Philo 11 Exercise Booklet

Philosophy 11 Course Outline I. Fundamental concepts in logic 1. Reasoning and inference 2. Deductive argument 3. Inductive reasoning and fallacy 4. Proof, logical consequence and consistency II. Formal logic 5. Formal languages 6. Propositional calculi 7. Principles of deduction 8. Proof procedures and strategies 9. Conditional proof III. Semantic techniques 10. Interpreting propositional calculi 11. Truth-functions 12. Argument forms and arguments 13. Proving invalidity using truth tables IV. Quantification theory 14. Predicate logic 15. Quantification 16. Quantification rules 17. Syllogisms and beyond 18. First-order semantics 19. Models for invalidity V. Advanced topics 20. Relations 21. Identity 22. Truth trees VI. Logic, mathematics and philosophy References: Copi, Irving et al. Introduction to Logic. Copi, Irving. Symbolic Logic. De Villa et al. Lohika at Pangangatwiran. Hodges, Wilfrid. Logic. Hurley, Patrick. A Concise Introduction to Logic. 1

Logic puzzles 1. On a certain train, the crew consists of the brakeman, the fireman and the engineer. Their names listed alphabetically are Jones, Robinson, and Smith. On the train are also three passengers with corresponding names, Mr. Jones, Mr. Robinson, and Mr. Smith. The following facts are known: a. Mr. Robinson lives in Detroit. b. The brakeman lives halfway between Detroit and Chicago. c. Mr. Jones earns exactly $20,000 a year. d. Smith once beat the fireman at billiards. e. The brakeman’s next-door neighbor, one of the three passengers mentioned, earns exactly three times as much as the brakeman. f. The passenger living in Chicago has the same name as the brakeman. What was the engineer’s name? 2. Five couples were married last week, each on a different weekday. From the information provided, determine the woman (one is Cathy) and man (one is Paul) who make up each couple, as well as the day on which each couple was married. a. Anne was married on Monday, but not to Wally. b. Stan’s wedding was on Wednesday. c. Rob was married on Friday, but not to Ida. d. Vern (who married Fran) was married the day after Eve. 3. Of three prisoners in a certain jail, one had normal vision, the second had only one eye, and the third was totally blind. The jailor told the prisoners that, from three white hats and two red hats, he would select three and put them on the prisoners’ heads. None could see what color hat he wore. The jailor offered freedom to the prisoner with normal vision if he could tell what color hat he wore. To prevent a lucky guess, the jailor threatened execution for any incorrect answer. The first prisoner could not tell what hat he wore. Next the jailor made the same offer to the one-eyed prisoner. The second prisoner could not tell what hat he wore either. The jailor did not bother making the offer to the blind prisoner, but he agreed to extend the same terms to that prisoner when he made the request. The blind prisoner said: I do not need to have my sight; From what my friends with eyes have said, I clearly see my hat is _____! How did he know? Sudoku

