Chapter 4 Answers to Exercises

Chapter 4 Answers to Exercises Exercises 4.a. Translate the following simple and complex English sentences into claims of propositional logic. If the claim is complex, also identify the major operator. 1. It is either a cat or a skunk. (C v S) Disjunction 2. It flies. F 3. If it flies, it is either a bird or a plane. (F (B v P)) Conditional 4. It looks very strange for a bird or plane. L 5. Either it is a bird and it is deformed or it is a plane and it is very small. ((B & D) v (P & S)) Disjunction 6. If it is a bird and it is deformed, then someone either burned it or painted it. ((B & D) (B v I)) Conditional 7. It is not a plane. ~P Negation 8. If someone burned it, it would not be hopping around so happily. (B ~H) Conditional 9. It is hopping around happily. H 10. If it was not burned, then someone must have painted it. (~B I) Conditional

Exercises 4.b. Using the interpretations provided, translate the following claims of propositional logic into English: 11. [T = I throw the ball; W = The window will break.] (T W) If I throw the ball, then the window will break. 12. [R = You re a Republican; D = You re a Democrat.] (((R v D) & ~R) D) If you re either Republican or a Democrat and you re not a Republican, then you re a Democrat. 13. [R = It is raining; S = The sidewalks are wet.] (((R S) & R) S) If it is the case that, if it is raining, then the sidewalks are wet, and the sidewalks are wet, then it is the case that it is raining. 14. [S = It snowed; R = The roof collapsed.] (((S R) & ~R) ~S) If it is the case that, if it snowed, then the roof collapsed, and it is not the case that the roof collapsed, then it did not snow. 15. [R = It is raining; B = I bring my umbrella; W = I get wet.] ((R & ~B) W) If it is raining and I did not bring my umbrella, then I get wet. 16. [B = The dog will bite; F = The dog is friendly; P = I will pet the dog.] ((B v F) & ((F P) & (B ~P))) Either the dog will bite or it is friendly and, if the dog is friendly, then I will pet it, and if it will bite, then I will not pet it. 17. [P = I pay for the ticket; K = I get kicked out.] (~(~P K) ~P) If it is not the case that if I do not pay for the ticket, then I will get kicked out, then I will not pay for the ticket. 18. [P = I am a professor; D = I have a Ph.D.; T = I have teaching experience.] (P (D & T)) I am a professor if and only if I have a Ph.D. and I have teaching experience. 19. [P = He went to the park; R = He went to the restaurant; S = He is on the swings; J = He is on the jungle gym.] ((P v R) & ((P (S v J))) He went to the park or the restaurant, and if he went to the park, then he is either on the swings or the jungle gym.

20. [D = She is a (medical) doctor; M = She went to medical school; L = She is licensed; I = She is immoral; H = She is a hack.] ((D M) & ((~L (I v H))) She is a medical doctor if and only if she went to medical school, and if she is not licensed, then she is either immoral or she is a hack. Exercises 4.c. Translate each of the following into claims of propositional logic 1. The tide is coming in (T). Moreover, it s getting dark (G). (T & G) 2. Unless you want to spend the night in jail (W), you should come home (H). (W v H) OR (~W H) 3. Passing 130 semester hours of courses in the right distribution (P) is necessary and sufficient for graduating college (G). (P G) 4. You can have candy (C) or bananas (B), but not both. ((C v B) & ~(C & B)) 5. He s a lawyer (L), but he is also a nice guy (N), and if you ask nicely (A), he might even work for free (M). ((L & N) & (A M)) 6. You are a male (M) just if you have a Y chromosome (Y), but you re a man (A) just in case you re over 18 (O). ((M Y) & (A O)) 7. If it is not the case that I m held accountable for my actions (~A), then there is no real reason to care about morality (~R). (~A ~R) 8. You shouldn t lie (~L). However, if an innocent person s life is at stake (I) and lying would prevent her death (P), then you should lie (L). (~L & ((I & P) L)) 9. Number 8 contains a contradiction (C). It cannot be the case that both You should lie (L) and Sometimes, you should lie (~L). (C & ~(L & ~L)) 10. The president of the company is neither good (~G) nor prudent (~P). Furthermore, if he doesn t stop complaining (~S), we will replace him (R). ((~G & ~P) & (~S R))

