Math 218 Spring 2010 Homework 4 Solutions Section 1.5 Note: For #10 I have written out the solutions in more detail than you would be required to give. 10) For each of these sets of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises. a) If I play hockey, then I am sore the next day. h s I use the whirlpool if I am sore. s w I did not use the whirlpool. w h: I play hockey. s: I am sore. w: I use the whirlpool. 1. w Premise 2. s w Premise 3. s Modus tollens from (1) and (2) I am not sore. 4. h s Premise 5. h w Hypothetical Syllogism from (2) and (4) If I play hockey, I use the whirlpool. 6. h Modus tollens from (3) and (4) OR from (1) and (5) I did not play hockey. b) If I work, it is either sunny or partly sunny. x(w (x) (S(x) P (x))) I worked last Monday or I worked last Friday. W (Monday) W (Friday) It was not sunny on Tuesday. S(Tuesday) It was not partly sunny on Friday. P (Friday) W (x): I work on x. S(x): It is sunny on x. P (x): It is partly sunny on x. Domain for all is {days of the week}. 1. W (Monday) W (Friday) Premise 2. W (x) (S(x) P (x)) Premise 3. P (Friday) Premise 4. W (Monday) (S(Monday) P (Monday)) Universal instantiation from (2) If I work Monday it is sunny or partly sunny on Monday.
5. W (Friday) (S(Friday) P (Friday)) Universal instantiation from (2) If I work Friday it is sunny or partly sunny on Friday. 6. W (Friday) S(Friday) Disjunctive syllogism from (3) and (5) If I work Friday it is sunny on Friday. 7. S(Monday) P (Monday) S(Friday) P (Friday) Modus ponens from (1), (4), and (6) It was either sunny or partly sunny on Monday or sunny on Friday. c) All insects have six legs. x[i(x) L(x)] Dragonflies are insects. x(d(x) I(x)) Spiders do not have six legs. x(s(x) L(x)) Spiders eat dragonflies. x((s(x) D(y) E(x, y)) I(x): x is an insect. D(x): x is a dragonfly. L(x): x has six legs. S(x): x is a spider. E(x, y): x eats y. Domain for all is {animals}. 1. x[i(x) L(x)] Premise 2. I(c) L(c) Universal instantiation from (1) If c is any insect then c has six legs. 3. x(d(x) I(x)) Premise 4. D(c) I(c) Universal instantiation from (3) (may use the same c since both are for all statements and c is arbitrary) If c is any dragonfly then c is an insect. 5. D(c) L(c) Hypothetical syllogism from (2) and (4) If c is any dragonfly then c has six legs. 6. x(d(x) L(x)) Universal generalization from (5) All dragonflies have six legs, or just Dragonflies have six legs. 7. x(s(x) L(x)) Premise 8. S(c) L(c) Universal instantiation from (7) If c is any spider then c does not have six legs. 9. L(c) I(c) Contrapositive of (2) If any c does not have six legs then c is not an insect. 10. S(c) I(c) Hypothetical syllogism from (8) and (9) If c is any spider then c is not an insect.
11. x(s(x) I(x)) Universal generalization from (10) All spiders are not insects, or just Spiders are not insects. d) Every student has an internet account. x(s(x) I(x)) Homer does not have an internet account. I(Homer) Maggie has an internet account. I(Maggie) S(x): x is a student. I(x): x has an internet account. Domain for both is {people}. 1. x(s(x) I(x)) Premise 2. S(Homer) I(Homer)) Universal instantiation from (1) If Homer is a student then Homer has an internet account. 3. I(Homer) Premise 4. S(Homer) Modus tollens from (2) and (3) Homer is not a student. Note: You may NOT conclude that Maggie is a student because that would be affirming the conclusion. The condition does not say that if you have an internet account you must be a student. e) All foods that are healthy to eat do not taste good. x(h(x) G(x)) Tofu is healthy to eat. H(tofu) You only eat what tastes good. x(e(x) G(x)) You do not eat tofu. E(tofu) Cheeseburgers are not healthy to eat. H(cheeseburger) H(x): x is healthy to eat. G(x): x tastes good. E(x): You eat x. Domain for all is {foods}. 1. x(h(x) G(x)) Premise 2. H(tofu) G(tofu) Universal instantiation from (1) 3. H(tofu) Premise 4. G(tofu) Modus ponens from (2) and (3) Tofu does not taste good. 5. x(e(x) G(x)) Premise
6. E(c) G(c) Universal instantiation from (5) If you eat any food c it tastes good and if any food c tastes good you eat it. 7. H(c) G(c) Universal instantiation from (1) Any food c that is healthy does not taste good. 8. E(c) G(c) Contrapositive of (6) 9. H(c) E(c) Hypothetical syllogism from (7) and (8) If any food c is healthy you don t eat it. 10. x(h(x) E(x)) Universal generalization from (9) You don t eat healthy foods. Note: Can also conclude that you don t eat tofu, but that is one of the premises anyway. Also, note H(cheeseburger) is a false hypothesis so no conclusions can be drawn (if a food is not healthy it may or may not taste good). f) I am either dreaming or hallucinating. d h I am not dreaming. d If I am hallucinating, I see elephants running down the road. h e d: I am dreaming. h: I am hallucinating. e: I see elephants running down the road. 1. d Premise 2. d h Premise 3. h Disjunctive syllogism I am hallucinating. 4. h e Premise 5. e Modus ponens from (3) and (4) I see elephants running down the road. 16) For each of these arguments determine whether the argument is correct or incorrect and explain why. a) Everyone enrolled in the university has lived in a dormitory. Mia has never lived in a dormitory. Therefore, Mia is not enrolled in the university. Correct: universal instantiation and modus tollens. b) A convertible car is fun to drive. Isaac s car is not a convertible. Therefore, Isaac s car is not fun to drive. Incorrect: denying the hypothesis is used.
c) Quincy likes all action movies. Quincy likes the movie Eight Men Out. Therefore, Eight Men Out is an action movie. Incorrect: affirming the conclusion is used. d) All lobstermen set at least a dozen traps. Hamilton is a lobsterman. Therefore, Hamilton sets at least a dozen traps. Correct: universal instantiation and modus ponens. 20) Determine whether these are valid arguments. a) If x is a positive real number, then x 2 is a positive real number. Therefore, if a 2 is positive, where a is a real number, then a is a positive real number. Incorrect: affirming the conclusion is used. b) If x 2 0, where x is a real number, then x 0. Let a be a real number with a 2 0; then a 0. Correct: modus ponens (and possibly universal instantiation). 24) Identify the error or errors in this argument that supposedly shows that if x(p (x) Q(x)) is true then xp (x) xq(x) is true. 1. x(p (x) Q(x)) Premise 2. P (c) Q(c) Universal instantiation 3. P (c) Simplification from (2) Error: Simplification is from an statement 4. xp (x) Universal generalization from (3) 5. Q(c) Simplification from (2) Error: Simplification is from an statement 6. xq(x) Universal generalization from (5) 7. x(p (x) xq(x)) Conjunction from (4) and (6) Error: Conjunction yields an statement Also, to apply universal generalization, it must be made clear that the c value that was generalized referred to any arbitrary c-value. Also, there is clearly an extra x in the last line that shouldn t be there but I actually think that may have been a typo.