Deductive
Logic and Venn Diagrams
Deduction
and Validity
Before getting to Venn Diagrams we should simply review here
some basic points about deductive arguments generally and validity in particular. Deductive
arguments are arguments wherein the conclusion is necessarily true (assuming
true premises and a valid form). In other words, it is impossible to
have a situation where: (1) the premises of the argument are true, and (2) the
form of the argument is valid, and (3) the conclusion is false. The reason for
this is very simple: the conclusion of a deductive argument does not contain
any new information it is already contained (in some implicit form) in the
premises itself.
Further, we can see from the above that the concept of
validity is very important for deductive arguments. The conclusion is
guaranteed to be true only if the form of the argument is valid and the
premises are true. Hence, we must know how to determine whether or not a
deductive argument is valid. NOTE: validity and invalidity apply only
to deductive arguments. Inductive arguments are neither valid nor invalid.
So, what is validity? Questions of validity are purely
formal. In testing for validity we are not in any way concerned with the
actual content of an argument. We are only concerned with its form the way in
which the premises are supposed to provide support for the conclusion. So, the
first step in testing for validity will always involve some process of abstraction
that is, some process wherein we ignore the content and focus only on the
form of the argument. Abstraction will almost always involve, in this context,
the replacement of particular contents with “variables” usually
letters (A, B, C, D). Second, we then arrange the
variables in the same form as they had in the argument itself that is, we
arrange them according to the specific form of the argument.
Example 1:
1. All
cows are animals 2.
Betsy is a cow. 3. Therefore, Betsy is an animal. 
Note the
“content” of this argument. What are we talking about? Betsy, cows and
animals. So, we assign variables. Cows = C Animals =
A Betsy = B 
Now,
arrange the variables in the same form as they have in the argument and we
get: 1. All C
are A 2. B
is C 3. Therefore, B is A. 
Once we
have the argument translated into variable form we are going to ask a simple question:
given that the premises are true, does the conclusion necessarily follow? There
are, generally, two different ways of answering this question, one is
impossibly hard, the other is very easy.
The impossibly hard method of answering this question is to
go through every possible combination of true premises that meet the form we
are testing and determine if it always gives us a true conclusion. What we are
looking for is simple: can we find an argument where the premises are true, but
the conclusion is false. If we can, then the form is invalid. So, for example,
we can replace C with Humans, A with Mortal, and B with Socrates, getting: All
humans are mortal, Socrates is a human, therefore
Socrates is mortal. Then we can replace C with Dog, A with Mammal, B with
Daisy, and this can go on forever. So, while this method works, it takes a very
very long time.
The easy method is to simply ask yourself:
will this form yield true conclusions given true premises no matter what the
specific content is? Testing this question is simple. Again, there are two
possibilities here. If you have categorical syllogisms then you test
this question by using Venn Diagrams. If you have compound statements
using logical operators, then you use Truth Tables. We are going
to look only at Venn Diagrams here, but the basic principle is the same:
assuming that the premises are true, does the conclusion necessarily follow?
Venn
Diagrams and Validity
A Venn Diagram (V) is a clear cut method for determining the
validity or invalidity of any form of categorical
Syllogism. A categorical syllogism is a deductive argument that
relates (through patterns of inclusion and exclusion) categories of things and
that contains two (and only two) premises. (Of course, a categorical syllogism
may have one explicit and one implicit or hidden premise.) Now, given that
categorical syllogisms articulate relations of inclusion and exclusion between
categories, we can determine that there are four (and only four) basic
categorical claims. These are:
Universal Affirmative. 
This is a
statement that claims that everything that falls within one category also