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6 3 2 5 8 7 2 5

1 6 4 8 2

5

3 1

4

I G D E A

C 8

G A B I E

9 8 7 5 4 7 6

D A F

H C F D G

D C F E H 2

B

Exercises in Logic Argument analysis 1. On the average, one out of every 2 billion people is killed by a falling meteor. I am confident, therefore, that I will not be killed by a falling meteor. 2. Kohlrabi must be nutritious because it is a vegetable and most vegetables are nutritious. 3. Andoni is an old man. Very few old men fly hang gliders. Therefore, Andoni does not fly hang gliders. 4. Either Peter loves carrots or he is not a rabbit. Peter is a rabbit. Therefore, Peter loves carrots. 5. The plane would not have crashed into the sea if Superman were real. But the plane did crash into the sea killing all its passengers. Therefore, Superman is not real. 6. There has never been any human casualty caused by falling debris from man-made satellites plunging back to the earth. Therefore, when a Russian satellite plunges back to earth next week, there will be no human casualties. 7. Though many people have passed the threshold of death, none has ever returned to tell about it. It is unlikely then, that anyone ever will. 8. My father was born in China, yours in France. So we can’t be brothers. 9. Roger is running swiftly. Therefore, Roger is running. 10. You’ll get wet if you go out. It’s raining. 11. Geraniums are not frost hardy. There are frequent frosts in Iceland. So geraniums are not native to Iceland. 12. Oil reserves are finite. Accordingly, since consumption is continuing at a rapid rate, eventually the world will run out of oil. 13. There are more people than hairs on any one person’s head. Therefore, two people have the same number of hairs on their heads. 14. Backward time travel will never be developed and used extensively by human beings at any time in the future. For if it were to be, then travelers from the future would be likely to visit our own time and times already past. But since such visits should be readily detectable and since we have no evidence of them, we can only conclude that the don’t occur. 15. Since most people sleep about a third of the time and people on different parts of the earth sleep at different times, at any given moment someone is asleep. Fallacies 1. The ether really exists; as a matter of fact, no one has definitely shown that it does not. 2. Everyone believes that Orestes killed Aegisthus. Therefore, he must be guilty. 3. The Golden Rule is basic to every system of ethics ever devised, and everyone accepts it in some form or other. It is therefore an undeniably sound moral principle. 4. Exercise is excellent for health. Therefore, Kumar, who has a serious heart ailment ought to take more exercise. 5. Studies show that during exams, students generally perform better under pressure. Therefore, they ought to be put under pressure all the time so that they will be better and more efficient. 6. Rock music must be better than classical music, since so many more stations play rock music. 7. Last night, I met the most beautiful man I ever saw. I believe that he is an angel because he told me so; and an angel would never lie. 8. God exists because the Bible tells us so. I know that the Bible is true because it is God’s word. 9. Wife: “Have you resolved to lead a straight life this year?” Husband: “Yes.” Wife: “So you now admit that you were hiding things behind my back last year!”

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10. We should legalize gambling because we can’t stop it. It is an integral part of living; a man gambles every time he crosses a street or takes a wife. 11. What is natural is good; but to make mistakes is natural. Therefore, to make mistakes is good. 12. Teenager: “But why can’t I be allowed to go in? Sure it says ‘Children of members under 18 are not allowed,’ but my father is already 40 years old!” 13. There is not a single cell in me that is more than seven years old. Therefore, I’m not more than seven years old. 14. Cooks have been preparing food for generations, so our cook must be a real expert. 15. The force of gravitation is nothing but a small deviation from no force at all, becoming appreciable only when huge quantities of matter are involved. Suppose a ball were made of all the gold that has been mined in the world, say 30,000 tons; and the ball would be about 46 feet in diameter. If it were placed in space, away from other attracting forces, a 200 pound man sitting on it would weigh the equivalent of 1/1000 of an ounce on the earth. A cricket could easily lift him and a frog could kick him completely away from the ball of gold. Since men are not usually so easily diverted from gold we may conclude, in the manner of Aesop, that the force of avarice greatly exceeds that of gravitation. Logical structure of compound propositions 1. Cadmium is toxic but zinc is not. 2. Both nickel and cobalt are ferromagnetic. 3. It is not the case that both aluminum and magnesium burn in air. 4. Aluminum and magnesium both do not burn in air. 5. Either platinum or aluminum does not dissolve in hydrochloric acid. 6. Neither platinum nor aluminum dissolves in hydrochloric acid. 7. Either sodium reacts with chlorine and potassium reacts with iodine or calcium reacts with bromine. 8. Zirconium does not absorb neutrons unless cadmium does. 9. Either copper or sulfur is a good conductor of electricity, but it is not the case that both of them are. 10. Lithium reacts violently with water if sodium does. 11. Lithium reacts violently with water if and only if sodium does. 12. Lithium reacts violently with water only if sodium does. 13. Gallium’s having a low melting point implies that indium and thallium do, too. 14. That osmium absorbs hydrogen is a sufficient condition for iridium to do so. 15. That osmium absorbs hydrogen is a necessary condition for iridium to do so. 16. If actinium is radioactive, then if thorium is, so is protactinium. 17. Kung sosobra ang kumakalat na pera, siyempre tataas ang bilihin. 18. Siguradong iiwas ang mga investor kung tutuloy na magkakagulo sa rehiyon. 19. Mayroong demokrasya kung ganoon din at ganoon lamang na mayroong malayang karapatang bumoto. 20. If he uses good bait, then he catches the legal limit provided the fish are biting. He uses good bait and catches the legal limit. Therefore, the fish are biting. 21. We will have a gale unless the wind changes direction. The wind will change direction only if the anticyclone moves further south. Indeed, the anticyclone will move further south. Therefore, we will have a gale. 22. If the anti-terrorism bill becomes law, then we will lose some of our freedoms. Martial law will be declared only if terrorists go unpunished. Neither will the anti-terrorism bill become law nor will martial law be declared. Therefore, it is not both the case that we will lose some of our freedoms and terrorists will go unpunished.

