Another method of symbolizing categorical propositions is the use of the Venn diagram.
John Venn, who introduced the method (thus the name Venn Diagram) used two overlapping circles to represent the relationship between two classes. Consider the diagram below.
The shaded portion represents a class that has no members.
The area with an “X” signifies that the class has at least one member.
is read as “S but not P” and this represents the class of things that are part of S but are not part of P.
is read as “not S but P” and this represents the class of things that are part of P but are not part of S.
is read as “S but P” and this represents the class of things that are both parts of S and P.
The diagrams below are used to represent the four standard types of categorical propositions.
The shaded area of the Venn diagram above represents a class that has no members. In the Venn diagram for a universal affirmative (A) proposition, the area “S but not P” is shaded to indicate that all members of S are members of P. Thus, we say, “All S are P”.
The shaded area of the Venn diagram above represents a class that has no members. In the Venn diagram for a universal negative proposition (E), the area SP is shaded to indicate that this class has no members. Thus, we say, “No S are P”.
A particular affirmative (I) proposition asserts that there is at least one member of S that is a member of P. This is diagrammed by placing an “X” in the area SP, which is common to the two classes. Thus, in the diagram above, we place an “X” in the area SP.
A particular negative (O) proposition asserts that there is at least one member of S that is not a member of P. Thus, in the diagram above, we place the “X” on the area “S but not P” to indicate that indeed there is at least one member of S that is not a member of P.
Venn Diagram and Existential Import
In traditional or Aristotelian logic, one assumes that universal affirmative (A) and universal negative (E) propositions have existential import. Thus, in the example “All angels are holy”, one assumes that there are angels and that all of them are holy. However, in applying the Venn diagram, one does not make this assumption. In the Venn diagram, all angels are said to be holy only if there are indeed angels. But the Venn diagram for a universal affirmative (A) proposition does not contain an area in which there is a symbol to show that there is an angel. Hence, the propositions “All angels are holy” or “No angels are holy” are non-existential propositions. In the first place, there are no angels in reality. The Venn diagram below will demonstrate this point.
As we can see, both application of traditional rules and use of the Venn diagram presuppose that only particular affirmative (I) and particular negative (O) propositions have existential import. Thus, a Venn diagram for the particular affirmative (I) proposition “Some angels are holy” contains an “X” to show that there is at least one angel that is holy. Please see the diagram below.
Venn Diagram and Test of Validity
A Venn diagram can be used to show the validity of categorical syllogisms. Three intersecting circles are needed to diagram a categorical syllogism, one circle for each class. The following rules will be observed in testing the validity of syllogism using a Venn diagram:
- The universal premise should be diagrammed first if the argument also contains a particular premise;
- The letter “X” should be placed on the line dissecting an area if the whole area is so designated in the premise;
- Only the premises should be diagrammed; and
- If the conclusion is self-evident in the diagram, then the argument or syllogism is valid.
Let us consider the example below, which is already in its standard form.
Example 1:
How do we determine the validity of the syllogism above using a Venn diagram?
First, we need to draw three intersecting circles (that is, circles for S, P, and M) and then number the areas by starting at the center, and then clockwise. Please see the Venn diagram of this syllogism below.
Now that we have drawn three intersecting circles, each for S, P, and M, our next task is to diagram the syllogism above.
Let us start with the first premise, that is, “All M are P”. It must be noted that since the premise talks about the circles for M and P only, so we will imagine that the circle for S does not exist; hence, we will diagram M and P only. Now, since the premise says “All M are P”, that is, all members of M are members of P, then we will shade areas 5 and 6 to show that all members of M, which are areas 1 and 4, are part of P. The Venn diagram of the syllogism above now looks like this:
After we diagrammed the first premise, let us proceed to diagram the second premise, which reads “All S are M”. This time, the premise talks about S and M only, so we will imagine that the circle for P does not exist. Now, since the premise says “All S are M”, that is, all members of S are members of M, then we will shade areas 7 and 2 to show that indeed all members of S, which is area 1, are part of M. The diagram now looks like this:
Since the Venn diagram of the above syllogism is now complete, let us proceed to analyze the diagram to determine whether the syllogism is valid or invalid. As rule #3 says, we diagram only the premises; hence, we do not diagram the conclusion. And as rule #4 says, the argument or syllogism is valid if the conclusion is self-evident in the Venn diagram.
