Solutions Chapter 6

CHAPTER 6

Categorical Logic: Syllogisms

The starred items are also contained in the Answer Key in the back of The Power of Logic.

Exercise 6.1

Part A: Standard Form

*1. 1. All novels are books.

2. Some works of art are books.

So, 3. Some works of art are novels.

2. Already in standard form

3. Already in standard form

*4. 1. Some beautiful things are paintings.

2. All sculptures are beautiful things.

So, 3. Some sculptures are not paintings.

5. 1. All famous short stories are interesting stories.

2. Some short stories are not interesting stories.

So, 3. Some short stories are not famous short stories.

6. 1. No millionaires are poor persons.

2. Some artists are millionaires.

So, 3. Some artists are not poor persons.

*7. 1. All sadists are mean persons.

2. All art critics are mean persons.

So, 3. All art critics are sadists.

8. 1. All metaphors are words.

2. All metaphors are figures of speech.

So, 3. All figures of speech are words.

9. Already in standard form

*10. Already in standard form

11. 1. Some comedians are prophets.

2. Some comedians are poets.

So, 3. Some poets are prophets.

12. 1. All novelists are authors.

2. All authors are insightful persons.

So, 3. All insightful persons are novelists.

*13. 1. No aspiring actors are saints.

2. Some aspiring actors are not egoists.

So, 3. Some egoists are saints.

14. Already in standard form

15. 1. No ancient Greek poets are persons who are alive.

2. All persons who are alive are artists.

So, 3. No artists are ancient Greek poets.

Part B: Mood and Figure

*1. Second figure: IAI (not valid)

2. Third figure: III (not valid)

3. Fourth figure: AEE (valid)

*4. Third figure: EOI (not valid)

5. First figure: EAE (valid)

6. Fourth figure: EAE (not valid)

*7. Second figure: AII (not valid)

8. Second figure: OOO (not valid)

9. Third figure: OAO (valid)

*10. First figure: IAO (not valid)

11. Second figure: AOO (valid)

12. First figure: EIO (valid)

*13. Second figure: AAA (not valid)

14. Third figure: AAA (not valid)

15. First figure: AII (valid)

Part C: Putting Syllogisms into Standard Form

*1. 1. All cowboys are persons who love horses.

2. Some farmers are not persons who love horses.

So, 3. Some farmers are not cowboys. Second figure: AOO (valid)

2. 1. All operas are events that include singers.

2. No rodeos are operas.

So, 3. No rodeos are events that include singers. First figure: AEE (invalid)

3. 1. No rock stars are rodeo stars.

2. Some rock stars are persons who love country music.

So, 3. Some persons who love country music are not rodeo stars.
Third figure: EIO (valid)

*4. 1. No cowards are bull riders.

2. Some bull riders are fools.

So, 3. Some fools are not cowards. Fourth figure: EIO (valid)

5. 1. All cowgirls are talented riders.

2. No cowgirls are city slickers.

So, 3. No city slickers are talented riders. Third figure: AEE (invalid)

6. 1. Some sheriffs are gunslingers.

2. All gunslingers are drifters.

So, 3. Some drifters are sheriffs. Fourth figure: IAI (valid)

*7. 1. No cattle rustlers are good guys.

2. All cowboys in white outfits are good guys.

So, 3. No cowboys in white outfits are cattle rustlers. Second figure: EAE (valid)

8. 1. All barkeeps who serve rotgut are bad guys.

2. Some barkeeps who serve rotgut are persons who will not live long.

So, 3. Some persons who will not live long are bad guys. Third figure: AII (valid)

9. 1. All good movies are westerns.

2. All movies worth seeing are good movies.

So, 3. All movies worth seeing are westerns. First figure: AAA (valid)

*10. 1. All bulls are animals that are hard to ride.

2. Some broncos are not bulls.

So, 3. Some broncos are not animals that are hard to ride. First figure: AOO (invalid)

11. 1. All cattle barons are wealthy landowners.

2. No wealthy landowners are buckaroos.

So, 3. No buckaroos are cattle barons. Fourth figure: AEE (valid)

12. 1. No bounty hunters are sodbusters.

