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.
ATS
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:
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.