Now, the conclusion says “All S are P”. As we can see in the Venn diagram of the syllogism above, the conclusion “All S are P” is perfectly diagrammed, that is, it is self-evident. In fact, since areas 2, 6, and 7 are shaded, then they do not exist anymore. What is left now of the class S is area 1, which all belongs to P. Thus, the above syllogism is valid.
Let us consider another example.
Example 2:
Let us draw three intersecting circles for this syllogism, each for S, M, and P, and then number the areas by starting at the center, and then clockwise. As rule #1 says, we will diagram first the universal premise if the syllogism also contains a particular premise. Since the first premise in the syllogism above is particular, while the second premise is universal, then we will diagram first the second premise, that is, “All M are S”.
The second premise says “All M are S”, so we will shade areas 4 and 5 to show that all members of M, which are areas 1 and 6, are part of S. The Venn diagram of the syllogism above will now look like this:
Let us proceed to diagram the second premise, which says “Some M are P”. Since this is a particular proposition, then we will not use the shading method; instead, we will place an “X” on the designated area. Since the premise says “Some M are P”, and since area 4 is already shaded, then it does not exist anymore. Thus, we will place the “X” on area 1 to show that indeed there is at least one member of M that is a member of P. The Venn diagram of the syllogism above now looks like this:
The Venn diagram of the syllogism above is now complete. Let us proceed to determine the validity of this syllogism.
Again, rule #3 says, we diagram only the premises; hence, we do not diagram the conclusion. And as rule #4 says, the argument or syllogism is valid if the conclusion is self-evident in the Venn diagram. Now, the conclusion of the syllogism above says “Some S are P”, and if we look at the Venn diagram above, there is an “X” on area 1, which indicates that there is at least one member of S that is a member of P. Hence, the conclusion which reads “Some S are P” is perfectly diagrammed in the Venn diagram above; indeed, the conclusion is self-evident. Therefore, the above syllogism is valid.
Let us consider another example.
Example 3:
Let us draw three intersecting circles for this syllogism, each for S, M, and P, and then number the areas by starting at the center, and then clockwise.
As we already know, we will diagram the universal premise first, which is “All S are M”. Thus, the Venn diagram of the syllogism above now looks like this:
Then let us diagram the second premise, which says “Some M are P”. Since areas 1 and 4 are so designated in the premise, then we will place the “X” on the line that dissects areas 1 and 4 to show that the whole area is so designated. Hence, the Venn diagram of the syllogism above now looks like this:
Since the “X” is on the line that dissects areas 1 and 4, this gives us an inconclusive reading of the conclusion. For this reason, the above syllogism is invalid. Indeed, the conclusion is not self-evident; it is not perfectly diagrammed.
Let us analyze one more example:
Example 4:
Let us draw three intersecting circles for this syllogism, each for S, M, and P, and then number the areas by starting at the center, and then clockwise.
Let us first diagram the first premise, which reads “All M are P. The Venn diagram of the above syllogism now looks like this:
Then let us diagram the second premise, which reads “No S are M”. The Venn diagram of the above syllogism now looks like this:
The conclusion of the above syllogism, which reads “No S are P”, asserts that no members of S should be a member of P. But area 2 of the Venn diagram above, which is also an area of SP, is not shaded. Thus, the Venn diagram above does not perfectly diagram the conclusion; indeed, the conclusion is not self-evident. Therefore, the above syllogism is invalid.