Chapter 1 Exercise Answers
Contents
Chapter 1 2
1.1 Exercises – Page 10 2
Chapter 2 3
2.1 Exercises – page 14 3
2.2 Exercises – page 24 3
2.3 Exercises – page 39 6
2.4 Exercises – page 55 11
2.5 Exercises – page 65 14
2.5 Exercises – page 69 22
Chapter 3 25
3.4 and 3.5 Exercises – pages 78 and 93 25
3.9 Exercises – page 100 27
3.9 Exercises – page 103 28
3.10 Exercises – page 110 30
3.11 Exercises – page 122 33
Chapter 4 36
4.3 Exercises – page 129 36
4.5 Exercises – page 138 38
4.7 Exercises – page 148 39
Chapter 5 43
5.1 Exercises – page 155 43
Chapter 6 45
6.2 Exercises – page 170 45
6.3 Exercises – page 179 50
Chapter 1
1.1 Exercises – Page 10
A. Check the text for definitions.
B. Statements
1. Yes.
2. No – question.
3. Yes (but we don’t know whether it’s true or not).
4. Yes (but people don’t agree about whether it’s true or not).
5. Yes (but the truth might change depending on who “I” refers to).
6. No – exclamation.
7. Yes (but false).
8. Yes.
9. No – exclamation.
10. Yes.
C. Arguments.
1. Valid; if the premises were true, the conclusion would follow necessarily. Sound; the premises are true (If I give you false premises, they will be obviously false).
2. Valid; if it were true that ALL humans have three heads and that Sid were a human, it would follow necessarily that Sid would have three heads. Unsound; it is obviously false that all humans have three heads.
3. Valid; if the premises were true, the conclusion would follow necessarily. Sound; the premises are not obviously false.
4. Invalid. Even if I always carry my umbrella when it rains, I might carry it at other times, too (maybe I just like it or I use it for a sunshade). So the fact that I have my umbrella doesn’t guarantee that it is raining. Soundness doesn’t apply to invalid arguments.
5. Valid; if the premises were true, the conclusion would follow necessarily. Sound; the premises are not obviously false.
6. Invalid. Since the premise only states that most cats have 3 heads, Tibby may be one of the cats that doesn’t have 3 heads, so the conclusion doesn’t necessarily follow from the premises. Soundness doesn’t apply to invalid arguments.
7. Valid; if the premises were true, the conclusion would follow necessarily. Sound; the premises are all true.
8. Valid; if the premises were true, the conclusion would follow necessarily. Unsound; it is obviously false that all mammals have green fur.
9. Invalid; mammals are not the only animals that have lungs, so having lungs is not enough to guarantee that something is a mammal (and in fact, it’s true that iguanas have lungs but are not mammals). Soundness doesn’t apply to invalid arguments.
10. Invalid; mammals are not the only animals that have lungs, so having lungs is not enough to guarantee that something is a mammal, even though it is in fact true that all dogs have lungs and are mammals. This is a case of an argument having true premises AND a true conclusion but being invalid – this is possible! Soundness doesn’t apply to invalid arguments.
Chapter 2
2.1 Exercises – page 14
Identify the quantifier, the subject term, the verb, and the predicate term in the following categorical statements:
| |Quantifier |Subject Term |Verb |Predicate Term |
|All dogs are mammals. |All |Dogs |are |Mammals |
|Some mammals are dogs. |Some |Mammals |are |Dogs |
|No mammals are fish. |No |Mammals |are |Fish |
|Some people are Americans. |Some |People |are |Americans |
|All Americans are people. |All |Americans |are |People |
2.2 Exercises – page 24
|All dogs are mammals. |[pic] |
| | |
|(NOTE: The letters you choose for the terms don’t really matter – | |
|If you use different letters, it’s fine (but they should be | |
|something recognizable to me!)) | |
|No dogs are fish. |[pic] |
|No fish are dogs. |[pic] |
|Some dogs are Labrador retrievers. |[pic] |
|All Labrador retrievers are dogs. |[pic] |
|Some dogs are NOT Labrador retrievers. |[pic] |
|Some children are adorable. |[pic] |
|Some children are NOT adorable. |[pic] |
|Some adorable things are children. |[pic] |
|Some adorable things are NOT children. |[pic] |
|All firefighters are brave people. |[pic] |
|No firefighters are cowards. |[pic] |
|Some firefighters are men. |[pic] |
|Some firefighters are women. |[pic] |
|Some firefighters are not men. |[pic] |
|Some firefighters are not women. |[pic] |
|All presidents of the U.S. are American citizens (terms: presidents|[pic] |
|of the U.S. and American citizens). | |
|Some American citizens are presidents of the U.S. |[pic] |
|No French citizens are presidents of the U.S. |[pic] |
|No presidents of the U.S. are French citizens. |[pic] |
|All residents of southern California are people who live in |[pic] |
|earthquake zones. | |
|Some people who live in earthquake zones are residents of southern |[pic] |
|California. | |
|Some people who live in earthquake zones are not residents of |[pic] |
|southern California. | |
|Some cars are fast machines. |[pic] |
2.3 Exercises – page 39
|Original |New |Same/ |
| | |Different |
|All humans are things with hearts (Obversion) | |Same |
|All H are T |No H are non-T | |
|[pic] | | |
| |[pic] | |
|Some humans are bald people (Conversion) | |Same |
|Some H are B |Some B are H. | |
| | | |
|[pic] |[pic] | |
|No children are good listeners (Contraposition) | |Different |
|No C are G |No non-G are non-C | |
|[pic] | | |
| |[pic] | |
|Some children are artists (Converstion) | |Same |
|Some C are A |Some A are C | |
|[pic] | | |
| |[pic] | |
|No ducks are geese | |Same |
|(Obversion) |All D are non-G | |
|No D are G | | |
|[pic] |[pic] | |
|Some ducklings are ugly things (Contraposition) | |Different |
|Some D are U |Some non-U are non-D | |
|[pic] | | |
| |[pic] | |
|Some ducklings are not ugly things (Contraposition) | |Same |
|Some D are not U |Some non-U are not non-D | |
|[pic] | | |
| |[pic] | |
|All students are hard workers (Obversion) | |Same |
|All S are H |No S are non-H | |
|[pic] | | |
| |[pic] | |
|All philosophers are intelligent people (Obversion) | |Same |
|All P are I |No P are non-I | |
|[pic] | | |
| |[pic] | |
|All intelligent people are philosophers (Obversion) | |Same |
|All I are P |No I are non-P | |
|[pic] | | |
| |[pic] | |
|No politicians are honest people (Conversion) | |Same |
|No P are H |No H are P | |
| | | |
|[pic] |[pic] | |
|Some politicians are honest people (Conversion) | |Same |
|Some P are H |Some H are P | |
|[pic] | | |
| |[pic] | |
|Some politicians are honest people (Contraposition) | |Different |
|Some P are H |Some non-H are non-P | |
|[pic] | | |
| |[pic] | |
|Some politicians are not honest people (Conversion) | |Different |
|Some P are not H |Some H are not P | |
|[pic] | | |
| |[pic] | |
|Some politicians are not honest people (Contraposition) | |Same |
|Some P are not H |Some non-H are not non-P | |
|[pic] | | |
| |[pic] | |
2.4 Exercises – page 55
|Some cats are good pets |[pic] |Evaluation: |
|All good pets are soft pets | |The conclusion would be diagrammed if there |
|(Some cats are soft pets | |were an X in area 3 OR 6 |
| | |There is an X in area 3 |
| | |The conclusion is diagrammed |
| | |The argument is valid. |
|Some cats are soft pets |[pic] |Evaluation: |
|All good pets are soft pets | |The conclusion would be diagrammed if there |
|(Some cats are good pets | |were an X in area 3 OR 6 |
| | |There is not an X in area 3 or 6 (the X is on |
| | |the line) |
| | |The conclusion is NOT diagrammed |
| | |The argument is INvalid. |
|Some cats are soft pets |[pic] |Evaluation: |
|All soft pets are good pets | |The conclusion would be diagrammed if there |
|(Some cats are good pets | |were an X in area 3 OR 6 |
| | |There is an X in area 3 |
| | |The conclusion is diagrammed |
| | |The argument is valid. |
|All Dobermans are biters |[pic] |Evaluation: |
|No biters are good pets | |The conclusion would be diagrammed if there |
|(No Dobermans are good pets | |were shading in areas 3 AND 6 |
| | |There is shading in areas 3 and 6 |
| | |The conclusion is diagrammed |
| | |The argument is valid. |
|Some Dobermans are biters |[pic] |Evaluation: |
|No biters are good pets | |The conclusion would be diagrammed if there |
|(No Dobermans are good pets | |were shading in areas 3 AND 6 |
| | |There is not shading in areas 3 AND 6 |
| | |The conclusion is not diagrammed |
| | |The argument is invalid. |
|Some Dobermans are biters |[pic] |Evaluation: |
|No biters are good pets | |The conclusion would be diagrammed if there |
|(Some Dobermans are not good pets | |were an X in area 2 OR 5 |
| | |There is an X in area 2 |
| | |The conclusion is diagrammed |
| | |The argument is valid. |
|Some Dobermans are not biters |[pic] |Evaluation: |
|No biters are good pets | |The conclusion would be diagrammed if there |
|(Some Dobermans are good pets | |were an X in area 3 OR 6 |
| | |There is not an X in area 3 or 6 (the X is on |
| | |the line) |
| | |The conclusion is not diagrammed |
| | |The argument is invalid. |
|Some Dobermans are not biters |[pic] |Evaluation: |
|All biters are poor pets | |The conclusion would be diagrammed if there |
|(Some Dobermans are not poor pets | |were an X in area 2 or 5 |
| | |There is not an X in area 2 or 5 (the X is on |
| | |the line). |
| | |The conclusion is not diagrammed |