Exercises 4.d. Translate the following complex claims into propositional logic: 1. If you want to run for president (R), you must be both at least 35 years old (T) and a naturally born citizen of the United States (N). (R (T & N)) 2. The writing is on the wall (W) and the check is in the mail (C), and if this is true, then you only have two choices: pack it in now (P) or stick around for the long haul (S). ((W & C) & ((W & C) (P v S))) 3. You shouldn t be telling me what to do (~T) unless you think you can take responsibility for my running this company into the ground (R) or for turning it into a Fortune 500 company (F). (~T v (R v F)) OR (~(R v F) ~T) 4. If it s not the case that you re sick (~Y) and it is not the case that your kids (~K) or spouse is sick (~S), then you should be getting dressed for work (D) or risk not getting that promotion (R). ((~Y & (~K & ~S)) (D v R)) 5. [T]he proscribing any citizen as unworthy the public confidence by laying on him an incapacity of being called to offices of trust or emolument, unless he profess or renounce this or that religious opinion, is depriving him injudiciously of those privileges and advantages to which, in common with his fellow-citizens, he has a natural right (From A Bill for Establishing Religious Freedom, Thomas Jefferson, 1779). If you regard someone as unworthy for public office (U) if it is not the case that they renounce a belief (R), then you are depriving that person of a right (no civil right: ~C). ((U v R) ~C) OR ((~R U) ~C) 6. All that up to the present time I have accepted as most true and certain I have learned either from the senses or through the senses; but it is sometimes proved to me that these senses are deceptive, and it is wiser not to trust entirely to anything by which we have once been deceived. (From Meditations on First Philosophy, by René Descartes, meditation 1.) I have accepted my senses as true (A), and my senses are sometimes deceptive (D), and it is not wise to accept anything deceitful (~W). (A & (D & ~W)) 7. [During the Enlightenment] Opposition to belief in God increased because it was used by princes ruling by God s grace, by cardinals, bishops, and priests as a means of preventing the diffusion of the light of reason and of keeping the people in tutelage and servitude. (From Does God Exist? by Hans Küng, A.II.4.)

During the Enlightenment, opposition to belief in God increased because it was used by princes ruling by God s grace as a means of preventing the diffusion of the light of reason and of keeping the people in tutelage and servitude (G), and by cardinals as a means of preventing the diffusion of the light of reason and of keeping the people in tutelage and servitude (C), and by bishops as a means of (B), and by priests as a means of (P). (G & (C & (B & P))) 8. If my letters are condemned in Rome (R), then what I condemn in them is condemned in heaven (I). (From Pensées, fr. 919, by Blaise Pascal trans. W. F. Trotter.) (R I) 9. The framework of bones being the same in the hand of a man, wing of a bat, fin of the porpoise, and leg of the horse, the same number of vertebrae forming the neck of the giraffe and of the elephant, and innumerable other such facts, at once explain themselves on the theory of descent with slow and slight successive modifications. (From The Origin of Species, 1 edition, by Charles Darwin, Chapter XIV.) The theory of decent with slow and slight modifications explains the framework of bones (B), and the same number of vertebrae (V), and innumerable other such facts (F). (B & (V & F)) The concept of explanation is difficult to translate into symbolic logic because it could express a causal relation, a logical relation, or an evidential relation. Here it seems to stand as evidence, in which case, this may be better expressed as an argument than as a claim: 1. The theory of decent with slow and Explanation (If this is true, we would successive modifications. expect claims 2, 3, and 4 to be true.) 2. The framework of bones 3. The same number of vertebrae 4. Innumerable other such facts. 10. Well, sir, it s surprising (S), but it well may be a fact that neither of them does know exactly what that bird is (~K), and that nobody in all this whole wide sweet world knows what it is (~W), saving and excepting only your humble servant, Caspar Gutman, Esquire (C). (From The Maltese Falcon, by Dasheill Hammett, Chapter 11.) It is surprising and, neither of them and nobody else knows what the bird is, and Caspar knows. (S & ((~K & ~W) & C)) Note except here is not being used as a conditional ( except when ). Caspar is saying that, it may be the case that no one knows except him, which is a conjunction: no one knows but him.

Exercises 4.e. Using the translation guide, translate the following sentences of propositional logic into English: 1. [D = It is the dog; B = It is the bird; F = It is the fish.] (~D (B v F)) Unless it is the dog, it is either the bird or the fish. OR If it is not the dog, then it is either the bird or the fish. 2. [H = It is hot; S = I am sick; T = The thermostat is working.] ((~H & S) v ~T) Either it is not hot and I am sick, or the thermostat is not working. 3. [A = It is always raining; D = It is at least depressing; F = It is fun.] ((~A D) & (D ~F)) If it is not always raining, then it is at least depressing, and if it is at least depressing, then it s not fun. 4. [C = I am cleaning the bathroom; W = You are washing dishes; S = You are scrubbing the floors.] (C (W v S)) If I am cleaning the bathroom, then either you are washing dishes or scrubbing the floors. 5. [B = We re going to Bonnaroo; A = Annie can go; D = You agree to drive half the time.] (B (A D)) If we re going to Bonnaroo, then Annie can go if and only if you agree to drive half the time. Exercises 4.f. For the following three claims, construct your own translation guide for each, then translate into English: 1. ((P Q) & R) P: I m in pain. Q: I will quit. R: Running is painful. If I m in pain, then I will quit, and running is painful. 2. ((S v L) ~H) S: Sylvia has been arrested. L: Sylvia is lying. H: Sylvia is coming home. If Sylvia has been arrested or is lying, she is not coming home.

