Philosophy 5320

Links

• Syllabus
• Notes on probability, part I

Schedule (please read all the listed material, if any, for the indicated date)

Homework

Submit electronically whatever can be submitted electronically, and put everything else in the instructor's mailbox.

• Due January 19:
  • LPL 1.2 [0], 1.3 [4], 1.4 [12], 1.7 [7], 1.9 [3], 1.11 [5], 1.13 [5], 1.14 [3], 1.17 [3], 1.22 [3], 2.1 [12]
• Due January 26:
  • LPL 2.9 [3], 2.14 [4], 2.18 [3], 2.23 [4], 2.27 [4], 3.3 [3], 3.21 [6](you don't have to do all of 3.22, but you should do the check step in 3.22 before submitting), 4.12-18 [4 each], 4.34 [3], 4.43 [5]. Show that nor is truth-table complete, where A nor B holds iff (A or B) does not hold. [5] Bonus: Suppose that * is a truth-table-complete binary operator. Show that the truth-table of * must be such that T*T is F, and F*F is T. [4]
• Due February 2:
  • LPL 5.6 [4], 5.13 [3], 6.5 [4], 6.6 [5], 6.18 [4], 6.19 [6], 6.20 [6]. Bonus: Suppose that * is a truth-table-complete binary operator. Show that * is nand or nor. (Hint: You may assume the Bonus from the previous problem set. Observe that there are only four binary operators * such that T*T is F and F*F is T.) [4]
• Due February 9:
  • LPL 6.28 [3], 6.29 [5], 6.30 [6], 6.34 [3], 6.38 [3], 6.41 [5], 6.42 [6], 8.23 [3], 8.24 [4], 8.27 [7], 8.32 [5], 8.36 [7], 8.46 [6], 8.51 [3].
• Due February 16:
  • LPL 9.3, 9.19, 9.25, 11.13, 11.21, 11.22, 11.27. Bonus: Let F* be a propositional logic system just like Fitch with the exception that negation-elimination is not permitted. Show that (a) soundness holds for F*, but (b) completeness does not hold for F*. (Note: You may use soundness and completeness for Fitch if you like. Assume classical logic in the metalanguage.)
• Due February 23:
  • LPL 10.5, 10.17, 10.19, 10.22, 10.29, 12.4, 12.5, 12.23, 13.3, 13.8.
• Due March 2:
  • LPL 13.17, 13.18, 13.32, 13.33, 13.36, 13.37, 13.50, 14.28.
• Due March 16:
  • 15.4, 15.11, 15.24, 15.32, 15.35, 15.40, 15.50, 15.63, 15.67, 15.68 [you may hand the last three in on March 23]. Bonus: 15.19, 15.69.
• Due March 23:
  • 15.63, 15.67, 15.68 (if not yet done).
• Due March 30:
  • 16.15. Consider the following inductive definition: (a) any two-letter sequence consisting of a consonant followed by a vowel is a cv-sequence (note: only a, e, i, o and u are deemed to be vowels; everything else, including y is deemed to be a consonant); (b) any sequence of letters formed by conjoining two cv-sequences is a cv-sequence; (c) all cv-sequences are formed by using the rules in (a) and (b). Prove that in every cv-sequence the number of vowels is equal to the number of consonants. Bonus: Use mathematical induction to prove the pigeon-hole principle ("Dirichlet Box Principle"), that if there are n boxes and m pieces of paper to put into the boxes, with m>n, then at least one of the boxes will have to contain at least two pieces of paper.
• Due April 6:
  • 1. (a) Consider a circular spinner, marked in degrees around the edge, from 0 (inclusive) to 360 (non-inclusive), where the spinner has an equal chance of ending up pointing in any direction. Suppose that the spinner is spun twice, independently. Find an appropriate set of situations or sample space for evaluating the probabilities of propositions like (i) The first spin landed at exactly 47.2; (ii) The two spins landed within 10 degrees of one another; (iii) The first spin landed within 15 degrees of 270 and the second landed within 3 degrees of 123; and (iv) The second spin landed within 10 degrees of 15 or within 5 degrees of 0.
  • (b) Give intuitive arguments for what the probabilities of (i), (ii) and (iii), respectively, should be.
  • 2. Informally prove that if A and B are disjoint events, then P(A intersect B)=0. ("intersect" is the intersection symbol)
  • 3. Informally prove that if P(p)>0 relative to some set S of situations, then the proposition p is possible relative to S.
  • 4. (a) Suppose that you have three fair dice tossed independently and simultaneously. What is the relevant sample space for looking at outcomes?
  • (b) What is the probability that all the dice will show the same number, and why?
  • 5. Give an example, different from those given in class, of a case where there is a proposition p that is almost surely false (i.e., P(p)=0), but nonetheless intuitively p can be true.
  • Bonus 1: Let X be the set of all natural numbers. Show that there is no probability measure defined on all the subsets of X such that P({n})=P({m}) for all natural numbers n and m.
  • Bonus 2: Show that it is possible for A to be independent of C, B to be independent of C, but A intersect B to fail to be independent of C.
  • Bonus 3: Suppose there are n independent fair dice tossed simultaneously. What is the probability that they all give the same answer, and why?
• Due April 13:
  • 1. Choose two of the following fallacies of probabilistic reasoning, and give a plausible sounding philosophical argument (in any area of philosophy) that commits the fallacy. Then supply plausible numbers for all the relevant probabilities and conditional probabilities, and do a Bayesian calculation to see if the argument can be salvaged. In the following "p makes q likely" means P(q|p)>1/2, and something holds "probably" provided that it holds with probability greater than 1/2 given the evidence. If really stuck for a philosophical argument, give a plausible sounding argument in another discipline.
    • p is true; p incrementally confirms q; q entails r; therefore, p incrementally confirms r
    • p is true; p entails q; q makes r likely; therefore, probably r
    • p is true; p entails q; q incrementally confirms r; therefore, probably r
    • p is true; q is true; p makes r likely; q makes r likely; therefore, probably r
    • p or r is true; p makes r likely; q makes r likely; therefore, probably r
  • 2. Kahneman and Tversky gave people the following story: “Linda is 31-years-old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.”  They asked if it's more likely that (a) Linda is a bank teller, or (b) a bank teller and feminist? 85% of respondents said that (b) was more likely. Prove from the axioms of probability that the respondents were wrong on a literal reading of the question. Give a charitable explanation of the mistake.
  • 3. Consider the hypothesis H that all ravens are black. Intuitively, finding a raven and noticing that it's black is evidence for H. Use Bayes' theorem, along with reasonable assumptions, to show this.
  • Bonus: Consider the hypothesis H* that all non-black things are non-ravens. By parallel with problem 3, finding a non-black thing and noticing that it's not a raven should be evidence for H*. For instance, looking at a non-blac sock and noticing that it's a non-raven seems to be evidence for H*. But H* is logically equivalent to the hypothesis H in 3. So, it seems that looking at a non-black sock and noticing that it's not a raven is evidence for H. But that's absurd. Find a plausible way to resolve this paradox.
• Due April 20:
  • Konyndyk 1.5.2.d, 2.1.e.C, 2.1.h.A (yes or no for each suffices), 2.1.h.B, 2.3.1-3, 2.3.g.A, 2.4.b.3 (in S4). Bonus: 2.4.b.5, 2.4.b.7 (in S4).
• Due 10 am (strict) on May 7:
  • Konyndyk 2.5.b.1,4,10, 2.5.d.A.1-3 (in S5). Bonus: 2.5.d.A.5.