7.3.1.1 Learning Categorical Syllogisms

Syllogism exercise examples

Learning Categorical Syllogisms

I. Follow these seven steps for complete analysis of syllogisms:

1. identify the conclusion; if a premise is missing (i.e., if it’s an enthymeme), determine what the missing premise should say.

2. put all statements into standard form (quantifier/ subject category/ “to be”/ predicate category)

3. put the syllogism in standard form:  major premise first; three terms each occurring twice. If there are four or five terms, and some are the negation of others, use conversion, obversion or contraposition to rearrange and reduce the number of terms to three.

4. name the mood and figure

5. diagram and read

6. confirm with the rules

7. if invalid, provide a counterexample.

 

II. Exercises. 

1. Only drivers have accidents and Joe has never had one, so he mustn’t be a driver.

2. Only lawbreakers go to jail, so King must have been a lawbreaker.

3. All philosophy majors are good at logic because all good logic students are jazz lovers and all philosophy majors are jazz haters.

4. Only stupid people smoke.  It destroys your lungs.

5. If the butler did it, he had to have known either the victim or her husband, so he couldn’t have done it, since he knew neither of them.

6. Fennel is better than cumin because it’s more expensive.

7. All invalid syllogisms break a rule, so Plato’s is valid since there’s no rule it breaks.

8. Only valid syllogisms are sound, and Russell’s is not invalid, so it’s sound.

9. No valid syllogism has two negative premises, but since Spinoza’s has two negatives it must be invalid.

10. Only syllogisms with distributed middles are valid, so Hegel’s is invalid.

11. Anyone who can analyze a syllogism can see this one is valid, and I can see that.

12. Some intelligible statements are true because all unintelligible ones are meaningless even though some false statements are meaningful.

13. According to the Pew poll, 43% of Americans 18-29 years old don’t know what the Supreme Court decided last week on the Affordable Care Act. But if that’s true, they’re very susceptible to manipulation by the campaign ads.

14. We are sadly mistaken if we feel that the election of Albert Boutrell as mayor will bring the millennium to Birmingham. While Mr. Boutrell is a much more gentle person than Mr. Connor, they are both segregationists, dedicated to maintenance of the status quo.

15. All segregation statutes are unjust because segregation distorts the soul and damages the personality.

Solutions

Solutions

1. Only drivers have accidents and Joe has never had one, so he mustn’t be a driver.

All people who have accidents are drivers.

All people who are Joe are people who never had an accident.

No people who are Joe are drivers

AAE1?  No, because there are four terms in this rendition. “People who never have accidents” and “People who have accidents” are the complement of one another, so Obversion should be used on the minor premise, making it “No people who are Joe are people who have accidents.”  So it’s AEE1.  valid

 

2. Only lawbreakers go to jail, so King must have been a lawbreaker.

Missing premise: King went to jail.

All people who go to jail are lawbreakers.

All people who are King are people who went to jail

All people who are King are lawbreakers.

AAA1 valid

 

3. All philosophy majors are good at logic because all good logic students are jazz lovers and all philosophy majors are jazz haters.

All people who are good at logic are jazz lovers

All Phil majors are jazz haters.   Obvert to:     No Phil majors are jazz lovers

All Phil majors are people who are good at logic

AEA 2    Negative premise without a negative conclusion. Invalid

 

 

4. Only stupid people smoke.  It destroys your lungs.

All people who are destroying their lungs are stupid people

All smokers are people who are destroying their lungs

All smokers are stupid people.

AAA 1 valid

 

5. If the butler did it, he had to have known either the victim or her husband, so he couldn’t have done it, since he knew neither of them.

All doers of the deed are people who knew either the victim or her husband.

No person= the Butler are people who knew either the victim or her husband.

No person = the Butler is the doer of the deed.

