Venn Diagrams for Categorical Syllogisms

The technique of Venn diagrams for categorical syllogisms is based on the fact that in a valid syllogism, the conclusion asserts no more than what is already contained, implicitly, in the premises. If the conclusion asserts more than that, it does not follow from the premises, and the syllogism is invalid. The technique is to diagram the premises, and then see whether anything would have to be added in order to diagram what the conclusion asserts. If so, the syllogism is invalid; if not, it is valid.

Example:

No M is P No horned animal is a carnivore
All S is M All moose are horned animals


No S is P No moose is a carnivore

The first step is to diagram the major premise, using the circles representing M (horned animals) and P (carnivores). So we shade out the area of overlap between M and P.

The second step is to add the minor premise to our diagram, using the circles representing S and M. Since this is an A proposition, we shade out the region of S outside M.

The final step is to examine the completed diagram of the premises and determine whether it contains the information asserted by the conclusion. The conclusion asserts that no S is P. Thus it requires that the overlap between S and P be shaded out, and the premises taken together do shade out that region. So the syllogism is valid.

For a syllogism to be valid, the combined diagram must contain all the information asserted by the conclusion. It may contain more information, but it cannot omit anything.

Now let's try a syllogism with a particular premise.

Example:

No M is P
Some M are S
___________
Some S are not P

First we diagram the major premise.

Second we diagram the minor premise.

Notice that we diagrammed the major premise first. This is not required logically, but whenever there is a particular and a universal premise, it is best to diagram the universal one first. By diagramming the universal premise first, we have shaded out one of the subregions, so now we know that the X for the other premise must go outside the P circle. And that's useful information, it means that at least one S is not P. Since that is what the conclusion asserts, the argument is valid.

If a syllogism is invalid, a Venn diagram will reveal that fact in one of two ways. The combined diagram for the premises will either fail to shade out an area excluded by the conclusion, or it will fail to put an X where the conclusion requires one.

Example:

All P are M
All S are M


All S are P

The Venn diagram reveals the invalidity by failing to shade out the right areas.

In the combined diagram, the area of P outside M has been shaded to represent the major premise, and the area of S outside M has been shaded to represent the minor. But one area in the region of S outside P--the one indicated by the arrow--has not been shaded. Thus, the premises leave open the possibility that some S are not P; they do not guarantee that all S are P. So the conclusion does not follow; the syllogism is invalid.

Now let's examine another case in which the invalidity is revealed by the placement of Xs.


All P are M
Some S are M
Some S are P

Notice that the X is on the line between two subregions of the overlap between S and M. Locating the X on the line means: I know something is both an S and an M, but I don't know whether it is also a P or not. But the conclusion does assert that some S are P. For the premises to justify this assertion, they would have to give us an X in the area of overlap between S and P. But all they tell us is: there's an S that may or may not be a P. The conclusion doesn't follow.


Comprehension Questions
1 The technique of Venn diagrams is based on the fact that in a valid syllogism the conclusion
a) asserts no more than what is already contained, implicitly, in the premises
b) asserts more than what is contained in the premises
2 For a syllogism to be valid, the combined diagram must
a) contain all information asserted by the conclusion
b) either fail to shade out an area excluded by the conclusion, or fail to put an x where the conclusion requires one.
3 For a syllogism to be invalid, the combined diagram must
a) contain all information asserted by the conclusion
b) either fail to shade out an area excluded by the conclusion, or fail to put an x where the conclusion requires one.
4 Which of the choices represent the Venn diagrams of the given categorical syllogism?

All of Shakespeare's dramas are in blank verse
Some great plays are in blank verse


Some great plays are Shakespeare's dramas
a) c)
b) d)
5 According to the Venn diagrams, the previous syllogism is valid or invalid?
a) valid
b) invalid
6 Which of the choices represent the Venn diagrams of the given categorical syllogism?

All vertebrates reproduce sexually
All vertebrates are animals


All animals reproduce sexually
a) c)
b) d)
7 According to the Venn diagrams the previous syllogism is valid or invalid?
a) valid
b) invalid
8 Which of the choices represent the Venn diagrams of the given categorical syllogism?

No A is B
Some A are not C


Some C are not B
a) c)
b) d)
9 According to the Venn diagrams the previous syllogism is valid or invalid?
a) valid
b) invalid
10 Which of the choices represent the Venn diagrams of the given categorical syllogism?

No horned animals are carnivores
All moose are horned animals


No moose are carnivores
a) c)
b) d)
11 According to the Venn diagrams the previous syllogism is valid or invalid?
a) valid
b) invalid


Enthymemes

© Copyright 1998, W.W. Norton & Co.