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Categorical Syllogisms
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You will be able to :
Describe a standard form categorical
syllogism
Recognize the terms of a syllogism
Identify the mood and figure of a syllogism
Use the Venn Diagram technique for testing
syllogisms
List and describe the syllogistic rules and
syllogistic fallacies
List the 15 valid forms of the categorical
syllogism
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A syllogism is a deductive argument in which a
conclusion is inferred from two premises.
Every syllogism has three terms: a major term, a
minor term, and a middle term.
The major term is the predicate of the syllogism;
the minor term is the subject; and the middle
term appears in both premises but not in the
conclusion.
A categorical syllogism is a deductive
argument consisting of 3 categorical
propositions that together contain exactly 3
terms, each of which occurs in exactly 2 of the
constituent propositions.
No heroes are cowards
 Some soldiers are cowards
 Therefore, some soldiers are not heroes
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A categorical syllogism is in standard
form when its propositions appear in the
order major premise, minor premise, and
conclusion
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To identify the terms by name, look at
the conclusion:
 Some heroes are not soldiers
 Major term – term that appears as the
predicate (heroes)
 Minor term – term that appears as the subject
(soldiers)
 Middle term – term that never appears in the
conclusion (cowards)
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Major premise
› Contains the major term (heroes)
› No heroes are cowards
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Minor premise
› Contains the minor term (soldiers)
› Some soldiers are cowards
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Order of standard form
› The major premise is stated first
› The minor premise is stated second
› The conclusion is stated last
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Its mood is determined by the three
letters identifying the types of its four
propositions (A, E, I, and O). There are 64
possible different moods.
› No heroes are cowards (E proposition)
› Some soldiers are cowards (I proposition)
› Therefore, some soldiers are not heroes (O
propostion)
› Mood - EIO
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The figure of a syllogism is determined by the position
of the middle term in its premises.
There are four possible figures:
First Figure - middle term is the subject of the major
premise and the predicate term of the minor premise
Second Figure – middle term is the predicate term of
major and minor premises
Third Figure – middle term is the subject of both
premises
Fourth Figure – middle term is the predicate term of
the major premise and the subject term of the minor
premise
First Figure
M—P
S—M
.:S—P
Secod Figure
P—M
S—M
.:S—P
Third Figure
M—P
M—S
.:S—P
Fourth Figure
P—M
M—S
.:S—P
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Back to our example
› No heroes are cowards (E)
› Some soldiers are cowards (I)
› Therefore, some soldiers are not heroes (O)
› Middle term (cowards) appears as predicate in
both premises – this makes it a 2nd figure
› Figure and mood together determine a
categorical syllogism’s logical form.
› The logical form of this syllogism is EIO-2
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Since all 64 moods can appear in all four
figures, there are 256 standard form
categorical syllogisms. Only 15 are valid.
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A valid syllogism is valid by virtue of its
form alone
› AAA-1 syllogisms are always valid
 All M is P
 All S is M
 Therefore, all S is P
They are valid regardless of subject matter
All Greeks are human
All Athenians are Greek
Therefore, all Athenians are human
All AAA-1 syllogisms are valid.
All M is P.
All S is M.
.:All S is P.
All sticking with HUM200 are smart
people.
All reading this are sticking with HUM
200.
.:All reading this are smart people.
Deductive logic aims to discriminate
between valid and invalid arguments.
 The validity or invalidity of a syllogism is
entirely a function of its form or structure.
 At times, mere inspection is enough to
determine that an argument is valid.
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By logical analogy we can use syllogisms' mood and figure to show the
validity (or invalidity) of other syllogisms.
Liberals want universal
health insurance.
Some in the
administration want
All P are M.
universal health
Some S are M.
insurance.
.:Some S are P.
.:Therefore, some in the
administration are
liberals.
All rabbits run fast.
Some horses run fast.
.:Therefore some horses
are rabbits.
The conclusion known to be false (some horses are rabbits) proves
that all (AII–2) syllogisms are invalid.
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When mere inspection will not reveal whether an argument is
valid or invalid, Venn diagrams can be used to test for validity.
In addition, there are six essential rules for standard form
categorical syllogisms—and six corresponding fallacies which
occur when these rules are broken:
Rule 1: A syllogism must contain exactly three terms, each of
which is used in the same sense.
Rule 2: The middle term must be distributed in at least one
premise.
Rule 3: If either term is distributed in the conclusion, then it must
be distributed in the premises.
Rule 4: No syllogism can have two negative premises.
Rule 5: If either premise is negative, the conclusion must be
negative.
Rule 6: No syllogism with a particular conclusion can have two
universal premises.
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If S is Swedes, P is peasants and M is musician then:
› Spm represents all Swedes who are not peasants or
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musicians
SPm represents all Swedes who are peasants but not
musicians
sPm represents all peasants who are not Swedes or
musicians
spM – are all musicians who are not Swedes or peasants
sPM – are all peasants who are musicians but not
Swedes
SpM – are all Swedes who are musicians but not
peasants
SPM – are all Swedes who are musicians and peasants
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Venn Diagram for all
M are P
You black out/shade
out all the M that is
not in the P since it
only exists in the P
Mp=0
Venn Diagram for All
M are P and All S are
M
 "All M is P" (Mp=0) and
"All S is M" (Sm=0)
 Shade out all areas of
M that are not in P,
then shade out all
areas of S that are not
in M – leaving only a
small area of
convergence
 This is an AAA-1 valid
syllogism
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Invalid syllogisms give
invalid diagrams:
 All dogs are mammals.
 All cats are mammals.
 Therefore, all cats are
dogs.