falls within some other category. E.G. “All
humans are animals.” Note here that we are saying there cannot be something that
is human and is not animal, or that everything we would put in the “human”
category, we would also put in the “animal” category. Note further, that this
is not a reversible relationship. “All A are B” does
not equal “All B are A.” We can call this the relation of Universal Inclusion. 
Universal Negative. 
This is a
statement that claims that nothing within one category also falls within some
other category. In other words, to put something in one category excludes it
from some other category. E.G. “No
human is reptilian.” Note here that we are saying that you cannot put
something in both the “human” category and at the same time in the “reptile”
category. Note that this is reversible: “No A are B” is the same as “No B are
A.” We can call this the relation of Universal Exclusion. 
Particular Affirmative. 
This is a
statement that claims there are some members of one category that also are
members (or fall within) another category. E.G.
“Some humans are rational.” Note here that we are saying that some of the
things that belong to the category of “human” also belong to the category of
“rational.” Further note: there is no implication that if some humans are
rational that some humans are not rational. We can call this the relation of Partial Inclusion. 
Particular Negative. 
This is a
statement that claims that there are some members of one category that are
not members (or do not fall within) another category. E.G. “Some
humans are not reptilian.” Note here that we are saying that there are some
members of the category of “human” that do not fall within the category of
“reptile.” Again, the fact that some humans are not reptilian does not imply
that some humans are. We can call this the relation of Partial Exclusion. 
These four
basic claims form a network of necessary relations. The truth/falsity of one
type of claim about some content will necessarily imply the truth/falsity of
another type of claim about the same object. This set of necessary relations is
known as the “square of opposition” (of which there are two versions). See your
textbook, page 165 for an explanation of the square of opposition. We will not
spend any time on it here. But, as we work through the process of diagramming a
categorical syllogism we will see these relations emerge.
Diagramming Categorical Syllogisms (DCS).
The first step in DCS is very simple. Since arguments are never
written in a nice easy script, it is often necessary to rewrite the argument
into a form that can be easily analyzed and evaluated. We have already seen
this with casting, and we will see this here as well. The key for categorical
syllogisms is to identify the categories being used, and the relations being
drawn between those categories.
Step One: 
First translate
the argument into a categorical structure. This involves identifying the
categories that are being related to one another, and the manner of relation
(Universal Affirmation, Universal Negation, Particular Affirmation, and
Particular Negation). 
Generally
speaking this is easy enough to do. Look at the following examples:
Most apples are red. 
We identify
the categories here as “apples” and “things that are red.” Then, we note that
“most” means the same as “some.” 
Some apples are things
that are red. (The formal relations here are bolded, the categories are underlined.) 
Many people are not as stupid as politicians would have us
believe. 
We
identify the categories here as “people” and “those things that are as stupid
as politicians would have us believe.” NOTE: that we do not include
the not here as part of the category. You should always leave
negations (not, no) outside of categories. Then we see that “many” here means
the same as “some” and since we see a “not” here we obviously have a “some
are not” or particular negation. 
Some people are not among
those things that are as stupid as politicians would have us believe. Rather clumsy, I know, but it helps in the diagramming. 
There are no politicians that are honest. 
The
categories here are clearly “politicians” and “things that are honest”, the
“no” indicates we have a Universal Negation. 
No politicians are things that are honest. 
Everyone who works in 
The
categories are “people who work in 
All people who work in 
On the
whole, this process is rather simple and with practice you will become an
expert at it. There are, however, a few simple rules to keep in mind.