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Formal proofs (rules of inference) 1. A B | (A v C) v B

2. A e B A |BvC

3. A e B (B e C) v (C e A) |Ae C

4. (A e B) v (C e D) A |BvD

5. A v (B e A) ~A v C | ~B

6. A v B (A v C) e D |Av D

7. A e B BeC De E AvD |CvE

8. (A v B) e C (C v B) e [A e (D / E)] Av D |D/ E

9. A e B ~C e (D e E) C v (A v D) ~C |BvE

10. A e ~B ~A e (C e ~B) (~D v ~C) e ~~B ~D | ~C

11. (A e B) v (C e D) (B e E) v (D e F) A |EvF

12. (A v B) e (C v D) (C v D) e E A |E

13. (A e B) v (C e D) EeF (A v E) v (C v G) |BvF

14. (~A v ~B) e (C e B) BeA ~A | ~C

15. A e B BvC (C v ~A) e (D v ~A) ~B |D

16. (A v B) e C (C v D) e E DvA ~D |E

Formal proofs (with replacement rule) 1. ~P |PeQ

2. P |Qe P

3. P e (Q e R) | Q e (P e R)

4. P e (Q v R) |PeQ

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5. P e Q | P e (Q v R)

6. P e Q | (P v R) e Q

7. (P v Q) e R |PeR

8. P e Q Pe R | P e (Q v R)

9. P e Q ReQ | (P v R) e Q

10. P e Q PvQ |Q

11. (P v Q) e ~R R | ~P

12. P v (Q v R) R v ~P |R

13. (P v Q) v R ~P v ~Q |R

14. P / Q Q/ R |P/R

15. (P v Q) e (R v S) ~R | ~P

16. (P v ~Q) e R ~(Q v R) | ~P

17. P e Q ~P e R ~Q e ~R |Q

18. P e (Q v R) R e (S v T) ~S |PeQ

19. (P e ~Q) e R ~(P v Q) | R v ~Q

20. (~P e Q) v (R e Q) ~(~R v P) |Q

21. ~~P e Q ~P e R S e ~R Sv T |Q

22. (P v Q) / (~R v ~S) (P v Q) e T Te U Ue V | (R v S) v V

23. P / (Q v R) ~R |P/Q

24. [(P v Q) e ~R] v [~R e ~(S v T)] ~U v (Q v S) | ~V

25. P / Q Q / (R v S) R / (P v S) PvS |PvS

26. P e Q Q e [P e (R v S)] R/S ~(R v S) | ~P

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27. (P e Q) v (Q e R) (R e S) v (S e T) ~T [~(Q v T) e U] v (U e P) |R

28. (P e Q) e (R / S) (T e ~U) e (Q / ~U) {[(Te ~U)v(V/W)]v (RvS)} e [(V/ W)e (Pe Q)] (T e ~U) v (V / W) RvS | (Q / ~U) v (R / S)