| | |The argument is invalid. |
|Some Dobermans are not biters |[pic] |Evaluation: |
|All poor pets are biters | |The conclusion would be diagrammed if there |
|(Some Dobermans are not poor pets | |were an X in area 2 or 5 |
| | |There is an X in area 5 |
| | |The conclusion is diagrammed |
| | |The argument is valid. |
|Some Dobermans are biters |[pic] |Evaluation: |
|Some biters are good pets | |The conclusion would be diagrammed if there |
|(Some Dobermans are good pets | |were an X in area 3 or 6 |
| | |There is not an X in area 3 or 6 (both Xs are |
| | |on the lines) |
| | |The conclusion is not diagrammed |
| | |The argument is invalid. |
|Some horses are black things |[pic] |Evaluation: |
|Some things in California are black things | |The conclusion would be diagrammed if there |
|(Some horses are things in California | |were an X in area 3 or 6 |
| | |There is not an X in area 3 or 6 (both Xs are |
| | |on the lines) |
| | |The conclusion is not diagrammed |
| | |The argument is invalid. |
|All BMWs are fast cars. |[pic] |Evaluation: |
|Some BMWs are not safe cars. | |The conclusion would be diagrammed if there |
|( Some safe cars are not fast cars. | |were an X in area 2 or 5 |
| | |There is not an X in area 2 or 5 |
| | |The conclusion is not diagrammed |
| | |The argument is invalid. |
|No television shows are things that promote |[pic] |Evaluation: |
|children’s imaginations. | |The conclusion would be diagrammed if there |
|Some television shows are things that claim to | |were an X in area 2 or 5 |
|promote children’s imaginations. | |There is an X in area 2 |
|( Some things that claim to promote children’s | |The conclusion is diagrammed |
|imaginations are not things that promote | |The argument is valid. |
|children’s imaginations. | | |
|Some American politicians are people who take |[pic] |Evaluation: |
|bribes. | |The conclusion would be diagrammed if there |
|No people who take bribes are people that should | |were an X in area 2 or 5 |
|be leaders of a democracy. | |There is not an X in area 2 or 5 |
|( Some people that should be leaders of a | |The conclusion is not diagrammed |
|democracy are not American politicians. | |The argument is invalid. |
|All people who watch Lost are people who believe |[pic] |Evaluation: |
|in the paranormal. | |The conclusion would be diagrammed if there |
|All people who watch Lost are people who believe | |were shading in areas 2 AND 5 |
|in time travel. | |There is not shading in area 5 |
|(All people who believe in time travel are people | |The conclusion is not diagrammed |
|who believe in the paranormal. | |The argument is invalid. |
|All people who want to protect helpless animals |[pic] |Evaluation: |
|are caring people. | |The conclusion would be diagrammed if there |
|All people who want to protect fetuses in mothers’| |were shading in areas 2 AND 5 |
|wombs are caring people. | |There is not shading in area 2 |
|( All people who want to protect helpless animals | |The conclusion is not diagrammed |
|are people who want to protect fetuses in the | |The argument is invalid. |
|mothers’ wombs. | | |
|All corporations are things that want to control |[pic] |Evaluation: |
|the political process through advertising. | |The conclusion would be diagrammed if there |
|All things that want to control the political | |were shading in areas 2 AND 5 |
|process through advertising are things that want | |There is shading in areas 2 and 5 |
|to undermine democracy. | |The conclusion is diagrammed |
|( All corporations are things that want to | |The argument is valid. |
|undermine democracy. | | |
|All countries that have universal health care are |[pic] |Evaluation: |
|advocaters of socialism. | |The conclusion would be diagrammed if there |
|No advocaters of socialism are admired by the | |were shading in areas 3 AND 6 |
|United States. | |There is shading in areas 3 and 6 |
|(No countries that have universal health care are | |The conclusion is diagrammed |
|admired by the United States | |The argument is valid. |
2.5 Exercises – page 65
|All children (C) are cute people (T) and some children (C) are funny people (F). |[pic] |
|You have two premises: | |
|All C are T | |
|Some C are F | |
| | |
|Missing conclusion: Some T are F or Some F are T. Conversion works on particular | |
|affirmative statements; these both mean the same thing. | |
|No fish are air-breathers and all salmon are fish. |[pic] |
|You have two premises: | |
|No F are A | |
|All S are F | |
| | |
|Missing conclusion No S are A or No A are S. Conversion works on universal negative | |
|statements; these both mean the same thing. | |
|All Republicans (R) are critics of health care reform (C) and some Californians (F) are|[pic] |
|Republicans (R). | |
|You have two premises: | |
|All R are C | |
|Some F are R | |
| | |
|Missing conclusion: Some C are F or Some F are C. Conversion works on particular | |
|affirmative statements; these both mean the same thing. | |
|No Republicans are correct thinkers about health care and Bob is a republican |[pic] |
|You have 2 premises | |
|No R are C | |
|All B are R | |
| | |
|Missing conclusion: No B are C or No C are B. Conversion works on universal negative | |
|statements; these both mean the same thing. | |
|All dogs (D) have hair (H) because all dogs (D) are mammals (M) |Diagram with All H are M: |
|You have a premise and the conclusion |[pic] |
|All D are M | |
|_________________ |This diagram/argument is invalid, but the correct |
|( All D are H |argument/diagram must be valid, so this cannot be the |
| |missing premise. |
|Conclusion is universal, so both premises must b universal | |
|Conclusion is affirmative, so both premises must be affirmative |Diagram with All M are H: |
|So missing premise must be universal affirmative with M and H; either All M are H or | |
|All H are M |[pic] |
| |This diagram/argument is valid, so All M are H must be the|
|All M are H |correct premise. |
| | |
| | |
|Some mammals are black because some dogs are black |Diagram with All M are D |
|Premise missing | |
|Some D are B |[pic] |
|_________________ |This diagram/argument is invalid, so this cannot be the |
|( Some M are B |missing premise. |
| | |
|The conclusion is particular so there must be one and only one particular premise. | |
|The conclusion is affirmative so both premises must be affirmative. | |
|So the other premise must be an affirmative universal with D and M; either All M are D | |
|or All D are M |Diagram with All D are M |
| |[pic] |
|All D are M |This diagram/argument is valid, so All D are M must be the|
| |correct premise. |
|All happy children (H) are well-fed children (W) and some African children (A) are |[pic] |
|not well-fed children (W) | |
|You have two premises: | |
|All H are W | |
|Some A are not W | |
| | |
|Missing conclusion: The easiest way to determine what it is is to draw a diagram. | |
| | |
|Diagrammed conclusion: Some A are not H (if you have the H and the A on opposite | |
|sides, that’s okay – it should still read “Some A are not H”). | |
|Some Democrats are not capitalists because some Democrats are supporters of health care|[pic] |
|reform. | |
|You have a premise and a conclusion. | |
|Some D are S | |
|______________ | |
|(Some D are not C | |
| | |
|A particular conclusion means you have one and only one particular premise | |
|A negative conclusion means one premise must be negative. | |
|So the missing premise must be a universal negative with S and C | |
|Note – conversion works on universal negatives; No S are C means the same as No C are S| |
|– so it doesn’t matter which way you write it. | |
| | |
|No S are C OR No C are S | |
|Some Japanese cars are not safe vehicles because no safe vehicles are recalled |[pic] |
|vehicles. | |
|You have a premise and a conclusion | |
|No S are R | |
|_____________ | |
|(Some J are not S | |
| | |
|The conclusion is particular, so there must be a particular premise. | |
|The conclusion is negative so there must be a negative premise. | |
|There is a negative premise already, so the other premise must be an affirmative | |
|particular premise with J and R. | |
|Note – conversion works with particular affirmatives, so Some J are R and Some R are J | |
|mean the same thing. | |
| | |
|Some J are R OR Some R are J | |
|Some Japanese car companies should go out of business because they have caused the | |
|death of unsuspecting drivers. | |
|You have a premise and the conclusion | |
|Some J are D | |
|___________ | |
|(Some J are O |Diagram with All O are D |
| | |
|The conclusion is particular so there must be one and only one particular premise. |[pic] |
|The conclusion is affirmative, so both premises must be affirmative. |This diagram/argument is invalid, so this cannot be the |
|The other premise must be a universal affirmative, using O and D; All O are D or All D |missing premise. |
|are O. | |
|. |Diagram with All D are O |
|All D are O | |
| |[pic] |
| | |
| | |
| |This is a valid diagram/argument, so this has to be the |
| |missing premise. |
2.5 Exercises – page 69
1) a) All carnivores are dangerous animals
b) All African lions are brave animals
c) All brave animals are carnivores
d) Therefore, all African lions are dangerous animals.