2. Some outlaws are sodbusters.

So, 3. Some outlaws are not bounty hunters. Second figure: EIO (valid)

*13. 1. No trail bosses are hired hands.

2. Some ranchers are hired hands.

So, 3. Some ranchers are trail bosses. Second figure: EII (invalid)

14. 1. All horse thieves are persons who will be hanged.

2. Some bandits are not horse thieves.

So, 3. Some bandits are not persons who will be hanged. First figure: AOO (invalid)

15. 1. Some sheep ranchers are not fast guns.

2. All sheep ranchers are honest citizens.

So, 3. Some honest citizens are not fast guns. Third figure: OAO (valid)

Part D: Constructing Syllogisms

*1. 1. No ducks are geese.

2. Some birds are ducks.

So, 3. Some birds are not geese. Valid

2. 1. All men are humans.

2. No birds are humans.

So, 3. No birds are men. Valid

3. 1. Some birds are ducks.

2. All birds are living things.

So, 3. Some living things are ducks. Valid

*4. 1. No ducks are men.

2. All men are living things.

So, 3. No living things are ducks. Invalid

5. 1. All ducks are birds.

2. All pintails are ducks.

So, 3. No pintails are birds. Invalid

6. 1. No ducks are geese.

2. Some birds are geese.

So, 3. Some birds are not ducks. Valid

*7. 1. Some birds are not ducks.

2. All birds are living things.

So, 3. Some living things are not ducks. Valid

8. 1. Some birds are ducks.

2. All ducks are living things.

So, 3. Some living things are birds. Valid

9. 1. No geese are ducks.

2. No men are geese.

So, 3. All men are ducks. Invalid

*10. 1. All ducks are birds.

2. Some living things are not birds.

So, 3. Some living things are not ducks. Valid

Exercise 6.2

Part A: Venn Diagrams and Standard Form

*1. Some ancient philosophers are persons who believed in the unreality of change.

2. Some acts of killing are not acts of murder.

3. No divine beings are limited beings.

*4. All persons who deserve harsh treatment from the IRS are tax-dodgers.

5. Some current musical hits are not things that will be hits next year.

6. All persons who believe in reincarnation are persons who believe in life after death.

*7. No chlorofluorocarbons are things that are good for the ozone layer.

8. Some corporations are things that cheat the government.

9. All persons who should be incarcerated are violent offenders.

*10. No physical entities are spiritual entities.

Part B: Venn Diagrams and Arguments

*1. Some chairs are not thrones. So, some thrones are not chairs.

2. All scallops are mollusks. Hence, no scallops are nonmollusks.

3. All minds are brains. Therefore, all nonbrains are nonminds.

*4. Some married persons are persons who have attachment disorders. Thus, some persons who have attachment disorders are married persons.

5. No laypersons are priests. It follows that all laypersons are nonpriests.

6. Some political philosophers are egalitarians. Accordingly, some nonegalitarians are things that are not political philosophers.

*7. No elephants are beetles. Consequently, no nonbeetles are nonelephants.

Note regarding the diagram of the conclusion: The area of overlap between nonbeetles and nonelephants is the area outside BOTH circles; hence, the area outside both circles should be shaded in.

8. No Pickwickian interpretations are obvious interpretations. So, no obvious interpretations are Pickwickian interpretations.

9. Some rays are devilfish. Hence, some devilfish are not nonrays.

*10. Some wines are not merlots. Therefore, some nonmerlots are not nonwines.

11. All acts of torture are immoral acts. It follows that all immoral acts are acts of torture.

12. Some vipers are not copperheads. Consequently, some vipers are noncopperheads.

*13. Some mammals are edentulous animals. Thus, all mammals are edentulous animals.

14. Some boring events are colloquia. Accordingly, some boring events are not colloquia.

15. Some theists are not predestinarians. Therefore, no theists are predestinarians.

Exercise 6.3

Part A: Argument Forms

*1. 1. All M are P.

2. Some M are not S.

So, 3. Some S are not P.