Conditional proof 1. P e Q (Q v R) e S |PeS

2. P e Q (R e Q) e S |PeS

3. P e ~Q ~(R v ~P) | R e ~Q

4. (~P e Q) e R (R v S) e T |PeT

5. P e (Q v R) S e (Q v T) | (P v S) e Q

6. P v (Q v ~R) (P v Q) e (S v ~R) |ReS

7. ~P | (P v Q) e R

8. P e ~(P v Q) |PeR

9. (P v Q) e ~R R v (S v T) P/ Q |PeS

10. P e (Q e R) Q e (R e S) | P e (Q e S)

11. P e Q | (R v P) e (Q v R)

12. P | (Q e R) e [(R e S) e (Q e S)]

13. P e (~Q e R) Q v (R e S) | P e (~Q e S)

14. (P v Q) / R Pe Q |P/R

15. (P e Q) e (R e S) (~T v ~U) e (S e V) (P e T) v (T e P) ~(~T e U) |ReV

16. P e Q ~Q ~P e R ReS |SvT

17. P e (Q v R) S e (R v T) ~R | ~(Q v T) e ~(P v S)

18. P e (Q e R) (P e ~Q) e (S v R) ~R |S

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19. P e Q Q e [(R e ~~R) e S] |PeS

20. (P e Q) v (R e S) (Q v S) e {[T e (T v U)] e (P v R)} |P/R

21. P v (Q e R) [Q e (Q v R)] e (S v T) (S e P) v (T e U) |PvU

22. [(P e Q) v (R e S)] [(Q v S) e {[T e (U e T)] e (P v R)}] |P/R

23. | P e (Q e P)

24. | (P e Q) e [P e (P v Q)]

Proving invalidity 1. P |PeQ

2. P e Q ~P | ~Q

3. P e Q Qe R |ReP

4. P e (Q e R) Q e (R e S) |PeS

5. P v Q QvR |PvR

6. P / Q Qe R |PvR

7. P e (Q e R) Pv R |Q

8. (P e Q) v (R e S) ~(Q v S) | ~(P v S)

9. (P v Q) e R PvQ |R

10. P / (Q v R) ~P | ~R

11. P e Q Qe R ~R v (~S v P) |S

12. (P e Q) v (R e ~S) (~P v ~R) e (T v U) ~Q v U | T e ~R

13. (~P e Q) v R (R e Q) v ~Q | ~(Q e P)