3. ((A B) v (C D) A: All is well. B: The best have won. C: Crowds are angry. D: Defeat is imminent. If all is well, then the best have won, or if the crowds are angry, then defeat is imminent. Exercises 4.g. Reformulate the following arguments, removing extraneous material and rhetorical devices, then translate them into propositional logic: 1. A government-sponsored health care program requires heavily taxing all citizens. Excessive taxation is no different than stealing from hardworking people. Stealing from people is immoral. Therefore, government-sponsored health care is immoral. 1. A government-sponsored health care program requires taxing all citizens. 2. Excessive taxation is no different from stealing from people. 3. Stealing from people is immoral. 4. Therefore, government-sponsored health care is immoral. 1. G 2. E 3. S 4. I All the claims in this argument are simple claims, and are, therefore, translated using only one capital letter. 2. If within the same church miracles took place on the side of those in error, this would lead to error. The schism is obvious, the miracle is obvious, but schism is a clearer sign of error than a miracle is of truth: therefore the miracle cannot lead to error. But apart from schism error is not as obvious as a miracle; therefore the miracle would lead to error. (From Pensées, fr. 878, by Blaise Pascal, trans. A. J. Krailsheimer.) 1. If miracles occurred in favor of people expressing false claims (O), the miracle would lead the church to believe false claims (F). 2. The schism is obvious (S) and the miracle is obvious (M), and if there is a schism (S), then the church would not be led to believe false claims (~F). 3. Therefore, the miracle cannot lead the church to believe false claims (~F). 4. If there is no schism (~S), then premise 1 is true (O F). 5. Therefore, if there is no schism, but there is a miracle on behalf of those who teach false claims, the miracle would lead the church to believe false claims. 1. (M F) 2. ((S & M) & (S ~F)) 3. ~F 4. (~S (M F)) 5. ((~S & M) F) As it stands, Pascal s argument is contradictory. He says, if a miracle occurs on behalf of those who teach false claims, then it will lead the church to believe false claims. He also says, if there is a miracle and a schism, the church will not be led to believe false claims.

From this we can derive (F & ~F), which cannot be true. To resolve this contradiction, Pascal needs an additional premise that says: (M F) unless S. 3. When we put together three things first, the natural attraction between opposite sexes; secondly, the wife s entire dependence on the husband, every privilege or pleasure she has being either his gift, or depending entirely on his will; and lastly, that the principle object of human pursuit, consideration, and all objects of social ambition, can in general be sought or obtained by her only through him, it would be a miracle if the object of being attractive to men had not become the polar star of feminine education and formation of character. (From The Subjection of Women, by John Stuart Mill, Chapter 1.) [Hint: Mill is arguing that the education system of his day encourages women to be wholly dependent on men. His conclusion has a bit of rhetorical flourish: Given these cultural facts, how could you expect any different?] 1. There is a natural attraction between opposite sexes (N). 2. A wife depends entirely on the husband (for financial and social support) (D). 3. All of the objects of a woman s social ambition must be pursued through her husband (O). 4. If 1, 2, and 3 are true, it is likely that attractiveness to men will become the primary focus of female education (((N & D) & O) A) 5. Thus, it is likely that attractiveness to men will become the primary focus of female education (A). 1. N 2. D 3. O 4. (((N & D) & O) A) 5. A 4.... [O]ur civil rights have no dependence on our religious opinions (~(C R)), any more than our opinions in physics (P) or geometry (G); and therefore the proscribing any citizen as unworthy the public confidence by laying on him an incapacity of being called to offices of trust or emolument (U), unless he profess or renounce this or that religious opinion (R), is depriving him injudiciously of those privileges and advantages to which, in common with his fellow-citizens, he has a natural right (~C)... (From A Bill for Establishing Religious Freedom, Thomas Jefferson, 1779). 1. ~(C R) 2. ~(C P) 3. ~(C G) 4. ((~R U) ~C) [5. ~(~R U)] Note: Dependence relationships are tricky to translate into symbolic logic. It is easiest (though not always correct) to think of them as necessary conditions. If religious belief is a necessary condition of civil rights, we would express it like this: (C R). If they do not, we simply negate this claim: ~(C R). Also, the conclusion here is implicit: We should not consider someone unworthy of public office because of a religious belief: ~(~R U). 5. 1. If it could be demonstrated that any complex organ existed which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break

