Abstract: The Venn Diagram technique is shown for
typical as well as unusual syllogisms. The problem of existential
import is introduced by means of these diagrams.
I. One good method to test syllogisms rapidly is the Venn Diagram
technique. This class assumes you are already familiar with
diagramming categorical propositions. If not, you might wish to
review how to diagram standard form statements here:
Venn Diagrams.


A. The syllogism is a two premiss
argument having three terms, each of which is used twice in the
argument.


B. Each term (major, minor, and middle
terms) can be represented by drawing a circle. Anything inside the
circle is designated as a member of the class named by that term and
anything outside the circle is something else.


C. Since a syllogism is valid if and only if the premisses
entail the conclusion, diagramming the premisses will exhibit the
logical geography of the conclusion.
If the syllogism is invalid, diagramming the premisses
will not show that conclusion necessarily follows from the premises. 

D. Since we have three classes,
we expect to have three overlapping circles.
A diagram similar to this one was first used by John Venn
[Symbolic
Logic (London: Macmillan, 1818), 105]: 






1. The area within the denoted circle
represents where members of the class would be, and the area outside
the circle represents all other individuals (the complementary class).
The various areas of the diagram are noted above.




2. Shading represents the knowledge no individual
exists in the class; for example, this diagram shows no
“Yeti” exists within the circle.
Empty space represents the fact that no information is known whether
any thing exists in that area of the class.



3. An “X” in the circle represents “at least one “thing”
” exists in the class of things, and so “X” here corresponds
with the word “some.”

II. Some typical examples of syllogisms are shown
here by their mood and figure.


A. EAE1 


1. The syllogism has an E statement for
its major premiss, an A statement for its minor premiss, and an
E statement for its conclusion. By convention the conclusion is labeled
with S (the minor term) being the subject and P (the major term)
being the predicate.
The position of the middle term is the “lefthand wing” in this
illustration of the syllogistic figures:



2. The form of this syllogism written out is
No M is P.
All S is M
No S is P.



3. We diagram one premiss at a time on the diagram of
three interlocking circles. So, as shown on the diagram below, we superimpose
the major premiss E statement diagram “No M is P”
on the area represented by the interlocking circles P and M.
Next, we superimpose the A minor premiss “All S is M”
on the interlocking circles S and M.
Note, in the diagram below, how the area in
common between S and P (called “the lens of the S
and P classes) has been completely shaded out indicating that “No
S is P.”
No individual X can exist within
the SP lens area because our diagram shows that area must be empty. In a
sense, the conclusion “automatically” results from diagramming only
the two premisses. We do not pencil in the conclusion — we merely read it out
by observing the result of sketching the two premises.




This diagram illustrates that all syllogisms of the form EAE1 are valid
because the diagram shows that if there are any Ss then they are in a
completely separate area than Ps.


B. AAA1 


1. This syllogism is composed entirely of
“A” statements with the Mterms arranged in the
“lefthand wing” as was the syllogism above. 


2. Its form is written out as
All M is P.
All S is M.
All S is P.



3. Note, after diagramming the figure below, how
the only unshaded area of S is within the area of all three classes. The
important thing to notice is that this area of S is entirely within
the P class.
Hence, we can conclude that if there are any Ss they have to be also in
the P class. 



This diagrams shows all AAA1 syllogisms are always valid. In ordinary
language the AAA1 and the (above) EAE1 syllogisms are by far the
most frequently used everyday syllogistic argument forms.


C. AII3 


1. The AII3 syllogism has the
Mterms arranged in the subject position — the right
side of the orange flying brick diagram illustration above. 


2. This syllogism sets up as
All M is P.
Some M is S.
Some S is P.



3. When diagramming the minor premise of
this syllogism, notice how you are “forced” to put the
X from the minor premiss in the area of the diagram shared by all
three classes.
The “X” cannot go on the Pline because the
shading indicates that part of the SM lens area is empty. This
“logical” forcing enables you to readoff the conclusion, that
“Some S is P” because the X is in the union
of S, P and M.



4. This syllogism is a good example why …
The universal premiss should be diagrammed before a particular premiss is
diagrammed.
If we were to diagram the particular premiss first, the
“X” would go on the line. Then, we would have to move
it when we diagram the universal premiss because the universal premiss empties
an area where the “X” could have been. 





D. AII2 


1. The AII2 has the M terms in the
predicate of both premisses. 


2. The syllogism is written out as …
All P is M.
Some S is M.
Some S is P.



3. The diagram below shows that the
“X” could be in the SMP area or
in the SPM area. Since we do not know exactly which area it is in,
we put the “X” on the line as shown to indicate that the
X might be in either place, but we can't be sure. Thus, the conclusion
can't be read off, so the argument is invalid.
When an X is on a line, we do not know with certainty exactly
where it is. So, when we go to read the conclusion, we do not know
for sure whether the X is in the P class or not. Since the
conclusion cannot be read with certainty, the AII2 syllogism is judged
invalid. 





E. The final syllogism
described here, the EAO4 raises some interesting problems. 


1. Notice that in this syllogism there are
universal premisses with a particular conclusion. 


2. Its form is expanded as …
No P is M.
All M is S.
Some S is not P.



3. And its diagram is rather easily drawn as …







4. When we try to read the conclusion, we see
that there is no “X” in the SMP class. We must
conclude that the syllogism is invalid because we cannot readoff “Some S is
not P.” There is no X anywhere on the diagram.



5. However, if we independently knew that at least one
individual in the the class represented by M exists, then that one M
would have to be in the SMP class. That M would have
to be a S's as well. Hence, we know that some S's are not P's!
In other words, the EOA4 syllogism is valid if we knew ahead of time the
additional premiss “M exists.”



6. Most contemporary logicians have concluded
that we should not assume any class exists unless we have some form of
independent evidence. 



a. However, we often want to speculate about theoretical entities without
assuming their existence. 



b. For example, in science and mathematics, our logic will
apply when talking about circles, points, frictionless planes, and freely falling bodies
even though these entities do not physically exist. 



c. This diagram illustrates the contemporary topic called the
problem of existential import. When can we reasonably conclude something exists? How
does this conclusion affect our theory of logical validity?