14. ~P e (~Q e ~R) ~(R v ~P) e S |SeQ

15. (P v Q) e R R e (Q v S) P e (~T e Q) (T e P) e ~S |Q/ R

16. P v (Q v R) (P v Q) v (S / ~T) (S e ~T) e (T v ~U) (U e V) v (V e T) (Q e R) e V |V

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Propositional arguments 1. If we receive radio signals from outer space, then extraterrestrial beings exist. We do not receive such signals. Therefore, extraterrestrials do not exist. 2. Godzilla will destroy Tokyo unless he makes it to New York. Godzilla makes it to New York. Therefore, Godzilla will not destroy Tokyo. 3. If the Pampanga Dragons win at Bacolod they will win the National Championship. Therefore if the Pampanga Dragons win at Bacolod, then if they continue to rebound well then they will win the National Championship. 4. This creature is warm-blooded if and only if it is either a mammal or a bird. But it is clearly not a mammal. So it must be a warm-blooded bird. 5. Only if the burglars were sophisticated professionals could they have successfully dismantled the alarm system. Hence they were not sophisticated professionals, for they were unable to dismantle it. 6. If CJ studies, she receives good grades. If CJ does not study, then she enjoys college. If CJ does not receive good grades, then she does not enjoy college. Thus, CJ received good grades. 7. He has to play to win or play clean, but he cannot do both. His conscience will be easy if he plays clean. If he plays to win and his conscience is uneasy, he will be a miserable man. He will play to win. So he will be a miserable man. 8. If either wages or prices are raised, there will be inflation. If there is inflation, then either Congress must regulate it or people will suffer. If the people suffer, Congressmen will be unpopular. Congress will not regulate inflation, and Congressmen will not be unpopular. Therefore, wages will not rise. 9. Either he has his old car repaired or he buys a new car. If he has his old car repaired, then he will owe a lot of money to the garage. If he owes a lot of money to the garage, then he will not soon be out of debt. If he buys a new car, then he must borrow money from the bank, and if he must borrow money from the bank, then he will not soon be out of debt. Either he will soon be out of debt or his creditors will force him into bankruptcy. Therefore, his creditors will force him into bankruptcy. 10. If God were willing to prevent evil but unable to do so, he would be impotent. If he were able to prevent evil but unwilling to do so, he would be malevolent. Evil can exist only if God is either unwilling or unable to prevent it. There is evil. If God exists, he is neither impotent nor malevolent. Therefore God does not exist. 11. The rain in Spain slows down the train as it crosses the plain. The main train in Spain never crosses the plain if the engineer develops a pain from the falling rain. If the engineer does not develop a pain from the falling rain, then either he gets a pain from Lady Jane or the rain in Spain does not slow down the train as it crosses the plain. It follows that the main train in Spain crosses the plain if and only if the engineer does not get a pain from Lady Jane. 12. Dust particles from Mt. Pinatubo’s eruption encircled the earth’s atmosphere. Therefore, either global temperatures will go down or they will not. 13. Kapag uuwi si Rico, siguradong tuloy ang lakad sa Coron kung ganoon din at ganoon lamang na walang masyadong aasikasuhin dito. Kung alinman sa uuwi si Emer o si Ricia, siguradong uuwi si Rico. Uuwi si Ricia at wala namang masyadong aasikasuhin dito. Samakatwid, matutuloy ang lakad sa Coron. 14. Maayos ang takbo ng negosyo kung ganoon din at ganoon lamang na malaki ang kita nito. Alinman sa bagsak ang dolyares o malaki ang kita ng negosyo. Kung bagsak ang dolyares, walang order ang natatanggap. Maraming order na natatanggap. Samakatwid, hindi maayos ang takbo ng negosyo. 15. May utang si Ka Mando. Walang ani ngayong tag-init at delikado ang ani sa tag-ulan dahil sa La Niña. Kung alinman sa walang ani ngayong tag-init o tataas ang halaga ng pataba, siguradong sasama ang lagay ng kooperatiba. Kung alinman sa sasama ang lagay ng kooperatiba o magkakagalit ang mga kasapi ng sangguniang baranggay, siguradong magtatago si Ka Mando. Samakatwid, parehong hindi totoong may utang o magtatago si Ka Mando.

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Logical structure (predicate) 1. Snakes are reptiles. 2. Sparrows are not mammals. 3. Snakes are not all poisonous. 4. Snake bites are sometimes fatal. 5. The common cold is never fatal. 6. Children are present. 7. Nurses are always considerate. 8. Diplomats are not always rich. 9. No boy scout ever cheats. 10. Only executives have secretaries. 11. There are honest politicians. 12. All that glitters is not gold. 13. Not all feminists are radicals. 14. No one but a criminal lives in a penitentiary. 15. A child pointed his finger at the emperor. 16. Masayahin ang mga Pilipino. 17. Hindi dumating ang lahat ng panauhin. 18. Ang bawat tao ay may sariling paninindigan. 19. Ang ilan sa kanyang kasama ay hindi nanindigan. 20. Walang bilanggong walang kasalanan. 21. Malalim na ang gabi ngunit wala pa siya. 22. Si Maria ay masarap magluto at matamis magmahal. 23. Kung maganda ang panahon, siguradong papalaot si Andres. 24. Habang ang lahat ng tala sa langit ay nagniningning, mahimbing ang tulog ng bawar hayop sa sansinukob. 25. Some feminists are warm and considerate. 26. Some hackers are clever but not smart. 27. Only policemen and firemen are both indispensable and underpaid. 28. A gladiator wins if and only if he is lucky. 29. Any person is a coward who deserts. 30. Among snakes, only copperheads and rattlers are poisonous. Quantificational arguments 1. All rabbits love carrots. Roger is a rabbit. Therefore, Roger loves carrots. 2. No athletes are bookworms. Nancy is a bookworm. Therefore, Nancy is not an athlete. 3. To be a swindler is to be a thief. None but the underprivileged are thieves. Therefore, swindlers are always underprivileged. 4. None but the brave deserve the fair. Only soldiers are brave. Therefore, the fair are deserved only by soldiers. 5. All jesters are knaves. No knaves are lucky. Therefore, no jesters are lucky. 6. No gamblers are happy. Some idealists are happy. Therefore, some idealists are not gamblers. 7. There are no uniforms that are not washable.. There are no washable velvets. Therefore, there are no velvet uniforms. 8. Lahat ng mga Capampangan ay masarap magluto. Si Mameng ay isang Capampangan. Kaya siguradong masarap magluto si Mameng.

