Chapter 16:Venn Diagrams

Venn Diagrams (pp. 159-160)

• Venn diagrams represent the relationships between classes of objects by way of the relationships among circles.

• Venn diagrams assume the Boolean interpretation of categorical syllogisms.

• Shading an area of a circle shows that it is empty.

• Placing an X in an area of a circle shows that there is at least one thing that is contained in the class represented by that area.

Venn Diagrams (pp. 159-160)

• For universal propositions, shade (draw lines through) the areas that are empty. - All S are P. - All P are S.

- No S are P. - No P are S.

Venn Diagrams (pp. 159-160)

• For particular propositions, place an X in the area that is inhabited. - Some S are P. - Some P are S.

- Some S are not P. - Some P are not S.

Venn Diagrams for Syllogisms (pp. 162-166)

• To test a syllogism by Venn diagrams, you diagram the premises to see whether the conclusion is also diagrammed.

• This requires three interlocking circles, one for each term:

Venn Diagrams for Syllogisms (pp. 162-166)

• This divides the diagram into eight distinct regions (a line over a term means “not”):

Venn Diagrams for Syllogisms (pp. 162-166)

• Diagram the premises to see whether you have diagrammed the conclusion.– You should always set up the diagram in the same

way: upper left circle for the minor term; upper right circle for the major term; bottom circle for the middle term.

– If by diagramming the premises you have diagrammed the conclusion, the argument is valid.

– If by diagramming the premises you have not diagrammed the conclusion, the argument is invalid.

Venn Diagrams for Syllogisms (pp. 162-166)

• If you have both a universal premise and a particular premise, you should diagram the universal premise first: this will sometimes “force” the X into a determinate region.

• If you have a particular premise and the X is not forced into a determinate section of the diagram, it goes “on the line.” The line in question is always the line of the circle not mentioned in the premise.

• It might be helpful to draw a separate, two-circle diagram of the conclusion; but never add anything to the three-circle diagram other than the diagrams of the premises.

Venn Diagrams: Examples(pp. 162-166)

• Consider the following syllogism:All logicians are critical thinkers.

All philosophers are logicians.

All philosophers are critical thinkers.

• Where L represents the middle term and C represents the major term, and P represents the minor term, the diagram for the major premise looks like this:

Venn Diagrams: Examples(pp. 162-166)

Now you diagram the minor premise on the same diagram:

Venn Diagrams: Examples(pp. 162-166)

• If you’re so inclined, compare the diagram for the conclusion alone.

• Since the premises require that all of P that is outside of C is shaded, we have diagrammed the conclusion in diagramming the premises. The argument is valid.

Venn Diagrams: Examples(pp. 162-166)

• If you find the process a bit odd, consider an argument of the following form:

All P are M.

No M are S.No S are P.

Draw a two circle diagram for each of the premises:

Venn Diagrams: Examples(pp. 162-166)

• Roll them together to form a three-circle diagram:

• You have diagrammed the conclusion by diagramming the premises. The argument form is valid.

Venn Diagrams: Examples(pp. 162-166)

• Consider the following syllogism: No arachnids are cows.

All spiders are arachnids.

No spiders are cows.• Let S represent the minor term (spiders),

C represent the major term (cows), and A represent the middle term (arachnids). Since both premises are universals, let us begin by diagramming the major premise. We shade the area were S and C overlap:

Venn Diagrams: Examples(pp. 162-166)

Now diagram the minor premise on the same diagram:

Compare the diagram for the conclusion alone, if you wish:

By diagramming the premises we have diagrammed the conclusion. The argument is valid.

Venn Diagrams: Examples(pp. 162-166)

• Consider the following syllogism:Some lizards are reptiles.All reptiles are beautiful beasts. Some beautiful beasts are lizards.

• Here we have a particular premise and a universal premise. When you have both, you diagram the universal premise first. “Why?” you ask. The X for representing the particular should always go into a determinate area if possible. If you diagram the universal first, the X is forced into a determinate area of the diagram:

Venn Diagrams: Examples(pp. 162-166)

Then add the X.

The argument is valid. The diagram shows that there is at least one thing (X) that is a beautiful lizard, so the argument is valid.

Venn Diagrams: Examples(pp. 162-166)

• If you’d diagrammed the particular premise first the X would have gone on the line, since the X goes on the line except when the area on one side of the line is shaded. So, if you’d diagrammed the particular premise first, the diagrams would look like this:

• It is bad form to have an X on the line if the area on one side of the line is shaded. You would have to erase and place it in the unshaded area.

Venn Diagrams: Examples(pp. 162-166)

• Most syllogistic forms are invalid. Consider the following:All P are M.

All M are S.

All S are P.

• Diagram the major premise, then diagram the minor premise on the same diagram:

We have diagrammed “All P are M,” which is not the conclusion. So the argument form is invalid.

Venn Diagrams: Examples(pp. 162-166)

• Consider an argument of the following form:All M are P.

No M are S.

No S are P.

• An area has been shaded twice. So, we haven’t diagrammed the conclusion. The argument form is invalid

Venn Diagrams: Examples(pp. 162-166)

• Consider the following syllogism:Some aardvarks are not sheep, and no sheep are trumpets, so all aardvarks are trumpets.

After making sure there are exactly three terms, you could represent the form as follows:

No S are T.

Some A are not S.

All A are T.

Venn Diagrams: Examples(pp. 162-166)

• You diagram the major premise, since it’s universal:

• Now you diagram the particular. The X has to be in A and outside of S. Since it could be in either of two areas, neither of which is shaded, you place the X on the T circle that divides A into two parts. It looks like this:

TSA

Venn Diagrams: Examples(pp. 162-166)

The X is on the line. That is sufficient to show that the argument form is invalid. If you prefer, you could compare the top two circles to the two-circle diagram for the conclusion. You’d notice that you have not diagrammed the conclusion. (The diagram for a universal is always a strictly shady affair.)

Venn Diagrams: Examples(pp. 162-166)

• Consider the following:All mice are rodents, so some mice are

bothersome beasts, since some rodents are bothersome beasts.

• There are three terms, so we may set out the form as follows:

Some R are B.

All M are R

Some M are B.

Venn Diagrams: Examples(pp. 162-166)

• This time the major premise is a particular, and the minor premise is a universal. So, we diagram the minor premise first:

• Now we diagram the major, placing an X in the area where B and R overlap. The X goes on the line:

Venn Diagrams: Examples(pp. 162-166)

The argument is invalid.

Venn Diagrams: Examples(pp. 162-166)

• In summary:– Make sure you have exactly three terms.– If there is a universal premise and a particular

premise, diagram the universal premise first.– If neither of the areas where the X could go is

shaded, the X goes on the line. – No syllogism whose diagram places an X on

the line or results in double-shading is valid.– It is valid if and only if shading the premises

results in shading the conclusion.