Logic and Reasoning Practice Final Exam Spring 2015
Name _________________________________ Section Number ________
The final examination is worth 100 points.
1. (5 points) What is an argument? Explain what is meant when one says that logic is the normative study of arguments. In what way(s) is the standard for assessing the goodness of an argument different for deductive and inductive arguments?
2. (5 points) Fill in the truth tables below, where (P % Q ) = ((P → ~Q ) ∨ (~P → Q )):
P Q (P ∧ Q) (P ∨ Q) (P → Q) (P % Q)
Do ONE of the following two problems. Only the first completed problem will be graded.
3. (5 points) Use truth tables to decide whether the argument below is valid or invalid.
((P ∧ ~Q ) → Q ) (Q →~R ) ~R
4. (5 points) Use truth tables to classify the two sentences below (tautologous, self- contradictory, or contingent) and then compare them by saying which and how many of the following categories apply: equivalent, contradictory, consistent, and inconsistent.
~(~(~(P ∧ ~Q ) ∧ ~P ) ∧ ~P ) ~(P → P )
5. In the following problems, you are asked to prove some alternative rules of inference.
a. (3 points) Addition { P } ⊢ (P ∨ Q ) [3 lines]
b. (4 points) Contraposition { (P → Q ) } ⊢ (~Q → ~P ) [5 lines]
c. (3 points) Disjunctive Syllogism { (P ∨ Q ), ~P } ⊢ Q [11 lines]
6. (5 points) Translate the following argument into our zeroth-order formal language and then give a proof to show that the argument is valid.
John is not allowed to argue unless Michael has paid.
If John argues, then John is allowed to argue.
John argues.
Michael has paid.
7. (5 points) Suppose that Billy loves Suzy, Suzy loves Jane, and Jane loves only herself.
Use a directed graph to represent the relation x loves y as it applies to Billy, Suzy, and Jane. Is the relation an equivalence relation? Justify your answer.
8. Answer the questions below about small worlds, models, and validity. For Parts (a) and (b), consider a small world consisting of three objects—named a, b, and c—such that for predicates F and G, all of the following are true: Fa, Fb, ~Fc, ~Ga, ~Gb, and Gc.
a. (2 points) Is the small world a model for the sentence ((∀x)Fx → (∀y)Gy)?
Explain your answer.
b. (2 points) Is the small world a model for the sentence (∀x)(Fx → Gx)?
Explain your answer.
c. (1 point) What, if anything, can you say about the logical relationship(s) between the sentences in Parts (a) and (b)? Explain your answer.
9. (5 points) For this problem, you may purchase up to two hints for one point per hint. Give a natural deduction proof [10 lines] to show the following:
{ } ⊢ ((∃x)~Dx ∨ (∀y)Dy)
10. (5 points) Translate the following argument into our formal language and then give a proof to show that the argument is valid. Given that the argument is valid, are you rationally compelled to believe that ethics is not worth studying? Explain your answer.
Every principle in ethics has some counter-example.
If so, then particularism is true.
Ethics is worth studying only if particularism is not true.
Ethics is not worth studying.
11. Answer the questions below about sets and set theory.
a. (3 points) How many elements are in the set A = {x, y, {x, y, {w}}, z}?
b. (3 points) If B is the set {x, w, {y}, z}, what is the intersection (A ∩ B) of A and B?
c. (2 points) List all of the subsets of the set C = {α, β, λ}.
d. (2 points) Explain why the empty set is a subset of every set.
12. Answer the questions about probability below by referring to the following diagram:
a. (3 points) What is the probability that a randomly-chosen object is shaded?
b. (3 points) What is the probability that a randomly-chosen object is an unshaded triangle?
c. (2 points) What is the probability that a randomly-chosen object is either an unshaded triangle or a shaded circle?
d. (2 points) What is the probability that a randomly-chosen object is a triangle given that it is shaded?
13. Answer the following conceptual questions about probability.
a. (4 points) Suppose your friend Rudy says he thinks there is a 50 percent chance that a Democrat will be elected President in 2016 and that it is twice as likely that a Republican will be elected. Is Rudy’s statement coherent? Explain your answer.
b. (3 points) Rudy assigns probability 0.8 to the sentence (∀x)Fx. What probability should Rudy assign to the sentence ~(∃x)~Fx?
Explain your answer.
c. (3 points) One day, Rudy tells you about a woman he met named Jane. Jane is a registered member of the Democratic Party and regularly contributes to a blog discussing gender bias in hiring practices.
Rudy says that he thinks it is less likely that Jane is a bank teller than it is that Jane is a feminist and a bank teller. Is Rudy’s claim reasonable or not? Explain your answer.
Do ONE of the following two problems. Only the first completed problem will be graded.
14. (5 points) John Connor is constantly on the run from terminator cyborgs. In his experience, about one person in a thousand is a terminator cyborg. John has noticed that terminators are much more likely than ordinary humans to say, “Thank you for
explaining,” after receiving an answer to a question. His corresponding credences are 0.8 and 0.2. One day, John overhears someone nearby say, “Thank you for explaining,” what degree of belief should he have that the person he overheard is a terminator cyborg?
15. (5 points) Your doctor is a Bayesian who once took a very good logic course as an undergraduate. You go in to see her with a severe headache and a cough. She thinks there is a 75% chance that you only have tension headaches, a 20% chance that you have meningitis, and a 5% chance that you have something else. She collects a sample of cerebrospinal fluid and tests it using gram staining. If you do not have meningitis, the test will be negative with probability one. But even if you do have meningitis, the test has a 40% chance of coming back negative. Suppose the test comes back negative. What is the probability that you have meningitis?
16. (5 points) Compare and contrast the three interpretations of probability discussed in lecture. Do philosophical debates about how to interpret probability matter? If so, how? If not, why not?
17. (5 points) Suppose Talan has a box full of colored blocks. One in five of the blocks are red. If Talan pulls out seven blocks—one at a time—and puts each block back before pulling out another, what is the probability that he pulls out exactly four red blocks?
18. (5 points) Suppose you have two trick coins. One has a bias of 1/3 for heads, and the other has a bias of 2/3 for heads. You cannot tell them apart just by looking, and you have forgotten which one is which. So, you decide to flip one of them several times and make a guess at which it is. You flip the coin ten times and observe heads come up six times.
Set up the formula that you need in order to calculate the probability that you have been flipping the coin that has a 1/3 probability of heads on any given flip. Do not calculate a numerical answer. How would your answer change if you had picked your coin out of a bag of 20 coins in which all but one had a bias of 1/3 for heads?
BONUS #1: The Drinker Paradox. (5 points) Give a natural deduction proof to show the following:
{ } ⊢ (∃x)(Dx → (∀y)Dy)
You might find it helpful to use the result of Problem #9 in your proof.
If Dx = “x drinks at the Pig and Whistle,” what does the sentence that you are being asked to prove say in English? Now that you have translated the sentence, does it seem plausible to you? If not, what do you think has gone wrong? Explain your answers.
BONUS #2: Bayesian Statistics. (5 points) Give a conceptual account of how a Bayesian makes inferences about the world. Illustrate with a data-generating process that might plausibly be modeled with a binomial distribution.