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9. Ang mga makata ay malikhain. Ang ibang makata ay mga intelektwal. Samakatwid, may mga intelektwal na malikhain. 10. Ang mayayaman lamang ang may kotse. Hindi lahat ng may kotse ay marunong magmaneho. Kung gayon, may mayayamang hindi marunong magmaneho. 11. Acids and bases are chemicals. Vinegar is an acid. Therefore, vinegar is a chemical. 12. Tigers are fierce and dangerous. Some tigers are beautiful. Therefore, some dangerous things are beautiful. 13. Teachers are either enthusiastic or unsuccessful. Teachers are not all unsuccessful. Therefore, there are enthusiastic teachers. 14. A communist is either a fool or a knave. Fools are naive. Not all communists are naive. Therefore, some communists are knaves. 15. Bananas and grapes are fruits. Fruits and vegetables are nourishing. Therefore, bananas are nourishing. 16. All professors are learned. All learned professors are savants. Therefore, all professors are learned savants. 17. Argon compounds and sodium compounds are either oily or volatile. Not all sodium compounds are oily. Therefore, some sodium compounds are volatile. 18. No employee who is either slovenly or discourteous can be promoted. Therefore, no discourteous employee can be promoted. 19. Bees and wasps sting if they are angry or frightened. Therefore, any bee stings if it is angry. 20. A woman without a husband is unhappy unless she has a boyfriend. Daphne is a happy woman. Therefore, either she is not without a husband or she has a boyfriend. Invalidity 1. All revolutionaries are brave. All heroes are brave. Therefore, all revolutionaries are heroes. 2. All mercenaries are undependable. No guerillas are mercenaries. Therefore, no guerillas are undependable. 3. All generals are handsome. Some intellectuals are handsome. Therefore, some generals are intellectuals. 4. All collies are affectionate. Some collies are watchdogs. Therefore, all watchdogs are affectionate. 5. Some malcontents are noisy. Some officials are not noisy. Therefore, no officials are malcontents. 6. Some physicians are quacks. Some quacks are not responsible. Therefore, some physicians are not responsible. 7. If anything is metallic, then it is breakable. There are breakable ornaments. Therefore, there are metallic ornaments. 8. All zebras wear stripes, and all prisoners wear stripes; therefore some zebras are prisoners. 9. Some philosophers are mathematicians. Russell is a philosopher. Hence, Russell is a mathematician. 10. Some lovely things are little things, for some puppies are little things and most puppies are lovely things. Relational propositions 1. Achilles is the father of Neoptolemus. 2. The Parthenon is taller than the Acropolis. 3. Aeschylus wrote “Agamemnon”. 4. Apollo unrequitedly loves Daphne. 5. Medea is a friend of either Creon or Aegeus. 6. Laches knows everything. 7. Laches knows something.

