Solutions to Counter-example exercises

1. All oranges are fruits and all oranges are citrus, so all citrus are fruits.


All O are F

All O are C

so All C are F



all cats are mammals

all cats are animals

so all animals are mammals


2. No actors are politicians, so no politicians are insensitive, since no actors are insensitive.


No A are P

No A are I

so No P are I




No cats are Collies

No cats are Dogs

so No Collies are Dogs


3. Some students are people who shower every day, because some students are people with jobs and some people with jobs shower every day.



Some J are P          Some animals are birds

Some S are J          Some dogs are animals

so Some S are P     so some dogs are birds.




4. All Boy Scouts are Christians so all Christians know how to tie square knots, since all Boy Scouts know how to tie square knots.



All B are C

All B are S

so All C are S




all cats are animals

all cats are mammals

all animals are mammals



5. All people who are envious are either spiteful or greedy, so everyone is greedy, since some envious people are spiteful.



All P who are E are either S or G

Some P who are E are S

So All P are G



all animals that are mammals are either male or female.

some animals that are mammals are male,

so all animals are female.


6. All people who assist others in suicide are guilty of murder. Accordingly some compassionate people are not guilty of murder inasmuch as some people who assist others in suicide are compassionate.



All A are G                                                             All dogs are animals

Some A are C                                                         Some dogs are poodles

so Some C are not G                                             Some poodles are not animals


7. Some farm workers are not people who are paid decent wages, because no illegal aliens are people paid decent wages and some illegal aliens are not farm workers.


No I are D                                                      No dogs are cats

Some I are not F                                           Some dogs are not Tabbies

so Some F are not D                                     Some Tabbies are not cats


8. All community colleges with low tuition are either schools with large enrollments or institutions supported by taxes. Therefore all community colleges are institutions supported by taxes.




all C with L are either S or I                                  All cats with legs are either male or female

so all C are I                                                             so all cats are female



9. If energy taxes are increased, then either the deficit will be reduced or global warming will be taken seriously. If the deficit is reduced, inflation will be checked. Therefore if energy taxes are increased, inflation will be checked.


If I, then R or G

If R, then C

so if I, then C



If you’re in Congress you’re either a Senator or a Rep

If you’re a Senator, you have a 6 year term

so if you’re in Congress, you have a 6 year term.


No politicians are true statesmen, for some are people who will stop at nothing to win an election, and no one who will do that is a true statesman.

If human rights are recognized, then civilization flourishes, and if equality prevails, then civilization flourishes, so if human rights are recognized, equality prevails.

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