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Introduction to Logic
Venn Diagrams
Categorical Syllogisms

"Geometriae practicae novae et auctae tractatus I[-IV] ..." by Schwenter, Daniel, 1585-1636. NOAA PHOTO LIbrary

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 quickly syllogisms is the Venn Diagram technique. This class assumes you are already familiar with diagramming categorical propositions. You might wish to review these now:  Venn Diagrams.
A. A 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 a circle.
C. Since a syllogism is valid if and only if the premisses entail the conclusion, diagramming the premisses will reveal the logical geography of the conclusion in a valid syllogism. If the syllogism is invalid, then diagramming the premisses is insufficient to show the conclusion must follow.
D. Since we have three classes, we expect to have three overlapping circles.
Blank Venn Diagram
1. The area in 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 area of the diagram are noted above.
2. Shading represents the knowledge that no individual exists in that area. Empty space represents the fact that no information is known about that area.
3. An "X" represents "at least one (individual)" and so 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."
2. The form written out is

No M is P.

All S is M

No S is P.

3. Note, in the diagram below, how the area in common between S and P has been completely shaded out indicating that "No S is P." The conclusion has been reached from diagramming only the two premisses. All syllogisms of the form EAE-1 are valid.
Diagram of EAE-1 Syllogism
B. AAA-1
1. This syllogism is composed entirely of "A" statements with the M-terms arranged in the "left-hand wing" as well.
2. Its form is written out as

All M is P.

All S is M.

All S is P.

3. Note, in the diagram below, how the only unshaded area of S is in all three classes. The important thing to notice is that this area of S is entirely within the P class. Hence, the AAA-1 syllogism is always valid. In ordinary language the AAA-1 and the EAE-1 syllogisms are by far the most frequently used.
Diagram of AAA-1 Syllogism
C. AII-3
1. The AII-3 syllogism has the M-terms arranged in the subject position--the right side of the brick.
2. This syllogism sets up as

All M is P.

Some M is S.

Some S is P.

3. When diagramming the 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 this part of the SM area is empty. This "logical" forcing enables you to read-off the conclusion, "Some S is P."
4. This syllogism is a good example why the universal premiss should be diagrammed before diagramming a particular premiss. 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.

3. 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. 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 where it is. Since the conclusion cannot be read with certainty, the AII-2 syllogism is 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 written out 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."
5. However, if we know that M exists, all the members of M have to be in the SMP class. These M's are 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 know ahead of time the additional premiss "M exists."
6. Most contemporary logicians have concluded that we should not assume any class exists unless we have evidence.
a. We want to talk 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?

Check your understanding of Venn Diagrams with a Quiz on Testing Syllogisms

Readings: Venn Diagrams

A Survey of Venn Diagrams is an extension of the combinatorial properties of the diagrams by Frank Ruskey at the University of Victoria.

Readings: Existential Fallacy

Bertrand Russell; Hugh MacColl, “The Existential Import of Propositions,” Mind New Series 14 no. 55 (July, 1905), 392-402. DOI: 10.1093/mind/xiv.3.398">

Does This Syllogism by Russell Show That Aristotelian Logic Doesn't Work,” Philosophy Stack Exchange: A discussion about Bertrand Russell's example criticism of Aristotle's syllogisms with this argument:
“If I were to say: ‘All golden mountains are mountains, all golden mountains are golden, therefore some mountains are golden,” my conclusion would be false, though in some sense my premisses would be true.”

Bertrand Russell, “Aristotle's Logic,” in A History of Western Philosophy (1946 London: Taylor and Francis, 2005), 190.

Gyula Klima, “Existential Import and the Square of Opposition,” in John Buridan (Oxford: Oxford University Press, 2009), 143-157. DOI: 10.1093/acprof:oso/9780195176223.003.0006

Terence Parsons, “The Traditional Square of Opposition,” The Stanford Encyclopedia of Philosophy (Summer, 2017).

Abraham Wolf, The Existential Import of Categorical Predication (Cambridge University Press, 1905).

Michael Wreen, “Existential Import,” Crítica: Revista Hispanoamericana de Filosofía 16 no. 47 (August, 1984), 59-64.

Joseph S. Wu, The Problem of Existential Import (From George Boole to P.F. Strawson) Notre Dame Journal of Formal Logic X no. 4 (October 1969), 415-424. DOI: 10.1305/ndjfl/1093893792

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