V Diagram

Introduction to Logic
Venn Diagrams
Categorical Syllogisms

Abstract: The Venn Diagram technique is shown for typical as well as unusual syllogisms. The problem of existential import is introduced by means of these diagrams.

I. One good method to test syllogisms rapidly is the Venn Diagram technique. This class assumes you are already familiar with diagramming categorical propositions. If not, you might wish to review how to diagram standard form statements here: Venn Diagrams.

A. The syllogism is a two premiss argument having three terms, each of which is used twice in the argument.

B. Each term (major, minor, and middle terms) can be represented by drawing a circle. Anything inside the circle is designated as a member of the class named by that term and anything outside the circle is something else.

C. Since a syllogism is valid if and only if the premisses entail the conclusion, diagramming the premisses will exhibit the logical geography of the conclusion.

If the syllogism is invalid, diagramming the premisses will not show that conclusion necessarily follows from the premises.

D. Since we have three classes, we expect to have three overlapping circles.

A diagram similar to this one was first used by John Venn [Symbolic Logic (London: Macmillan, 1818), 105]:
Blank Venn Diagram

1. The area within the denoted circle represents where members of the class would be, and the area outside the circle represents all other individuals (the complementary class). The various areas of the diagram are noted above.

A circle with its interior shaded2. Shading represents the knowledge no individual exists in the class; for example, this diagram shows no “Yeti” exists within the circle.

A circle with its blankEmpty space represents the fact that no information is known whether any thing exists in that area of the class.

A circle with with an “x” in its center3. An “X” in the circle represents “at least one “thing” ” exists in the class of things, and so “X” here corresponds with the word “some.”

II. Some typical examples of syllogisms are shown here by their mood and figure.

A. EAE-1

1. The syllogism has an E statement for its major premiss, an A statement for its minor premiss, and an E statement for its conclusion. By convention the conclusion is labeled with S (the minor term) being the subject and P (the major term) being the predicate.

The position of the middle term is the “left-hand wing” in this illustration of the syllogistic figures:

A circle with with an “x” in its center

1. The form of this syllogism written out is

No M is P.
All S is M
No S is P.

1. We diagram one premiss at a time on the diagram of three interlocking circles. So, as shown on the diagram below, we superimpose the major premiss E statement diagram “No M is P” on the area represented by the interlocking circles P and M.

Next, we superimpose the A minor premiss “All S is M” on the interlocking circles S and M.

Note, in the diagram below, how the area in common between S and P (called “the lens of the S and P classes) has been completely shaded out indicating that “No S is P.”

No individual X can exist within the SP lens area because our diagram shows that area must be empty. In a sense, the conclusion “automatically” results from diagramming only the two premisses. We do not pencil in the conclusion — we merely read it out by observing the result of sketching the two premises.
Diagram of EAE-1 Syllogism

This diagram illustrates that all syllogisms of the form EAE-1 are valid because the diagram shows that if there are any Ss then they are in a completely separate area than Ps.

B. AAA-1

1. This syllogism is composed entirely of “A” statements with the M-terms arranged in the “left-hand wing” as was the syllogism above.
2. Its form is written out as

All M is P.
All S is M.
All S is P.

1. Note, after diagramming the figure below, how the only unshaded area of S is within the area of all three classes. The important thing to notice is that this area of S is entirely within the P class.

Hence, we can conclude that if there are any Ss they have to be also in the P class.
Diagram of AAA-1 Syllogism

This diagrams shows all AAA-1 syllogisms are always valid. In ordinary language the AAA-1 and the (above) EAE-1 syllogisms are by far the most frequently used everyday syllogistic argument forms.

C. AII-3

1. The AII-3 syllogism has the M-terms arranged in the subject position — the right side of the orange flying brick diagram illustration above.
2. This syllogism sets up as

All M is P.
Some M is S.
Some S is P.

1. When diagramming the minor premise of this syllogism, notice how you are “forced” to put the X from the minor premiss in the area of the diagram shared by all three classes.

The “X” cannot go on the P-line because the shading indicates that part of the SM lens area is empty. This “logical” forcing enables you to read-off the conclusion, that “Some S is P” because the X is in the union of S, P and M.

1. This syllogism is a good example why … The universal premiss should be diagrammed before a particular premiss is diagrammed. If we were to diagram the particular premiss first, the “X” would go on the line. Then, we would have to move it when we diagram the universal premiss because the universal premiss empties an area where the “X” could have been. Diagram of AII-3 Syllogism

D. AII-2

1. The AII-2 has the M terms in the predicate of both premisses.
2. The syllogism is written out as …

All P is M.
Some S is M.
Some S is P.

1. The diagram below shows that the “X” could be in the SMP area or in the SPM area. Since we do not know exactly which area it is in, we put the “X” on the line as shown to indicate that the X might be in either place, but we can't be sure. Thus, the conclusion can't be read off, so the argument is invalid.

When an X is on a line, we do not know with certainty exactly where it is. So, when we go to read the conclusion, we do not know for sure whether the X is in the P class or not. Since the conclusion cannot be read with certainty, the AII-2 syllogism is judged invalid.
Diagram of AII-2 Syllogism

E. The final syllogism described here, the EAO-4 raises some interesting problems.

1. Notice that in this syllogism there are universal premisses with a particular conclusion.
2. Its form is expanded as … No P is M. All M is S. Some S is not P.

3. And its diagram is rather easily drawn as …
   Diagram of EAO-4 Syllogism

4. When we try to read the conclusion, we see that there is no “X” in the SMP class. We must conclude that the syllogism is invalid because we cannot read-off “Some S is not P.” There is no X anywhere on the diagram.

5. However, if we independently knew that at least one individual in the the class represented by M exists, then that one M would have to be in the SMP class. That M would have to be a S's as well. Hence, we know that some S's are not P's!

In other words, the EOA-4 syllogism is valid if we knew ahead of time the additional premiss “M exists.”

1. Most contemporary logicians have concluded that we should not assume any class exists unless we have some form of independent evidence. a. However, we often want to speculate about theoretical entities without assuming their existence. b. For example, in science and mathematics, our logic will apply when talking about circles, points, frictionless planes, and freely falling bodies even though these entities do not physically exist. c. This diagram illustrates the contemporary topic called the problem of existential import. When can we reasonably conclude something exists? How does this conclusion affect our theory of logical validity?