ADDING ANOTHER CATEGORY
Now we're ready to add another
category to our universe. When we do, we'll find that the number of
distinct areas increases, from four to eight. Again using colors
to identify these areas, the blue, yellow, and red areas are the unshared
portions of each of the three categories. The violet area is the shared
region of the blue and red circles; the green area is the shared region of the
blue and yellow circles; and the orange area is the shared region of the
red and yellow
circles. The central gray area is the region shared by all three circles, and
the white area is the portion of the universe outside all of these categories.
Venn stopped at three categories,
because that's the largest number of both partially and fully overlapping yet distinct categories that can be diagrammed
in two dimensions using only circles. In a threedimensional
medium, using spheres instead of flat circles (each sphere centered at
one of the four corners of a tetrahedron, also called a threesided
pyramid), a fourth sphere could be
merged with the other three, and several more areas of overlap would
appear. In two dimensions, more categories could be diagrammed,
but only if some of them are given odd shapes. The addition of a fourth
category would increase the number of distinct areas from eight to
sixteen (including the external universe). But in any case,
clarity would suffer, and the intuitive informationconveying advantage
of a simple diagram would be lost.
Even so, a threecategory universe
offers plenty of logical possibilities—in part because it's ideal for
depicting the classic threeterm syllogism. The subject and
predicate terms are those of the conclusion. The remaining
category is the syllogism's middle term,
represented by the topcenter
circle.
IMPORTANT: When using a Venn diagram to illustrate a syllogism,
we diagram the premises only, not the conclusion. Our purpose
is to use the diagram to see whether or not the premises actually do
imply the conclusion, without presupposing it. After we diagram
the premises, we observe whether the markings on the completed diagram
correspond to the conclusion. If they do, then the syllogism is
valid; if not, then the syllogism is invalid.
Let's try this syllogism for a
start:
All artists are eccentric.
Some painters are artists.
So, some painters are eccentric.
The third statement is the
conclusion; its subject term is "painters," and its predicate term is
"eccentric." As with the twocategory diagrams, the conclusion's
subject term corresponds to the S category in the threecategory
diagram, and its predicate term corresponds to the P category.
Thus, we would expect the conclusion, "Some painters are eccentric," to
be represented in the completed Venn diagram by something—indicated by
either a cross (x) or a crossborder mark—somewhere in the overlap area
between the S and P circles. We'll make a mental
note of this for now, and we'll see if this turns out to be the case
once we diagram the two premises.
The first premise also contains the predicate term of the conclusion,
"eccentric," so this must be the major premise. The
second premise contains the conclusion's subject term, so this must be
the minor premise. In addition, the major and
minor premises share the term "artists," so this is the middle term,
which corresponds to the M category.
So now we must figure out which
areas of the diagram are occupied by something, and which must be
vacant. Universal statements (forms A and E) tend to narrow down
the range of possibilities more than particular statements (forms I and
O), so beginning with a premise that's a universal statement is usually the quickest way to
fill in a diagram without having to do any backtracking.
First, the only universal statement
in this syllogism is the major premise, "All artists are eccentric."
"Artists" is the middle term. It indicates that anything that
isn't eccentric isn't an artist, so we shade out all parts of the M
circle that don't overlap the "eccentric" category, which, as we've
said, is the P circle.
Second, to account for the minor
premise, "Some painters are artists," we look for any areas in the
"painters" category (the S circle) which overlap the "artists"
category (the M circle). We find there are two of these;
however, one of them is already shaded out, and thus vacant.
There's only one such area—the central region—which hasn't been
eliminated from consideration,
so we place an x there. And indeed, the location of the x
in the SP overlap area reaffirms the conclusion, "Some painters are eccentric."
(The syllogism doesn't tell us anything about painters who aren't
artists—i.e., whether they are or aren't eccentric—so we can neither
mark nor shade out any area outside the M [artists] category.)
This is a pattern associated with
a syllogism with a mood of AII (i.e., the major premise is an Aform
statement, and the minor premise and conclusion are Iform statements).
[Incidentally, when dealing with mood, which is specified by a
threeletter code corresponding to the A, E, I, or O forms of the
syllogism's major and minor premises and its conclusion, respectively,
it's important to remember that, in some text fonts, the uppercase
letter I (i) closely resembles a lowercase L (l). Be careful
not to confuse the mood AII (AII)
with the word All (All).]
NOTE: The
remainder of this panel deals largely with various configurations of
form, mood, and figure. The discussion can get rather dry, and
some readers might find their eyes glazing over. This is okay.
Don't try to memorize everything. The important thing is to become
aware that some configurations are valid and some are not. In most
cases, the distinction between validity and invalidity, if not
immediately obvious, requires just a little close examination and
thinkingthrough of the syllogism, This is where knowing how to
use Venn diagrams as a visual aid to categorical reasoning can be
extremely helpful.
For threestatement syllogisms,
each of whose statements might take one of four forms, there are 4^{3}
(4 ´ 4 ´ 4),
or 64, possible moods. Further complicating the matter is an
additional factor called figure,
which has to do with the sequence of major (P), minor (S),
and middle (M) terms in each of the premises. (While the
sequences in the premises can vary, the sequence
in the conclusion is always the minor term as subject and
the major term as predicate.) This gets a bit deeper into the
topic than most people want or need to go, so we won't bother to
discuss it in detail. However, we'll list the four figures for any
readers who might be interested.
Figure 