2. 1. No P are M.

2. Some M are not S.

So, 3. Some S are P.

3. 1. No M are P.

2. Some M are S.

So, 3. Some S are not P.

*4. 1. Some P are M.

2. Some S are M.

So, 3. Some S are P.

5. 1. All P are M.

2. Some M are not S.

So, 3. Some S are not P.

6. 1. All P are M.

2. Some S are not M.

So, 3. Some S are not P.

*7. 1. No P are M.

2. Some M are S.

So, 3. Some S are P.

8. 1. All P are M.

2. No S are M.

So, 3. No S are P.

9. 1. No P are M.

2. No S are M.

So, 3. No S are P.

*10. 1. All M are P.

2. No S are M.

So, 3. No S are P.

Part B: Categorical Syllogisms

*1. 1. All Athenians are Greeks.

2. Some humans are not Athenians.

So, 3. Some humans are not Greeks.

2. 1. All sentient things are rights-holders.

2. All animals are sentient things.

So, 3. All animals are rights-holders.

3. 1. No evil things are good things.

2. All serial killers are evil things.

So, 3. No serial killers are good things.

*4. 1. All liars are self-deceived persons.

2. All liars are wicked persons.

So, 3. All wicked persons are self-deceived persons.

5. 1. All persons without a conscience are happy persons.

2. Some criminals are persons without a conscience.

So, 3. Some criminals are happy persons.

6. 1. All persons who have faith are virtuous persons.

2. Some highly moral persons are not persons who have faith.

So, 3. Some highly moral persons are not virtuous persons.

*7. 1. No human beings are omniscient beings.

2. Some divine beings are human beings.

So, 3. Some divine beings are not omniscient beings.

8. 1. Some wars are things ordained by God.

2. All great evils are wars.

So, 3. Some great evils are things ordained by God.

9. 1. All acts worth doing are acts worth doing well.

2. Some hobbies are not acts worth doing.

So, 3. Some hobbies are not acts worth doing well.

*10. 1. All brain events are physical events.

2. No mental events are physical events.

So, 3. No mental events are brain events.

11. 1. Some philosophical views are not views worth considering.

2. All philosophical views are views that have been held by geniuses.

So, 3. Some views that have been held by geniuses are not views worth considering.

12. 1. No wicked persons are persons utterly without a conscience.

2. All wicked persons are deeply confused individuals.

So, 3. No deeply confused individuals are persons utterly without a conscience.

*13. 1. All similarity statements are metaphorical statements.

2. All statements are similarity statements.

So, 3. All statements are metaphorical statements.

14. 1. All morally permissible acts are acts that conform to the categorical imperative.

2. Some acts of suicide are acts that conform to the categorical imperative.

So, 3. Some acts of suicide are morally permissible acts.

15. 1. All obligatory acts are acts that maximize utility.

2. Some acts that maximize utility are not prescribed by the Ten Commandments.

So, 3. Some acts prescribed by the Ten Commandments are not obligatory acts.

*16. 1. No acts foreknown by God are free acts.

2. Some acts are acts foreknown by God.

So, 3. Some acts are not free acts.

17. 1. All moral acts are acts approved of by God.

2. Some acts of killing are acts approved of by God.

So, 3. Some acts of killing are moral acts.

18. 1. All divine beings are omnipotent beings.

2. No human beings are omnipotent beings.

So, 3. No human beings are divine beings.

*19. 1. All unhappy persons are persons who have inner conflicts.

2. Some successful comedians are unhappy persons.

So, 3. Some successful comedians are persons who have inner conflicts.

20. 1. All persons who have walked over others to get to the top are evil persons.

2. Some tycoons are persons who have walked over others to get to the top.

So, 3. Some tycoons are evil persons.

21. 1. Some trees are maples.

2. Some trees are oaks.

So, 3. Some oaks are maples.

22. 1. No balalaikas are banjos.

2. Some balalaikas are beautiful things.

So, 3. Some beautiful things are not banjos.

23. 1. All tyrants are mendacious persons.

2. All tyrants are liars.

So, 3. All liars are mendacious persons.