Reordering:
All A are B
All B are C
All C are D
Therefore, all A are D
Intermediate conclusions (becoming the next premise) (You should draw diagrams; I haven’t)
All A are B
All B are C
All A are C
All A are C
All C are D
All A are D
2) a) All children are entertaining individuals
b) Some artistic people are beautiful people
c) All beautiful people are people who deserve praise
d) All artistic people are children
e) Therefore, some entertaining people are people who deserve praise.
Reordering
All B are D
Some A are B
All A are C
All C are E
Some E are D
Intermediate conclusions (becoming the next premise) (You have to draw diagrams; I haven’t)
All B are D
Some A are B
Some A are D
Some A are D
All A are C
Some D are C
Some D are C
All C are E
Some E are D
3) a) All dogs are animals
b) All Boeing 767s are jets
c) No animals are things made by humans
d) All jets are things made by humans
e) Therefore, no Boeing 767s are dogs
Reordering:
All D are A
No A are H
All J are H
All B are J
No B are D
Intermediate conclusions (becoming the next premise) (You have to draw diagrams; I haven’t)
All D are A
No A are H
No H are D
No H are D
All J are H
No J are D
No J are D
All B are J
No B are D
4) a) All Democrats are people liberal about social issues
b) All people liberal about social issues are people who are in favor of abortion rights
c) Some people who are conservative about economic issues are people who favor increased government regulation of banks
d) No people who are in favor of abortion rights are people who are conservative about economic issues
e) Therefore, some people who favor increased government regulation of banks are not Democrats
All D are L
All L are A
No A are C
Some C are I
Some I are not D
All D are L
All L are A
All D are A
All D are A
No A are C
No D are C
No D are C
Some C are I
Some I are not D
5) a) All go-getters are students
b) Some lazy people are underemployed people
c) No students are lazy people
d) All youthful optimists are go-getters
e) Some unemployed people are not youthful optimists
Reordering:
All Y are G
All G are S
No S are L
Some L are U
Some U are not Y
All Y are G
All G are S
All Y are S
All Y are S
No S are L
No Y are L
No Y are L
Some L are U
Some U are not Y
Chapter 3
3.4 and 3.5 Exercises – pages 78 and 93
“Mary sings”, “Bob sings”, and “Mary dances” are all TRUE.
“Bob dances” is FALSE.
1. Mary sings and Bob dances.
M & D
T F F
2. Mary and Bob sing and dance.
(M & B) & (C & D)
T T T F T F F
3. If Mary sings, then she dances.
M → C
T T T
4. If Mary doesn’t sing then Bob dances.
~M → D
FT T F
5. If Bob sings or Mary sings, then Mary dances.
(B ( M) → C
T T T T T
6. If Bob sings or Mary sings, then Mary dances and Bob dances.
(B ( M) → (C & D)
T T T F T F F
7. If Bob sings or dances, then Mary sings but she doesn’t dance.
(B ( D) → (M & ~C)
T T T F T F FT
8. If Bob sings then Mary sings, and if Bob dances then Mary dances.
(B → M) & (D → C)
T T T T F T T
9. If Mary sings then Bob sings, and if Bob dances then Mary dances.
(M → B) & (D → C)
T T T T F T T
10. If Mary doesn’t sing then Bob doesn’t sing, and if Bob dances then Mary doesn’t dance.
(~M → ~B) & (D → ~C)
FT T FT T F T FT
11. If Mary doesn’t dance then Bob doesn’t dance, and if Bob doesn’t sing then Mary doesn’t sing.
(~C → ~D) & (~B → ~M)
FT T TF T FT T FT
12. If Bob doesn’t sing then Mary sings and dances.
~B → (M & C)
FT T T T T
13. If Mary and Bob dance they both sing.
(C & D) → (M & B)
T F F T T T T
14. Mary sings, and, if Bob doesn’t sing or dance, then she dances.
M & (~(B ( D) → C) OR M & ((~B & ~D) → C)
T T F T T F T T T T FT F FT T T
15. If Bob doesn’t sing and Bob doesn’t dance then Mary sings and dances.
(~B & ~D) → (M & C) OR ~(B ( D) → (M & C)
FT F TF T T T T F T T F T T T T
16. If Bob doesn’t dance then Mary sings and dances; and Bob doesn’t dance.
(~D → (M & C)) & ~D
TF T T T T T TF
17. If Mary or Bob sing, then Mary dances; and if Mary sings and Bob doesn’t sing, then Bob dances.
((M ( B) → C) & ((M & ~B) → D)
T T T T T T T F FT T F
18. If, if Bob dances then Mary sings, then if Mary doesn’t dance, then Bob sings.
(D → M) → (~C → B)
F T T T FT T F
19. It is not the case that if Mary and Bob both sing then they both dance; however, if Mary doesn’t sing and Bob doesn’t sing, then neither of them dance.
~[(M & B) → (C & D)] & [(~M & ~B) → (~C & ~D)]
T T T T F T F F T FT F FT T FT F TF
20. If it is not the case that if Mary sings, then Bob dances, then it not the case that either Mary and Bob sing or neither Mary nor Bob dance.
~(M → D) → ~[(M & B) ( (~C & ~D)]
T T F F F F T T T T FT F TF
3.9 Exercises – page 100
1. A B (A & B) & (A & ~B)
T T T T T F T F FT
T F T F F F T T TF
F T F F F F F F FT
F F F F F F F F TF All false – Contradiction.
2. A B (A → B) → (A & ~B)
T T T T T F T F FT
T F T F F T T T TF
F T F T T F F F FT
F F F T F F F F TF Both Ts and Fs - Contingent
3. A B (A ( B) ( (~A ( ~B)
T T T T T T FT F FT
T F T T F T FT T TF
F T F T T T TF T FT
F F F F F T TF T TF All true – Tautology.
4. A B C ((A → B) → C) & ~C
T T T T T T T T F FT
T T F T T T F F F TF
T F T T F F T T F FT
T F F T F F T F T TF
F T T
F T F There are now both Ts and Fs on the table, so
F F T it can be declared contingent.
F F F
5. M D B (~M & D) → (~B → ~D)
T T T FT F T T FT T FT
T T F FT F T T TF F FT
T F T FT F F T FT T TF
T F F FT F F T TF T TF
F T T TF T T T FT T FT There are now both Ts
F T F TF T T F TF F FT and Fs on the table, so
F F T it can be declared
F F F contingent.
3.9 Exercises – page 103
Consistency
Use a full truth table to test each of the following sets of statements for consistency. Be able to explain what makes the statements consistent or inconsistent.
NOTE: For these problems, I have placed the Ts and Fs directly under the statement letters
1. A & ~B A ( B A → ~B
T F FT T T T T F FT
T T TF T T F T T TF The statements are ALL true here. The set is
F F FT F T T F T FT consistent.