AEE2  valid

 

 

 

6. Fennel is better than cumin because it’s more expensive.

All things that cost more than cumin are things that are better than cumin

All things that are fennel are things that cost more than cumin

All things that are fennel are things that are better than cumin

AAA1

 

 

7. All invalid syllogisms break a rule, so Plato’s is valid since there’s no rule it breaks.

All invalid syllogisms are rule breakers

No syllogisms=Plato’s are rule breakers

All syllogisms= Plato’s are valid syllogisms.  Obvert to  No syllogisms=Plato’s are invalid syll.

AEE2  valid

 

 

8. Only valid syllogisms are sound, and Russell’s is not invalid, so it’s sound.

All sound syllogisms are valid syllogisms

No syllogisms=Russell’s are invalid syllogisms  Obvert to    All R are valid syllogisms

All syllogisms=Russell’s are sound syllogisms

A SV

E R~V

A RS

This has four terms; the easiest fix is to obvert the minor premise to A RV. That makes it an  AAA2, which has an undistributed middle term. Invalid.

 

Counterexample:

All cats are animals   T

All dogs are animals  T

All dogs are cats        F

 

9. No valid syllogism has two negative premises, but since Spinoza’s has two negatives it must be invalid.

No valid syllogisms are syllogisms with two negative premises

All syllogisms = Spinoza’s are syllogisms with two negative premises

All syllogisms = Spinoza’s are invalid syllogisms

E VN

A SN

A S~V

That makes four terms; the easiest fix is to obvert the conclusion to E SV.  So it’s an EAE 2, which is valid.

 

10. Only syllogisms with distributed middles are valid, so Hegel’s is invalid.

All valid syllogisms are syllogisms with distributed middles.

No syllogisms= Hegel’s are syllogisms with distributed middles.

All syllogisms= Hegel’s are invalid syllogisms

A VD

E HD

A H~V     This has four terms, and the easiest solution is to obvert the conclusion to E HV No syllogisms=Hegel’s are valid syllogisms.

AEE 2  is valid.

 

11. Anyone who can analyze a syllogism can see this one is valid, and I can see that.

All people who can analyze syllogisms are people who can see this one is valid.

All people=me are people who can see this one is valid

All people=me are people who can analyze syllogisms.

(An enthymeme with an unstated but obvious conclusion.)

A AP

A MP

A MA

This is AAA2, which is invalid, because it has an undistributed middle term.

Counterexample:

All dogs are animals T

All cats are animals  T

All cats are dogs      F

 

12. Some intelligible statements are true because all unintelligible ones are meaningless even though some false statements are meaningful.

I  ~T  M

A  ~I  ~M    Contrapose to    A  M I

I   I T           Obvert to:          O  I ~T

So it is IAO 4  (invalid) , not IAI 2.     Negative conclusion without negative premise. Illicit major term also

Counterexample:

Some animals are cats   T

All cats are mammals   T

Some mammals are not animals    F

 

13. According to the Pew poll, 43% of Americans 18-29 years old don’t know what the Supreme Court decided last week on the Affordable Care Act. But if that’s true, they’re very susceptible to manipulation by the campaign ads.

All people who are ignorant of the SC ruling on the ACA are people susceptible to manipulation by campaign ads.

Some people are people who are ignorant of the SC ruling on the ACA.

Some people are people susceptible to manipulation by campaign ads.

 

A  I S

I  P I

I  P S

AII -1  valid

 

14. We are sadly mistaken if we feel that the election of Albert Boutrell as mayor will bring the millennium to Birmingham. While Mr. Boutrell is a much more gentle person than Mr. Connor, they are both segregationists, dedicated to maintenance of the status quo.

No segregationists are people committed to correcting what is wrong with Birmingham.

All people = Boutrell are segregationists

No people = Boutrell are people committed to correcting what is wrong with Birmingham.

EAE 1  Valid
15. All segregation statutes are unjust because segregation distorts the soul and damages the personality.

All laws that distort the soul and damage the personality are unjust laws.

All segregation statutes are laws that distort the soul and damage the personality.

All segregation statutes are unjust laws.

AAA 1

 

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