24. 1. All aphorisms are apothegms.

2. All epigrams are aphorisms.

So, 3. All epigrams are apothegms.

25. 1. All Saint Bernards are large dogs.

2. Some large dogs are not brown dogs.

So, 3. Some brown dogs are not Saint Bernards.

Exercise 6.4

Part A: Argument Forms

1. 1. No M are P.

2. All S are M.

So, 3. Some S are not P.

2. 1. All P are M.

2. No S are M.

So, 3. Some S are not P.

3. 1. All M are P

2. All M are S.

So, 3. Some S are P.

*4. 1. All M are P.

2. All M are S.

3. At least one M exists.

So, 4. Some S are P.

5. 1. No M are P.

2. All M are S.

So, 3. Some S are not P.

6. 1. No M are P.

2. All M are S.

3. At least one M exists.

So, 4. Some S are not P.

*7. 1. No S are P.

So, 2. Some non-P are not non-S.

8. 1. All P are M.

2. No M are S.

So, 3. Some S are not P.

9. 1. All P are M.

2. All M are S.

So, 3. Some S are P.

*10. 1. No S are P.

2. At least one S exists.

So, 3. Some non-P are not non-S.

Part B: Testing Arguments

*1. 1. All persons who never make mistakes are admirable persons.

2. All ideal humans are persons who never make mistakes.

So, 3. Some ideal humans are admirable persons.

2. 1. No plants are animals.

2. All weeds are plants.

So, 3. Some weeds are not animals.

3. 1. No perfect circles are perfect squares.

2. Some perfect circles are objects of beauty.

So, 3. Some objects of beauty are not perfect squares.

*4. 1. All persons who advocate the use of overwhelming nuclear force are persons who lack moral sensibility.

2. All persons who advocate the use of overwhelming nuclear force are persons who should not serve as world leaders.

3. At least one person who advocates the use of overwhelming nuclear force exists.

So, 4. Some persons who should not serve as world leaders are persons who lack moral sensibility.

5. 1. All lions are cats.

2. All cats are mammals.

So, 3. Some mammals are lions.

6. 1. No terrorists who use nuclear weapons are good persons.

2. All terrorists who use nuclear weapons are persons who mean well.

3. At least one terrorist who uses nuclear weapons exists.

So, 4. Some persons who mean well are not good persons.

*7. 1. All sycophants are flatterers.

2. All flatterers are disgusting persons.

3. At least one flatterer exists.

So, 4. Some disgusting persons are sycophants.

8. 1. Some highly educated persons are sybarites.

2. All sybarites are poor role models.

So, 3. Some poor role models are highly educated persons.

9. 1. No oaks are elms.

2. All oaks are trees.

3. At least one elm exists.

So, 4. Some trees are not elms.

*10. 1. No members of the IRA are members of the IRS.

So, 2. It is false that all members of the IRA are members of the IRS.

11. 1. All great inventors are slightly odd persons.

2. At least one great inventor exists.

So, 3. Some slightly odd persons are great inventors.

12. 1. No anarchists are Republicans.

2. At least one anarchist exists.

So, 3. Some non-Republicans are not nonanarchists.

*13. 1. All scarlet things are red things.

So, 2. It is false that no scarlet things are red things.

14. 1. No cities are nations.

2. At least one city exists.

So, 3. Some cities are not nations.

15. 1. All persons with perfect memories are persons who remember everything.

So, 2. Some persons with perfect memories are persons who remember everything.

*16. 1. It is false that some Germans are Zoroastrians.

So, 2. Some Germans are not Zoroastrians.

17. 1. It is false that some vampires are living things.

2. At least one vampire exists.

So, 3. Some vampires are not living things.

18. 1. All persons who do not care about social justice are heartless persons.

So, 2. It is false that no persons who do not care about social injustice are heartless persons.

*19. 1. No kangaroos are karate experts.

So, 2. Some kangaroos are not karate experts.

20. 1. It is false that some tyrants are not humans.

2. At least one tyrant exists.

So, 3. Some tyrants are humans.