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8. Whoever loves God is never lost. 9. Calchas advises only princes. 10. Athena advises some heroes. 11. Ajax avenged some of his friends. 12. Nasa kanluran ng Pampanga ang Bulkang Pinatubo. 13. Kinamayan ni Jose sina Gardo at Cris. 14. Ang nagbabala kay Miriam ay maaaring si Eddie o si Joseph. 15. Kung maganda si Angela, iibigin siya ni Manuel. Multiply general propositions 1. Everyone loves someone. 2. Everyone loves everyone. 3. Someone loves everyone. 4. Someone loves someone. 5. No one loves everyone. 6. No one loves anyone. 7. If Dido did not love Aeneas, then nobody loved anybody. 8. Every person can sell something or other. 9. Some people can sell anything. 10. Everybody hurts sometimes. 11. Some people break everything they touch. 12. Every person admires some people he or she meets. 13. All UP students can read any of the books in the main library. 14. God only helps those who help themselves. 15. No one learns anything unless he teaches it to himself. 16. There will never be a time when a romantic will fall in love with a logician. 17. Only someone who hasn’t sinned is permitted to cast stones at those who have. 18. Anyone who promises everything to everyone is certain to disappoint somebody. 19. If any freshman fails, then some senior ought to tutor him or her. 20. If someone is too noisy, then everyone in the room will be annoyed with him. Relational arguments 1. Romeo loves either Diana or Juliet. But Romeo does not love Englishwomen. Diana is an Englishwoman. Therefore, Romeo loves Juliet. 2. Ptah is an Egyptian god, and he is the father of all Egyptian gods. Therefore, he is the father of himself. 3. All of Hillary’s children are unmarried. Rhodippe is a daughter of Hillary. Everyone who is a daughter of someone is a child of that person. Therefore, Rhodippe is unmarried. 4. Whoever is a friend of either Michael or Paul will receive a gift. If Michael has any friends, then Eileen is one of them. Therefore, if Anne is a friend of Michael, then Eileen will receive a gift. 5. None of Ockham’s followers like any realist. Any of Ockham’s followers likes at least one of Hobbes’ followers. I am a follower of Ockham. Therefore, some of Hobbes’ followers are not realists. 6. Any professional can outplay any amateur. Phaidrias is a professional but he cannot outplay Drakes. Therefore, Drakes is not an amateur. 7. Whoever won the match was congratulated. Anyone who congratulated Marty would have patted him on the back. Nobody patted Marty on the back. Therefore, Marty did not win the match. 8. Hector is a person. Furthermore, Hector is smarter than any person in the class. Since no person is smarter than himself, it follows that Hector is not in the class.

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9. A horse is an animal. Therefore, whoever owns a horse owns an animal. 10. Ang bawat inibig ni Arnel ay galit kay Tita. Hindi galit si Tita sa kaniyang sarili. Kung gayon, hindi inibig ni Arnel si Tita. Identity 1. Dr. Seuss is Theodore Geisel. 2. Hermann Hesse is not Andre Gide. 3. Aphrodite is the Goddess Cypris. 3. Only Art loves Betsy. 4. The only opera written by Beethoven is Fidelio. 5. All moons except Titan are without an atmosphere. 6. Every recent Pope except John Paul II and Benedict XVI was Italian. 7. Hydrogen is the lightest element. 8. The painter of “The Starry Night” was Van Gogh. 9. Desdemona was Othello’s wife. 10. Everyone admires the most intelligent person in the world. 12. Charmides’ father admires him. 13. There is at most one god. 14. Exactly one person is rich. 15. Exactly two persons are rich. 16. Mars has at least two moons. 17. Cerberus has three heads. 18. Every state elects two senators. 19. No more than two visitors are permitted. 20. If Virgil isn’t the most intelligent student, then there is only one student who is more intelligent than he is. Arguments involving identity 1. The man who committed the crime was in the apartment. Now whoever was in the apartment was in town. Obviously, anybody who was abroad was not in town. Brian was abroad. Therefore, Brian was not the man who committed the crime. 2. A Zen master knows the sound of one hand clapping. Strymodoros is a Zen master. The man I saw at the casino does not know the sound of one hand clapping. Therefore, the man I saw at the casino cannot be Strymodoros. 3. The Andromeda Galaxy is the nearest galaxy to us. M31 is a galaxy. It is not the case that the Andromeda Galaxy is nearer to us than M31. Therefore, M31 is the same as the Andromeda Galaxy. 4. Some of Jane Collier’s novels are interesting. The only novel Jane Collier wrote is “The Cry”. Therefore, “The Cry” is interesting. 5. The dog that bit the letter carrier is a large terrier. Ajax is a small dog. Therefore, Ajax did not bite the letter carrier. 6. Any fish can swim faster than any smaller one. Therefore, if there is a largest fish, then there is a fastest fish. Truth-trees 1. A e B A |BvC