down. (Charles Darwin, The Origin of Species, 6th ed., p. 154). 2. An irreducibly complex system cannot be produced directly by slight, successive modifications of a precursor system (Michael Behe, Darwin s Black Box, p. 39.) 3. [T]he blood-clotting system fits the definition of irreducible complexity. (Michael Behe, Darwin s Black Box, p. 86.) 4. Faced with such complexity beneath even simple phenomena, Darwinian theory falls silent. (Michael Behe, Darwin s Black Box, p. 97). 1. If any organ (or other biological phenomenon) exists that cannot be explained by evolution, then evolution is false. 2. If a system is irreducibly complex, it cannot be explained by evolution. 3. The blood-clotting system is an irreducibly complex system. 4. Therefore, evolution is false. 1. (~P ~E) 2. (I ~P) 3. I 4. ~E 6. 1. Brigid bought a newspaper called The Call, then disappeared. 2. Brigid and Joel Cairo are both after the Maltese Falcon. 3. Joel Cairo had the same copy of The Call and had ripped out a section listing the arrival of ships into the local port. 4. The section Cairo had ripped out included a ship called the La Paloma and five other ships. 5. The La Paloma burned last night. 6. Therefore, a good place to look for clues as to Brigid s disappearance would be the La Paloma. (Adapted from The Maltese Falcon, by Dasheill Hammett.) 1. Brigid bought newspaper X (N) and disappeared (D). 2. Both Brigid (B) and Joel are looking for the Maltese Falcon (J). 3. Joel bought newspaper X (X) and ripped out section R (R) [4. If Brigid had newspaper X (N), she also had section R (R).] 5. If you have section R (R), you might know about the La Paloma (L) and five other ships (F). 6. The La Paloma burned last night (T). 7. Therefore, where the La Paloma burned would be a good place to look for clues about Brigid (C). 1. (N & D) 2. (B & J) 3. (X & R) [4. (N R)] 5. (R (L & F)) 6. T 7. C This argument would need a few more premises to be cogent. For instance, we would need a premise connecting the likelihood of finding clues to the burning of the La Paloma, and knowledge of the La Paloma to the disappearance of Brigid, and a premise expressing our desire to find Brigid and maybe a few others. But, roughly, we get the idea.

7.... [T]he fact that restricting access to abortion has tragic side effects does not, in itself, show that the restrictions are unjustified, since murder is wrong regardless of the consequences of prohibiting it. (Mary Anne Warren, On the Moral and Legal Status of Abortion, 1973.) 1. If an act is murder (M), then it is wrong (W) and should be prohibited (P). 2. If it is the case that, if murder is prohibited (I), then it leads to bad consequences (B), then it is still the case that, if it is murder (M), it is wrong (W) and should be prohibited (P). [2. If it is the case that, if some act X is an abortion (X), then it is murder (M), then it is wrong (W) and should be prohibited (P).] 3. Therefore, it is not the case that, if we prohibit abortion (A), then there are tragic consequences (B), entails that it is not wrong (~W) should not be prohibited (~P). 1. (M (W & P)) 2. ((I B) (M (W & P))) [2. ((X M) (W & P))] 3. ~((A B) (~W & ~P)) The idea is that, until the question of whether abortion is murder is settled, pointing out the tragic side effects of abortion is not sufficient to permit it, since if it is murder, then it should be prohibited regardless of its consequences. Maybe there are other reasons abortion should not be prohibited, but its prohibition s bad consequences aren t among them. 8. According to the Humanitarian theory [of punishment], to punish a man because he deserves it is mere revenge and, therefore, barbarous and immoral. My contention is that this doctrinereally means that each one of us, from the moment he breaks the law, is deprived of the rights of a human being. The reason is this. The Humanitarian theory removes from Punishment the concept of Desert. But the concept of Desert is the only connecting link between punishment and justice. It is only as deserved or undeserved that a sentence can be just or unjust. (C. S. Lewis, The Humanitarian Theory of Punishment, God in the Dock, pp. 287 88.) 1. If the Humanitarian theory is true, then punishment is not something that is deserved. 2. If punishment is not something that is deserved, then it is unjust. 3. A punishment is just only if it is deserved. [4. If you have a right to justice and commit a crime, then a punishment is deserved.] 5. If the Humanitarian theory is true, you do not have a right to justice. 1. (H ~D) 2. (~D ~J) 3. (J D) [4. ((R & C) D)] 5. (H ~R) Lewis implies here (and argues elsewhere) that humans have a right to justice, so this is an implicit premise of the argument. The argument is that, any theory that entails that your punishment has nothing to do with your desert, also entails that you have no right to justice. Since Lewis believes this is false, he also believes the Humanitarian theory of punishment is false.