F F TF F F F F T TF
2. A → (A ( B) (A & ~B) → (A & B) A & B
T T T T T T F FT T T T T T T T The statements are ALL true here.
T T T T F T T TF F T F F T F F The set is consistent.
F T F T T F F FT T F F T F F T
F T F F F F F TF T F F F F F F
Inconsistent – There are NO rows (horizontal) on which the statements are ALL true.
3. A ( (A ( B) A & (A & ~B) A → B
T T T T T T F T F FT T T T
T T T T F T T T T TF T F F
F T F T T F F F F FT F T T
F F F F F F F F F TF F T F
Inconsistent – There are NO rows (horizontal) on which the statements are ALL true.
4. A ( (B → C) (A & B) → ~C B & ~C
T T T T T T T T F FT T F FT
T T T F F T T T T TF T T TF All true – the set is consistent
T T F T T T F F T FT F F FT
T T F T F T F F T TF F F TF
F T T T T F F T T FT T F FT
F T T T F F F T T TF T T TF
F T F T T F F F T FT F F FT
F F F F F F F F T TF F F TF
5. B → (C & D) ~B & (C & D) B ( (C ( D)
T T T T T FT F T T T T T T T T
T F T F F FT F T F F T T T T F
T F F F T FT F F F T T T F T T
T F F F F FT F F F F T T F T F
F T T T T TF T T T T F T T F T All true – the set is consistent
F T T F F TF F T F F F T T T F
F T F F T TF F F F T F T F T T
F T F F F TF F F F F F F F F F
Validity
NOTE: For these problems, I have placed the Ts and Fs directly under the statement letters
1. A & ~B A ( B ( A → ~B
T F FT T T T T F FT
T T TF T T F T T TF
F F FT F T T F T FT
F F TF F F F F T TF
Valid – there are NO rows on which the two premises are true and the conclusion is false.
2. A → (A ( B) (A & ~B) → (A & B) ( A & B
T T T T T T F FT T T T T T T T
T T T T F T T TF F T F T T F F
F T F T T F F FT T F F T F F T
F T F F F F F TF T F F F F F F
Invalid – There is a row (the 3rd and/or 4th row) on which the premises are true and the conclusion is false. Either row is sufficient.
3. A ( (A ( B) A & (A & ~B) A → B
T T T T T T F T F FT T T T
T T T T F T T T T TF T F F
F T F T T F F F F FT F T T
F F F F F F F F F TF F T F
Invalid – There is a row (the 2nd row) on which the premises are true and the conclusion is false.
4. A ( (B → C) (A & B) → ~C ( B & ~C
T T T T T T T T F FT T F FT
T T T F F T T T T TF T T TF
T T F T T T F F T FT F F FT
T T F T F T F F T TF F F TF
F T T T T F F T T FT T F FT
F T T T F F F T T TF T T TF
F T F T T F F F T FT F F FT
F F F F F F F F T TF F F TF
Invalid – There is a row (the 3rd row, among others) on which the premises are true and the conclusion is false.
5. B → (C & D) ~B & (C & D) ( B ( (C ( D)
T T T T T FT F T T T T T T T T
T F T F F FT F T F F T T T T F
T F F F T FT F F F T T T F T T
T F F F F FT F F F F T T F T F
F T T T T TF T T T T F T T F T
F T T T F TF F T F F F T T T F
F T F T T TF F F F T F T F T T
F T F F F TF F F F F F F F F F
Valid – there are NO rows on which the premises are true and the conclusion is false.
3.10 Exercises – page 110
1. A ( (B → C) (A & B) → ~C B & ~C
T T T T T FT T F FT
T T T T F T T T T TF T T TF
T F T T F FT F F FT
T F F T F TF F F TF
F T T F T FT T F FT
F T F F T TF T T TF
F F T F F FT F F FT
F F F F F TF F F TF
Consistent. The third statement is true only in the 2nd and 6th rows. The first and second statements are also true on the second row, so there is a row on which all the statements are true.
2. B → (C & D) ~B & (C → D) B ( (C ( D)
T T T FT F T T T T T
T T F FT F T F T T F
T F T FT F F T T F T
T F F FT F F F T F F
F T T T T TF T T T T F T T T T
F T F TF F T F F F T F
F F T TF T F T T F F T
F F F TF T F T F F F F
Consistent. The second statement is true only in the 5th, 7th, and 8th rows. Both other statements are also true on the 5th row.
3. A & (B & ~C) B ( (~A ( ~C) (C → A) & ~B
T F T F FT T FT FT T T FT
T T T T TF T T FT TF F T T F FT
T F F F FT F FT FT T T TF
T F F F TF F FT TF F T TF
F F T FT T TF FT T F FT
F F T TF T TF TF T F FT
F F F FT F TF FT F F TF
F F F TF F TF TF F F TF
Inconsistent. The first statement is only true in the 2nd row. On that row the third statement is false. So there are NO rows on which all of the statements are true.
4. (B & A) → (A → ~C) (A ( B) → C (B & C) → (A & B)
T T T F T F FT T T T T T T T
T T T T T T TF T T T F F T F T T
F F T T T F FT T T F T T F F T T T F F
F F T T T F TF T F F F F T F
T F F T F T FT F T T T T F T
T F F T F T TF F T F T F F T
F F F T F T FT F F T F T F F
F F F T F T TF F F F F F F F
Consistent – all three statements are true on the third line.
5. (A ( B) → (B → ~C) ((A & B) → ~C) → B (A & B) & ~C
T T T FT T T FT T T T F FT
T T T T T T TF T T T T TF T T T T T TF
T F F FT T F FT F T F F F FT
T F F TF T F TF F T F F F TF
F T T FT F T FT T F F T F FT
F T T TF F T TF T F F T F TF
F F F FT F F FT F F F F F FT
F F F TF F F TF F F F F F TF
Consistent – all three statements are true on the second line.
Truth Tables for Validity
1. A ( (B → C) (A & B) → ~C ( B & ~C
T T T T T T F FT T F FT
T T F T T TF T T TF
T T F T T F F T FT F F FT
T F F T F TF F F TF
F T T F T FT T F FT
F T F F T TF T T TF
F F T F F FT F F FT
F F F F F TF F F TF
Invalid – on the 3rd row, the premises are true while the conclusion is false.
2. B → (C & D) ~B & (C → D) ( B ( (C ( D)
T T T FT T T T T T T
T T F FT T F T T T F
T F T FT F T T T F T
T F F FT F F T T F F
F T T TF T T F T T T T
F T F TF T F F T T T F
F F T TF F T F T F T T
F T F F TF T F T F F F F F F
Invalid – In row 8, the premises are true while the conclusion is false.
3. A & (B & ~C) B ( (~A ( ~C) ( (C → A) & ~B
T F T F FT T FT FT T T F FT
T T T TF T T FT T TF F T F FT
T F FT F FT FT T T T T TF
T F TF F FT TF F T T T TF
F T FT T TF FT T F F FT
F T TF T TF TF F F F FT
F F FT F TF FT T F F F TF
F F TF F TF TF F T F T TF
Invalid – on the second row, both premises are true while the conclusion is false.
4. (B & A) → (A → ~C) (A ( B) → C ( (B & C) → (A & B)
T T T FT T T T T T T T T T T
T T T TF T T F T F F T T T T
F T T FT T F T F F T T T F
F T T TF T F F F F F T T F
T F F T F T FT F T T T T T T T F F F T
T F F TF F T F T F F T F F T
F F F FT F F T F F T T F F F
F F F TF F F F F F F T F F F
Invalid – on the 5th row, the premises are true while the conclusion is false.
5. (A ( B) → (B → ~C) ((A & B) → ~C) → B ((A & B) & ~C
T T T F T F FT T T T F FT T T T T F FT
T T T TF T T TF T T T T T TF
T F F FT T F F T FT F F T F F FT
T F F TF T F F T TF F F T F F F TF
F T T F T F FT F F T T FT T T F T F FT
F T T T T T TF F F T T TF T T F F T F TF
F F F FT F F FT F F F F F FT
F F F TF F F TF F F F F F TF
Invalid – on the 6th row the premises are all true while the conclusion is false.