2. P e (Q e R) Pv R |Q

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3. A v (B e A) ~A v C | ~B

4. A e B BeC De E AvD |CvE

5. P e (Q e R) | Q e (P e R) 6. A e B ~C e (D e E) C v (A v D) ~C |BvE

7. (A v B) e C (C v D) e E DvA ~D |E

8. P e Q Qe R ~R v (~S v P) |S

9. (A v B) e C (C v B) e [A e (D / E)] Av D |D/ E

10. P e (Q v R) R e (S v T) ~S |PeQ

11. (P v Q) e (R v S) ~P e (T e ~T) ~R | ~T

12. (P v Q) e R R e (Q v S) P e (~T e Q) (T e P) e ~S |Q/ R

13. P e Q Q e [P e (R v S)] R/S ~(R v S) | ~P

14. P | (Q e R) e [(R e S) e (Q e S)]

15. | P e (Q e P)

16. | (P e Q) e [P e (P v Q)]

17. | [P e (Q v R)] / [(P e Q) v (P e R)]

18. | (P e Q) e [~(Q v R) e ~(R v P)] Solution E I G C D H B A F

C F A B I E H D G

H D B G F A I C E

A C D I G B E F H

B H E D A F C G I

I G F E H C A B D

G E H A B D F I C

F A I H C G D E B

D B C F E I G H A

14

Principles of logic Rules of inference 1. Modus Ponens (MP): p e q, p | q 2. Modus Tollens (MT): p e q, ~q | ~p 3. Disjunctive Syllogism (DS): p v q, ~p | q 4. Simplification (Simp): p v q | p 5. Conjunction (Conj): p, q | p v q 6. Hypothetical Syllogism (HS): p e q, q e r | p e r 7. Addition (Add): p | p v q 8. Constructive Dilemma (CD): (p e q) v (r e s), p v r | q v s Replacement 9. Double Negation (DN): p / ~~p 10. De Morgan’s Theorem (DM): ~(p v q) / (~p v ~q) and ~(p v q) / (~p v ~q) 11. Commutation (Com): (p v q) / (q v p) and (p v q) / (q v p) 12. Association (Assoc): [p v (q v r)] / [(p v q) v r] and [p v (q v r)] / [(p v q) v r] 13. Distribution (Dist): [p v (q v r)] / [(p v q) v (p v r)] and [p v (q v r)] / [(p v q) v (p v r)] 14. Transposition (Trans): (p e q) / (~q e ~p) 15. Implication (Impl): (p e q) / (~p v q) 16. Exportation (Exp): [(p v q) e r] / [p e (q e r)] 17. Tautology (Taut): p / (p v p) and p / (p v p) 18. Equivalence (Equiv): (p / q) / [(p e q) v (q e p)] and (p / q) / [(p v q) v (~p v ~q)] Conditional proof (Conditional introduction) 19. If P | Q, then | P e Q. Quantification rules 20. Equivalences: (œx)Nx / ~(›x)~N x and (›x)Nx / ~( œx)~Nx 21. Universal Instantiation (UI): (œ x)(Nx) | N< , (where < is any individual symbol) 22. Universal Generalization (UG): N y | (œ x)(Nx), (where (1) y is not a constant, (2) y is not free in a line obtained by EI, (3) y is not free in an assumed premise within whose scope Ny) 23. Existential Instantiation (EI): (›x)(N x) | N