Chapter 6

Categorical Syllogisms

Objectives

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

Syllogisms

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.

Categorical Syllogisms Examples

No heroes are cowards Some soldiers are cowards Therefore, some soldiers are not

heroes

A categorical syllogism is in standard form when its propositions appear in the order major premise, minor premise, and conclusion

Identification of terms

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)

Terms, continued

Major premise› Contains the major term (heroes)› No heroes are cowards

Minor premise› Contains the minor term (soldiers)› Some soldiers are cowards

Order of standard form› The major premise is stated first› The minor premise is stated second› The conclusion is stated last

Mood of the Syllogism

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

Figure of the Syllogism

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

Figure of the Syllogism

First Figure Secod Figure Third Figure Fourth Figure

M—PS—M.:S—P

P—MS—M.:S—P

M—PM—S.:S—P

P—MM—S.:S—P

Figure of the Syllogism 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

Figure of the Syllogism

Since all 64 moods can appear in all four figures, there are 256 standard form categorical syllogisms. Only 15 are valid.

The Formal Nature of Syllogistic Logic

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 matterAll Greeks are humanAll Athenians are GreekTherefore, all Athenians are human

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.

All AAA-1 syllogisms are valid.

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.

Liberals want universal health insurance.Some in the administration want universal health insurance..:Therefore, some in the administration are liberals.

All P are M.Some S are M..:Some S are P.

All rabbits run fast.Some horses run fast..:Therefore some horses are rabbits.

By logical analogy we can use syllogisms' mood and figure to show the validity (or invalidity) of other syllogisms.

The conclusion known to be false (some horses are rabbits) proves that all (AII–2) syllogisms are invalid.

Venn Diagrams

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.

Ven

n D

iag

ram

s

If S is Swedes, P is peasants and M is musician then: › Spm represents all Swedes who are not peasants or

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

Venn Diagrams

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 Diagrams

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

Venn Diagrams

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

Venn Diagrams

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)

Venn Diagram

(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)

Venn Diagrams

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

Rules and Fallacies

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.

Rules and Fallacies

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.

Rules and Fallacies 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› All tigers are mammals› All mammals are animals› Therefore, all animals are tigers

Rules and Fallacies

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.

Rules and Fallacies

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› All crows are birds› Some wolves are not crows› Some wolves are birds

Rules and Fallacies

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

Rules and FallaciesRule 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 processillicit 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

15 Valid Forms

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.

The 15 Valid Forms

First Figure Second Figure Third Figure Fourth Figure

M-PS-M

P-MS-M

M-PM-S

P-MM-S

1. AAA-1Barbara

5. AEE-2Camestres

9. AII – 3Datisi

13. AEE-4Camenes

2. EAE -1Celarent

6. EAE -2Cesare

10. IAI – 3Dismasis

14. IAI-4 Dimaris

3. AII-1Darii

7. AOO – 2Baroko

11. EIO-3Ferison

15. EIO – 4Fresison

4. EIO -1Ferio

8. EIO -2Festino

12. OAO -3Bokardo

Questions for Discussion

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.

Essay

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?