3.11 Exercises – page 122
The indirect method for consistency
1. A & (B & ~C) B ∨ (~A ∨ ~C) (C → A) & ~B
T T T T TF T T FT T TF F T T T FT
There is only one way for the first statement to be true; if A and B are true and C is false (so ~C is true). With these truth values, the third statement cannot be true. Since it was not possible to make all of the statements true at the same time, they are inconsistent.
2. (A ∨ B) → (B → ~C) ((A & B) → C) → ~B (A & B) & ~C
T T T T T T TF T T T F F T FT T T T T TF
The statements all turn out to be true on this assignment (making the third statement true first). The set is consistent.
3. (B & A) → (A → C) ~[(A & B) → C] (B & C) → (A & B)
T T T T T F F T T T T F F T T F T T T T
For the middle statement to be true (the tilde is the main operator), the conditional must be false. That means A and B must be true and C false. This makes the third statement true, but the first statement false. It is impossible to make all of the statements true at the same time, so the statements are inconsistent.
4. A → B (B → C) & ~(A ∨ E) D & (E ∨ B) A → E
T T T T T T T F T T T T T T T T T T T
F T T T T T T T F F F T T F T T F T F
F T F
The first conditional will be true in THREE ways; if A and B are both true; if A is false and B is true, and if A and B are both false. If A and B are true, the second statement comes out false. That line DOESN’T show consistency. But if A is false and B is true, the statements all come out true. Since all we need is one line on which all of the statements are consistent, this line is enough to show that they’re consistent. We don’t have to look at the third line.
5. A → B (A & C) & (B → ~D) ~D → C D ∨ ~C
T T T T T T T T T TF TF T T F T FT
F T T F F ? F
F T F F F ? F
The first conditional will be true if A and B are both true, if A is false and B is true, or if they are both false. If A and B are both true, then D ∨ ~C comes out false. If A is false and B is true, the second statement comes out false, whatever C is. Similarly, if A and B are false, the second statement comes out false. We can’t make all of the statements true, so the set is inconsistent.
The indirect method for validity
1. NOTE: The problem in the book is incorrect – the last B in the conclusion should be ~B
(B & A) → (A → ~C) (A ∨ B) → ~C ∴ (B & C) → (A → ~B)
T T T T FT T T T F T F FT
There is only one way for the conclusion to be false. On this truth-value assignment, the second premise is not true. So there are no truth value assignments on which the premises are true and the conclusion is false, so the argument is valid.
2. (A ∨ B) → (B → ~C) ((A & B) → C) → B ∴(B → A) ∨ ~C
F T T F T F FT F F T T T T T T F F F FT
There is only one way for the conclusion to be false. On this truth value assignment, the first premise is false. It is not possible to have true premises and a false conclusion, so the argument is valid.
3. (A & C) → B ~C → D (D & B) ∨ C A → D ∴ A &B
T F F T F TF T T T F F T F T T T T F F
F F T T T FT T F T T T T F T F F F T
F F F
The conclusion will be false if A is true and B false, if A is false and B true, or if A is false and B is false. In the first case, one of the premises is also false. In the second case, however, all of the premises can be made true (there is actually more than one way, but one is all we need). So there is a case in which the premises are true and the conclusion is false, so the argument is invalid. We don’t have to complete the last line.
4. C → A D → A E → A (A ∨ ~C) → B D ∨ E ∴ A & B
T T T T T T T T T T T F T T T T F F
F T F F T F F T F F T TF T T F T F F F T
F T F F T F F T F F T TF T F F T F F F F
The conclusion can be false in any of three ways. On each of these truth value assignments, a premise comes out false. Hence, there is no way to make the premises true while the conclusion is false, so the argument is valid. (NOTE: Any statements or statement letters that are not filled out will not change the result and so don’t matter).
5. B ∨ C C → (D & ~E) (D → E) ∨ (A & G) G & H (H & D) → C ∴ A & B
F T T T T T T TF T F F T T T T T T T T T T T T T F F
F F T
F F F
There are three ways for the conclusion to be false. On the first truth value assignment, all of the premises come out to be true. Since there is now a row on which the premises are all true and the conclusion is false, the argument can be declared invalid, and the second and third row need not be checked.
Chapter 4
4.3 Exercises – page 129
A. 1. A ( (C ( D) Given
2. ~A & ~C Given /( D
3. ~A 2, &E
4. ~C 2, &E
5. C ( D 3,1 DS
6. D 4,5 DS
A. 1. ~A & ~B Given
2. A ( ~D Given
3. B ( (D ( ~E) Given /( ~E
4. ~A 1, &E
5. ~B 1, &E
6. ~D 2,4 DS
7. D ( ~E 3,5 DS
8. ~E 6,7 DS
B. 1. P ( ((Q & ~R) ( S) Given
2. ~S & ~P Given
3. R ( T Given /( T
4. ~S 2, &E
5. ~P 2, &E
6. (Q & ~R) ( S 1,5 DS
7. Q & ~R 4,6 DS
8. ~R 7, &E
9. T 3,8 DS
C. 1. (D ( E) & (F ( G) Given
2. H ( (~D & ~G) Given
3. ~H Given
/( E AND F
4. D ( E 1, &E
5. F ( G 1, &E
6. ~D & ~G 2,3 DS
7. ~D 6, &E
8. ~G 6, &E
9. E 4,7 DS
10. F 5,8 DS
D. 1. (~A & ~B) & (B ( (C ( A)) Given /(C
2. ~A & ~B 1, &E
3. ~A 3, &E
4. ~B 3, &E
5. B ( (C ( A) 1, &E
6. C ( A 4,5 DS
7. C 3,6 DS
E. 1. [(H ( ~I) & (H ( ~J)] & (~H & ~K) Given
2. K ( [(I ( ~L) & (J ( ~M)] Given
3. L ( N Given
4. M ( O Given /( N AND O
5. (H ( ~I) & (H ( ~J) 1, &E
6. H ( ~I 5, &E
7. H ( ~J 5, &E
8. ~H & ~K 1, &E
9. ~H 8, &E
10. ~K 8, &E