Major
Premise 

Minor
Premise 

Conclusion 
1st 

M  P 

S  M 

S  P 
2nd 

P  M 

S  M 

S  P 
3rd 

M  P 

M  S 

S  P 
4th 

P  M 

M  S 

S  P 
So, multiplying the 64 possible
moods by the 4 figures, we get a total of 256 possible
configurations of terms in a syllogism. However, as we'll be
relieved to learn, it turns out that all but twelve of these are
invalid! Moreover, the rules for Venn diagrams apply only to the
modern system (in which the possibility of an empty category exists), and this further reduces the diagrammable number of valid
configurations to eight. (Not so bad after all!)
The only mood
that is valid in all
figures is EIO:
Figure 

Pattern of Premises in EIO Mood 
1st 

No M are P, and some S
are M, so some S are not P. 
2nd 

No P are M, and some S
are M, so some S are not P. 
3rd 

No M are P, and some M
are S, so some S are not P. 
4th 

No P are M, and some M
are S, so some S are not P. 
Let's take a look at how this mood
appears in a Venn diagram, using the firstfigure sequence. If we'd
like a concrete example, this one will do:
No shales are limestones.
Some fossilbearing rocks are shales.
So, some fossilbearing rocks are not limestones.
The major premise, "No M are
P," requires us to shade out any areas where the M (shales) and
P (limestones) circles overlap. The minor premise, "Some S are M,"
is marked by placing an x in the only remaining area where the
S (fossilbearing rocks) and M (shales) circles overlap. Since this
occupied area isn't shared
by the P (limestones) circle, the conclusion, "Some S
(fossilbearing rocks) are not P (limestones),"
is visually affirmed.
Other moods that can be valid
(depending on figure)
are: AAA, AEE, AII, AOO, EAE, IAI, and OAO. Some additional moods, which
are invalid by modern standards, but which can be valid under the
Aristotelian system are: AAI, AEO, and EAO. From this, if
we're observant, we can detect some general points which might come in
handy at some juncture.

A valid conclusion can be argued
only if at least one of the premises is a universal (A or E)
statement.

A valid conclusion having a universal
form (A or E) can be argued only from all universal premises
(moods AAA, AEE, and EAE).

A particular (I or O) conclusion argued
from two universal (A and / or E) premises can be valid only under the Aristotelian
(noemptycategories) system.
We won't bother to diagram all
twelve potentially valid moods, nor even just the eight that modern
thought accepts. But let's try one more, to demonstrate a pattern
produced when both premises are universal
statements:
All whales are mammals.
No mammals are fish.
So, no whales are fish.
First, we identify each statement's
function in the syllogism. The conclusion is "no whales are
fish;" the subject term is "whales" and the predicate term is "fish."
This time, the premise that contains the conclusion's predicate term is the second
one, "No mammals are fish," so this is the major premise. But to
determine the syllogism's logical configuration, we need to consider the
statements according to the standard form—major premise, minor premise,
and conclusion, in that order.
No mammals are fish.
All whales are mammals.
So, no whales are fish.
The major premise is an E
statement; the minor premise has an A form; and the conclusion is
another E form. So this syllogism has mood EAE, and we'd expect the
diagrammed pattern to be different from the AII and EIO syllogisms we
diagrammed before. Let's see. The subject term is "whales,"
the predicate term is "fish," and the middle term is "mammals."
Since both premises are universal, let's just diagram them in order.
The major premise relates the
middle term "mammals" to the predicate term "fish," declaring them
mutually exclusive. So, we begin by shading out (in red) the
overlapping areas between the middle and predicate categories, to
mark this region as vacant.
The minor premise relates the
subject term "whales" to the middle term "mammals,"
with a claim that all
members of the category of whales are also in the category of mammals.
In other words, there are no whales that are not mammals. So, we
shade out (in blue) any areas of the subject category (whales) that
don't overlap the middle category (mammals).
From the completed diagram, we can visually observe
that the only potentially occupied area of the "whales" category (the
S circle) is the unshaded portion that overlaps the "mammals"
category (the M circle) but not the "fish" category (the P
circle). Since both of the premises of this syllogism are
universals, the only areas marked are those which are ruled out as
unoccupied.
Like most people, Aristotle would have reasoned
that, if there is such a thing as all whales, then surely some whales must exist in the remaining presumably
nonvacant portion of the S circle. But Venn, being a modern
thinker who accepts zero as a possible value for all, does
not make such an assumption. Lacking a clear stipulation that some whales exist, we
can't jump to the conclusion that there must be whales in the remaining portion of
the S circle.
But suppose that, besides the syllogism, we have
additional information that at least one whale exists. All whales
must occupy the S circle, and they can occupy only an unshaded,
nonempty area. There's only one area in the diagram that
satisfies both requirements: where the S (whales) circle overlaps
the M (mammals), but not the P (fish). So, that's where we place the X.
And indeed, this corresponds to the conclusion, "No whales are fish."
Other unshaded areas might or might not have things in them, but those
things are not whales.
HINT: If you find yourself
using lots of Venn diagrams as aids to thinking or communication,
consider creating a printable page of blank diagrams (rectangle and
circles only, with no shading or occupancy markings) to use as a handy
template. If you wish, you may use the following sample as a
prototype. Highlight and copy the entire contents of the box
below, and insert / paste it into a wordprocessor page, with margins
set to 0.5 inches (1.27 cm.). Then save the file, and print it to
use as a sketch sheet in your projects.
Subject:
____________________________________
S =
________________________________________
M = _______________________________________
P = ________________________________________
Valid? Yes / No (circle one)

Subject:
____________________________________
S =
________________________________________
M = _______________________________________
P = ________________________________________
Valid? Yes / No (circle one)

Subject:
____________________________________
S =
________________________________________
M = _______________________________________
P = ________________________________________
Valid? Yes / No (circle one)

Subject:
____________________________________
S =
________________________________________
M = _______________________________________
P = ________________________________________
Valid? Yes / No (circle one)

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