Exercise 6.5

Part A: Enthymemes

*1. 1. No certainties are propositions that should be rejected.

2. All self-evident propositions are certainties.

So, 3. No self-evident propositions are propositions that should be rejected.

2. 1. All virtues are beneficial traits.

2. No vices are beneficial traits.

So, 3. No vices are virtues.

3. 1. All humans are rational animals.

2. No ducks are rational animals.

So, 3. No ducks are humans.

*4. 1. All simple substances are indestructible entities.

2. All atoms are simple substances.

So, 3. All atoms are indestructible entities.

5. 1. All envious persons are persons who want others to fail.

2. No good persons are persons who want others to fail.

So, 3. No good persons are envious persons.

6. 1. No liars are praiseworthy individuals.

2. Some liars are human beings.

So, 3. Some human beings are not praiseworthy individuals.

*7. 1. All rational statements are scientific statements.

2. No aesthetic judgments are scientific statements.

So, 3. No aesthetic judgments are rational statements.

8. 1. All value judgments are relative judgments.

2. All verdicts rendered in courts of law are value judgments.

So, 3. All verdicts rendered in courts of law are relative judgments.

9. 1. All perfect beings are beings that have every virtue.

2. Some gods are not beings that have every virtue.

So, 3. Some gods are not perfect beings.

*10. 1. All rational beliefs are beliefs that are proportioned to the available evidence.

2. Some beliefs about aliens are not beliefs that are proportioned to the available evidence.

So, 3. Some beliefs about aliens are not rational beliefs.

11. 1. No matters of faith are provable beliefs.

2. Some beliefs about life after death are matters of faith.

So, 3. Some beliefs about life after death are not provable beliefs.

12. 1. No bears are wolves.

2. Some grizzlies are bears.

So, 3. Some grizzlies are not wolves.

*13. 1. All harmful traits are forms of laziness.

2. All vices are harmful traits.

So, 3. All vices are forms of laziness.

14. 1. All composite substances are substances that have parts.

2. No souls are substances that have parts.

So, 3. No souls are composite substances.

15. 1. All evil things are things to be avoided.

2. Some evil things are pleasurable things.

So, 3. Some things to be avoided are pleasurable things.

Part B: More Enthymemes

*1. 1. All events that are foreseen by God are predetermined events.

2. All events are events that are foreseen by God.

So, 3. All events are predetermined events.

2. 1. All events that are predetermined are necessary events.

2. All events are events that are predetermined.

So, 3. All events are necessary events.

3. 1. All events are necessary events.

2. All sins are events.

So, 3. All sins are necessary events.

*4. 1. No necessary events are events that can be avoided.

2. All justly punished acts are events that can be avoided.

So, 3. No justly punished acts are necessary events.

5. 1. All beings that are omnipotent and perfectly good are beings that create the best of all possible worlds.

2. Some deities are beings that are omnipotent and perfectly good.

So, 3. Some deities are beings that create the best of all possible worlds.

6. 1. All composites are aggregates of simple substances.

2. At least one composite exists.

So, 3. Some composites are aggregates of simple substances.

*7. 1. All things that can be annihilated are things that can come apart.

2. No things that have no parts are things that can come apart.

So, 3. No things that have no parts are things that can be annihilated.

8. 1. No things that have the power to help everyone but help only some are fair things.

2. Some deities are things that have the power to help everyone but help only some.

So, 3. Some deities are not fair things.

9. 1. All beings who punish persons who do their best are unjust beings.

2. No perfectly good beings are unjust beings.

So, 3. No perfectly good beings are beings who punish persons who do their best.

*10. 1. All events are occurrences caused by a deity.

2. All sins are events.

So, 3. All sins are occurrences caused by a deity.

Exercise 6.6

Part A: Removing Term-Complements

*1. 1. No M are P.

2. All S are M.

So, 3. No S are P.

2. 1. No L are P.

2. Some L are D.

So, 3. Some D are not P.

3. 1. All M are P.

2. Some S are M.

So, 3. Some S are P.

*4. 1. All S are P.

2. No P are C.

So, 3. No C are S.

