explain why "Some S is not P" does not entail "Some S is P." venn_o.gif (1466 bytes)
III. For practice, diagram the following statements. The diagrams provided in the answers are only partially correct since they are generic A, E, I, or O diagrams. Your answers should have the subject and predicate classes clearly labeled.
1. All orchids are Cattyleas.

2. All ribosomes are structures in the cytoplasm.

3. No Zurich gnomes are peddlers.

4. No rap is hard rock.

5. Some musicians are not pianists.

6. Some dragsters are not funny cars.

7. Some Presidents are forgetful persons.

8. Some sculptors are not painters.

V. The use of Venn Diagrams to evaluate two-premiss arguments is explained in the class concerning  Syllogistic Venn Diagrams.

Recommended Background Reading:

John Venn--a short biography in the MacTutor History of Mathematics Archive by created by John J. O'Connor and Edumnd F. Robertson at the University of St. Andrews.Return to Logic Homepage    

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