Rules for
Translation (See
textbook, pages 161162)
Rule One 
If you
have a statement that reads: “All people are not stupid” or any form where
you have “All” linked with “are not” then this cannot be translated. This is because
such claims can be translated as either “Some people are not stupid” or “No
people are stupid.” 
Rule Two 
When you
have an “individual” named in the argument (like Socrates)
treat the name as a category. Hence, Socrates becomes “people who are Socrates.” 
Rule Three 
Adjectives
become categories. Thus, “Fred is smart”, becomes “All people who
are named Fred are people who are smart.” 
Rule Four 
Turn
verbs into categories as well. Thus, “Fish swim” becomes “All fish
are things that swim.” 
Rule Five 
Categories
should always be positive there is no such thing as a negative category.
Hence, negations should always be part of the form of the translation, never
the category itself. Two examples here: A. “There
are many things that are not human made.” This becomes: “Some things
are not things made by humans.” B. “There
are many things that are unpleasant.” This becomes “Some things are
things that are unpleasant.” Notice
that in (B) we leave “unpleasant” alone this is because “unpleasant” is not
a negation it is not a negation of the whole category but stands as a
category separate from those things that are pleasant. Again, this takes
practice, but you will get the hang of it. 
After completing
this step, you then want to replace the categories with variables (letters):
Step Two: 
Replace categories with variables. 
This is a
relatively simple step and many people as they become better and better at this
process skip step one altogether and proceed to step two. Very simply it
involves assigning some variable to each of the categories. Usually it is best
to pick a variable (pick a letter) which is related to the category itself.
Thus, use “C” to stand for “cat” and “D” to stand for “dog.” Then, you should
rewrite the argument in variable form, much like we did in the first example:
1. All
cows are animals 2. Betsy
is a cow. 3. Therefore, Betsy is an animal. 
Note the “content”
of this argument. What are we talking about? Betsy, cows and animals. So, we
assign variables. Cows = C Animals =
A Betsy = B 
Now,
arrange the variables in the same form as they have in the argument and we
get: 1. All C
are A 2. B is C 3. Therefore, B is A. 
Then, after step two, we are ready for the process of diagramming.
The process of diagramming is again very simple. We use overlapping circles to
represent the various categories and their interrelation. Since all categorical
syllogisms will have three (and only three) categories, there will always be
three circles.
(See Fig. 1)
Fig 1. 
The “seven domains”
found in this figure are important. Each represents a different possibility of
inclusion and exclusion. We shall look at each in turn:
Domain One 
This domain is where you put something which is “A,” but is
in neither of the other circles, “B” or “C.” 
Domain Two 
This domain is where you put something which is in both
“A” and “B” but not in “C.” 
Domain Three 
This domain is where you put something which is in “B” but
not in “A” or “B.” 
Domain Four 
This domain is where you put something that is both “A”
and “C” but not in “B.” 
Domain Five 
This domain is where you put something that is in all
three circles. 
Domain Six 
This domain is where you put something that is both in “B”
and “C” but not in the “A” circle. 
Domain Seven 
This domain is where you put
something that is only in “C” but is neither in “A” nor “B.” 
Notice that
these domains all deal with relations of inclusion and exclusion. When diagramming
specific claims (e.g. All A are B, No A are B, Some A
are B, Some A are not B) we use two (and only two) basic methods: Shading and Putting an X.
Shading
and Putting and X
Shading is only used when dealing with “All” and “No” claims
(Universal affirmation and negation), putting an X is used only when dealing
with “Some are” and “some are not” claims (Particular affirmation and
negation). This gives us then, Steps three and four (with a few corresponding
rules):
Step Three 
Shade in all the “All” or “No” claims found in the
argument. Note that you should always start with the “all” or “no” claims. Never
begin diagramming with a “some” or “some are not” claim. 
Rules for Shading 
If you shade in a section you are literally saying that
“nothing can go in there” 

