Venn Diagram Technique for testing
syllogisms
•We have used two-circle Venn diagrams
to represent standard-form categorical
propositions. In order to test a
categorical syllogism by the method of
Venn diagrams, one must first represent
both of its premises in one diagram.
That will require drawing three
overlapping circles, for the two premises
of a standard-form syllogism contain
three different terms-minor term, major
term, and middle term.
• Let’s start with this example:
• No doctors are professional wrestlers.
All cardiologists are doctors.
 No cardiologists are professional wrestlers.
• Since this argument, like all standard-form
categorical syllogisms, has three category-terms
(in this case, “cardiologists,” “professional
wrestlers,” and “doctors”), we need three
interlocking circles rather than two to represent
the three categories. The minor term will go on
the top left, the major term will go on the the top
right, and the middle term will be the bottom
circle.
• The diagram for our example is as follows:
The first premise states that no doctors are
professional wrestlers. To represent this claim, we
shade that part of the Doctors circle that overlaps
with the Professional wrestlers circle, as follows:
• The second premise states that all cardiologists are
doctors. To represent this claim, we shade that
part of the Cardiologists circle that does not
overlap with the Doctors circle:
• We now have all the information we need to see
whether the argument is valid. The conclusion
tells us that no cardiologists are professional
wrestlers. This means that the area where the
Cardiologists and Professional wrestlers circles
overlap is shaded, that is, empty. We look at the
diagram to see if the area is shaded, and we see
that it is indeed shaded. That means that the
conclusion is implicitly “contained in” (i.e.
follows logically from) the premises. Thus, the
argument is shown to be valid.
• Let’s look at a second example:
• All snakes are reptiles.
All reptiles are cold-blooded animals.
 All snakes are cold-blooded animals.
• Next, we diagram the first premise. The premise
states that all snakes are reptiles. We represent
this information by shading the area of the Snakes
circle that does not overlap with the Reptiles
circle.
• Next, we diagram the second premise. The second
premise states that all reptiles are cold-blooded
animals. We represent this claim by shading that
part of the Reptiles circle that does not overlap
with the Cold-blooded animals circle.
• Finally, we look to see if the information
contained in the conclusion is depicted in the
diagram. The conclusion tells us that all snakes
are cold-blooded animals. This means that the
part of the Snakes circle that does not overlap with
the Cold-blooded animals circle should be
completely shaded. Inspection of the diagram
shows that this is indeed the case. So, the
argument is valid.
• Let’s look at a third example:
• Some Baptists are coffee-lovers.
All Baptists are Protestants.
 Some Protestants are coffee-lovers.
• Notice that this example includes two “some”
statements. Diagramming “some” statements is a
little trickier than diagramming “all” or “no”
statements. As we have seen, “some” statements
are diagrammed by placing Xs rather than by
shading. Most mistakes in Venn diagramming
involve incorrect placement of an X.
• To avoid such mistakes, remember the following
rules:
• 1. If the argument contains one “all” or “no”
statement, this statement should be diagrammed
first. In other words, always do any necessary
shading before placing an X. If the argument
contains two “all” or “no” statements, either
statement can be done first.
• 2. When placing an X in the area, if one part of the
area has been shaded, place the X in the unshaded
part. Examples:
x
3. When placing an X in an area, if one part of the
area has not been shaded, place the X precisely
on the line separating the two parts. Example:
x
• Back to the third example:
• Some Baptists are coffee-lovers
All Baptists are Protestants.
 Some Protestants are coffee-lovers.
• First, we draw and label our three circles:
• Next, we need to decide which premise to diagram
first. Should it be the “some” premise or the “all”
premise? Rule one states that we should start with
the “all” premise:
• Now we can diagram the first premise, which
states that some Baptists are coffee-lovers. To
represent this claim, we place an X in the area of
the Baptists circle that overlaps with the Coffeelovers circle. Part of this area, however, is shaded.
This means that there is nothing in that area. For
that reason, we place the X in the unshaded
portion of the Baptists circle that overlaps with the
Coffee-lovers circle, as follows:
• Finally, we inspect the completed diagram to see
if the information contained in the conclusion is
represented in the diagram. The conclusion states
that some Protestants are coffee-lovers. This
means that there should be an X in the area of the
Protestants circle that overlaps with the Coffeelovers circle. A glance at the diagram show that
there is an X in this area. Thus, the argument is
valid.
• So far, all the categorical syllogisms we have
looked at have been valid. But Venn diagrams can
also show when a categorical syllogism is invalid.
Here is one example:
• All painters are artists.
Some magicians are artists.
 Some magicians are painters.
• First, we draw and label our three circles:
• Since the premise begins with “all” and the second
premise begins with “some,” we diagram the first
premise first. The first premise states that all
painters are artists. To depict this claim, we shade
that part of the Painters circle that does not
overlap with the Artists circle:
• Next, we enter the information of the second
premise, the claim that some magicians are artists.
To represent this claim, we place an X in that
portion of the Magicians circle that overlaps with
the Artists circle. That area, however, is divided
into two parts (the areas here marked “1” and “2”),
and we have no information that warrants placing
the X in one of these areas rather than the other.
In such cases, we place the X precisely on the line
between the two sections, as follows:
• The X on the line means that we have no way of
knowing from the information given whether the
magician-who-is-an-artist is also a magician-whois-a-painter.
• The conclusion states that some magicians are
painters. This means that there should be an X
that is definitely in the area where the Magicians
and Painters overlap. There is an X in the
Magicians circle, but it dangles on the line
between the Artists circle and the Painters circle.
We don’t know whether it is inside or outside the
Painters circle. For that reason, the argument is
invalid.