 (All S is M. All P is M. All
S is P. AAA-2)
 All cats are clearly not
dogs as seen in the
diagram
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Diagram the
universal premise first
if the other premise is
particular (ii)
All artists (M) are
egoists (P)
Some artists (M) are
paupers (S)
Therefore, some
paupers (S) are
egoists (P)
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(i) All great scientists
(P) are college
graduates (S)
Some professional
athletes (M) are
college graduates
(S)
Therefore, some
professional athletes
(M) are great
scientists (P)
Label the 3 circles of a Venn Diagram
with the syllogism’s 3 terms
 Diagram both premises, starting with the
universal premise
 Inspect the diagram to see whether the
diagram of the premises contains a
diagram of the conclusion
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Rule 1 . Avoid using 4 terms (even
unintentionally)
› Power tends to corrupt
› Knowledge is power
› Knowledge tends to corrupt
› Although this seems to have 3 terms, it
actually has 4 since the word power is being
used in 2 different ways. In the first sense it
means control over things or people; in the
second it means the ability to control things.
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Rule 2. Distribute the middle term in at least one
premise.
If the middle term is not distributed into at least one
premise, the connection required by the conclusion
cannot be made.
Fallacy of the undistributed middle:
› All sharks are fish
› All salmon are fish
› Therefore, all sharks are salmon
› The middle term is what connects the major and minor
terms. If the middle term is not distributed, then the major
and minor terms might be related to different parts of the
M class, thus giving no common ground between the S
and P.
Rule 3. Any term distributed in the
conclusion must be distributed in the
premises.
 When the conclusion distributes a term
that was undistributed in the premises, it
says more about the term than the
premise did.
 Fallacy of illicit process
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› All tigers are mammals
› All mammals are animals
› Therefore, all animals are tigers
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Rule 4. Avoid two negative premises.
2 premises asserting exclusion cannot provide
the linkage that the conclusion asserts.
Fallacy of the exclusive premises
› No fish are mammals
› Some dogs are not fish
› Some dogs are not mammals
› If the premises are both negative then the
relationship between P and S is denied. The
conclusion cannot, therefore, say anything in a
positive manner. That information goes beyond what
is contained in the premises.
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Rule 5. If either premise is negative, the
conclusion must be negative.
Class inclusion can only be stated by
affirmative propositions
 Fallacy of drawing an affirmative
conclusion from a negative premise
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› All crows are birds
› Some wolves are not crows
› Some wolves are birds
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Rule 6. From two universal premises no
particular conclusion may be drawn.
Universal propositions have no existential
import
Particular propositions have existential import
Cannot draw a conclusion with existential
import from premises that do not have
existential import
Existential fallacy
› All mammals are animals
› All tigers are mammals
› Some tigers are animals
Rule
Fallacy Avoided
Rule 1. Avoid four terms.
the fallacy of four terms
Rule 2. Distribute the middle in at
least one premise.
the fallacy of the undistributed
middle
Rule 3. Any term distributed in the
conclusion must be distributed in
the premises
the fallacy of illicit process
illicit process of the major term
(illicit major)
illicit process of the minor term
(illicit minor)
Rule 4. Avoid two negative
premises.
the fallacy of exclusive premisses
Rule 5. If either premise is
negative, the conclusion must be
negative.
the fallacy of drawing an
affiermative conclusion from a
negative premiss
Rule 6. From two universal
premises no particular conclusion
may be drawn.
the existential fallacy
There are 64 possible moods
 There are 4 possible figures
 There are 64x4 = 256 possible logical
forms
 Only 15 are valid
 It is possible, through a process of
elimination, to prove that only these 15
forms can avoid violating all six basic
rules.
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First Figure
Second Figure
Third Figure
Fourth Figure
M-P
S-M
P-M
S-M
M-P
M-S
P-M
M-S
1. AAA-1
Barbara
5. AEE-2
Camestres
9. AII – 3
Datisi
13. AEE-4
Camenes
2. EAE -1
Celarent
6. EAE -2
Cesare
10. IAI – 3
Dismasis
14. IAI-4
Dimaris
3. AII-1
Darii
7. AOO – 2
Baroko
11. EIO-3
Ferison
15. EIO – 4
Fresison
4. EIO -1
Ferio
8. EIO -2
Festino
12. OAO -3
Bokardo
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1. “All good stereos are made in Japan, but no good stereos are
inexpensive; therefore, no Japanese stereos are inexpensive.”
Rewrite this syllogism in standard form, and name its mood and
figure.
2. Come up with a random list of four possible moods; then, pick
one of the four figures and use it to produce four different
syllogisms. Are any of the syllogisms valid?
3. What is the method of logical analogies? Apply it to this
argument to see if it is valid: “No logic professors are successful
politicians, because no conceited people are successful
politicians, and some logic professors are conceited people.”
4. Write out AOO-3 using S and P as the subject and predicate
terms and M as the middle term. (You may need to refer to the
chart of the four syllogistic figures.)
5. Using the syllogistic form in question #4 (or any other form, if
you like) construct a Venn diagram to test it for validity.
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1. Take a current editorial from a major newspaper (such as The New
York Times) and find a categorical syllogism in it. Then, decide what its
form is, and (using one of the methods for testing validity) label it as valid
or invalid.
2. Describe how Venn diagrams can be used to test the validity of a
standard form categorical syllogism. Then, give an example of one valid
and one invalid form and show how the diagram makes the status of
each clear. (Be sure to mark the premises in the right order!)
3. Explain the steps in one of the cases of the deduction of the 15 valid
forms of the categorical syllogism.
4. Two of the six essential rules for the formation of the standard-form
syllogism concern themselves with the distribution of terms. Explain what
distribution means and why these two rules are necessary. What fallacies
result, for instance, when these rules are broken?
5. Two of the six essential rules for the formation of the standard-form
syllogism discuss the quality of categorical propositions. What are these
two rules, and which fallacies result when they are broken?