11. ~I 6,9 DS
12. ~J 7,9 DS
13. (I ( ~L) & (J ( ~M) 2,10 DS
14. I ( ~L 13, & E
15. J ( ~M 13, &E
16. ~L 11,14 DS
17. ~M 12,15 DS
18. N 3,16 DS
19. 0 4,17 DS
F. 1. (M & N) ( (~P & ~Q) Given
2. ~(M & N) Given
3. (P ( R) & (Q ( S) Given
/(R AND S
4. ~P & ~Q 1,2 DS
5. ~P 4, &E
6. ~Q 4, &E
7. P ( R 3, &E
8. Q ( S 3, &E
9. R 5,7 DS
10. S 6,8 DS
4.5 Exercises – page 138
A. 1. A → B
2. A
3. B → C /( C
4. B 1,2 MP
5. C 3,4 MP
B. 1. D ( E
2. ~D
3. E → F /(F
4. E 1,2 DS
5. F 3,4 MP
C. 1. A → C
2. ~C
3. A ( B /(B
4. ~A 1,2 MT
5. B 3,4 DS
D. 1. C ( E
1. ~E
2. C → ~D
3. D ( B /(B
4. C 1,2 DS
5. ~D 3,5 MP
6. B 4,6 DS
E. 1. C & D
2. C → E
3. D → F
4. (E & F) → G /(G
5. C 1, &E
6. D 1, &E
7. E 2,5 MP
8. F 3,6 MP
9. E & F 7,8 &I
10. G 4,9 MP
F. 1. J → K
2. K → L
3. L → M
4. M → N
5. J /(N
6. K 1,5 MP
7. L 2,6 MP
8. M 3,7 MP
9. N 4,8 MP
G. 1. A ( B
2. ~A
3. B → ~C
4. D → C /(~D
5. B 1,2 DS
6. ~C 3,5 MP
7. ~D 4,6 MT
H. 1. ~A ( B
2. ~B
3. C → A
4. C ( D /( D
5. ~A 1,2 DS
6. ~C 3,5 MT
7. D 4,5 DS
I. 1. ~A ( ~B
2. A
3. C → B
4. D → C /(~D
5. ~B 1,2 DS
6. ~C 3,5 MT
7. ~D 4,6 MT
4.7 Exercises – page 148
A. 1. A ( B
2. A
3. B ( (C & D)
4. D ( E /( E
5. B 1,2 MP
6. C & D 3,5 MP
7. D 6 &E
8. E 4,7 MP
B. 1. (D → A) & (C → B)
2. ~A
3. ~B /( ~D & ~C
4. D ( A 1 &E
5. C ( B 1 &E
6. ~D 2,4 MT
7. ~C 3,5 MT
8. ~D & ~C 6,7 &I
C. 1. [(D & (E ∨ F)) ( G] ∨ B
2. ~(B ∨ G)
3. D & (H ( (E ∨ F)) /( ~H
4. ~B & ~G 2 DeM
5. ~B 4 &E
6. ~G 4 &E
7. (D & (E ∨ F)) ( G 1,5 DS
8. ~(D & (E ∨ F)) 6,7 MT
9. ~D v ~(E v F) 8 DeM
10. D 3 &E
11. H ( (E v F) 3 &E
12. ~(E v F) 9,10 DS
13. ~H 11,12 MT
D. 1. A
2. (A ( B) → C
3. C → D /( D
4. A ∨ Β 1, ∨Ι
5. C 2,4 MP
6. D 3,5 MP
E. 1. E
2. [(E ∨ M) & (F ∨ Q)] ( G
3. F /( G
4. E ∨ Μ 1 ∨Ι
5. F ∨ Q 3 ∨Ι
6. (E ( M) & (F ∨ Q) 4,5 &I
7. G 2,6 MP
F. 1. A → (B & C)
2. A
3. (B ( E) → D /( D
4. B & C 1,2 MP
5. B 4 &E
6. B ( E 5 ∨I
7. D 3,6 MP
G. 1. (B ∨ C) ( D
2. ~D
3. B ∨ E
4. C ∨ F /( E & F
5. ~(B ∨ C) 1,2 MT
6. ~B & ~C 5 DeM
7. ~B 6 &E
8. ~C 6 &E
9. E 3,7 DS
10. F 4,8 DS
11. E & F 9,10 &I
H. 1. (A ( B) → C
2. ~C
3. D → A
4. B ( E /( ~D & E
5. ~(A ∨ B) 1,2 MT
6. ~A & ~B 5 DeM
7. ~A 6 &E
8. ~B 6 &E
9. ~D 3,7 MT
10. E 4,8 DS
11. ~D & E 9,10 &I
I. 1. M
2. (M ∨ N) ( ~(P & Q)
3. P & (R ( Q) /( ~R
4. M ∨ N 1 ∨I
5. ~(P & Q) 2,4 MP
6. ~P ∨ ~Q 5 DeM
7. P 3 &E
8. R ( Q 3 &E
9. ~Q 6,7 DS
10. ~R 8,9 MT
J. 1. (A & B) → C
2. ~C & A
3. D → B /(~D
4. ~C 2 &E
5. A 2 &E
6. ~(A & B) 1,4 MT
7. ~A ∨ ~B 6 DeM
8. ~B 5,7 DS
9. ~D 3 8 MT
K. 1. (A ( B) → ~(C ( D)
2. A
3. D ( E
4. (E ( Z) → F /( F
5. A ∨ B 2 ∨I
6. ~(C ∨ D) 1,5 MP
7. ~C & ~D 6 DeM
8. ~D 7 &E
9. E 3,8 DS
10. E ∨ Z 9 ∨I
11. F 4,10 MP
L. 1. ~A & ~B
2. ~(A ( B) → ~(C ( D)
3. C ( E
4. ~F → D
5. (E & F) → G /(G
6. ~(A ∨ B) 1 DeM
7. ~(C ∨ D) 2,6 MP
8. ~C & ~D 7 DeM
9. ~C 8 &E
10. ~D 8 &E
11. E 3,9 DS
12. F 4,10 MT
13. E & F 11,12 &I
M. 1. ~A & ~B
2. (C & D) → (A ( B)
3. D /(~C
4. ~(A ∨ B) 1 DeM
5. ~(C & D) 2,4 MT
6. ~C ∨ ~D 5 DeM
7. ~C 3,6 DS
N. 1 ~A
2. ~(A & B) → C /( C
3. ~A ∨ ~B 1 ∨I
4. ~(A & B) 3 DeM
5. C 2,4 MP
O. 1. ~A
2. (C ( D) → (A & B)
3. (F & G) → (C ( D)
4. F /(~G
5. ~A ∨ ~B 1 ∨I
6. ~(A & B) 5 DeM
7. ~(C ∨ D) 2,6 MT
8. ~(F & G) 3,7 MT
9. ~F ∨ ~G 8 DeM
10. ~G 4,9 DS
P. 1. (B & C) ( (E & F)
2. ~E
3. B
4. ~C ( D /( D
5. ~E ∨ ~F 2 ∨I
6. ~(E & F) 5 DeM
7. ~(B & C) 1,6 MT
8. ~B ∨ ~C 7 DeM
9. ~C 3,8 DS
10. D 4,9 MP
Q. 1. (A ( B) ( C
2. ~C
3. A ( D
4. E ( B
5. (D & ~B) ( F /( F
6. ~(A ( B) 1,2 MT
7. A & ~B 6 NCR
8. A 7 &E
9. ~B 7 &E
10. D 3,8 MP
11. D & ~B 9,10 &I
12. F 5,11 MP
R. 1. A
2. ~B
3. C ( (A ( B)
4. C ∨ D /( D
5. A & ~B 1,2 &I
6. ~(A ( B) 5 NCR
7. ~C 3,6 MT
8. D 4,7 DS
S. 1. [(A ∨ B) ( C] ( D
2. ~(D ∨ A) /( B
3. ~D & ~A 2 DeM
4. ~D 3 &E
5. ~A 3 &E
6. ~[(A ∨ B) ( C] 1,4 MT
7. (A ∨ B) & ~C 6 NCR
8. A ∨ B 7 &E
9. B 5,8 DS
T. 1. [(A & B) ∨ (~B ∨ ~D)] ( (E & F)
2. ~E /( ~(D ( A)
3. ~E ∨ ~F 2 ∨I
4. ~(E & F) 3 DeM
5. ~[(A & B) ∨ (~B ∨ ~D)] 1,4 MT
6. ~(A & B) & ~(~B ∨ ~D) 5 DeM
7. ~(A & B) 6 &E
8. ~(~B ∨ ~D) 6 &E
9. ~A ∨ ~B 7 DeM
10. B & D 8 DeM
11. B 10 &E
12. D 10 &E
13. ~A 9,11 DS
14. D & ~A 12,13 &I
15. ~(D ( A) 14 NCR
U. 1. (~A ∨ B) ( C
2. D ( (A ( B)
3. ~C /( ~D
4. ~(~A ∨ B) 1,3 MT
5. A & ~B 4 DeM
6. ~(A ( Β) 5 NCR
7. ~D 2,6 MT
V. 1. A
2. (A ∨ B) ( (C ( ~A)
3. C ∨ D /( D
4. A ∨ B 1 ∨I
5. C ( ~A 2,4 MP
6. ~C 1,5 MT
7. D 3,7 DS
W. 1. B ( C
2. ~(A ( C)
3. D ( (A ( B) /( ~D
4. A & ~C 2 NCR
5. A 4 &E
6. ~C 4 &E
7. ~B 1,6 MT
8. A & ~B 5,7 &I
9. ~(A ( B) 8 NCR
10. ~D 3,9 MT
X. 1. (A & (B ( C)) ( D
2. ~E ( C
3. ~(A ( D) /( E
4. A & ~D 3 NCR
5. A 4 &E
6. ~D 4 &E
7. ~(A & (B ( C)) 1,6 MT
8. ~A ∨ ~(B ( C) 7 DeM
9. ~(B ( C) 5,8 DS
10. B & ~C 9 NCR
11. ~C 10 &E
12. E 2,11 MT
Y. 1. ((A ∨ B) ( C) ( ((E & (F & G))
2. B ( C
3. ~E /( A
4. ~E ∨ ~(F & G) 3 ∨I
5. ~(E & (F & G)) 4 DeM
6. ~((A ∨ B) ( C) 1,5 MT
7. (A ∨ B) & ~C 6 NCR
8. A ∨ B 7 &E
9. ~C 7 &E
10. ~B 2,9 MT
11. A 8, 10 DS
Chapter 5
5.1 Exercises – page 155
The following inductive arguments have very common inductive argument forms. See if you can identify the kind of argument each is (you won’t be tested on this). Then, for each argument, answer these questions:
1. Assign a probability to the conclusion, based on the premises (i.e., “this conclusion is ____% probable, based on these premises). Explain why you chose that number. Is the argument strong or weak? (NOTE: I am more interested in the number you give the probability and your reasons for giving the number than I am on the answer “strong” or “weak”).