5. 1. Some M are not P.

2. All M are S.

So, 3. Some S are not P.

6. 1. No M are E.

2. All S are E.

So, 3. No S are M.

*7. 1. All M are P.

2. All S are M.

So, 3. All S are P.

8. 1. All J are A.

2. All S are J.

So, 3. All S are A.

9. 1. No M are P.

2. Some S are M.

So, 3. Some S are not P.

*10. 1. Some S are M.

2. All M are E.

So, 3. Some E are S.

Part B: Standard Form

*1. 1. Some A are B.

2. All B are C. Sub 1: Some A are C.

3. All C are D.

So, 4. Some D are A.

2. 1. No H are E.

2. All G are H. Sub 1: No G are E.

3. Some F are G.

So, 4. Some F are not E.

3. 1. All M are N.

2. All N are L. Sub 1: All M are L.

3. Some K are not L.

So, 4. Some K are not M.

*4. 1. No E are C.

2. All D are E. Sub 1: No D are C.

3. All B are D. Sub 2: No B are C.

4. Some A are B.

So, 5. Some A are not C.

5. 1. All B are D.

2. All D are M. Sub 1: All B are M.

3. No F are M.

So, 4. No F are B.

6. 1. No U are T.

2. All S are T. Sub 1: No S are U.

3. All R are S. Sub 2: No R are U.

4. All P are R.

So, 5. No P are U.

*7. 1. All A are B.

2. No C are B. Sub 1: No C are A.

3. Some D are not C.

So, 4. Some D are not A.

This sorites is invalid.

8. 1. All P are Q.

2. All Q are R. Sub 1: All P are R.

3. All R are S. Sub 2: All P are S.

4. No T are S. Sub 3: No T are P.

5. All T are V.

So, 6. No V are P.

This sorites is invalid.

9. 1. All F are G.

2. All G are H. Sub 1: All F are H.

3. No J are H. Sub 2: No J are F.

4. Some K are J.

So, 5. Some K are not F.

*10. 1. Some B are A.

2. All B are C. Sub 1: Some C are A.

3. All C are D.

So, 4. Some D are A.

Part C: Sorites

*1. 1. All A are T.

2. No T are P. Sub 1: No P are A.

3. Some D are P.

So, 4. Some D are not A.

2. 1. No D are R.

2. All P are D. Sub 1: No P are R.

3. All V are P.

So, 4. No V are R.

3. 1. All W are F.

2. All F are P. Sub 1: All W are P.

3. All P are V. Sub 2: All W are V.

4. Some H are not V.

So, 5. Some H are not W.

*4. 1. All Z are D.

2. All D are M. Sub 1: All Z are M.

3. No E are M. Sub 2: No E are Z.

4. Some P are E.

So, 5. Some P are not Z.

5. 1. All V are S.

2. All S are U. Sub 1: All V are U.

3. Some U are C.

So, 4. Some C are V.

This sorites is invalid.

6. 1. No H are C.

2. Some P are C. Sub 1: Some P are not H.

3. All P are S.

So, 4. Some S are not H.

*7. 1. All A are P.

2. All P are R. Sub 1: All A are R.

3. Some F are not R.

So, 4. Some F are not A.

8. 1. No M are I.

2. Some M are E. Sub 1: Some E are not I.

3. All E are R.

So, 4. Some R are not I.

9. 1. No S are G.

2. All M are S. Sub 1: No M are G.

3. All H are M. Sub 2: No H are G.

4. Some R are H.

So, 5. Some R are not G.

*10. 1. All P are R.

2. No B are R. Sub 1: No B are P.

3. All C are B. Sub 2: No C are P.

4. Some F are C.

So, 5. Some F are not P.

Exercise 6.7

Part A: Forms

*1. EEE. Fourth figure. Violates Rule 4: invalid.

2. AEE. First figure. Violates Rule 3 (illicit major): invalid.

3. AAA. Third figure. Violates Rule 3 (illicit minor): invalid.

*4. AAA. Second figure. Violates Rule 2 (undistributed middle term): invalid.