Hence, if we have a claim that says: “All A are B” we are
going to shade in all of the “A” circle except where
it overlaps the “B” circle (this includes the places where it overlaps the
“C” circle). By doing this we are saying that you cannot put something into
the “A” circle without at the same time putting it into the “B” circle. 

If you have a claim that says “No A are B” then we would
shade in all of the area where “A” and “B” overlap (including where both
overlap with “C”) in order to indicate that you cannot put something in the
place where “A” and “B” overlap. 


Step Four 
If there are any “some” or “some are not” premises, put an
“X” where the premise indicates an “X” should be. 
Rules for “X” 
For “some .... are...” claims,
you should put an “X” in the overlap between the two categories
indicated. 

For “some ... are not....” claims, you should put an “X” outside
of the overlap between those claims. 

If there is more than one domain in which the “X”
might go, then you put the “X” on the line between the two domains. 

Any time there is an “X” on the line, then
you necessarily have an invalid argument. 


Step Five 
Now, you check for validity. Note, that you only diagram
the premises. After having diagramed the premises, the conclusion
should be evident. If it is not evident, if you have to do more work to make
the conclusion evident, then the argument is clearly invalid. 
This will
be easier if we look at some examples.
A simple
example first: [Note: I cannot shade
or put an X in these diagrams do to
the limits of word and my knowledge of how to manipulate it –so, try to figure
out how to do these by reading the descriptions in the third column.]
1. All A
are B 2. All
B are C 3. All A are C 

1. Only
diagram the premises (1 & 2) 2. Start
with “All” or “No” claims (universal claims) only universal claims, so no
problem there. 3. Shade
Universal claims. 4. This is a clearly valid claim because the conclusion
the idea that All A are C is clearly indicated here there is no way to
put something into the A circle without also putting it into the C circle,
since only Domain 5 is open to the A circle and this includes the C circle. 
1. All A
are B 2. No
C are B 3. No A
are C 

1. Only
diagram the premises (1 & 2) 2. Start with
“All” or “No” claims (universal claims) only universal claims, so no
problem there. 3. Shade
Universal claims. 4. Once again, this is a clearly valid claim because the
conclusion the idea that No A are C is clearly
indicated here. There is no place where the A and the C circle overlap that
has not been shaded in. 
1. No A
are B 2. All
B are C 3. No A
are C 

1. Only
diagram the premises (1 & 2) 2. Start with
“All” or “No” claims (universal claims) only universal claims, so no
problem there. 3. Shade
Universal claims. 4. Now, notice, that the conclusion claims that No A are
C, this would mean that the area between A and C is completely shaded in
(domains 4 & 5), but it is not. Hence, the conclusion is not contained in
the premises, so this argument is INVALID. 
1. No A
are B 2.
Some Bare C 3. Some C
are not A 

1. Only
diagram the premises (1 & 2) 2. Start
with universal claims (premise 1). 3. Shade
universal claims: here we shade out the area between A and B to indicate that
there is no possibility of something being both A and B (domains 2 & 5) 4. Now
put in the X. Note we are saying that some B are C.
So the X should go into the overlap between B and C. There is only one open
domain in the overlap between B and C (domain 6). So we put an X there. 5. Now, is the conclusion, that some C are not A here?
Where is the X? Is it in the C? Yes. Is it in the A? No. Well, that’s it.
That is what the conclusion calls for. So, this is indeed a Valid argument. 
1. Some A
are B. 2. All
B are C. 3. Some A are C. 

1. Only
diagram the premises (1 & 2) 2. Start
with the universal claims here that is premise 2. 3. Shade
universal claim: shade all area of B that is outside of C (domains 2 &
3). This shows that something cannot be B without at the same time being C. 4. Put in
the X (premise 1). Now, the X should go in A and in B. So, it must go in the
overlap. However, there is only one place in the overlap that has not been
shaded out domain 5. So, we put the X there. 5. Is this valid? The conclusion says that Some A are C. With an X in domain 5 we are saying the same thing
so this is valid. 
1. Some A
are B 2. All
C are B 3. Some A
are C. 

1. This
looks like the last one, but is not. 2. Only
diagram the premises, 1 & 2 3. Start with
universal claims, again 2. 4.
Diagram universal claims shade in area of C outside of B (domains 4 &
7). 5. Now
put an X. The claim is that you need to put an X between A and B. There are two
possibilities: domain 2 and domain 5. Are either shaded out? No. So the X
could go into either area since it is not clear which area into which it
should go, we put the X on the line between the two
areas. 6. This is invalid because there is an X on the line. 
Okay. Now you just have to
practice this. Good luck.