2. Is the argument cogent or uncogent? Why? (If you’re not sure whether statements are true or not, just choose one and state that in your answer. If statements are about imaginary people, take them to be true.)
1) My brother said that the democrats are sure to win the next election. Therefore, the democrats will win.
a) This conclusion is about 10% probable, based on the premises. No information is given about why the brother knows anything about politics. This is a weak argument
b) All weak arguments are uncogent
2) Michael Jordan (the famous basketball player) said on a TV ad: “Hanes underwear are the best.” So Hanes underwear must be the best.
a) This conclusion is about 10% probable, based on the premises. Michael Jordan is a basketball player, not an expert on underwear. This is a weak argument
b) All weak arguments are uncogent
3) A Gallup poll reported that 67% of Americans favor abolishing the death penalty. Therefore, my neighbor Sally would support abolishing the death penalty.
a) This conclusion is 67% probable, based on the premise. Gallup polls are reputable. This is a relatively strong argument
b) Assuming the premise is true the argument is cogent
4) 80% of the bald people in the world are men. Pat is bald. Therefore, Pat is a man.
a) This conclusion is 80% probable, based on the premises. This is a strong argument
b) Assuming the 1st premise is true (I don’t know), the argument is cogent.
5) 80% of humans are over 10 feet tall. Pat is a human. Therefore, Pat is over 10 feet tall.
a) This conclusion is 80% probable, based on the premises. This is a strong argument
b) The 1st premise is clearly false; the argument is uncogent.
6) I left my car headlights on last night. Therefore, this morning, my car battery will be dead.
a) Most of the time leaving car headlights on will drain the car battery overnight. This conclusion is more than 50% probable, based on the premises. This is a strong argument
b) Assuming the premise is true, the argument is cogent.
7) Every fall, during the month of September, Canadian geese fly over my house headed southward. It is now September. I expect to see the geese any day now.
a) This conclusion is more than 50% probable, based on the premises. This is a strong argument
b) The truth of the 1st premise is pretty well established; the argument is cogent.
8) Every January, a herd of elephants migrates through Los Angeles, headed for Mexico. It is now January. We should expect to see the herd of elephants any day now.
a) This conclusion is more than 50% probable, based on the premises. This is a strong argument
b) The 1st premise is clearly false, the argument is uncogent.
9) A whole school of sardines got stranded in Long Beach Harbor and died last month. We should expect to see schools of sardines dying in the harbor every month now.
a) This did happen once, but one event is not enough to establish a prediction. The conclusion is very improbable, based on the premises. This is a weak argument
b) All weak arguments are uncogent.
10) Bob was just honorably discharged from the Navy Seals and is healthy. He can do 100 pushups easily. Bill was also just honorably discharged from the Navy Seals and is healthy. He’ll be able to do 100 pushups as well.
a) This conclusion is more than 50% probable, based on the premises. This is a strong argument
b) Assuming the premises are true, the argument is cogent.
11) When a black cat crosses your path, it causes you to have particularly good luck. A black cat crossed my path this morning. So I’m expecting good luck any time.
a) If the 1st premise were true, then the conclusion would be more than 50% probable, based on the premises. This is a strong argument.
b) The 1st premise is clearly false, the argument is uncogent.
Chapter 6
6.2 Exercises – page 170
Use the 4-step analysis method to evaluate these arguments. Some of them are NOT fallacies.
1. Mary is a figure skater. Most figure skaters are physically fit. Therefore, Mary is physically fit.
a. The conclusion is, Mary is physically fit.
b. The arguer tries to convince us of the conclusion by giving the evidence that Mary is a figure skater and most (not all) figure skaters are physically fit.
c. The evidence is relevant and sufficient to make the conclusion probable (but not true for certain!)
d. This is a strong inductive argument.
2. Mary is a bad figure skater because she falls down during every performance, has never won a medal, and is not graceful.
a. The conclusion is, Mary is a bad figure skater.
b. The arguer tries to convince us of the conclusion by citing evidence to show the probability of the conclusion.
c. The evidence is relevant and sufficient to make the conclusion probable (but not true for certain!)
d. This is a strong inductive argument.
3. Bill says we should support his investment program. But Bill was arrested last year for beating his wife. Therefore, we should reject Bill’s advice.
a. The conclusion is to reject Bill's advice about the investment program.
b. The arguer tries to convince us by saying that Bill has beat his wife.
c. This is not a good argument. The evidence provided is irrelevant. Bill's beating his wife has nothing to do with his investment program.
d. This is an ad hominem abusive fallacy (some people think it is a Red Herring. In one sense it is, but Ad Hominem is better, because this is a specific form)
4. Bill says that we should support his investment program. He says that he knows we cheated on our taxes last year and that he has a good friend in the IRS.
a. The conclusion is that we should support Bill's program.
b. The arguer tries to convince us by (implying) a threat to turn us in to the IRS.
c. The argument is weak. The evidence/threat is irrelevant to the issue of Bill's program.
d. This is an appeal to force fallacy.
5. Bill says that we should support his investment program, because if we don’t, he’ll lose all the money he’s invested in it and will go bankrupt.
a. The conclusion is to support Bill's program.
b. The arguer tries to convince us by trying to make us feel sorry about what will happen to Bill if his program fails.
c. This is a weak argument. The evidence presented has nothing to do with whether the program is a good program or not.
d. This is an appeal to pity fallacy.
6. Bill says we should support his investment program. But Bill is a fraud. He has been in jail twice for promoting phony investment schemes. Therefore, we should reject Bill’s advice.
a. The conclusion is not to support Bill's program.
b. The arguer tries to convince us by citing evidence that Bill is a fraud.
c. This is a strong argument. The evidence presented makes it likely that Bill is a fraud, in which case investing would not be a wise idea.
a. This is an ad hominem abusive form, but not a fallacy.
7. Bill says that we should invest in his investment program. But I don’t think it’s such a good idea. The stock market is a volatile enterprise. It has gone up and down over and over through its lifetime. Sometimes it’s up and sometimes it’s down. It has crashed at least twice that we know of.
a. The conclusion is not to support Bill's program.
b. The arguer tries to convince us by talking about the ups and downs of the stock market.
c. This is a weak argument. The evidence presented has nothing to do with whether the program is a good program or not. It's not even clear that Bill's program is a stock market investment program.
d. This is a red herring fallacy.
8. Bill says we should invest in his investment program because everyone he knows has invested.
a. The conclusion is to invest in Bill's program.
b. The arguer tries to convince us by saying that “everyone” has invested.
c. This is a weak argument. The evidence presented has nothing to do with whether the program is a good program or not. We don’t know who “everyone” is or what the results.
d. This is an appeal to the people fallacy.
9. Bill says that we should invest in his investment program. He’s using the latest economic theories so it’s sure to succeed.
a. The conclusion is not to invest in Bill's program.
b. The arguer tries to convince us by talking about the “latest economic theories.”
c. This is a weak argument. Thtrere is no evidence about what they theoriesa are or why they are good ones.
d. This is an appeal to novelty fallacy.
10. Bill’s investment program is going to fail because he has not thought it out clearly, he has no concept of basic economics, and he doesn’t know anything about the stock market.
a. The conclusion is that Bill's program is going to fail.
b. The arguer tries to convince us by citing evidence that makes it likely that Bill's program will fail.
c. This is a strong argument. The evidence presented is relevant and sufficient to make it probable that the conclusion is true.
d. This is not a fallacy.
11. American Express ad: You're in a strange city. You've been robbed. What will you do? Who will you turn to? Carry American Express.
a. The conclusion in advertisements is often not stated clearly. In this case it is something like “American Express is good” or “You should carry American Express.”
b. The arguer tries to convince us by trying to make us feel afraid, and suggesting that American Express could help us not be afraid.
c. This is a weak argument. The evidence presented has nothing to do with whether a person will get robbed while travelling.
d. This is an appeal to force/fear fallacy.
12. A wine ad shows a room full of shiny, happy people, all drinking wine. The caption says, “Gallo – a part of your life.” This is an argument – evaluate it.
a. The conclusion is “You should buy Gallo wine.”
b. The arguer tries to convince us by trying to make us believe that if we buy Gallo wine, we will be happy, shiny people with great relationships.
c. This is a weak argument. The evidence presented has nothing to do with whether we will be happy, social people or not.
d. This is an appeal to the people fallacy.