5. EIO. First figure. Satisfies all five rules: valid.

6. EAO. Third figure. Violates Rule 5: invalid.

*7. AOO. Second figure. Satisfies all five rules: valid.

8. IAO. First figure. Violates Rules 2, 3, and 4: invalid.

9. AOO. First figure. Violates Rule 3 (illicit major): invalid.

*10. OOO. Second figure. Violates Rules 3 (illicit major) and 4: invalid.

11. AII. Second figure. Violates Rule 2 (undistributed middle): invalid.

12. EAE. Fourth figure. Violates Rule 3 (illicit minor): invalid.

*13. OAO. First figure. Violates Rule 2: invalid.

14. EOO. Third figure. Violates Rule 4 (two negative premises): invalid.

15. AEO. Fourth figure. Violates Rule 5: invalid.

*16. AAI. First figure. Violates Rule 5: invalid.

17. AAA. Fourth figure. Violates Rule 3 (illicit minor): invalid.

18. EAO. Second figure. Violates Rule 5: invalid.

*19. III. Third figure. Violates Rule 2: invalid.

20. IAI. Second figure. Violates Rule 2: invalid.

Part B: Valid or Invalid?

*1. Some M are P. No S are M. So, some S are not P. Mood: IEO. First figure. The syllogism violates Rule 3: invalid.

2. All M are P. No S are M. So, some S are P. Mood: AEI. First figure. Violates Rule 4 and Rule 5: invalid. (Since it violates Rule 5, it is invalid on the modern view of existential import, but if that were the only rule it violated, it would be valid on the traditional Aristotelian view. However, it also violates Rule 4 [having an affirmative conclusion with a negative premise], and so is invalid on the traditional Aristotelian view.)

3. No P are M. Some S are M. So, some S are not P. Mood: EIO. Second figure. Satisfies all five rules: valid.

*4. All P are M. No S are M. So, no S are P. Mood: AEE. Second figure. There is an equivocation on the word "animals." In the second premise "animals" means "nonhuman animals." In the first premise "animals" means (roughly) "living organisms capable of moving about." The syllogism thus violates Rule 1: invalid.

5. All M are P. No S are M. So, no S are P. Mood: AEE. First figure. Violates Rule 3 (illicit major): invalid.

6. No P are M. Some M are S. So, some S are not P. Mood: EIO. Fourth figure. Satisfies all five rules: valid.

*7. All M are P. All S are M. So, some S are P. Mood: AAI. First figure. Violates Rule 5: hence invalid on the modern view. But since it does not violate any other rules, it is valid on the traditional Aristotelian view.

8. All M are P. Some M are S. So, some S are P. Mood: AII (nominally). Third figure (nominally). If this were the correct form of the syllogism, it would be valid. But there is an equivocation on the term "banks," which is used in the sense of the bank of a river in the first premise, and in the sense of a financial institution in the second premise, so that this argument does not have three and only three terms, but four terms, and thus it violates Rule 1. In effect, it is not really a syllogism. (Note that it is possible for an argument with four or more terms to be valid, though that isn't the case with this example.)

9. All P are M. All S are M. So, all S are P. Mood: AAA. Second figure. Violates Rule 2: invalid.

*10. Some M are not P. All S are M. So, some S are not P. Mood. OAO: First figure. The syllogism violates Rule 2: invalid.

11. No P are M. All S are M. So, some S are not P. Mood: EAO. Second figure. Violates Rule 5: invalid on the modern (Boolean) view, but since it does not violate any other rules, it is valid on the traditional Aristotelian view.

12. Some M are P. All S are M. So, some S are P. Mood: IAI. First figure. Violates Rule 2 (undistributed middle term): invalid.

13. All P are M. Some M are not S. So, some S are not P. Mood: AOO. Fourth figure. Violates Rule 2: invalid.

14. Some M are P. No S are M. So, some S are not P. Mood: IEO. First figure. Violates Rule 3 (illicit major): invalid.

15. No M are P. All M are S. So, no S are P. Mood: EAE. Third figure. Violates Rule 3 (illicit minor): invalid.

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