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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.