13. It can’t be so wrong for me to have the occasional affair. After all, almost no animals in nature are strictly monogamous.
a. The conclusion is that it is not wrong to have “the occasional affair.”
b. The arguer tries to convince us by stating that almost no animals in nature are monogamous.
c. This is a weak argument. There’s no evidence presented about why what’s true of animals should be true of humans.
d. This is a naturalistic fallacy.
14. It can’t be so wrong for me to cheat on my boyfriend occasionally, so long as he never finds out about it. It’s not hurting anyone.
a. The conclusion is that it is okay to cheat on her boyfriend (so long as…)
b. The arguer tries to convince us by talking about the case in which the boyfriend doesn’t find out.
c. This is a weak argument. It reduces morality to the consequences, but there is more to morality than just consequences.
d. This is an appeal to consequences fallacy.
15. Marriage has always been defined as between a man and a woman. That’s enough reason to keep it that way.
a. The conclusion is to define marriage as between a man and a woman.
b. The arguer tries to convince us by noting that marriage “has always been” defined this way.
c. This is a weak argument. The fact that people do things for long periods of time doesn’t make them right (slavery was practices for a long time as well – that didn’t make it right).
d. This is an appeal to tradition fallacy.
16. The idea that marriage must be between a man and a woman is ancient and outdated. We have to get our laws up to date.
a. The conclusion is to change the laws about whom can be married.
b. The arguer tries to convince us by stating that the idea that marriage must be between a man and woman is ancient and outdated.
c. This is a weak argument. The fact that an idea is old or ancient does not make it wrong. The law against murder is a very old law; that doesn’t make it wrong.
d. This is an appeal to novelty fallacy.
17. There is no strong argument for keeping marriage limited to opposite genders. So it’s time to change it.
a. The conclusion is to change whom should get married.
b. The arguer tries to convince us by claiming that there’s no reason not to change it.
c. This is a weak argument. Lack of support for one position does not constitute support for the opposite position.
d. This is an appeal to ignorance fallacy.
18. There is no strong argument for changing the definition of marriage. So we have to keep it what it is.
a. The conclusion is not to change the definition of marriage.
b. The arguer tries to convince us by claiming that there’s no reason to change it.
c. This is a weak argument. Lack of support for one position does not constitute support for the opposite position.
d. This is an appeal to ignorance fallacy.
19. Why should gay marriage be allowed? Because there’s nothing wrong with it, that’s why!
a. The conclusion is that gay marriage should be allowed.
b. The arguer tries to convince us by stating that there is nothing wrong with gay marriage.
c. This is a weak argument. The arguer just repeats the same idea – that something should be allowed is another way of saying that there is nothing wrong with it. No independent evidence is provided.
d. This is a begging the question (circular) fallacy.
20. Abortion should be allowed because lots of women will go out and have abortions anyway, and many will be harmed by illegal back-alley abortions.
a. The conclusion is that abortion should be allowed.
b. The arguer tries to convince us by discussing the consequences of not allowing abortions.
c. This is a weak argument. It reduces morality to the consequences, but there is more to morality (and legality) than just consequences.
d. This is an appeal to consequences fallacy
21. Senator Jones says that we should raise taxes to help balance the budget. But we all know what Senator Jones is up to. She just wants to be re-elected and she knows that she won’t be re-elected if she doesn’t vote for increased taxes.
a. The conclusion is to reject Sen. Jones’ idea to raise taxes.
b. The arguer tries to convince us by discussing a (bad) motivation Sen. Jones might have for raising taxes.
c. This is a weak argument. Whether this is Sen. Jones’ motivation is irrelevant. The issue is, what evidence or arguments are there favoring or not favoring raising taxes.
d. This is an ad hominem circumstantial fallacy.
22. Senator Jones says that we should raise taxes to help balance the budget. And she is right. Raising taxes will help balance the budget.
a. The conclusion is unclear, but probably to support Jones.
b. The arguer tries to convince us by saying that Jones is right.
c. This is a weak argument. The reason given to support Jones is just a repetition of Jones’ argument.
d. This is a begging the question (circular) fallacy.
6.3 Exercises – page 179
1. Bill says that if we support his investment program, our investment will pay great dividends. Then we will become rich and can quit our jobs and live on a luxury yacht for the rest of our lives. I think we should do it.
a. The conclusion is to support Bill’s program.
b. The argues tries to convince us by talking about the good things that will happen
c. There is no evidence presented to support the claim that these things will actually happen. Weak argument.
d. This could be construed as a weak consequences argument, but is more specifically a weak Slippery Slope (even though the slope is in a positive direction).
2. Bill says that we should invest in his investment program. He says that Bob Jones, the famous Rock musician, looked at it and fully endorses it.
a. The conclusion is to support Bill’s program.
b. The argues tries to convince us by citing the opinion of Bob Jones
c. There is no basis for believing that Bob Jones knows anything about investment programs. Weak argument
d. Weak appeal to authority
3. Bill says that we should invest in his investment program. He says that Bob Jones, the well-known economist, looked at it and claims that it is very likely to be successful.
a. The conclusion is to support Bill’s program.
b. The argues tries to convince us by citing the opinion of Bob Jones
c. If Bob Jones is a well-known economist, he probably knows about investment programs. There is little information here, but it’s a relatively strong argument
d. Strong appeal to authority
4. Why is gay marriage wrong? Because it makes me sick, that’s why!
a. The conclusion is gay marriage is wrong.
b. The argues tries to convince us by talking about how the idea of gay marriage makes him or her feel
c. The fact that an idea makes someone feel a certain way is no evidence that the idea is true or false.
d. This could be construed as a begging the question – suppressed evidence fallacy, but it is also a weak egoist fallacy – the arguer is only citing his or her own opinion.
5. I spent a week traveling in Germany. All of the people in every restaurant I ate in were very friendly and welcoming. I think the Germans are a very friendly people.
a. The conclusion is that Germans people are friendly.
b. The arguer tries to convince us by mentioning the German people she or he met in restaurants while traveling in Germany for a week.
c. The question here is, are the people one would meet in restaurants while traveling in Germany for a week representative of the German people as a whole. It is not a terribly strong argument, but it is probably (slightly) over 50% probable.
d. Relatively strong generalization argument
6. If we raise taxes to decrease the budget deficit, then investors will have less money to invest in business. Then business will slow down, which means that people will make even less money, which will mean a worse economy AND less tax revenue. Raising taxes is a bad idea!
a. The conclusion is not to raise taxes.
b. The argues tries to convince us by talking about the bad things that will happen
c. If we could predict what would happen if we change taxes, this argument would have been settled long ago. Predictions in economic systems are impossible.
d. This is a weak slippery slope argument.
7. If we cut back on government spending to decrease the budget deficit, then we will have less money to spend on social programs for the poor, needy, and uneducated. That means that we will be harming the people that need us most, and a less educated workforce will mean a worse economy. We must continue to help the people that need us!
a. The conclusion is not to cut back on government spending.
b. The argues tries to convince us by talking about the bad things that will happen
c. If we could predict what would happen if we change government spending, this argument would have been settled long ago. Predictions in economic systems are impossible.
d. This is a weak slippery slope argument.
8. My brother said that there are three workers from Guatemala working on the construction crew that he is on. I guess the Guatemalans are taking over our construction industry.
a. The conclusion is that Guatemalans are taking over the construction industry.
b. The argues tries to convince us by mentioning three Guatemalan workers on a construction crew.
c. Three Guatemalans on one crew are in no way enough to generalize to a whole industry.
d. This is a weak generalization argument.
9. I forgot to turn the headlights off on my car last night (and it doesn’t turn them off by itself!). I expect the battery to be dead this morning.
a. The conclusion is that the car battery will be dead.
b. The argues tries to convince us by mentioning the fact that he or she left the headlights on
c. There is a strong (causal) connection between headlights being left on and batteries being dead. This is a strong argument
d. Strong causal argument
10. Last May, a school of sardines got stranded in Long Beach harbor and all died. I guess we should expect this to happen every May now.
a. The conclusion is that sardines will get stranded in Long Beach harbor and die every year.
b. The argues tries to convince us by mentioning one instance of sardines being stranded in Long Beach harbor and dying.
c. This argument is weak. One incident of something happening is not enough to claim a causal connection or to generalize to the future.
d. This could be either a weak causal argument or a weak generalization argument.
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