Logic and formal reasoning final exam
This final assignment consists of five questions, worth 50 points each, whose answers are to take the form of 1-3 page essays (hence five to fifteen pages aggregate). The intention is to leave considerable discretion in the distribution of effort across questions. Each question requires some thought, but any reasonable effort to engage with the matters raised by these questions will earn partial credit. (Naturally, the more coherent and insightful the thought and the more strongly connected are the answers to the material presented in the readings and recordings assigned for the course, the more points will be earned).
1)Suppose that a set of examination questions is sent to a class of 40 students, the responses to which are due a couple of weeks thereafter. Among the submitted responses, there is a subset of more than 20 submissions that exhibit a striking similarity of content and verbal formulation, a similarity all the more remarkable given the fact that most of the answers are incorrect, some incorrect in a quite bizarre way . Some flavor of two of the most egregious of these answers are captured by the following illustration, carefully crafted to give an indication of the various dimensions along which the responses go radically wrong: Question: “Pick a prime number between 100 and 1,000, and, using the method discussed in class, calculate its square root to three decimal places”. In response, 20 students select the number 64, and all 20 give the number 9.32184 as its square root.
Approaching this puzzle in accordance with the mode of analysis embodied in Bayes’ Theorem, how would one go about evaluating the relative probabilities of hypotheses such as H1)Each student selected the number and derived this answer independently of the others; H2)The students worked together to gain an understanding of the indicated method of extracting square roots, and then each proceeded independently to select a number and work out its square root; H3)The students all utilized a uniquely easily accessible source to understand the method for the extraction of roots, but then fell into the same natural mistake in calculating the square root.
Discuss whether any of these hypotheses provide plausible explanations of these responses. Are there other a priori plausible hypotheses that might account for this response? For each of the hypotheses H1, H2, and H3, construct a (less extreme) variant of this illustration that makes that hypothesis plausible.
2)Summarize the argument developed in Sections 1-5 of Carl Hempel’s Studies in the Logic of Confirmation, specifically that part of the paper than deals with Nicod’s characterization of the confirmation relation and the example of the Ravens that undermines it. Discuss the following situations:
a)The urn in front of you either contains 100 white balls (H1), or 75 white balls and 25 black balls (H2). Assume that these are the only two hypotheses that are at play. Does the blind extraction of a white ball confirm H1? H1 and H2? Neither? Discuss.
b)Granted the logical equivalence of “All ravens are black” and “All non-black objects are non-ravens”, what accounts for our intuition (or illusion) that the observation of a white piece of chalk confirms the latter but not the former statement?
c)Granted that any observation that confirms one of the two universal generalizations must confirm the other, why is it that the observation of a white piece of chalk confirms (or perhaps just seems to confirm) the two logically equivalent generalizations to a lesser degree than does the observation of a black raven? Can you define a universe in which the observation of a white piece of chalk confirms “All ravens are black” to a greater degree than does the observation of a black raven?
3)There are three desserts A, B and C, where it is assumed that the agent X prefers both A and B to C.
Suppose a coin in our possession is biased 2-to-1 in favor of Heads (i.e., it has a 2/3 chance of landing Heads and a 1/3 chance of landing Tails when tossed), and that (concerning this coin) the statement “The coin lands Heads” functions as an ethically neutral proposition for X (that is, a proposition, whatever its probability, to whose truth or falsity X is indifferent).
a)Design a series of pairs of lotteries (utilizing this biased coin) that demonstrates that X’s preference for A over B is more than two times but less than ten times his preference for B over C.
b)What role is played by the supposition of the existence of a neutral proposition in the employment of a scheme of lotteries in calibrating an agent’s preferences?
4)In their paper Picking and Choosing, Edna Ullmann-Margalit and Sidney Morgenbesser argue that i)that there are no fundamental impediments to the existence of picking situations proper; ii)that picking situations are quite common; iii)that there are no systematic rules for the transformation of a choosing situation into a picking situation that do not involve the pick of one rule rather than another. Summarize their argument and evaluate their position.
You might consider, in particular, the following questions: Is there any reason that the chance device suggested by Nicholas Rescher for selecting among alternatives among which one is indifferent need to be unbiased? Can one argue that any selection situation, whether a choosing or a picking situation, invariably resolves at some stage of implementation into a picking situation?
5)This is a question about “Newcomb’s Problem”.
Two boxes, one transparent and visibly containing $1,000 (A), and one opaque (B) are placed before an agent X at time t. The contents of A are supposed fixed, the contents of B known by the agent to have been determined by the prior action of a highly accurate Predictor that has placed $1,000,000 in box B if it predicted that X will select only box B and has placed nothing in box B if it predicted that X will select both boxes.
A)Assume (for the sake of argument) a universal acceptance of the in fact completely discredited hypothesis that the strong statistical correlation between smoking and a host of serious diseases including lung cancer is accounted for by a genetic factor that is the common cause of both. In your view, can someone who believes that these statistics should not in themselves present a deterrent to smoking reconcile this view with an advocacy of the “one-box” solution to Newcomb’s Problem?
B)Suppose that the back of box B is transparent, and that a completely trustworthy and reliable friend of X is able to see whether $1,000,000 is in box B. Assume that the Predictor can predict what if anything the friend will say and what X will hear, and that it has factored this into its prior analysis and decision. If the friend were able to communicate with X, does it matter whether i)the friend simply recommends a selection (“Take both boxes!”) or ii)reveals the actual contents of box B to X (e.g., “Box B is empty”)? Why or why not? If the friend were able to announce out loud the contents of box B, would it be advantageous for X to place himself in a situation in which he is unable to make out what his friend says? Explain.